{"schema_version": "1.0", "item_id": "rcd2_6516fa8d135e2ec9", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "permutation patterns", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_aa5f64b267e35265", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5346D791A6AE", "name": "Siegel's lemma", "statement_latex": "max|X_j| <= (D^{-1}*sqrt(det(AA^T)))^(1/(N-M))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (max|X_j|) \\le x \\land x \\le ((D^{-1}*sqrt(det(AA^T)))^(1/(N-M)))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover max|X_j| <= (D^{-1}*sqrt(det(AA^T)))^(1/(N-M)).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover max|X_j| <= (D^{-1}*sqrt(det(AA^T)))^(1/(N-M)).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "max|X_j| <= (D^{-1}*sqrt(det(AA^T)))^(1/(N-M))", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover max|X_j| <= (D^{-1}*sqrt(det(AA^T)))^(1/(N-M)).\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_45f5f15dbf2d1e10", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_271c8e25fc8f8de7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis, measure theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4A564A0B747B", "name": "Monotone convergence theorem", "statement_latex": "lim_{n -> infinity} a_n = sup {a_n}", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((lim_{n → infinity} a_n),(sup {a_n}))=0", "to_expression": "lim_{n -> infinity} a_n = sup {a_n}", "justification": "zero discrete distance iff equality", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{n → infinity} a_n = sup {a_n}.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_2f1807533f51cdb2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "calculus of variations", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7F37206494D7", "name": "Isoperimetric inequality", "statement_latex": "4πA ≤ L^2", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (4πA) \\le x \\land x \\le (L^2)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover 4πA <= L^2.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover 4πA <= L^2.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "4πA ≤ L^2", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover 4πA <= L^2.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_178f41df361fb3f1", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "computational number theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_88fbe406628dae53", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematical economics", "subfield": "linear algebra", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1A4ADB6F0297", "name": "Hawkins–Simon condition", "statement_latex": "[I - A]x = d", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,([I - A]x))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(d))", "to_expression": "[I - A]x = d", "justification": "each argmin set is a singleton", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as [I - A]x = d.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_0a5c01f4782cae7e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "scattering theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0C5D6466FC1D", "name": "Levinson's theorem", "statement_latex": "φ(+∞) − φ(0) = −π(1/2*n0 + nb)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(φ(+∞) − φ(0)))+\\delta_D(z,(−π(1/2*n0 + nb))))=0", "to_expression": "φ(+∞) − φ(0) = −π(1/2*n0 + nb)", "justification": "zero loss iff some z equals both", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as φ(+∞) − φ(0) = −π(1/2*n0 + nb).\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_582c2e1f928a5b30", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic calculus", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_58173198312A", "name": "Boué–Dupuis formula", "statement_latex": "-log E[e^{-f(B)}] = inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)]", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((-log E[e^{-f(B)}]),(inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)]))=0", "to_expression": "-log E[e^{-f(B)}] = inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)]", "justification": "zero discrete distance iff equality", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as -log E[e^{-f(B)}] = inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)].\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_2c516c5d7d51b60a", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "time-reversal symmetry", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a6a5ee31b8f4014c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "linear algebra", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_08D175804776", "name": "Cayley–Hamilton theorem", "statement_latex": "p_A(A) = 0", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((p_A(A)),(0))=0", "to_expression": "p_A(A) = 0", "justification": "zero discrete distance iff equality", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p_A(A) = 0.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_c14d4eb07dcb7c30", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "automorphic forms", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_64D8463F3766", "name": "Fundamental lemma", "statement_latex": "SO_{\\gamma _{H}}(1_{K_{H}})=\\Delta (\\gamma _{H},\\gamma _{G})O_{\\gamma _{G}}^{\\kappa }(1_{K_{G}})", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(SO_{\\gamma _{H}}(1_{K_{H}}))=z\\;\\land\\;z=(\\Delta (\\gamma _{H},\\gamma _{G})O_{\\gamma _{G}}^{\\kappa }(1_{K_{G}}))", "to_expression": "SO_{\\gamma _{H}}(1_{K_{H}})=\\Delta (\\gamma _{H},\\gamma _{G})O_{\\gamma _{G}}^{\\kappa }(1_{K_{G}})", "justification": "common witness iff equality", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as SO_{\\\\gamma _{H}}(1_{K_{H}})=\\\\Delta (\\\\gamma _{H},\\\\gamma _{G})O_{\\\\gamma _{G}}^{\\\\kappa }(1_{K_{G}}).\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_685215e73ac45625", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "topology", "subfield": "differential topology", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_78c107bfca939729", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b9311af87fb01e96", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "Ramsey theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_05F0F9A888B8", "name": "Erdős–Szekeres theorem", "statement_latex": "n = (r - 1)(s - 1) + 1", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((n) \\in S \\land ((r - 1)(s - 1) + 1) \\notin S) \\lor (((r - 1)(s - 1) + 1) \\in S \\land (n) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover n = (r - 1)(s - 1) + 1.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "n = (r - 1)(s - 1) + 1", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover n = (r - 1)(s - 1) + 1.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_2bb1f082ed232f6d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "complex analysis", "subfield": "complex geometry", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_38a018c29468b21c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "combinatorics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_843AE41690C1", "name": "q-Vandermonde identity", "statement_latex": "binom(m+n, k)_q = sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "{(binom(m+n, k)_q)} = {(sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)})}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "{L} = {R} iff L = R by extensionality.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover binom(m+n, k)_q = sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "binom(m+n, k)_q = sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)}", "justification": "{L} = {R} iff L = R by extensionality.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover binom(m+n, k)_q = sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)}.\"]"}], "source_equivalence_identity": "{L} = {R} iff L = R by extensionality."}
{"schema_version": "1.0", "item_id": "rcd2_789110b6998162ad", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "complex analysis", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D3ABA1826867", "name": "Fundamental theorem of algebra", "statement_latex": "exists z in C such that p(z)=0", "assumptions": "p in C[x] is nonconstant."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\neg\\forall x\\in D,x\\notin \\{z\\in C:p(z)=0\\}", "to_expression": "exists z in C such that p(z)=0", "justification": "not all nonmembers iff member exists", "preconditions": "p in C[x] is nonconstant.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as exists z in C such that p(z)=0.\"]"}], "source_equivalence_identity": "not all nonmembers iff member exists"}
{"schema_version": "1.0", "item_id": "rcd2_73b688347bf5f292", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "topology", "subfield": "algebraic topology", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a93169124a661949", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "statistical mechanics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_9e87ce0a616d2b0c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "algebra", "subfield": "polynomials", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_235D1B5AD510", "name": "Routh–Hurwitz theorem", "statement_latex": "p - q = (1/π) Δ arg f(iy)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(p - q))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(1/π) Δ arg f(iy))", "to_expression": "p - q = (1/π) Δ arg f(iy)", "justification": "each argmin set is a singleton", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p - q = (1/π) Δ arg f(iy).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_f0e754c9afa94ce8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "quadratic forms", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_434E3E7B6A54", "name": "Sylvester's law of inertia", "statement_latex": "B = SAS^T", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(SAS^T) = (B)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover B = SAS^T.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "B = SAS^T", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover B = SAS^T.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_59770635374ce268", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "modular arithmetic", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c587f0569d461171", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "differential calculus", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_90fb5ad3a64575c2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "linear algebra", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_08D175804776", "name": "Cayley–Hamilton theorem", "statement_latex": "p_A(A) = 0", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(p_A(A)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(0))", "to_expression": "p_A(A) = 0", "justification": "each argmin set is a singleton", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p_A(A) = 0.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_077c39d731537779", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CCCCB5088787", "name": "Mean value theorem", "statement_latex": "f'(c) = (f(b)-f(a))/(b-a)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((f'(c)),z):z\\in D\\}=\\{((f(b)-f(a))/(b-a),z):z\\in D\\}", "to_expression": "f'(c) = (f(b)-f(a))/(b-a)", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f'(c) = (f(b)-f(a))/(b-a).\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_44ec684a1e75d0ea", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C3C168B97D27", "name": "Watson's quintuple product identity", "statement_latex": "prod_{n>=1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1})", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n \\\\in \\mathbb{Z}} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1})) - (prod_{n) \\le 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A-B <= 0 iff A <= B by adding B to both sides.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover prod_{n >= 1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1}).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A-B <= 0 iff A <= B by adding B to both sides.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover prod_{n >= 1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1}).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "prod_{n>=1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1})", "justification": "A-B <= 0 iff A <= B by adding B to both sides.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover prod_{n >= 1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1}).\"]"}], "source_equivalence_identity": "A-B <= 0 iff A <= B by adding B to both sides."}
{"schema_version": "1.0", "item_id": "rcd2_7b8c17a5b206ba08", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "multilinear algebra", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F538F79B0750", "name": "Lagrange's identity", "statement_latex": "(\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2 = \\sum_{i<j} (a_i b_j - a_j b_i)^2", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "(\\{((\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2)\\}\\times\\{(\\sum_{i<j} (a_i b_j - a_j b_i)^2)\\})\\cap\\{(z,z):z\\in D\\}=\\{(((\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2),(\\sum_{i<j} (a_i b_j - a_j b_i)^2))\\}", "to_expression": "(\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2 = \\sum_{i<j} (a_i b_j - a_j b_i)^2", "justification": "intersection is singleton iff pair diagonal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (\\\\sum a_k^2)(\\\\sum b_k^2) - (\\\\sum a_k b_k)^2 = \\\\sum_{i<j} (a_i b_j - a_j b_i)^2.\"]"}], "source_equivalence_identity": "intersection is singleton iff pair diagonal"}
{"schema_version": "1.0", "item_id": "rcd2_631d12eee454196e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "computer science", "subfield": "formal language theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_37D96EA4CFB7", "name": "Pumping lemma for regular languages", "statement_latex": "w = xyz", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(xyz) = (w)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover w = xyz.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "w = xyz", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover w = xyz.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_0e7af38f06ede5ec", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "probability theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_87bb705f98b17e33", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "computer science", "subfield": "probabilistic combinatorics", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_921bf0709a6b57d5", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum mechanics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_9bf3628c1d8d43e5", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_9acb92f9d30c46c1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "geometry", "subfield": "discrete geometry", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_DCDC0E01EC8B", "name": "Covering problem of Rado", "statement_latex": "F(X) = inf_S sup_I (|I|/|S|)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((F(X)),z):z\\in D\\}=\\{((inf_S sup_I (|I|/|S|)),z):z\\in D\\}", "to_expression": "F(X) = inf_S sup_I (|I|/|S|)", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as F(X) = inf_S sup_I (|I|/|S|).\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_70fdb8d1d666ba36", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "queueing theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D0BC02092015", "name": "Bartlett's theorem", "statement_latex": "mu(t) = integral_{-infinity}^{t} A(s)p(s,t) ds", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(mu(t)) \\in {(integral_{-\\infty}^{t} A(s)p(s,t) ds)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover mu(t) = integral_{-infinity}^{t} A(s)p(s,t) ds.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "mu(t) = integral_{-infinity}^{t} A(s)p(s,t) ds", "justification": "L in {R} iff L = R.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover mu(t) = integral_{-infinity}^{t} A(s)p(s,t) ds.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_7422e37231d22159", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "graph theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_74a6b64914341a33", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_617ACB7CFB1C", "name": "Specht's theorem", "statement_latex": "tr W(A, A*) = tr W(B, B*)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((tr W(A, A*)),(tr W(B, B*)))=0", "to_expression": "tr W(A, A*) = tr W(B, B*)", "justification": "zero discrete distance iff equality", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as tr W(A, A*) = tr W(B, B*).\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_3f15bb3547625ab4", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "harmonic analysis", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a5c5c58e4dcdbc44", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "geometric probability", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F57E82952899", "name": "Wendel's theorem", "statement_latex": "p_{n,N}=2^{-N+1}\\sum _{k=0}^{n-1}{\\binom {N-1}{k}}", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(p_{n,N}))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(2^{-N+1}\\sum _{k=0}^{n-1}{\\binom {N-1}{k}}))", "to_expression": "p_{n,N}=2^{-N+1}\\sum _{k=0}^{n-1}{\\binom {N-1}{k}}", "justification": "each argmin set is a singleton", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p_{n,N}=2^{-N+1}\\\\sum _{k=0}^{n-1}{\\\\binom {N-1}{k}}.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_0d7137fce72ac9fd", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "measure theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6ad53d2e747f6ed8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "statistical mechanics", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_63998BA24BBA", "name": "Fluctuation theorem", "statement_latex": "Pr(Σ_t=A) / Pr(Σ_t=-A) = e^(At)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(Pr(Σ_t=A) / Pr(Σ_t=-A))\\Longleftrightarrow z=(e^(At)))", "to_expression": "Pr(Σ_t=A) / Pr(Σ_t=-A) = e^(At)", "justification": "same equality fiber iff equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr(Σ_t=A) / Pr(Σ_t=-A) = e^(At).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_6bbf80a104b58d7e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "symmetric functions", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4E5DD778CBB5", "name": "Newton's identities", "statement_latex": "k e_k = \\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(k e_k = \\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i) \\lor (0 = 1)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "0=1 is false, so Phi_T or false iff Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "k e_k = \\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i", "justification": "0=1 is false, so Phi_T or false iff Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "0=1 is false, so Phi_T or false iff Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_468e7d3feb7d9dcf", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_303E62405496", "name": "Lorden's inequality", "statement_latex": "E(R_b) <= E(X^2) / E(X)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(E(X^2) / E(X)) - (E(R_b)) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover E(R_b) <= E(X^2) / E(X).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover E(R_b) <= E(X^2) / E(X).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "E(R_b) <= E(X^2) / E(X)", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover E(R_b) <= E(X^2) / E(X).\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_f00173161e99d273", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "algebraic graph theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0ABE50B0E12D", "name": "Expander mixing lemma", "statement_latex": "|e(S,T) - (d/n)|S||T|| <= λ√|S||T|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\le (|e(S,T) - (d/n)|S||T||)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (λ√|S||T|)\\}", "to_expression": "|e(S,T) - (d/n)|S||T|| <= λ√|S||T|", "justification": "lower-cut inclusion iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |e(S,T) - (d/n)|S||T|| ≤ λ√|S||T|.\"]"}], "source_equivalence_identity": "lower-cut inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_35f73e894d3b7db1", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e102fb4e28812817", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5346D791A6AE", "name": "Siegel's lemma", "statement_latex": "max|X_j| <= (D^{-1}*sqrt(det(AA^T)))^(1/(N-M))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(D^{-1}*sqrt(det(AA^T)))^(1/(N-M))\\le t\\Rightarrow (max|X_j|)\\le t", "to_expression": "max|X_j| <= (D^{-1}*sqrt(det(AA^T)))^(1/(N-M))", "justification": "converse uses t=Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as max|X_j| ≤ (D^{-1}*sqrt(det(AA^T)))^(1/(N-M)).\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_3ecfc438fcf52443", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "functional analysis", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_50a56834ca81a883", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "hypergraph theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_9085b3cfb3896817", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "measure theory", "subfield": "stochastic analysis", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FB5E9395CC7C", "name": "Malliavin's absolute continuity lemma", "statement_latex": "|∫ Dφ(y)(x) dμ(y)| ≤ C(x) ‖φ‖_{∞}", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((|∫ Dφ(y)(x) dμ(y)|)-(C(x) ‖φ‖_{∞}))=0", "to_expression": "|∫ Dφ(y)(x) dμ(y)| ≤ C(x) ‖φ‖_{∞}", "justification": "sup equals max(0,X-Y)", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |∫ Dφ(y)(x) dμ(y)| ≤ C(x) ‖φ‖_{∞}.\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
{"schema_version": "1.0", "item_id": "rcd2_7b1027512595c4fc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "topology", "subfield": "general topology", "primary_mechanism": "quantifier", "transformation_category": "counting_combinatorial", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_04953D782E69", "name": "Tube lemma", "statement_latex": "X × Y → X", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall k\\in\\mathbb{N}_{>0}\\exists N\\forall n\\ge N:d((X × Y),(X))<1/k", "to_expression": "X × Y → X", "justification": "1/k base is cofinal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as X × Y → X.\"]"}], "source_equivalence_identity": "1/k base is cofinal"}
{"schema_version": "1.0", "item_id": "rcd2_ec81744149f02ab6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "physics", "subfield": "general relativity", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_737BDAA0F832", "name": "Stewart–Walker lemma", "statement_latex": "ΔδT = 0", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(ΔδT) \\in {(0)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover ΔδT = 0.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "ΔδT = 0", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover ΔδT = 0.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_c9b0e71ecf54a060", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum field theory", "subfield": "lattice field theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7FD1BE25E27D", "name": "Nielsen–Ninomiya theorem", "statement_latex": "H = ∑_{x,y} ψ^†(x) F(x, y) ψ(y)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(∑_{x,y} ψ^†(x) F(x, y) ψ(y)) = (H)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover H = ∑_{x,y} ψ^†(x) F(x, y) ψ(y).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "H = ∑_{x,y} ψ^†(x) F(x, y) ψ(y)", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover H = ∑_{x,y} ψ^†(x) F(x, y) ψ(y).\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
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{"schema_version": "1.0", "item_id": "rcd2_f74035cf177e7249", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "calculus", "subfield": "integral calculus", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2122e6db519a8af6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "numerical linear algebra", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FBD08849D0F3", "name": "Kreiss matrix theorem", "statement_latex": "K(A) = sup_{|z|>1} (|z|-1) ||(z-A)^-1||", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(1} (|z|-1) ||(z-A)^-1||) - (K(A) = sup_{|z|) < 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A-B<0 iff A<B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover K(A) = sup_{|z| > 1} (|z|-1) ||(z-A)^-1||.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A-B<0 iff A<B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover K(A) = sup_{|z| > 1} (|z|-1) ||(z-A)^-1||.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "K(A) = sup_{|z|>1} (|z|-1) ||(z-A)^-1||", "justification": "A-B<0 iff A<B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover K(A) = sup_{|z| > 1} (|z|-1) ||(z-A)^-1||.\"]"}], "source_equivalence_identity": "A-B<0 iff A<B."}
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{"schema_version": "1.0", "item_id": "rcd2_f9a4eccce03d3244", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "extremal graph theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4D5DD853BD7", "name": "Szemerédi regularity lemma", "statement_latex": "|d(X,Y) - d(A,B)| <= ε", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(|d(X,Y) - d(A,B)|)}{\\exp(|d(X,Y) - d(A,B)|)+\\exp(ε)}\\le\\frac{1}{2}", "to_expression": "|d(X,Y) - d(A,B)| <= ε", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |d(X,Y) - d(A,B)| ≤ ε.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_84b1c2cae20a035b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "statistical mechanics", "subfield": "non-equilibrium thermodynamics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D06DBDBC4CC8", "name": "Crooks fluctuation theorem", "statement_latex": "P[σ(x(t))] / P[σ(x_tilde(t))] = e^σ[x(t)]", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(e^σ[x(t)]) = (P[σ(x(t))] / P[σ(x_tilde(t))])", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover P[σ(x(t))] / P[σ(x_tilde(t))] = e^σ[x(t)].\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "P[σ(x(t))] / P[σ(x_tilde(t))] = e^σ[x(t)]", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover P[σ(x(t))] / P[σ(x_tilde(t))] = e^σ[x(t)].\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
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{"schema_version": "1.0", "item_id": "rcd2_e83a8d62e59080c4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "graph theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_57B1E9BA3579", "name": "Crossing number inequality", "statement_latex": "cr(G) >= e^3 / (29n^2)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(cr(G)) - (e^3 / (29n^2)) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover cr(G) >= e^3 / (29n^2).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover cr(G) >= e^3 / (29n^2).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "cr(G) >= e^3 / (29n^2)", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover cr(G) >= e^3 / (29n^2).\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_72ee534b2b5f253c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_15a9ecb28deead8c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "convex geometry", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4F00365B64E5", "name": "Minkowski's first inequality for convex bodies", "statement_latex": "V_1(K, L) ≥ V(K)^((n-1)/n) V(L)^(1/n)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (V_1(K, L))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (V(K)^((n-1)/n) V(L)^(1/n))\\}", "to_expression": "V_1(K, L) ≥ V(K)^((n-1)/n) V(L)^(1/n)", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as V_1(K, L) ≥ V(K)^((n-1)/n) V(L)^(1/n).\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_b5f4dc91a83876d0", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "algebraic K-theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ff13a2579d616b23", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "matrix theory", "subfield": "linear algebra", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_cfd7cdeec1b1241e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "Ramsey theory", "subfield": "discrete mathematics", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4B3C8E83990", "name": "Schur's theorem", "statement_latex": "x + y = z", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(x + y))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(z))", "to_expression": "x + y = z", "justification": "each argmin set is a singleton", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as x + y = z.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_f4ef8332e7735833", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "point processes", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C81FC5D06BF2", "name": "Campbell's theorem", "statement_latex": "E[sum_{x in N} f(x)] = integral f(x) dLambda(x)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(E[sum_{x in N} f(x)])\\Longleftrightarrow z=(integral f(x) dLambda(x)))", "to_expression": "E[sum_{x in N} f(x)] = integral f(x) dLambda(x)", "justification": "same equality fiber iff equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as E[sum_{x in N} f(x)] = integral f(x) dLambda(x).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_1234f345bfb59706", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "commutative algebra", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_666997CDC49C", "name": "Noether normalization lemma", "statement_latex": "A = k[y_1, ..., y_m]", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(A)\\Longleftrightarrow z=(k[y_1, ..., y_m]))", "to_expression": "A = k[y_1, ..., y_m]", "justification": "same equality fiber iff equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as A = k[y_1, ..., y_m].\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_1e3f0a2a23275da4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "analysis", "subfield": "complex analysis", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_065364B1C6FB", "name": "Arakelyan's theorem", "statement_latex": "|g − f| < ε", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;t\\le (|g − f|)\\Rightarrow t<(ε)", "to_expression": "|g − f| < ε", "justification": "converse uses t=X", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |g − f| < ε.\"]"}], "source_equivalence_identity": "converse uses t=X"}
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{"schema_version": "1.0", "item_id": "rcd2_249b198b4c58de68", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "probabilistic combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7C9C39383770", "name": "Ajtai–Komlós–Tusnády theorem", "statement_latex": "P(C_1 n log n < inf_{σ ∈ S_n} sum_{k=1}^n |X_k - Y_{σ(k)}| < C_2 n log n) = 1", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((P(C_1 n log n < inf_{σ ∈ S_n} sum_{k=1}^n |X_k - Y_{σ(k)}| < C_2 n log n)),(1))=0", "to_expression": "P(C_1 n log n < inf_{σ ∈ S_n} sum_{k=1}^n |X_k - Y_{σ(k)}| < C_2 n log n) = 1", "justification": "zero discrete distance iff equality", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(C_1 n log n < inf_{σ ∈ S_n} sum_{k=1}^n |X_k - Y_{σ(k)}| < C_2 n log n) = 1.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_40998f4562131728", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "prime numbers", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_3da7d23001f0ab84", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "Mathematical analysis", "subfield": "Sequences and series", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F3C0470E95D9", "name": "Abel's inequality", "statement_latex": "|∑ a_k b_k| ≤ max|B_k|(|a_n| + |a_n - a_1|)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (max|B_k|(|a_n| + |a_n - a_1|))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|∑ a_k b_k|)\\}", "to_expression": "|∑ a_k b_k| ≤ max|B_k|(|a_n| + |a_n - a_1|)", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |∑ a_k b_k| ≤ max|B_k|(|a_n| + |a_n - a_1|).\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_7289f9c6b89f2d5a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "decision theory", "subfield": "information theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D757C95A0326", "name": "Blackwell's informativeness theorem", "statement_latex": "V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(V(p))\\Longleftrightarrow z=(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)))", "to_expression": "V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)", "justification": "same equality fiber iff equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_36e41882b93153bb", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal set theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ee1b94b34ca8c884", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "analytic number theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_669480de8db8b93a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_893C1D2D0065", "name": "Fermat's Last Theorem", "statement_latex": "a^n + b^n = c^n", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(c^n) = (a^n + b^n)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover a^n + b^n = c^n.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "a^n + b^n = c^n", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover a^n + b^n = c^n.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_a5ee81b743f3f06b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_7378785f670bb408", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "elementary number theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_677EBF92D0D7", "name": "Lifting-the-exponent lemma", "statement_latex": "nu_p(x^n - y^n) = nu_p(x - y) + nu_p(n)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((nu_p(x^n - y^n)),z):z\\in D\\}=\\{((nu_p(x - y) + nu_p(n)),z):z\\in D\\}", "to_expression": "nu_p(x^n - y^n) = nu_p(x - y) + nu_p(n)", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as nu_p(x^n - y^n) = nu_p(x - y) + nu_p(n).\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
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{"schema_version": "1.0", "item_id": "rcd2_c9615e8628adcd18", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "analysis", "subfield": "complex analysis", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_8edc7378192e73e1", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_01ebb063bcab1202", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "commutative algebra", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0EBFF1A397C6", "name": "Local criterion for flatness", "statement_latex": "Tor_1^A(C, M) = 0", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(Tor_1^A(C, M)))+\\delta_D(z,(0)))=0", "to_expression": "Tor_1^A(C, M) = 0", "justification": "zero loss iff some z equals both", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Tor_1^A(C, M) = 0.\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_b6304a0974b1996e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "computer algebra", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F94B0806E6BB", "name": "Landau–Mignotte bound", "statement_latex": "||h||_1 <= 2^n * ||f||_1", "assumptions": "h is a factor of the integer polynomial f of degree n under the standard Landau-Mignotte hypotheses; l1 coefficient norms are finite."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(2^n * ||f||_1) - (||h||_1) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "h is a factor of the integer polynomial f of degree n under the standard Landau-Mignotte hypotheses; l1 coefficient norms are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ||h||_1 <= 2^n * ||f||_1.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "h is a factor of the integer polynomial f of degree n under the standard Landau-Mignotte hypotheses; l1 coefficient norms are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ||h||_1 <= 2^n * ||f||_1.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "||h||_1 <= 2^n * ||f||_1", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "h is a factor of the integer polynomial f of degree n under the standard Landau-Mignotte hypotheses; l1 coefficient norms are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ||h||_1 <= 2^n * ||f||_1.\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_f18fc5ba4c266d89", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0EDEF2DF9656", "name": "Fundamental theorem of arithmetic", "statement_latex": "n = p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((n),z):z\\in D\\}=\\{((p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k}),z):z\\in D\\}", "to_expression": "n = p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k}", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as n = p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k}.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_da30e5005511db4a", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "time-reversal symmetry", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_770418179df8594f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "calculus", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_31f527bb023bf9d5", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "combinatorics", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e6eb6f1cbf99a736", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f9acada7511c3ae4", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_4523fcad08f2e5e3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D737B48D674F", "name": "Borell–TIS inequality", "statement_latex": "P(‖f‖_T > E(‖f‖_T) + u) ≤ exp(-u^2 / (2σ_T^2))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((P(‖f‖_T > E(‖f‖_T) + u))-(exp(-u^2 / (2σ_T^2))))=0", "to_expression": "P(‖f‖_T > E(‖f‖_T) + u) ≤ exp(-u^2 / (2σ_T^2))", "justification": "sup equals max(0,X-Y)", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(‖f‖_T > E(‖f‖_T) + u) ≤ exp(-u^2 / (2σ_T^2)).\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
{"schema_version": "1.0", "item_id": "rcd2_5382ea62265d4fe5", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "symmetric functions", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4E5DD778CBB5", "name": "Newton's identities", "statement_latex": "k e_k = \\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(0 = 0) \\Rightarrow (k e_k = \\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Since 0=0 is true, true=>Phi_T has the same truth value as Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "k e_k = \\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i", "justification": "Since 0=0 is true, true=>Phi_T has the same truth value as Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "Since 0=0 is true, true=>Phi_T has the same truth value as Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_098d8aa78120bd60", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5346D791A6AE", "name": "Siegel's lemma", "statement_latex": "max|X_j| <= (D^{-1}*sqrt(det(AA^T)))^(1/(N-M))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (D^{-1}*sqrt(det(AA^T)))^(1/(N-M))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (max|X_j|)\\}", "to_expression": "max|X_j| <= (D^{-1}*sqrt(det(AA^T)))^(1/(N-M))", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as max|X_j| ≤ (D^{-1}*sqrt(det(AA^T)))^(1/(N-M)).\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_b8b8c61a05ad8e47", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c2480dd155098c61", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "probabilistic combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c6164ffc1c9d8039", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "polyhedral combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_844aa96226d16c4a", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "Ramsey theory", "subfield": "discrete mathematics", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_099df776f4c70b0b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "matroid theory", "subfield": "combinatorics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_09298FD96036", "name": "Ingleton's inequality", "statement_latex": "ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4) ≤ ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4)) \\le x \\land x \\le (ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4) <= ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4) <= ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4) ≤ ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4)", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4) <= ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4).\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_35cd4fdcc8aba9fa", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic geometry", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d3deaa614f51dc6f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "prime numbers", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_71AA6DBCD731", "name": "Bonse's inequality", "statement_latex": "p_n# > p_{n+1}^2", "assumptions": "n>=4; p_n# is the product of the first n primes; p_{n+1} is the next prime."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(p_{n+1}^2)}{\\exp(p_{n+1}^2)+\\exp(p_n#)}<\\frac{1}{2}", "to_expression": "p_n# > p_{n+1}^2", "justification": "equiv exp(X)<exp(Y)", "preconditions": "n>=4; p_n# is the product of the first n primes; p_{n+1} is the next prime.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p_n# > p_{n+1}^2.\"]"}], "source_equivalence_identity": "equiv exp(X)<exp(Y)"}
{"schema_version": "1.0", "item_id": "rcd2_1405f6c2646f2d21", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "enumerative combinatorics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0a570f1952157d5a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_69A821A9D880", "name": "Hockey-stick identity", "statement_latex": "\\sum_{i=r}^{n} \\binom{i}{r} = \\binom{n+1}{r+1}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((\\sum_{i=r}^{n} \\binom{i}{r}),(\\binom{n+1}{r+1}))=0", "to_expression": "\\sum_{i=r}^{n} \\binom{i}{r} = \\binom{n+1}{r+1}", "justification": "zero discrete distance iff equality", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\sum_{i=r}^{n} \\\\binom{i}{r} = \\\\binom{n+1}{r+1}.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_e124b03b49483199", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4CEAA916D1E3", "name": "Riemann–Siegel formula", "statement_latex": "ζ(s) = Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((ζ(s)),z)=((Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s)),z)", "to_expression": "ζ(s) = Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s)", "justification": "pair equality forces first coordinates equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ζ(s) = Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_749e177b03ef2abf", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a200670a24462239", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "graph theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_07CDE34AC57A", "name": "Counting lemma", "statement_latex": "γ <= ε^h / 4h", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(ε^h / 4h)\\le t\\Rightarrow (γ)\\le t", "to_expression": "γ <= ε^h / 4h", "justification": "converse uses t=Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as γ ≤ ε^h / 4h.\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_0835f8b727e5b0a1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "Algebra", "subfield": "Lie algebras", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AE35A2B1281A", "name": "Jacobi identity", "statement_latex": "x * (y * z) + y * (z * x) + z * (x * y) = 0", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((x * (y * z) + y * (z * x) + z * (x * y)) \\in S \\land (0) \\notin S) \\lor ((0) \\in S \\land (x * (y * z) + y * (z * x) + z * (x * y)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover x * (y * z) + y * (z * x) + z * (x * y) = 0.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "x * (y * z) + y * (z * x) + z * (x * y) = 0", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover x * (y * z) + y * (z * x) + z * (x * y) = 0.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_1f6e409da9bc1263", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "modular arithmetic", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_6B373783C03D", "name": "Thue's lemma", "statement_latex": "ay ≡ x (mod m)", "assumptions": "All displayed arithmetic expressions are integers; the modulus is a positive integer."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(m) \\mid (ay) - (x)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "alpha ≡ beta mod m iff m divides alpha-beta.", "preconditions": "All displayed arithmetic expressions are integers; the modulus is a positive integer.", "inverse_mapping_note": "[\"Step 1: Substitute alpha, beta, and m from the variant formula.\",\"Step 2: Convert the witness, ideal-membership, quotient, residue, or distance condition to m | (alpha-beta).\",\"Step 3: Use the definition of congruence modulo m to recover ay ≡ x (mod m).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "ay ≡ x (mod m)", "justification": "alpha ≡ beta mod m iff m divides alpha-beta.", "preconditions": "All displayed arithmetic expressions are integers; the modulus is a positive integer.", "inverse_mapping_note": "[\"Step 1: Substitute alpha, beta, and m from the variant formula.\",\"Step 2: Convert the witness, ideal-membership, quotient, residue, or distance condition to m | (alpha-beta).\",\"Step 3: Use the definition of congruence modulo m to recover ay ≡ x (mod m).\"]"}], "source_equivalence_identity": "alpha ≡ beta mod m iff m divides alpha-beta."}
{"schema_version": "1.0", "item_id": "rcd2_7648dd35a6df9b3e", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6b53de7c5537125c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "matrix theory", "subfield": "linear algebra", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_6095A64AB713", "name": "Golden–Thompson inequality", "statement_latex": "tr(exp(A+B)) <= tr(exp(A)*exp(B))", "assumptions": "A and B are Hermitian matrices of the same finite size."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;t\\le (tr(exp(A+B)))\\Rightarrow t\\le (tr(exp(A)*exp(B)))", "to_expression": "tr(exp(A+B)) <= tr(exp(A)*exp(B))", "justification": "converse uses t=X", "preconditions": "A and B are Hermitian matrices of the same finite size.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as tr(exp(A+B)) ≤ tr(exp(A)*exp(B)).\"]"}], "source_equivalence_identity": "converse uses t=X"}
{"schema_version": "1.0", "item_id": "rcd2_300827e33f89e153", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "algebraic combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a6fd05dfbec06485", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic geometry", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ca03f0efd0245402", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C3C168B97D27", "name": "Watson's quintuple product identity", "statement_latex": "prod_{n>=1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1})", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(prod_{n) - (1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n \\\\in \\mathbb{Z}} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1})) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover prod_{n >= 1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1}).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover prod_{n >= 1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1}).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "prod_{n>=1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1})", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover prod_{n >= 1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1}).\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_e47f87070dd64530", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "large deviations theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_7e361fbb80397e3d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "functional analysis", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d8853283a8537c2d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "analysis", "subfield": "approximation theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_538DF6DD562D", "name": "Favard's theorem", "statement_latex": "y_{n+1}=(x-c_{n})y_{n}-d_{n}y_{n-1}", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((y_{n+1}),((x-c_{n})y_{n}-d_{n}y_{n-1}))\\in\\{(z,z):z\\in D\\}", "to_expression": "y_{n+1}=(x-c_{n})y_{n}-d_{n}y_{n-1}", "justification": "pair in diagonal iff X=Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as y_{n+1}=(x-c_{n})y_{n}-d_{n}y_{n-1}.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_8dcd19440a5ce6d6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "calculus of variations", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7F37206494D7", "name": "Isoperimetric inequality", "statement_latex": "4πA ≤ L^2", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((4πA)-(L^2),(-\\infty,0])=0", "to_expression": "4πA ≤ L^2", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as 4πA ≤ L^2.\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
{"schema_version": "1.0", "item_id": "rcd2_76599ef104cb11e3", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "differential equations", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_855284f1535804c8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "special functions", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CA3873AC4ACD", "name": "Ramanujan's master theorem", "statement_latex": "∫[0,∞] x^{s-1} f(x) dx = Γ(s) φ(-s)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(∫[0,∞] x^{s-1} f(x) dx))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(Γ(s) φ(-s)))", "to_expression": "∫[0,∞] x^{s-1} f(x) dx = Γ(s) φ(-s)", "justification": "each argmin set is a singleton", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∫[0,∞] x^{s-1} f(x) dx = Γ(s) φ(-s).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
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{"schema_version": "1.0", "item_id": "rcd2_7ddf97876a960330", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "field theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b46640b6330a25d9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "quadratic forms", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D2FCDF28ABB7", "name": "Pfister's sixteen-square identity", "statement_latex": "(x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2) = z_1^2 + z_2^2 + ... + z_{16}^2", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "((x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2)) \\in {(z_1^2 + z_2^2 + ... + z_{16}^2)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover (x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2) = z_1^2 + z_2^2 + ... + z_{16}^2.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "(x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2) = z_1^2 + z_2^2 + ... + z_{16}^2", "justification": "L in {R} iff L = R.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover (x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2) = z_1^2 + z_2^2 + ... + z_{16}^2.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_2215867f5b04b84b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a6582c8d1eca41c1", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "statistics", "subfield": "statistical hypothesis testing", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e5e25c8061ac473f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_758c8e2ae4f3ef92", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "linear algebra", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FF8D78866125", "name": "Binet–Cauchy identity", "statement_latex": "(sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i<j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i)", "assumptions": "All sums are finite over the same index range; entries lie in R."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: ((sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i) < x \\land x < (j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All sums are finite over the same index range; entries lie in R.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i < j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All sums are finite over the same index range; entries lie in R.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i < j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "(sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i<j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i)", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All sums are finite over the same index range; entries lie in R.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i < j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i).\"]"}], "source_equivalence_identity": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B."}
{"schema_version": "1.0", "item_id": "rcd2_e13adc3ef9aef1e1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "commutative algebra", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0EBFF1A397C6", "name": "Local criterion for flatness", "statement_latex": "Tor_1^A(C, M) = 0", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((Tor_1^A(C, M)))=F((0))", "to_expression": "Tor_1^A(C, M) = 0", "justification": "indicator of {X} separates unequal objects", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Tor_1^A(C, M) = 0.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_86a5ef4e458d58fe", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "number theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0b89a3a2e3711068", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9103F62922E0", "name": "Multidimensional Chebyshev's inequality", "statement_latex": "Pr(sqrt((X-μ)^T * V^-1 * (X-μ)) > t) <= N / t^2", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (Pr(sqrt((X-μ)^T * V^-1 * (X-μ)) > t)) \\le x \\land x \\le (N / t^2)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Pr(sqrt((X-μ)^T * V^-1 * (X-μ)) > t) <= N / t^2.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Pr(sqrt((X-μ)^T * V^-1 * (X-μ)) > t) <= N / t^2.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "Pr(sqrt((X-μ)^T * V^-1 * (X-μ)) > t) <= N / t^2", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Pr(sqrt((X-μ)^T * V^-1 * (X-μ)) > t) <= N / t^2.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_bd739e043572b828", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "analysis", "subfield": "moment problem", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_92651f1a3fe78c96", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c0ad860dceb1ce9e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "algebraic graph theory", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0ABE50B0E12D", "name": "Expander mixing lemma", "statement_latex": "|e(S,T) - (d/n)|S||T|| <= λ√|S||T|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(|e(S,T) - (d/n)|S||T||)+s=(λ√|S||T|)", "to_expression": "|e(S,T) - (d/n)|S||T|| <= λ√|S||T|", "justification": "nonnegative slack exists iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |e(S,T) - (d/n)|S||T|| ≤ λ√|S||T|.\"]"}], "source_equivalence_identity": "nonnegative slack exists iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_30aad218874b178e", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "numerical analysis", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_590978db48738271", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "enumerative combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E1416BDA0AD6", "name": "Pólya enumeration theorem", "statement_latex": "|Y^X/G| = (1/|G|) * Σ_{g∈G} m^{c(g)}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(|Y^X/G| = (1/|G|) * Σ_{g∈G} m^{c(g)}) \\land (0 = 0)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "|Y^X/G| = (1/|G|) * Σ_{g∈G} m^{c(g)}", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "0=0 is true, so Phi_T and true iff Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_1a2fe84d44de5eee", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "quadratic residues", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_286d8fdf65100b60", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "physics", "subfield": "general relativity", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_737BDAA0F832", "name": "Stewart–Walker lemma", "statement_latex": "ΔδT = 0", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(0) = (ΔδT)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover ΔδT = 0.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "ΔδT = 0", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover ΔδT = 0.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
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{"schema_version": "1.0", "item_id": "rcd2_146d9608421bec78", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis, measure theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4A564A0B747B", "name": "Monotone convergence theorem", "statement_latex": "lim_{n -> infinity} a_n = sup {a_n}", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(lim_{n -) - (\\infty} a_n = sup {a_n}) > 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover lim_{n - > infinity} a_n = sup {a_n}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover lim_{n - > infinity} a_n = sup {a_n}.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "lim_{n -> infinity} a_n = sup {a_n}", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover lim_{n - > infinity} a_n = sup {a_n}.\"]"}], "source_equivalence_identity": "A<B iff B-A>0 over finite real quantities."}
{"schema_version": "1.0", "item_id": "rcd2_c9fc0f2b6023e37d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4273AD2C24D7", "name": "Bayes' theorem", "statement_latex": "P(A|B) = P(B|A)*P(A)/P(B)", "assumptions": "A and B are events with P(B)>0; conditional probabilities are defined."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((P(A|B)),z):z\\in D\\}=\\{((P(B|A)*P(A)/P(B)),z):z\\in D\\}", "to_expression": "P(A|B) = P(B|A)*P(A)/P(B)", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "A and B are events with P(B)>0; conditional probabilities are defined.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(A|B) = P(B|A)*P(A)/P(B).\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_40d4bb8ebfcb9df2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "Number theory", "subfield": "Elliptic functions", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_78B717A820D4", "name": "Jacobi triple product", "statement_latex": "product_{m=1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2}) = sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (product_{m=1}^{\\infty} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2}) = sum_{n=-\\infty}^{\\infty} x^{n^{2}}y^{2n}))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "product_{m=1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2}) = sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n}", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_61966bc90de6f96a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "additive number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E3E323366217", "name": "Lagrange's four-square theorem", "statement_latex": "p = a^2 + b^2 + c^2 + d^2", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((p),(a^2 + b^2 + c^2 + d^2))\\in\\{(z,z):z\\in D\\}", "to_expression": "p = a^2 + b^2 + c^2 + d^2", "justification": "pair in diagonal iff X=Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p = a^2 + b^2 + c^2 + d^2.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_dddc589dfdb48be7", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "sieve theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_38b6b09f6624cea6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal set theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4D5A3A69C1D", "name": "Sauer–Shelah lemma", "statement_latex": "|F| <= sum_{i=0}^{d} binom(n,i)", "assumptions": "F is a family of subsets of an n-element set with VC dimension d."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (sum_{i=0}^{d} binom(n,i))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|F|)\\}", "to_expression": "|F| <= sum_{i=0}^{d} binom(n,i)", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "F is a family of subsets of an n-element set with VC dimension d.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |F| ≤ sum_{i=0}^{d} binom(n,i).\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_2ab8e252afef5656", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "polynomial ring theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6bb1f1ce0c42b297", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "physics", "subfield": "general relativity", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FAE91EE049E6", "name": "Lovelock's theorem", "statement_latex": "A^{\\mu \\nu} = aG^{\\mu \\nu} + bg^{\\mu \\nu}", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(A^{\\mu \\nu}))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(aG^{\\mu \\nu} + bg^{\\mu \\nu}))", "to_expression": "A^{\\mu \\nu} = aG^{\\mu \\nu} + bg^{\\mu \\nu}", "justification": "each argmin set is a singleton", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as A^{\\\\mu \\\\nu} = aG^{\\\\mu \\\\nu} + bg^{\\\\mu \\\\nu}.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_b0472fd45164da7f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "elementary algebra", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_40DB1B003904", "name": "Binomial theorem", "statement_latex": "(x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\neg ((x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k)) \\Rightarrow (0 = 1)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "not Phi_T => false is not(not Phi_T), equivalent to Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "(x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k", "justification": "not Phi_T => false is not(not Phi_T), equivalent to Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "not Phi_T => false is not(not Phi_T), equivalent to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_6957ad77583a247d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "measure theory", "subfield": "stochastic analysis", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_1591eca4703232ee", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "statistical mechanics", "subfield": "non-equilibrium thermodynamics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6920a50c11f9a1a0", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "vector calculus", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d81cca22e814c7d5", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9EF5E48E9DAF", "name": "Rolle's theorem", "statement_latex": "f'(c) = 0", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((f'(c)) \\in S \\land (0) \\notin S) \\lor ((0) \\in S \\land (f'(c)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f'(c) = 0.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f'(c) = 0", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f'(c) = 0.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_f44ce35a229452fc", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "field theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ec27ef9f12407d2f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "polynomial ring theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BF300C15A52D", "name": "Gauss's lemma (polynomials)", "statement_latex": "F(X) = a_0 + a_1X + ... + a_nX^n", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(F(X)) \\in {(a_0 + a_1X + ... + a_nX^n)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover F(X) = a_0 + a_1X + ... + a_nX^n.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "F(X) = a_0 + a_1X + ... + a_nX^n", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover F(X) = a_0 + a_1X + ... + a_nX^n.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_e34a659ee7de8e60", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "calculus of variations", "subfield": "Morse theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_47ac7dd844d9ca27", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F01A12C6F5E3", "name": "Taylor's theorem", "statement_latex": "f(x) = sum_{n=0}^k (f^(n)(a)/n!) * (x-a)^n + R_k(x)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (f(x) = sum_{n=0}^k (f^(n)(a)/n!) * (x-a)^n + R_k(x)))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f(x) = sum_{n=0}^k (f^(n)(a)/n!) * (x-a)^n + R_k(x)", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_b6ea7883ff0f7de8", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum optics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_303fe4d6a3a22f54", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "hypergraph theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_8e00308e4457f48a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "vector calculus", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_6CFB1B566265", "name": "Stokes' theorem", "statement_latex": "integral integral_{Sigma} (curl F) dot dSigma = integral_{partial Sigma} F dot dr", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((integral integral_{Sigma} (curl F) dot dSigma),z):z\\in D\\}=\\{((integral_{partial Sigma} F dot dr),z):z\\in D\\}", "to_expression": "integral integral_{Sigma} (curl F) dot dSigma = integral_{partial Sigma} F dot dr", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as integral integral_{Sigma} (curl F) dot dSigma = integral_{partial Sigma} F dot dr.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_3d38e3047c063a1b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal set theory", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_03FC7B5F26C8", "name": "Erdős–Ko–Rado theorem", "statement_latex": "|A| <= binom(n-1,r-1)", "assumptions": "A is an intersecting family of r-subsets of an n-element set; n>=2r."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(|A|)+s=(binom(n-1,r-1))", "to_expression": "|A| <= binom(n-1,r-1)", "justification": "nonnegative slack exists iff X≤Y", "preconditions": "A is an intersecting family of r-subsets of an n-element set; n>=2r.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |A| ≤ binom(n-1,r-1).\"]"}], "source_equivalence_identity": "nonnegative slack exists iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_fc4f74139aecc97f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "permutation patterns", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8A6780273F9B", "name": "Stanley–Wilf conjecture", "statement_latex": "lim_{n→∞} |S_n(β)|^(1/n)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\forall\\varepsilon>0\\exists N\\forall n\\ge N:d((lim_{n),(∞} |S_n(β)|^(1/n)))<\\varepsilon", "to_expression": "lim_{n→∞} |S_n(β)|^(1/n)", "justification": "definition of metric convergence", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{n→∞} |S_n(β)|^(1/n).\"]"}], "source_equivalence_identity": "definition of metric convergence"}
{"schema_version": "1.0", "item_id": "rcd2_fea8fa6a9bf57927", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_3ebb9d58d6991b9c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "number theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A54D9D349D9E", "name": "Candido's identity", "statement_latex": "[x^2 + y^2 + (x+y)^2]^2 = 2[x^4 + y^4 + (x+y)^4]", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,([x^2 + y^2 + (x+y)^2]^2))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(2[x^4 + y^4 + (x+y)^4]))", "to_expression": "[x^2 + y^2 + (x+y)^2]^2 = 2[x^4 + y^4 + (x+y)^4]", "justification": "each argmin set is a singleton", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as [x^2 + y^2 + (x+y)^2]^2 = 2[x^4 + y^4 + (x+y)^4].\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_4e05e6e971afe70b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "Combinatorics", "subfield": "Order theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B5F5A5581E5D", "name": "Kruskal's tree theorem", "statement_latex": "T1 <= T2", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((T1)-(T2))=0", "to_expression": "T1 <= T2", "justification": "sup equals max(0,X-Y)", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as T1 ≤ T2.\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
{"schema_version": "1.0", "item_id": "rcd2_745f2b838b445848", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_20033DA37507", "name": "Kolmogorov–Arnold representation theorem", "statement_latex": "f(x_1, ..., x_n) = Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p))", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(f(x_1, ..., x_n))\\Longleftrightarrow z=(Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p))))", "to_expression": "f(x_1, ..., x_n) = Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p))", "justification": "same equality fiber iff equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f(x_1, ..., x_n) = Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p)).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
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{"schema_version": "1.0", "item_id": "rcd2_94786ea0f58c47d1", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "analysis", "subfield": "approximation theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e5ccf4552122c3b1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "differential calculus", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F1B027EB50E5", "name": "Interior extremum theorem", "statement_latex": "f'(x_0) = 0", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((f'(x_0)) \\in S \\land (0) \\notin S) \\lor ((0) \\in S \\land (f'(x_0)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f'(x_0) = 0.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f'(x_0) = 0", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f'(x_0) = 0.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
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{"schema_version": "1.0", "item_id": "rcd2_bb75073fa4d523df", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_89D2662A27BE", "name": "Freiman's theorem", "statement_latex": "|A+A| <= K|A|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (|A+A|) \\le x \\land x \\le (K|A|)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |A+A| <= K|A|.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |A+A| <= K|A|.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "|A+A| <= K|A|", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |A+A| <= K|A|.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_b07b2456e3f48f68", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "extremal graph theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4D5DD853BD7", "name": "Szemerédi regularity lemma", "statement_latex": "|d(X,Y) - d(A,B)| <= ε", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(ε) - (|d(X,Y) - d(A,B)|) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |d(X,Y) - d(A,B)| <= ε.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |d(X,Y) - d(A,B)| <= ε.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "|d(X,Y) - d(A,B)| <= ε", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |d(X,Y) - d(A,B)| <= ε.\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_e175434d823d297c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "analysis", "subfield": "complex analysis", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_74fc124ba71fe3bd", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "p-adic analysis", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3B924C720B22", "name": "Krasner's lemma", "statement_latex": "|alpha - beta| < |alpha - alpha_i|", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(|alpha - beta|)}{\\exp(|alpha - beta|)+\\exp(|alpha - alpha_i|)}<\\frac{1}{2}", "to_expression": "|alpha - beta| < |alpha - alpha_i|", "justification": "equiv exp(X)<exp(Y)", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |alpha - beta| < |alpha - alpha_i|.\"]"}], "source_equivalence_identity": "equiv exp(X)<exp(Y)"}
{"schema_version": "1.0", "item_id": "rcd2_267a3499e0b79072", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "Kleinian groups", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C0AAA35160EF", "name": "Jørgensen's inequality", "statement_latex": "|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2| >= 1", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\le (1)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2|)\\}", "to_expression": "|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2| >= 1", "justification": "lower-cut inclusion iff X≤Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2| ≥ 1.\"]"}], "source_equivalence_identity": "lower-cut inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_0114e08515e297e6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "differential equations", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_891C2603FAE3", "name": "Grönwall's inequality", "statement_latex": "u(t) <= u(a) * exp(integral from a to t of beta(s) ds)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((u(t))-(u(a) * exp(integral from a to t of beta(s) ds)))=0", "to_expression": "u(t) <= u(a) * exp(integral from a to t of beta(s) ds)", "justification": "sup equals max(0,X-Y)", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as u(t) ≤ u(a) * exp(integral from a to t of beta(s) ds).\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
{"schema_version": "1.0", "item_id": "rcd2_ef14a9be9b7fc045", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "analysis", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A35964624017", "name": "AM–GM inequality", "statement_latex": "(x + y)/2 >= sqrt(x*y)", "assumptions": "x and y are nonnegative real numbers."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (sqrt(x*y)) \\le x \\land x \\le ((x + y)/2)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "x and y are nonnegative real numbers.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (x + y)/2 >= sqrt(x*y).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "x and y are nonnegative real numbers.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (x + y)/2 >= sqrt(x*y).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "(x + y)/2 >= sqrt(x*y)", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "x and y are nonnegative real numbers.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (x + y)/2 >= sqrt(x*y).\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
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{"schema_version": "1.0", "item_id": "rcd2_4f9eae4086a59844", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "algebraic graph theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0ABE50B0E12D", "name": "Expander mixing lemma", "statement_latex": "|e(S,T) - (d/n)|S||T|| <= λ√|S||T|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (λ√|S||T|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|e(S,T) - (d/n)|S||T||)\\}", "to_expression": "|e(S,T) - (d/n)|S||T|| <= λ√|S||T|", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |e(S,T) - (d/n)|S||T|| ≤ λ√|S||T|.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_adb316ecc62980d7", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f1ef9d18fbc32c29", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "analytic number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C97AADA78BC1", "name": "Vaughan's identity", "statement_latex": "-zeta'(s)/zeta(s) = F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(-z\\eta'(s)/z\\eta(s)) \\in {(F(s) - z\\eta(s)F(s)G(s) - z\\eta'(s)G(s) + (-z\\eta'(s)/z\\eta(s) - F(s))(1 - z\\eta(s)G(s)))}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover -zeta'(s)/zeta(s) = F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s)).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "-zeta'(s)/zeta(s) = F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s))", "justification": "L in {R} iff L = R.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover -zeta'(s)/zeta(s) = F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s)).\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_579ffa8de69966fc", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "algebraic graph theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_99fca654e73f5845", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "scattering theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_47254b34490cf471", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F902B76211EA", "name": "Cantelli's inequality", "statement_latex": "Pr(X - E[X] ≥ λ) ≤ σ^2 / (σ^2 + λ^2)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((Pr(X - E[X] ≥ λ))-(σ^2 / (σ^2 + λ^2)),(-\\infty,0])=0", "to_expression": "Pr(X - E[X] ≥ λ) ≤ σ^2 / (σ^2 + λ^2)", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr(X - E[X] ≥ λ) ≤ σ^2 / (σ^2 + λ^2).\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
{"schema_version": "1.0", "item_id": "rcd2_a1fdf1e4612aaeab", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "algebraic combinatorics", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_29f318412cc27773", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2DC4FABA16C6", "name": "Kantorovich inequality", "statement_latex": "(Σ p_i x_i) * (Σ p_i / x_i) <= (a+b)^2 / (4ab)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (a+b)^2 / (4ab)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (Σ p_i x_i) * (Σ p_i / x_i)\\}", "to_expression": "(Σ p_i x_i) * (Σ p_i / x_i) <= (a+b)^2 / (4ab)", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (Σ p_i x_i) * (Σ p_i / x_i) ≤ (a+b)^2 / (4ab).\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_f6c66c35132474ad", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "decision theory", "subfield": "information theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b7854044dce82b99", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematical economics", "subfield": "linear algebra", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_058a8f52ee8f9c8e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "symplectic geometry", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C62ED9627235", "name": "Williamson theorem", "statement_latex": "SMS^T = D ⊕ D", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(SMS^T) \\in {(D ⊕ D)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover SMS^T = D ⊕ D.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "SMS^T = D ⊕ D", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover SMS^T = D ⊕ D.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_8fcbb54bfcb9e63e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "Ramsey theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_05F0F9A888B8", "name": "Erdős–Szekeres theorem", "statement_latex": "n = (r - 1)(s - 1) + 1", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(n))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,((r - 1)(s - 1) + 1))", "to_expression": "n = (r - 1)(s - 1) + 1", "justification": "each argmin set is a singleton", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as n = (r - 1)(s - 1) + 1.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_6289cb7e9f580069", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "measure theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0acc44516c7149f8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "asymptotic geometric analysis", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0B6D1767746C", "name": "Milman's reverse Brunn–Minkowski inequality", "statement_latex": "vol(s φ K + t ψ L)^(1/n) ≤ C (s vol(φ K)^(1/n) + t vol(ψ L)^(1/n))", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(C (s vol(φ K)^(1/n) + t vol(ψ L)^(1/n)))\\le t\\Rightarrow (vol(s φ K + t ψ L)^(1/n))\\le t", "to_expression": "vol(s φ K + t ψ L)^(1/n) ≤ C (s vol(φ K)^(1/n) + t vol(ψ L)^(1/n))", "justification": "converse uses t=Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as vol(s φ K + t ψ L)^(1/n) ≤ C (s vol(φ K)^(1/n) + t vol(ψ L)^(1/n)).\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_67fbd20ea24f6652", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "representation theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1292421DFB1E", "name": "Schur's lemma", "statement_latex": "ρ_W(g) ∘ f = f ∘ ρ_V(g)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(ρ_W(g) ∘ f))+\\delta_D(z,(f ∘ ρ_V(g))))=0", "to_expression": "ρ_W(g) ∘ f = f ∘ ρ_V(g)", "justification": "zero loss iff some z equals both", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ρ_W(g) ∘ f = f ∘ ρ_V(g).\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_0bb400062a322092", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "diophantine approximation", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_221c70900a68b182", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "zeta and L-functions", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_09EFA5455472", "name": "Apéry's theorem", "statement_latex": "\\zeta (3)=\\sum _{n=1}^{\\infty }{\\frac {1}{n^{3}}}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((\\zeta (3)),(\\sum _{n=1}^{\\infty }{\\frac {1}{n^{3}}}))=0", "to_expression": "\\zeta (3)=\\sum _{n=1}^{\\infty }{\\frac {1}{n^{3}}}", "justification": "zero discrete distance iff equality", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\zeta (3)=\\\\sum _{n=1}^{\\\\infty }{\\\\frac {1}{n^{3}}}.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_e68e6fcb9c3c3e95", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "additive combinatorics", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_081265713252", "name": "Erdős–Szemerédi theorem", "statement_latex": "max(|A + A|, |A * A|) >= c|A|^{1 + ε}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(c|A|^{1 + ε})+s=(max(|A + A|, |A * A|))", "to_expression": "max(|A + A|, |A * A|) >= c|A|^{1 + ε}", "justification": "nonnegative slack exists iff X≤Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as max(|A + A|, |A * A|) ≥ c|A|^{1 + ε}.\"]"}], "source_equivalence_identity": "nonnegative slack exists iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_9782c3f81646f793", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "queueing theory", "subfield": "probability", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_91967E3D2C5F", "name": "Little's law", "statement_latex": "L = lambda*W", "assumptions": "The queueing system is stable and in steady state; L, lambda, and W are finite real quantities."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(L))+\\delta_D(z,(lambda*W)))=0", "to_expression": "L = lambda*W", "justification": "zero loss iff some z equals both", "preconditions": "The queueing system is stable and in steady state; L, lambda, and W are finite real quantities.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as L = lambda*W.\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_cdad94f6c7ef024b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3596847807A0", "name": "Coupon collector's problem", "statement_latex": "E(T) = n H_n", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(n H_n) = (E(T))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E(T) = n H_n.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "E(T) = n H_n", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E(T) = n H_n.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_c5953138de8f5c24", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic calculus", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_58173198312A", "name": "Boué–Dupuis formula", "statement_latex": "-log E[e^{-f(B)}] = inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)]", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(-log E[e^{-f(B)}]) \\in {(inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)])}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover -log E[e^{-f(B)}] = inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)].\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "-log E[e^{-f(B)}] = inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)]", "justification": "L in {R} iff L = R.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover -log E[e^{-f(B)}] = inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)].\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_00185b96d82bdb4d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "numerical analysis, finite element method", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7C7FF749E01A", "name": "Bramble–Hilbert lemma", "statement_latex": "inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)} <= C |u|_{W^{m,p}(\\Omega)}", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (C |u|_{W^{m,p}(\\Omega)})\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)})\\}", "to_expression": "inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)} <= C |u|_{W^{m,p}(\\Omega)}", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\\\Omega)} ≤ C |u|_{W^{m,p}(\\\\Omega)}.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_4ce6942f70e8c359", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "vector calculus", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_6CFB1B566265", "name": "Stokes' theorem", "statement_latex": "integral integral_{Sigma} (curl F) dot dSigma = integral_{partial Sigma} F dot dr", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(integral_{partial Sigma} F dot dr) = (integral integral_{Sigma} (curl F) dot dSigma)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover integral integral_{Sigma} (curl F) dot dSigma = integral_{partial Sigma} F dot dr.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "integral integral_{Sigma} (curl F) dot dSigma = integral_{partial Sigma} F dot dr", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover integral integral_{Sigma} (curl F) dot dSigma = integral_{partial Sigma} F dot dr.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_09d7af8736ffb62b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "coding theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8CDA9CA5B14E", "name": "Noisy-channel coding theorem", "statement_latex": "R < C", "assumptions": "R is the coding rate and C is the channel capacity, both finite real values. This row encodes the achievability-side rate condition."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "((R),(C))\\in\\{(u,v)\\in\\mathbb{R}^2:u\\le v\\}\\setminus\\{(u,u):u\\in\\mathbb{R}\\}", "to_expression": "R < C", "justification": "≤ plus not equal iff <", "preconditions": "R is the coding rate and C is the channel capacity, both finite real values. This row encodes the achievability-side rate condition.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as R < C.\"]"}], "source_equivalence_identity": "≤ plus not equal iff <"}
{"schema_version": "1.0", "item_id": "rcd2_2b7051a4fa0ffd6e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "calculus", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3967E74F7B6F", "name": "Increment theorem", "statement_latex": "Δy = f'(x)Δx + εΔx", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((Δy) \\in S \\land (f'(x)Δx + εΔx) \\notin S) \\lor ((f'(x)Δx + εΔx) \\in S \\land (Δy) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover Δy = f'(x)Δx + εΔx.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "Δy = f'(x)Δx + εΔx", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover Δy = f'(x)Δx + εΔx.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_e3d1f4af0bf8dfb2", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_565cc97c4deb6b64", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "enumerative combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_8252d8bb9905662a", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_46cd417f5422a549", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "martingale theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1E094F8EB368", "name": "Azuma's inequality", "statement_latex": "P(X_N - X_0 >= epsilon) <= exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2)))\\le t\\Rightarrow (P(X_N - X_0 ≥ epsilon))\\le t", "to_expression": "P(X_N - X_0 >= epsilon) <= exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2))", "justification": "converse uses t=Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(X_N - X_0 ≥ epsilon) ≤ exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2)).\"]"}], "source_equivalence_identity": "converse uses t=Y"}
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{"schema_version": "1.0", "item_id": "rcd2_5c4d330a2546c42b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_7ec59dbf79d5e812", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "commutative algebra", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AB6EC963565B", "name": "Artin–Rees lemma", "statement_latex": "I^nM ∩ N = I^{n-k}(I^kM ∩ N)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(I^nM ∩ N)\\}=\\{(I^{n-k}(I^kM ∩ N))\\}", "to_expression": "I^nM ∩ N = I^{n-k}(I^kM ∩ N)", "justification": "singletons equal iff elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as I^nM ∩ N = I^{n-k}(I^kM ∩ N).\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_c1475c7366973751", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic calculus", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4D5944961D8", "name": "Itô's lemma", "statement_latex": "dX_t = mu_t dt + sigma_t dB_t", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(dX_t)\\Longleftrightarrow z=(mu_t dt + sigma_t dB_t))", "to_expression": "dX_t = mu_t dt + sigma_t dB_t", "justification": "same equality fiber iff equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as dX_t = mu_t dt + sigma_t dB_t.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_dd08ff4579f68951", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "algebraic combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B8C05F31E817", "name": "Dyson conjecture", "statement_latex": "constant_term(prod_{1<=i!=j<=n} (1-t_i/t_j)^{a_i}) = (sum a_i)! / prod(a_i!)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((constant_term(prod_{1≤i≠j≤n} (1-t_i/t_j)^{a_i})))=F((sum a_i)! / prod(a_i!))", "to_expression": "constant_term(prod_{1<=i!=j<=n} (1-t_i/t_j)^{a_i}) = (sum a_i)! / prod(a_i!)", "justification": "indicator of {X} separates unequal objects", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as constant_term(prod_{1≤i≠j≤n} (1-t_i/t_j)^{a_i}) = (sum a_i)! / prod(a_i!).\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
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{"schema_version": "1.0", "item_id": "rcd2_e119c201ef5b9e2b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "prime numbers", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_71AA6DBCD731", "name": "Bonse's inequality", "statement_latex": "p_n# > p_{n+1}^2", "assumptions": "n>=4; p_n# is the product of the first n primes; p_{n+1} is the next prime."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;t\\le (p_{n+1}^2)\\Rightarrow t<(p_n#)", "to_expression": "p_n# > p_{n+1}^2", "justification": "converse uses t=X", "preconditions": "n>=4; p_n# is the product of the first n primes; p_{n+1} is the next prime.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p_n# > p_{n+1}^2.\"]"}], "source_equivalence_identity": "converse uses t=X"}
{"schema_version": "1.0", "item_id": "rcd2_ee7f220cda3e1eb3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2B79341A5DAC", "name": "Doob's martingale inequality", "statement_latex": "P[max_{1<=i<=n} X_i >= C] <= E[max(X_n, 0)] / C", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(P[max_{1≤i≤n} X_i ≥ C])-(E[max(X_n, 0)] / C)\\}=0", "to_expression": "P[max_{1<=i<=n} X_i >= C] <= E[max(X_n, 0)] / C", "justification": "positive part vanishes iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P[max_{1≤i≤n} X_i ≥ C] ≤ E[max(X_n, 0)] / C.\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_3a0721ed8d4330d8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_83A41482F4E9", "name": "Fekete's lemma", "statement_latex": "lim_{n→∞} (a_n / n) = inf_{n∈N} (a_n / n)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((lim_{n→∞} (a_n / n)),(inf_{n∈N} (a_n / n)))=0", "to_expression": "lim_{n→∞} (a_n / n) = inf_{n∈N} (a_n / n)", "justification": "zero discrete distance iff equality", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{n→∞} (a_n / n) = inf_{n∈N} (a_n / n).\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_b5bd8dd7f7bf366c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0EDEF2DF9656", "name": "Fundamental theorem of arithmetic", "statement_latex": "n = p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((n),z)=((p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k}),z)", "to_expression": "n = p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k}", "justification": "pair equality forces first coordinates equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as n = p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k}.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_15f9f0c82e8661a3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "calculus", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3967E74F7B6F", "name": "Increment theorem", "statement_latex": "Δy = f'(x)Δx + εΔx", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((Δy),z):z\\in D\\}=\\{((f'(x)Δx + εΔx),z):z\\in D\\}", "to_expression": "Δy = f'(x)Δx + εΔx", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Δy = f'(x)Δx + εΔx.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_72f605cd32bc38d5", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "additive number theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E3E323366217", "name": "Lagrange's four-square theorem", "statement_latex": "p = a^2 + b^2 + c^2 + d^2", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((p) \\in S \\land (a^2 + b^2 + c^2 + d^2) \\notin S) \\lor ((a^2 + b^2 + c^2 + d^2) \\in S \\land (p) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover p = a^2 + b^2 + c^2 + d^2.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "p = a^2 + b^2 + c^2 + d^2", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover p = a^2 + b^2 + c^2 + d^2.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
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{"schema_version": "1.0", "item_id": "rcd2_5cb880827a635b6e", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0b8f504013faf3c1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "physics", "subfield": "general relativity", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FAE91EE049E6", "name": "Lovelock's theorem", "statement_latex": "A^{\\mu \\nu} = aG^{\\mu \\nu} + bg^{\\mu \\nu}", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((A^{\\mu \\nu}),z):z\\in D\\}=\\{((aG^{\\mu \\nu} + bg^{\\mu \\nu}),z):z\\in D\\}", "to_expression": "A^{\\mu \\nu} = aG^{\\mu \\nu} + bg^{\\mu \\nu}", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as A^{\\\\mu \\\\nu} = aG^{\\\\mu \\\\nu} + bg^{\\\\mu \\\\nu}.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_280e2c3bbc518dc3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "statistical mechanics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_63998BA24BBA", "name": "Fluctuation theorem", "statement_latex": "Pr(Σ_t=A) / Pr(Σ_t=-A) = e^(At)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (Pr(Σ_t=A) / Pr(Σ_t=-A) = e^(At)))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "Pr(Σ_t=A) / Pr(Σ_t=-A) = e^(At)", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_5d5d1aa4a1ad5555", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "statistical mechanics", "subfield": "non-equilibrium thermodynamics", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D06DBDBC4CC8", "name": "Crooks fluctuation theorem", "statement_latex": "P[σ(x(t))] / P[σ(x_tilde(t))] = e^σ[x(t)]", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(P[σ(x(t))] / P[σ(x_tilde(t))]))+\\delta_D(z,(e^σ[x(t)])))=0", "to_expression": "P[σ(x(t))] / P[σ(x_tilde(t))] = e^σ[x(t)]", "justification": "zero loss iff some z equals both", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P[σ(x(t))] / P[σ(x_tilde(t))] = e^σ[x(t)].\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
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{"schema_version": "1.0", "item_id": "rcd2_d19a679e2ce9dff1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "calculus of variations", "subfield": "Morse theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5B2379DEDFD8", "name": "Morse–Palais lemma", "statement_latex": "f(x) = ⟨Aφ(x), φ(x)⟩", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(f(x)) \\in {(⟨Aφ(x), φ(x)⟩)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(x) = ⟨Aφ(x), φ(x)⟩.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f(x) = ⟨Aφ(x), φ(x)⟩", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(x) = ⟨Aφ(x), φ(x)⟩.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_73a5af48da79d130", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "algebraic combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B8C05F31E817", "name": "Dyson conjecture", "statement_latex": "constant_term(prod_{1<=i!=j<=n} (1-t_i/t_j)^{a_i}) = (sum a_i)! / prod(a_i!)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (constant_term(prod_{1\\lei\\nej\\len} (1-t_i/t_j)^{a_i}) = (sum a_i)! / prod(a_i!)))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "constant_term(prod_{1<=i!=j<=n} (1-t_i/t_j)^{a_i}) = (sum a_i)! / prod(a_i!)", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
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{"schema_version": "1.0", "item_id": "rcd2_2ec218b5561e147b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0C6AE95E0EE0", "name": "Hoeffding's inequality", "statement_latex": "P(S_n - E[S_n] >= t) <= exp(-2*t^2 / sum_i (b_i-a_i)^2)", "assumptions": "The summands are independent and bounded with X_i in [a_i,b_i]; t>=0; denominator is positive."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((P(S_n - E[S_n] ≥ t))-(exp(-2*t^2 / sum_i (b_i-a_i)^2)))=0", "to_expression": "P(S_n - E[S_n] >= t) <= exp(-2*t^2 / sum_i (b_i-a_i)^2)", "justification": "sup equals max(0,X-Y)", "preconditions": "The summands are independent and bounded with X_i in [a_i,b_i]; t>=0; denominator is positive.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(S_n - E[S_n] ≥ t) ≤ exp(-2*t^2 / sum_i (b_i-a_i)^2).\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
{"schema_version": "1.0", "item_id": "rcd2_055dc5a3c2c2ab1d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "numerical analysis", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ac9bef6b86564449", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "geometry, probability theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_99a5f36f9760e2f8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "statistical mechanics", "subfield": "non-equilibrium thermodynamics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D06DBDBC4CC8", "name": "Crooks fluctuation theorem", "statement_latex": "P[σ(x(t))] / P[σ(x_tilde(t))] = e^σ[x(t)]", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((P[σ(x(t))] / P[σ(x_tilde(t))]) \\in S \\land (e^σ[x(t)]) \\notin S) \\lor ((e^σ[x(t)]) \\in S \\land (P[σ(x(t))] / P[σ(x_tilde(t))]) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover P[σ(x(t))] / P[σ(x_tilde(t))] = e^σ[x(t)].\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "P[σ(x(t))] / P[σ(x_tilde(t))] = e^σ[x(t)]", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover P[σ(x(t))] / P[σ(x_tilde(t))] = e^σ[x(t)].\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_ed755a0ca774ac56", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "enumerative combinatorics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_30BF4DFA8E4C", "name": "MacMahon's master theorem", "statement_latex": "G(k1, ..., km) = [x1^k1 ... xm^km] Prod_{i=1 to m} (Sum_{j=1 to m} a_{ij} x_j)^k_i", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((G(k1, ..., km)),([x1^k1 ... xm^km] Prod_{i=1 to m} (Sum_{j=1 to m} a_{ij} x_j)^k_i))\\in\\{(z,z):z\\in D\\}", "to_expression": "G(k1, ..., km) = [x1^k1 ... xm^km] Prod_{i=1 to m} (Sum_{j=1 to m} a_{ij} x_j)^k_i", "justification": "pair in diagonal iff X=Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as G(k1, ..., km) = [x1^k1 ... xm^km] Prod_{i=1 to m} (Sum_{j=1 to m} a_{ij} x_j)^k_i.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_3075279b5b047b83", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "calculus", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a3cec44e404e947b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "asymptotic geometric analysis", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0B6D1767746C", "name": "Milman's reverse Brunn–Minkowski inequality", "statement_latex": "vol(s φ K + t ψ L)^(1/n) ≤ C (s vol(φ K)^(1/n) + t vol(ψ L)^(1/n))", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((vol(s φ K + t ψ L)^(1/n))-(C (s vol(φ K)^(1/n) + t vol(ψ L)^(1/n))))=0", "to_expression": "vol(s φ K + t ψ L)^(1/n) ≤ C (s vol(φ K)^(1/n) + t vol(ψ L)^(1/n))", "justification": "sup equals max(0,X-Y)", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as vol(s φ K + t ψ L)^(1/n) ≤ C (s vol(φ K)^(1/n) + t vol(ψ L)^(1/n)).\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
{"schema_version": "1.0", "item_id": "rcd2_7b4c374be1219534", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "decision theory", "subfield": "information theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D757C95A0326", "name": "Blackwell's informativeness theorem", "statement_latex": "V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(max_{a \\in A} sum_{omega \\in Omega} u(a, omega) p(omega)) = (V(p))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega).\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_4e89ca9740ffa424", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "enumerative combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
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{"schema_version": "1.0", "item_id": "rcd2_2bad49ccaa299d6c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
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{"schema_version": "1.0", "item_id": "rcd2_aa97f058c3d404fa", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_83A41482F4E9", "name": "Fekete's lemma", "statement_latex": "lim_{n→∞} (a_n / n) = inf_{n∈N} (a_n / n)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(lim_{n→∞} (a_n / n))\\Longleftrightarrow z=(inf_{n∈N} (a_n / n)))", "to_expression": "lim_{n→∞} (a_n / n) = inf_{n∈N} (a_n / n)", "justification": "same equality fiber iff equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{n→∞} (a_n / n) = inf_{n∈N} (a_n / n).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
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{"schema_version": "1.0", "item_id": "rcd2_9e48ca3285aba5b9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "special functions", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CA3873AC4ACD", "name": "Ramanujan's master theorem", "statement_latex": "∫[0,∞] x^{s-1} f(x) dx = Γ(s) φ(-s)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(∫[0,∞] x^{s-1} f(x) dx))+\\delta_D(z,(Γ(s) φ(-s))))=0", "to_expression": "∫[0,∞] x^{s-1} f(x) dx = Γ(s) φ(-s)", "justification": "zero loss iff some z equals both", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∫[0,∞] x^{s-1} f(x) dx = Γ(s) φ(-s).\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
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{"schema_version": "1.0", "item_id": "rcd2_b433f9059add20ed", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "numerical linear algebra", "primary_mechanism": "distance", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FBD08849D0F3", "name": "Kreiss matrix theorem", "statement_latex": "K(A) = sup_{|z|>1} (|z|-1) ||(z-A)^-1||", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists s \\in \\mathbb{R}_{>0}: (K(A) = sup_{|z|) = (1} (|z|-1) ||(z-A)^-1||) + s", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A<B iff s=B-A is positive and B=A+s.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover K(A) = sup_{|z| > 1} (|z|-1) ||(z-A)^-1||.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A<B iff s=B-A is positive and B=A+s.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover K(A) = sup_{|z| > 1} (|z|-1) ||(z-A)^-1||.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "K(A) = sup_{|z|>1} (|z|-1) ||(z-A)^-1||", "justification": "A<B iff s=B-A is positive and B=A+s.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover K(A) = sup_{|z| > 1} (|z|-1) ||(z-A)^-1||.\"]"}], "source_equivalence_identity": "A<B iff s=B-A is positive and B=A+s."}
{"schema_version": "1.0", "item_id": "rcd2_ec90574f7fe67a71", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "discrete mathematics", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2F917150CBA3", "name": "Pigeonhole principle", "statement_latex": "max_{1<=i<=m} |B_i| >= ceil(n/m)", "assumptions": "n objects are distributed among m nonempty boxes B_1,...,B_m; m>=1."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(max_{1≤i≤m} |B_i|)\\le t\\Rightarrow (ceil(n/m))\\le t", "to_expression": "max_{1<=i<=m} |B_i| >= ceil(n/m)", "justification": "converse uses t=Y", "preconditions": "n objects are distributed among m nonempty boxes B_1,...,B_m; m>=1.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as max_{1≤i≤m} |B_i| ≥ ceil(n/m).\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_666c4a881096a021", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "functional analysis", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B9B50BB53C3D", "name": "Orlicz–Pettis theorem", "statement_latex": "∑_{n=1}^∞ |x^*(x_n)| < ∞", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(∑_{n=1}^∞ |x^*(x_n)|)-(∞)\\}=0\\;\\land\\;(∑_{n=1}^∞ |x^*(x_n)|)\\ne (∞)", "to_expression": "∑_{n=1}^∞ |x^*(x_n)| < ∞", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∑_{n=1}^∞ |x^*(x_n)| < ∞.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
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{"schema_version": "1.0", "item_id": "rcd2_3decd7ae24db7aad", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "probabilistic method", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8F1DAFA4A16E", "name": "Lovász local lemma", "statement_latex": "e*p*(d+1) <= 1", "assumptions": "Symmetric Lovasz local lemma setting; each bad event has probability at most p and dependency degree at most d."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((e*p*(d+1))-(1),(-\\infty,0])=0", "to_expression": "e*p*(d+1) <= 1", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "Symmetric Lovasz local lemma setting; each bad event has probability at most p and dependency degree at most d.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as e*p*(d+1) ≤ 1.\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
{"schema_version": "1.0", "item_id": "rcd2_700a67dfdbd9946d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "hypergraph theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0E38B7F67353", "name": "Baranyai's theorem", "statement_latex": "\\binom{k}{r} \\frac{r}{k} = \\binom{k-1}{r-1}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\binom{k-1}{r-1}) = (\\binom{k}{r} \\frac{r}{k})", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\binom{k}{r} \\\\frac{r}{k} = \\\\binom{k-1}{r-1}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\binom{k}{r} \\frac{r}{k} = \\binom{k-1}{r-1}", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\binom{k}{r} \\\\frac{r}{k} = \\\\binom{k-1}{r-1}.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_9e0bc7945e4bcbd6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_303E62405496", "name": "Lorden's inequality", "statement_latex": "E(R_b) <= E(X^2) / E(X)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(E(R_b))-(E(X^2) / E(X))\\}=0", "to_expression": "E(R_b) <= E(X^2) / E(X)", "justification": "positive part vanishes iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as E(R_b) ≤ E(X^2) / E(X).\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_18175dc22bda2904", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AE174D33AD68", "name": "Eaton's inequality", "statement_latex": "P(|sum_{i=1}^n a_i X_i| >= k) <= 2 inf_{0 <= c <= k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((P(|sum_{i=1}^n a_i X_i| ≥ k))-(2 inf_{0 ≤ c ≤ k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz),(-\\infty,0])=0", "to_expression": "P(|sum_{i=1}^n a_i X_i| >= k) <= 2 inf_{0 <= c <= k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(|sum_{i=1}^n a_i X_i| ≥ k) ≤ 2 inf_{0 ≤ c ≤ k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz.\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
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{"schema_version": "1.0", "item_id": "rcd2_367a0dbc353a32cf", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "algebra", "subfield": "polynomials", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d207ffd3af625083", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "symplectic geometry", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_67abe8faa2906acb", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_4ff69d51c187d2cc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "discrete mathematics", "subfield": "combinatorics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AE455D38D038", "name": "Bregman–Minc inequality", "statement_latex": "per(A) <= prod_{i=1}^n (r_i!)^(1/r_i)", "assumptions": "A is an n by n 0-1 matrix; r_i is the sum of row i; terms with r_i=0 are handled by the standard zero-permanent convention."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (prod_{i=1}^n (r_i!)^(1/r_i))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (per(A))\\}", "to_expression": "per(A) <= prod_{i=1}^n (r_i!)^(1/r_i)", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "A is an n by n 0-1 matrix; r_i is the sum of row i; terms with r_i=0 are handled by the standard zero-permanent convention.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as per(A) ≤ prod_{i=1}^n (r_i!)^(1/r_i).\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_a5142e0a8d891ef2", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a9f7575d7e1b7944", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "functional analysis", "subfield": "measure theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E4917D5C3396", "name": "Sazonov's theorem", "statement_latex": "sum_{i in I} ||T(e_i)||_H^2 < +infinity", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(sum_{i in I} ||T(e_i)||_H^2)-(+infinity)\\}=0\\;\\land\\;(sum_{i in I} ||T(e_i)||_H^2)\\ne (+infinity)", "to_expression": "sum_{i in I} ||T(e_i)||_H^2 < +infinity", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{i in I} ||T(e_i)||_H^2 < +infinity.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
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{"schema_version": "1.0", "item_id": "rcd2_c06d1c84729e23ee", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3BD55A296126", "name": "Rank–nullity theorem", "statement_latex": "rank(T) + nullity(T) = dim(V)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((rank(T) + nullity(T)),z):z\\in D\\}=\\{((dim(V)),z):z\\in D\\}", "to_expression": "rank(T) + nullity(T) = dim(V)", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as rank(T) + nullity(T) = dim(V).\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_cead953d3b015cbf", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c5c38524366fd674", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_42828E58305E", "name": "Pascal's rule", "statement_latex": "\\binom{n-1}{k} + \\binom{n-1}{k-1} = \\binom{n}{k}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\binom{n}{k}) = (\\binom{n-1}{k} + \\binom{n-1}{k-1})", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\binom{n-1}{k} + \\\\binom{n-1}{k-1} = \\\\binom{n}{k}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\binom{n-1}{k} + \\binom{n-1}{k-1} = \\binom{n}{k}", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\binom{n-1}{k} + \\\\binom{n-1}{k-1} = \\\\binom{n}{k}.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_8cc03de33c4134cb", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal set theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4D5A3A69C1D", "name": "Sauer–Shelah lemma", "statement_latex": "|F| <= sum_{i=0}^{d} binom(n,i)", "assumptions": "F is a family of subsets of an n-element set with VC dimension d."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(|F|)-(sum_{i=0}^{d} binom(n,i))\\}=0", "to_expression": "|F| <= sum_{i=0}^{d} binom(n,i)", "justification": "positive part vanishes iff X≤Y", "preconditions": "F is a family of subsets of an n-element set with VC dimension d.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |F| ≤ sum_{i=0}^{d} binom(n,i).\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_9ee12cd145de9c24", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "enumerative combinatorics", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7EDAF21632BF", "name": "Labelled enumeration theorem", "statement_latex": "F(z,t)=e^{g(z)t}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(F(z,t)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(e^{g(z)t}))", "to_expression": "F(z,t)=e^{g(z)t}", "justification": "each argmin set is a singleton", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as F(z,t)=e^{g(z)t}.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_9aa23d5e8070b221", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "large deviations theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_64a84fa14fa7b249", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "inclusion-exclusion principle", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_58BBE6141124", "name": "Schuette–Nesbitt formula", "statement_latex": "sum_{k=0}^m c_k 1_{N=k} = sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((sum_{k=0}^m c_k 1_{N=k}))=F((sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k))", "to_expression": "sum_{k=0}^m c_k 1_{N=k} = sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k", "justification": "indicator of {X} separates unequal objects", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{k=0}^m c_k 1_{N=k} = sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_99fb5dd51ce82830", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "number theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_051344808DC5", "name": "Gauss congruence", "statement_latex": "sum_{d|n} mu(d)*a_{n/d} ≡ 0 (mod n)", "assumptions": "n is a positive integer; the displayed sum is an integer; mu is the Möbius function."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "(n)\\mid((sum_{d|n} mu(d)*a_{n/d})-(0))", "to_expression": "sum_{d|n} mu(d)*a_{n/d} ≡ 0 (mod n)", "justification": "A≡B mod M iff M divides A-B", "preconditions": "n is a positive integer; the displayed sum is an integer; mu is the Möbius function.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{d|n} mu(d)*a_{n/d} ≡ 0 (mod n).\"]"}], "source_equivalence_identity": "A≡B mod M iff M divides A-B"}
{"schema_version": "1.0", "item_id": "rcd2_8ea493be0b9aa015", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "elementary algebra", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_40DB1B003904", "name": "Binomial theorem", "statement_latex": "(x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(0 = 0) \\Rightarrow ((x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Since 0=0 is true, true=>Phi_T has the same truth value as Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "(x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k", "justification": "Since 0=0 is true, true=>Phi_T has the same truth value as Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "Since 0=0 is true, true=>Phi_T has the same truth value as Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_13e5034c5a067f28", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_7be90b34db9a6f1e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "zeta and L-functions", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_09EFA5455472", "name": "Apéry's theorem", "statement_latex": "\\zeta (3)=\\sum _{n=1}^{\\infty }{\\frac {1}{n^{3}}}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (\\z\\eta (3)=\\sum _{n=1}^{\\infty }{\\frac {1}{n^{3}}}))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\zeta (3)=\\sum _{n=1}^{\\infty }{\\frac {1}{n^{3}}}", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_0f5ce57a91a79c24", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum field theory", "subfield": "lattice field theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7FD1BE25E27D", "name": "Nielsen–Ninomiya theorem", "statement_latex": "H = ∑_{x,y} ψ^†(x) F(x, y) ψ(y)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((H) \\in S \\land (∑_{x,y} ψ^†(x) F(x, y) ψ(y)) \\notin S) \\lor ((∑_{x,y} ψ^†(x) F(x, y) ψ(y)) \\in S \\land (H) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover H = ∑_{x,y} ψ^†(x) F(x, y) ψ(y).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "H = ∑_{x,y} ψ^†(x) F(x, y) ψ(y)", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover H = ∑_{x,y} ψ^†(x) F(x, y) ψ(y).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_d1cfe2d363fc0504", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "scattering theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ac1fe9c976d729db", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "computer science", "subfield": "formal language theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_856cf38f1de11279", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "vector calculus", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_32b6f7fa49fb54cd", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "large deviations theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B2BA1A96D211", "name": "Varadhan's lemma", "statement_latex": "lim_{ε→0} ε log E[exp(φ(Z_ε)/ε)] = sup_{x∈X} (φ(x) - I(x))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(lim_{ε→0} ε log E[exp(φ(Z_ε)/ε)]))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sup_{x∈X} (φ(x) - I(x))))", "to_expression": "lim_{ε→0} ε log E[exp(φ(Z_ε)/ε)] = sup_{x∈X} (φ(x) - I(x))", "justification": "each argmin set is a singleton", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{ε→0} ε log E[exp(φ(Z_ε)/ε)] = sup_{x∈X} (φ(x) - I(x)).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_9f780673148e7eb4", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "physics", "subfield": "general relativity", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2a7953732f9b03a0", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "permutation patterns", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0f7211542e9d2983", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "representation theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CAF4E18AA4C8", "name": "Brauer's height zero conjecture", "statement_latex": "\\nu(\\chi(1)) = \\nu(|G:D|)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((\\nu(\\chi(1))) \\in S \\land (\\nu(|G:D|)) \\notin S) \\lor ((\\nu(|G:D|)) \\in S \\land (\\nu(\\chi(1))) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\nu(\\\\chi(1)) = \\\\nu(|G:D|).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\nu(\\chi(1)) = \\nu(|G:D|)", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\nu(\\\\chi(1)) = \\\\nu(|G:D|).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_56c9f9bae87c9944", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "combinatorics on words", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_08AF905E8677", "name": "Fine and Wilf's theorem", "statement_latex": "a_n = a_{n+gcd(h,k)}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((a_n) \\in S \\land (a_{n+gcd(h,k)}) \\notin S) \\lor ((a_{n+gcd(h,k)}) \\in S \\land (a_n) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover a_n = a_{n+gcd(h,k)}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "a_n = a_{n+gcd(h,k)}", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover a_n = a_{n+gcd(h,k)}.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
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{"schema_version": "1.0", "item_id": "rcd2_85697eddb2d8665d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine geometry", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2AEFBE8A2E69", "name": "Siegel's theorem on integral points", "statement_latex": "g > 0", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(0)-(g)\\}=0\\;\\land\\;(0)\\ne (g)", "to_expression": "g > 0", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as g > 0.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_c9194588b8b63e63", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "cryptography", "subfield": "cryptanalysis", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E35608C3DE5E", "name": "Piling-up lemma", "statement_latex": "ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn) = 2^{n-1}∏_{i=1}^{n}ε(Xi)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn)),(2^{n-1}∏_{i=1}^{n}ε(Xi)))=0", "to_expression": "ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn) = 2^{n-1}∏_{i=1}^{n}ε(Xi)", "justification": "zero discrete distance iff equality", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn) = 2^{n-1}∏_{i=1}^{n}ε(Xi).\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_8566a262f82dd678", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "quadratic forms", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_434E3E7B6A54", "name": "Sylvester's law of inertia", "statement_latex": "B = SAS^T", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(B))+\\delta_D(z,(SAS^T)))=0", "to_expression": "B = SAS^T", "justification": "zero loss iff some z equals both", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as B = SAS^T.\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_24d80d4d36782108", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "computer science", "subfield": "probabilistic combinatorics", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AA4BF27A6AAD", "name": "Algorithmic Lovász local lemma", "statement_latex": "Pr[A] <= x(A) * Product_{B in Gamma(A)} (1 - x(B))", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((Pr[A])-(x(A) * Product_{B in Gamma(A)} (1 - x(B))))=0", "to_expression": "Pr[A] <= x(A) * Product_{B in Gamma(A)} (1 - x(B))", "justification": "sup equals max(0,X-Y)", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr[A] ≤ x(A) * Product_{B in Gamma(A)} (1 - x(B)).\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
{"schema_version": "1.0", "item_id": "rcd2_3c665f7ca2b051ee", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "classical mechanics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BC55A9432F80", "name": "Helmholtz theorem (classical mechanics)", "statement_latex": "dS = (dE + PdV) / T", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(dS) \\in {((dE + PdV) / T)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover dS = (dE + PdV) / T.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "dS = (dE + PdV) / T", "justification": "L in {R} iff L = R.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover dS = (dE + PdV) / T.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
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{"schema_version": "1.0", "item_id": "rcd2_3b00bcfd8e98ddd4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "complex analysis", "subfield": "complex geometry", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1CFF4C1B7932", "name": "Andreotti–Grauert theorem", "statement_latex": "dim_C H^i(X, F) < infinity", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{>0}\\;\\text{such that}\\;(dim_C H^i(X, F))+s=(infinity)", "to_expression": "dim_C H^i(X, F) < infinity", "justification": "positive slack iff X<Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as dim_C H^i(X, F) < infinity.\"]"}], "source_equivalence_identity": "positive slack iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_7c963bcb56d47806", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal set theory", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_5515aee15671d8f4", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "symbolic dynamics", "subfield": "cellular automata", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e2351c47a6717814", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C95D5999DB6B", "name": "Nullity theorem", "statement_latex": "nullity A = nullity H", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(nullity A)=z\\;\\land\\;z=(nullity H)", "to_expression": "nullity A = nullity H", "justification": "common witness iff equality", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as nullity A = nullity H.\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_39238425ed0b1bed", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "prime numbers", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_71AA6DBCD731", "name": "Bonse's inequality", "statement_latex": "p_n# > p_{n+1}^2", "assumptions": "n>=4; p_n# is the product of the first n primes; p_{n+1} is the next prime."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(p_{n+1}^2)-(p_n#)\\}=0\\;\\land\\;(p_{n+1}^2)\\ne (p_n#)", "to_expression": "p_n# > p_{n+1}^2", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "n>=4; p_n# is the product of the first n primes; p_{n+1} is the next prime.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p_n# > p_{n+1}^2.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_acb5ce45214c45fc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CCCCB5088787", "name": "Mean value theorem", "statement_latex": "f'(c) = (f(b)-f(a))/(b-a)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((f'(c)))=F((f(b)-f(a))/(b-a))", "to_expression": "f'(c) = (f(b)-f(a))/(b-a)", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f'(c) = (f(b)-f(a))/(b-a).\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_5763f647425dd113", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "inner product spaces", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2EE2B752D03D", "name": "Polarization identity", "statement_latex": "\\operatorname{Re} \\langle x, y \\rangle = \\frac{1}{2} (\\|x + y\\|^2 - \\|x\\|^2 - \\|y\\|^2)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(\\operatorname{Re} \\langle x, y \\rangle))+\\delta_D(z,(\\frac{1}{2} (\\|x + y\\|^2 - \\|x\\|^2 - \\|y\\|^2))))=0", "to_expression": "\\operatorname{Re} \\langle x, y \\rangle = \\frac{1}{2} (\\|x + y\\|^2 - \\|x\\|^2 - \\|y\\|^2)", "justification": "zero loss iff some z equals both", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\operatorname{Re} \\\\langle x, y \\\\rangle = \\\\frac{1}{2} (\\\\|x + y\\\\|^2 - \\\\|x\\\\|^2 - \\\\|y\\\\|^2).\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_08079f15e4468385", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum field theory", "subfield": "axiomatic quantum field theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FEF97AED413B", "name": "Cluster decomposition property", "statement_latex": "S_βα → S_β1α1 * S_β2α2", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (S_βα → S_β1α1 * S_β2α2))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "S_βα → S_β1α1 * S_β2α2", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_11efef3c43870db7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_95775DC75EC2", "name": "Oppenheim conjecture", "statement_latex": "|Q(x)| < \\epsilon", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\\\varepsilon) - (|Q(x)|) > 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |Q(x)| < \\\\epsilon.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |Q(x)| < \\\\epsilon.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "|Q(x)| < \\epsilon", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |Q(x)| < \\\\epsilon.\"]"}], "source_equivalence_identity": "A<B iff B-A>0 over finite real quantities."}
{"schema_version": "1.0", "item_id": "rcd2_1fe7d8fe8747ac4e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "symplectic geometry", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C62ED9627235", "name": "Williamson theorem", "statement_latex": "SMS^T = D ⊕ D", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((SMS^T) \\in S \\land (D ⊕ D) \\notin S) \\lor ((D ⊕ D) \\in S \\land (SMS^T) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover SMS^T = D ⊕ D.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "SMS^T = D ⊕ D", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover SMS^T = D ⊕ D.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
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{"schema_version": "1.0", "item_id": "rcd2_b11d502bd3c59b9e", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_bb8bcdabe524a017", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3BD55A296126", "name": "Rank–nullity theorem", "statement_latex": "rank(T) + nullity(T) = dim(V)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(rank(T) + nullity(T)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(dim(V)))", "to_expression": "rank(T) + nullity(T) = dim(V)", "justification": "each argmin set is a singleton", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as rank(T) + nullity(T) = dim(V).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_680356474e4c0615", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "numerical analysis", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_DC75D2014574", "name": "Céa's lemma", "statement_latex": "||u - u_h|| <= (gamma / alpha) * ||u - v||", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;((gamma / alpha) * ||u - v||)\\le t\\Rightarrow (||u - u_h||)\\le t", "to_expression": "||u - u_h|| <= (gamma / alpha) * ||u - v||", "justification": "converse uses t=Y", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ||u - u_h|| ≤ (gamma / alpha) * ||u - v||.\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_8ba386921dddd51b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "prime numbers", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c7efa83bf8bb8aa7", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "algebra", "subfield": "combinatorics", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b2e7f49abda5e83f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "harmonic analysis", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_67b50aeed184a445", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_7b11dc25659205be", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5A3D942AED87", "name": "Markov's inequality", "statement_latex": "P(X >= a) <= E[X] / a", "assumptions": "X is a nonnegative random variable; a>0; E[X] is finite."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(E[X] / a)\\le t\\Rightarrow (P(X ≥ a))\\le t", "to_expression": "P(X >= a) <= E[X] / a", "justification": "converse uses t=Y", "preconditions": "X is a nonnegative random variable; a>0; E[X] is finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(X ≥ a) ≤ E[X] / a.\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_38a8a33e67d1be48", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "theoretical computer science", "subfield": "algebraic complexity theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_46E2A92077B9", "name": "Schwartz–Zippel lemma", "statement_latex": "Pr[P(r1, ..., rn) = 0] ≤ d/|S|", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(d/|S|) - (Pr[P(r1, ..., rn) = 0]) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Pr[P(r1, ..., rn) = 0] <= d/|S|.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Pr[P(r1, ..., rn) = 0] <= d/|S|.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "Pr[P(r1, ..., rn) = 0] ≤ d/|S|", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Pr[P(r1, ..., rn) = 0] <= d/|S|.\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_d6b6f61bc5bf1885", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "combinatorics", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A5B93C0856B4", "name": "Handshaking lemma", "statement_latex": "Σ_{v∈V} deg(v) = 2|E|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(Σ_{v∈V} deg(v)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(2|E|))", "to_expression": "Σ_{v∈V} deg(v) = 2|E|", "justification": "each argmin set is a singleton", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Σ_{v∈V} deg(v) = 2|E|.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_d773e239e2e4ddd7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "Ramsey theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8E938CA76307", "name": "Milliken's tree theorem", "statement_latex": "S_T^n = C_1 ∪ ... ∪ C_r", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "(\\{(S_T^n)\\}\\times\\{(C_1 ∪ ... ∪ C_r)\\})\\cap\\{(z,z):z\\in D\\}=\\{((S_T^n),(C_1 ∪ ... ∪ C_r))\\}", "to_expression": "S_T^n = C_1 ∪ ... ∪ C_r", "justification": "intersection is singleton iff pair diagonal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as S_T^n = C_1 ∪ ... ∪ C_r.\"]"}], "source_equivalence_identity": "intersection is singleton iff pair diagonal"}
{"schema_version": "1.0", "item_id": "rcd2_568d4eb829335103", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "p-adic analysis", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3B924C720B22", "name": "Krasner's lemma", "statement_latex": "|alpha - beta| < |alpha - alpha_i|", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (|alpha - b\\eta|) < x \\land x < (|alpha - alpha_i|)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |alpha - beta| < |alpha - alpha_i|.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |alpha - beta| < |alpha - alpha_i|.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "|alpha - beta| < |alpha - alpha_i|", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |alpha - beta| < |alpha - alpha_i|.\"]"}], "source_equivalence_identity": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B."}
{"schema_version": "1.0", "item_id": "rcd2_e15a9e13f2df41ea", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum optics", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BFC6929833DF", "name": "Optical equivalence theorem", "statement_latex": "\\operatorname{tr}(\\hat{\\rho} \\cdot g_{\\Omega}(\\hat{a}, \\hat{a}^{\\dagger})) = \\int f_{\\bar{\\Omega}}(\\alpha, \\alpha^*) g_{\\Omega}(\\alpha, \\alpha^*) d^2\\alpha", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(\\operatorname{tr}(\\hat{\\rho} \\cdot g_{\\Omega}(\\hat{a}, \\hat{a}^{\\dagger})))=z\\;\\land\\;z=(\\int f_{\\bar{\\Omega}}(\\alpha, \\alpha^*) g_{\\Omega}(\\alpha, \\alpha^*) d^2\\alpha)", "to_expression": "\\operatorname{tr}(\\hat{\\rho} \\cdot g_{\\Omega}(\\hat{a}, \\hat{a}^{\\dagger})) = \\int f_{\\bar{\\Omega}}(\\alpha, \\alpha^*) g_{\\Omega}(\\alpha, \\alpha^*) d^2\\alpha", "justification": "common witness iff equality", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\operatorname{tr}(\\\\hat{\\\\rho} \\\\cdot g_{\\\\Omega}(\\\\hat{a}, \\\\hat{a}^{\\\\dagger})) = \\\\int f_{\\\\bar{\\\\Omega}}(\\\\alpha, \\\\alpha^*) g_{\\\\Omega}(\\\\alpha, \\\\alpha^*) d^2\\\\alpha.\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_50369d21c415ea3d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F01A12C6F5E3", "name": "Taylor's theorem", "statement_latex": "f(x) = sum_{n=0}^k (f^(n)(a)/n!) * (x-a)^n + R_k(x)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(f(x) = sum_{n=0}^k (f^(n)(a)/n!) * (x-a)^n + R_k(x)) \\lor (0 = 1)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "0=1 is false, so Phi_T or false iff Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f(x) = sum_{n=0}^k (f^(n)(a)/n!) * (x-a)^n + R_k(x)", "justification": "0=1 is false, so Phi_T or false iff Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "0=1 is false, so Phi_T or false iff Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_6e70cb42866e735f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "polynomial factorization", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_299359E3C599", "name": "Sophie Germain's identity", "statement_latex": "x^{4}+4y^{4}=(x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2})", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((x^{4}+4y^{4}))=F((x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2}))", "to_expression": "x^{4}+4y^{4}=(x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2})", "justification": "indicator of {X} separates unequal objects", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as x^{4}+4y^{4}=(x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2}).\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_018154856f3daf04", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "special functions", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CA3873AC4ACD", "name": "Ramanujan's master theorem", "statement_latex": "∫[0,∞] x^{s-1} f(x) dx = Γ(s) φ(-s)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(∫[0,∞] x^{s-1} f(x) dx)=z\\;\\land\\;z=(Γ(s) φ(-s))", "to_expression": "∫[0,∞] x^{s-1} f(x) dx = Γ(s) φ(-s)", "justification": "common witness iff equality", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∫[0,∞] x^{s-1} f(x) dx = Γ(s) φ(-s).\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_badf58f37c147a02", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematical economics", "subfield": "linear algebra", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e516e7ec964bfd22", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "calculus of variations", "subfield": "Morse theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5B2379DEDFD8", "name": "Morse–Palais lemma", "statement_latex": "f(x) = ⟨Aφ(x), φ(x)⟩", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((f(x)) \\in S \\land (⟨Aφ(x), φ(x)⟩) \\notin S) \\lor ((⟨Aφ(x), φ(x)⟩) \\in S \\land (f(x)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(x) = ⟨Aφ(x), φ(x)⟩.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f(x) = ⟨Aφ(x), φ(x)⟩", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(x) = ⟨Aφ(x), φ(x)⟩.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_550a29aad94f4e93", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_089287f9f67ab480", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematical analysis", "subfield": "number theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_9e2f334c5051e805", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "topology", "subfield": "algebraic topology", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2878c8f968a5c67f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "time-reversal symmetry", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BACF4A4770A1", "name": "Kramers' theorem", "statement_latex": "T^2 = -1", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(T^2) \\in {(-1)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover T^2 = -1.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "T^2 = -1", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover T^2 = -1.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_7e77f6b3b43e183e", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "hypergeometric series", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_74a89e6a4966799e", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "zeta and L-functions", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f0b2bd3a1ec94e45", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "percolation theory", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2633491087ae9974", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "vector calculus", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6b5a8ab35187c6e1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "measure theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_448CEF326F89", "name": "Doob–Dynkin lemma", "statement_latex": "f = g ∘ T", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(f))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(g ∘ T))", "to_expression": "f = g ∘ T", "justification": "each argmin set is a singleton", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f = g ∘ T.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_ae9644319cbe8474", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "topology", "subfield": "combinatorial topology", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_26bd1eec0d2e1325", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "Kleinian groups", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C0AAA35160EF", "name": "Jørgensen's inequality", "statement_latex": "|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2| >= 1", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (1)\\}", "to_expression": "|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2| >= 1", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2| ≥ 1.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_723cf757a16646f7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "probabilistic combinatorics", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7C9C39383770", "name": "Ajtai–Komlós–Tusnády theorem", "statement_latex": "P(C_1 n log n < inf_{σ ∈ S_n} sum_{k=1}^n |X_k - Y_{σ(k)}| < C_2 n log n) = 1", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((P(C_1 n log n < inf_{σ ∈ S_n} sum_{k=1}^n |X_k - Y_{σ(k)}| < C_2 n log n)),z):z\\in D\\}=\\{((1),z):z\\in D\\}", "to_expression": "P(C_1 n log n < inf_{σ ∈ S_n} sum_{k=1}^n |X_k - Y_{σ(k)}| < C_2 n log n) = 1", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(C_1 n log n < inf_{σ ∈ S_n} sum_{k=1}^n |X_k - Y_{σ(k)}| < C_2 n log n) = 1.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
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{"schema_version": "1.0", "item_id": "rcd2_c6b7c69e38ce04eb", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematical economics", "subfield": "linear algebra", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1A4ADB6F0297", "name": "Hawkins–Simon condition", "statement_latex": "[I - A]x = d", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;(([I - A]x),z)=((d),z)", "to_expression": "[I - A]x = d", "justification": "pair equality forces first coordinates equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as [I - A]x = d.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_7abbcee1a3fcde91", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2B79341A5DAC", "name": "Doob's martingale inequality", "statement_latex": "P[max_{1<=i<=n} X_i >= C] <= E[max(X_n, 0)] / C", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(P[max_{1≤i≤n} X_i ≥ C])}{\\exp(P[max_{1≤i≤n} X_i ≥ C])+\\exp(E[max(X_n, 0)] / C)}\\le\\frac{1}{2}", "to_expression": "P[max_{1<=i<=n} X_i >= C] <= E[max(X_n, 0)] / C", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P[max_{1≤i≤n} X_i ≥ C] ≤ E[max(X_n, 0)] / C.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_c6654c3176563cbf", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "complex analysis", "subfield": "complex geometry", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A0642B3B222A", "name": "Schwarz lemma", "statement_latex": "|f(z)| <= |z|", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(|z|)\\le t\\Rightarrow (|f(z)|)\\le t", "to_expression": "|f(z)| <= |z|", "justification": "converse uses t=Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |f(z)| ≤ |z|.\"]"}], "source_equivalence_identity": "converse uses t=Y"}
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{"schema_version": "1.0", "item_id": "rcd2_b7a3d27efac4aa4d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "incidence geometry", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B3E12B4FAA4C", "name": "Loomis–Whitney inequality", "statement_latex": "|| \\prod_{j=1}^{d} g_j \\circ \\pi_j ||_{L^1} \\le \\prod_{j=1}^{d} ||g_j||_{L^{d-1}}", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\prod_{j=1}^{d} ||g_j||_{L^{d-1}}) - (|| \\prod_{j=1}^{d} g_j \\circ \\pi_j ||_{L^1}) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover || \\\\prod_{j=1}^{d} g_j \\\\circ \\\\pi_j ||_{L^1} <= \\\\prod_{j=1}^{d} ||g_j||_{L^{d-1}}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover || \\\\prod_{j=1}^{d} g_j \\\\circ \\\\pi_j ||_{L^1} <= \\\\prod_{j=1}^{d} ||g_j||_{L^{d-1}}.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "|| \\prod_{j=1}^{d} g_j \\circ \\pi_j ||_{L^1} \\le \\prod_{j=1}^{d} ||g_j||_{L^{d-1}}", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover || \\\\prod_{j=1}^{d} g_j \\\\circ \\\\pi_j ||_{L^1} <= \\\\prod_{j=1}^{d} ||g_j||_{L^{d-1}}.\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_84e0b1a9f005abf8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "field theory", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_ED1CB49A03EE", "name": "Steinitz's theorem", "statement_latex": "L = K(α)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(L)=z\\;\\land\\;z=(K(α))", "to_expression": "L = K(α)", "justification": "common witness iff equality", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as L = K(α).\"]"}], "source_equivalence_identity": "common witness iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_77d65605b6131c29", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_763D96A1B9F8", "name": "Rouché–Capelli theorem", "statement_latex": "rank(A) = rank([A|b])", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((rank(A)))=F((rank([A|b])))", "to_expression": "rank(A) = rank([A|b])", "justification": "indicator of {X} separates unequal objects", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as rank(A) = rank([A|b]).\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_6b2f18d6baaacbc1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_303E62405496", "name": "Lorden's inequality", "statement_latex": "E(R_b) <= E(X^2) / E(X)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (E(R_b)) \\le x \\land x \\le (E(X^2) / E(X))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover E(R_b) <= E(X^2) / E(X).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover E(R_b) <= E(X^2) / E(X).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "E(R_b) <= E(X^2) / E(X)", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover E(R_b) <= E(X^2) / E(X).\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
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{"schema_version": "1.0", "item_id": "rcd2_f6b71545fdcb7107", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "measure theory", "subfield": "stochastic analysis", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FB5E9395CC7C", "name": "Malliavin's absolute continuity lemma", "statement_latex": "|∫ Dφ(y)(x) dμ(y)| ≤ C(x) ‖φ‖_{∞}", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(|∫ Dφ(y)(x) dμ(y)|)+s=(C(x) ‖φ‖_{∞})", "to_expression": "|∫ Dφ(y)(x) dμ(y)| ≤ C(x) ‖φ‖_{∞}", "justification": "nonnegative slack exists iff X≤Y", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |∫ Dφ(y)(x) dμ(y)| ≤ C(x) ‖φ‖_{∞}.\"]"}], "source_equivalence_identity": "nonnegative slack exists iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_634364748a90a118", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "Sobolev spaces", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_35353A67F17E", "name": "Pólya–Szegő inequality", "statement_latex": "∫ |∇u*|^p dx <= ∫ |∇u|^p dx", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(∫ |∇u*|^p dx)-(∫ |∇u|^p dx)\\}=0", "to_expression": "∫ |∇u*|^p dx <= ∫ |∇u|^p dx", "justification": "positive part vanishes iff X≤Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∫ |∇u*|^p dx ≤ ∫ |∇u|^p dx.\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_0ebd7cb5404107f0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_936D29F3BF52", "name": "Hsu–Robbins–Erdős theorem", "statement_latex": "∑_{n>=1}P(|S_{n}|>εn)<∞", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;t\\le (∑_{n≥1}P(|S_{n}|>εn))\\Rightarrow t<(∞)", "to_expression": "∑_{n>=1}P(|S_{n}|>εn)<∞", "justification": "converse uses t=X", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∑_{n≥1}P(|S_{n}|>εn)<∞.\"]"}], "source_equivalence_identity": "converse uses t=X"}
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{"schema_version": "1.0", "item_id": "rcd2_21bc8da7af836da8", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "algebra", "subfield": "polynomials", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_1d460e500f94c730", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5795F6B3279F", "name": "Chain rule (probability)", "statement_latex": "P(A ∩ B) = P(B|A)*P(A)", "assumptions": "A and B are events with P(A)>0 for the displayed conditional form."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((P(A ∩ B)),z)=((P(B|A)*P(A)),z)", "to_expression": "P(A ∩ B) = P(B|A)*P(A)", "justification": "pair equality forces first coordinates equal", "preconditions": "A and B are events with P(A)>0 for the displayed conditional form.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(A ∩ B) = P(B|A)*P(A).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
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{"schema_version": "1.0", "item_id": "rcd2_bf6f6822a7fddd2c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "large deviations theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ec9c3d1531b1033a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "combinatorics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3596847807A0", "name": "Coupon collector's problem", "statement_latex": "E(T) = n H_n", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(E(T)) \\in {(n H_n)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E(T) = n H_n.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "E(T) = n H_n", "justification": "L in {R} iff L = R.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E(T) = n H_n.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_7287bdc4e2a4b3c6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "complex analysis", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B6DBB23DD9DF", "name": "Phragmén–Lindelöf principle", "statement_latex": "|f(z)| < M", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(|f(z)|)-(M)\\}=0\\;\\land\\;(|f(z)|)\\ne (M)", "to_expression": "|f(z)| < M", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |f(z)| < M.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_9221229f6927e058", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D13DCB7F188A", "name": "Kuznetsov trace formula", "statement_latex": "∑_{c≡0 mod N}c^{-r}K(m,n,c)g(4π√mn/c)=Integral transform+Spectral terms", "assumptions": "All displayed arithmetic expressions are integers; the modulus is a positive integer."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(∑_{c≡0 mod N}c^{-r}K(m,n,c)g(4π√mn/c))=z\\;\\land\\;z=(Integral transform+Spectral terms)", "to_expression": "∑_{c≡0 mod N}c^{-r}K(m,n,c)g(4π√mn/c)=Integral transform+Spectral terms", "justification": "common witness iff equality", "preconditions": "All displayed arithmetic expressions are integers; the modulus is a positive integer.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∑_{c≡0 mod N}c^{-r}K(m,n,c)g(4π√mn/c)=Integral transform+Spectral terms.\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_76ce52916005f7d8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "coding theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F6DC10428DD6", "name": "Z-channel capacity", "statement_latex": "cap(Z) = H(p/(1+2^s))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(cap(Z)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(H(p/(1+2^s))))", "to_expression": "cap(Z) = H(p/(1+2^s))", "justification": "each argmin set is a singleton", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as cap(Z) = H(p/(1+2^s)).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_5e3b3d503d3fca92", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "statistical mechanics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_63998BA24BBA", "name": "Fluctuation theorem", "statement_latex": "Pr(Σ_t=A) / Pr(Σ_t=-A) = e^(At)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((Pr(Σ_t=A) / Pr(Σ_t=-A)),(e^(At)))=0", "to_expression": "Pr(Σ_t=A) / Pr(Σ_t=-A) = e^(At)", "justification": "zero discrete distance iff equality", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr(Σ_t=A) / Pr(Σ_t=-A) = e^(At).\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_476bdef8512d54e1", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "approximation theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d12f5ba42d0e1c3d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "Combinatorics", "subfield": "Order theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B5F5A5581E5D", "name": "Kruskal's tree theorem", "statement_latex": "T1 <= T2", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (T2)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (T1)\\}", "to_expression": "T1 <= T2", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as T1 ≤ T2.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_952aa210a64a182e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_89D2662A27BE", "name": "Freiman's theorem", "statement_latex": "|A+A| <= K|A|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (K|A|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|A+A|)\\}", "to_expression": "|A+A| <= K|A|", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |A+A| ≤ K|A|.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_4d8d747a176517b4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "calculus", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3967E74F7B6F", "name": "Increment theorem", "statement_latex": "Δy = f'(x)Δx + εΔx", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(f'(x)Δx + εΔx) = (Δy)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover Δy = f'(x)Δx + εΔx.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "Δy = f'(x)Δx + εΔx", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover Δy = f'(x)Δx + εΔx.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_b45f83cfcaa3046f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "Ramsey theory", "subfield": "discrete mathematics", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_66e56adce49bf3cc", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f01e9eb16c28f1da", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "algebra", "subfield": "polynomials", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_1ede75f36b492471", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "number theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A54D9D349D9E", "name": "Candido's identity", "statement_latex": "[x^2 + y^2 + (x+y)^2]^2 = 2[x^4 + y^4 + (x+y)^4]", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;(([x^2 + y^2 + (x+y)^2]^2),z)=((2[x^4 + y^4 + (x+y)^4]),z)", "to_expression": "[x^2 + y^2 + (x+y)^2]^2 = 2[x^4 + y^4 + (x+y)^4]", "justification": "pair equality forces first coordinates equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as [x^2 + y^2 + (x+y)^2]^2 = 2[x^4 + y^4 + (x+y)^4].\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_ed9b13e997aa522e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "prime number theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BFDCE2A9DBB2", "name": "Bertrand's postulate", "statement_latex": "exists prime p such that n < p < 2*n", "assumptions": "n is an integer greater than 1."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (\\exists prime p such that n < p < 2*n))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "n is an integer greater than 1.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "exists prime p such that n < p < 2*n", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "n is an integer greater than 1.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_849631bb16675719", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "time-reversal symmetry", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BACF4A4770A1", "name": "Kramers' theorem", "statement_latex": "T^2 = -1", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((T^2))=F((-1))", "to_expression": "T^2 = -1", "justification": "indicator of {X} separates unequal objects", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as T^2 = -1.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_1df2771ddc898034", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9DBE3057770F", "name": "Kneser's theorem", "statement_latex": "|A+B|=|A+H|+|B+H|-|H|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((|A+B|),z):z\\in D\\}=\\{((|A+H|+|B+H|-|H|),z):z\\in D\\}", "to_expression": "|A+B|=|A+H|+|B+H|-|H|", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |A+B|=|A+H|+|B+H|-|H|.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_756ca1f51135fc19", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "algebra", "subfield": "combinatorics", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E24E6480323C", "name": "Multinomial theorem", "statement_latex": "(x_{1}+x_{2}+\\cdots +x_{m})^{n}=\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{(((x_{1}+x_{2}+\\cdots +x_{m})^{n}),z):z\\in D\\}=\\{((\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}}),z):z\\in D\\}", "to_expression": "(x_{1}+x_{2}+\\cdots +x_{m})^{n}=\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}}", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (x_{1}+x_{2}+\\\\cdots +x_{m})^{n}=\\\\sum _{k_{1}+k_{2}+\\\\cdots +k_{m}=n} \\\\binom{n}{k_1, k_2, \\\\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\\\cdots x_{m}^{k_{m}}.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_5895315e0da5a7fa", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C3C168B97D27", "name": "Watson's quintuple product identity", "statement_latex": "prod_{n>=1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1})", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(prod_{n) \\in (1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n \\\\in \\mathbb{Z}} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1})) + [0,\\infty)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover prod_{n >= 1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1}).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover prod_{n >= 1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1}).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "prod_{n>=1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1})", "justification": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover prod_{n >= 1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1}).\"]"}], "source_equivalence_identity": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_8904a14f6691a743", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "functional analysis", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_fd85631fedc306af", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "algebra", "subfield": "combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_24005F6E3998", "name": "Abel's binomial theorem", "statement_latex": "Σ [k=0 to m] (m choose k) * (w + m - k)^(m - k - 1) * (z + k)^k = w^(-1) * (z + w + m)^m", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (Σ [k=0 to m] (m choose k) * (w + m - k)^(m - k - 1) * (z + k)^k = w^(-1) * (z + w + m)^m))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "Σ [k=0 to m] (m choose k) * (w + m - k)^(m - k - 1) * (z + k)^k = w^(-1) * (z + w + m)^m", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_b5538e71221e6863", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_83A41482F4E9", "name": "Fekete's lemma", "statement_latex": "lim_{n→∞} (a_n / n) = inf_{n∈N} (a_n / n)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(lim_{n→∞} (a_n / n)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(inf_{n∈N} (a_n / n)))", "to_expression": "lim_{n→∞} (a_n / n) = inf_{n∈N} (a_n / n)", "justification": "each argmin set is a singleton", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{n→∞} (a_n / n) = inf_{n∈N} (a_n / n).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_5f438d6ffdc27107", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "computer science", "subfield": "probabilistic combinatorics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AA4BF27A6AAD", "name": "Algorithmic Lovász local lemma", "statement_latex": "Pr[A] <= x(A) * Product_{B in Gamma(A)} (1 - x(B))", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (x(A) * Product_{B in Gamma(A)} (1 - x(B)))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (Pr[A])\\}", "to_expression": "Pr[A] <= x(A) * Product_{B in Gamma(A)} (1 - x(B))", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr[A] ≤ x(A) * Product_{B in Gamma(A)} (1 - x(B)).\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_571ea2805cad316a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "Number theory", "subfield": "Elliptic functions", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_78B717A820D4", "name": "Jacobi triple product", "statement_latex": "product_{m=1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2}) = sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((product_{m=1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2})),(sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n}))=0", "to_expression": "product_{m=1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2}) = sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n}", "justification": "zero discrete distance iff equality", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as product_{m=1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2}) = sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n}.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_ee877ca0fe3e960a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "combinatorics", "subfield": "combinatorial geometry", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_23E48784FFFD", "name": "Littlewood–Offord problem", "statement_latex": "P(S_n in A) <= binom(n,floor(n/2))/2^n", "assumptions": "S_n is the signed sum in the Littlewood-Offord setup; A is an interval or target set as specified by the theorem; n is positive."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(binom(n,floor(n/2))/2^n)\\le t\\Rightarrow (P(S_n in A))\\le t", "to_expression": "P(S_n in A) <= binom(n,floor(n/2))/2^n", "justification": "converse uses t=Y", "preconditions": "S_n is the signed sum in the Littlewood-Offord setup; A is an interval or target set as specified by the theorem; n is positive.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(S_n in A) ≤ binom(n,floor(n/2))/2^n.\"]"}], "source_equivalence_identity": "converse uses t=Y"}
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{"schema_version": "1.0", "item_id": "rcd2_aa3e8fe89b61fd08", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "calculus", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_fc5bf0dffd93e787", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_038d0726487a858a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal combinatorics", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C321BE7114E8", "name": "Ahlswede–Khachatrian theorem", "statement_latex": "|F| <= max_{r: t+2r <= n} |F_{n,k,t,r}|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(|F|)}{\\exp(|F|)+\\exp(max_{r: t+2r ≤ n} |F_{n,k,t,r}|)}\\le\\frac{1}{2}", "to_expression": "|F| <= max_{r: t+2r <= n} |F_{n,k,t,r}|", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |F| ≤ max_{r: t+2r ≤ n} |F_{n,k,t,r}|.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_25a81df5efdec9b4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_DB1D004A0930", "name": "Cramer's rule", "statement_latex": "x_i = \\det(A_i) / \\det(A)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((x_i),z)=((\\det(A_i) / \\det(A)),z)", "to_expression": "x_i = \\det(A_i) / \\det(A)", "justification": "pair equality forces first coordinates equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as x_i = \\\\det(A_i) / \\\\det(A).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_faea42079c3bf9ef", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "field theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f217d2918a27aa75", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_69A821A9D880", "name": "Hockey-stick identity", "statement_latex": "\\sum_{i=r}^{n} \\binom{i}{r} = \\binom{n+1}{r+1}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\sum_{i=r}^{n} \\binom{i}{r}))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\binom{n+1}{r+1}))", "to_expression": "\\sum_{i=r}^{n} \\binom{i}{r} = \\binom{n+1}{r+1}", "justification": "each argmin set is a singleton", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\sum_{i=r}^{n} \\\\binom{i}{r} = \\\\binom{n+1}{r+1}.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_54c3638114213ea6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "multilinear algebra", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F538F79B0750", "name": "Lagrange's identity", "statement_latex": "(\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2 = \\sum_{i<j} (a_i b_j - a_j b_i)^2", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(j} (a_i b_j - a_j b_i)^2) - ((\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2 = \\sum_{i) > 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (\\\\sum a_k^2)(\\\\sum b_k^2) - (\\\\sum a_k b_k)^2 = \\\\sum_{i < j} (a_i b_j - a_j b_i)^2.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (\\\\sum a_k^2)(\\\\sum b_k^2) - (\\\\sum a_k b_k)^2 = \\\\sum_{i < j} (a_i b_j - a_j b_i)^2.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "(\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2 = \\sum_{i<j} (a_i b_j - a_j b_i)^2", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (\\\\sum a_k^2)(\\\\sum b_k^2) - (\\\\sum a_k b_k)^2 = \\\\sum_{i < j} (a_i b_j - a_j b_i)^2.\"]"}], "source_equivalence_identity": "A<B iff B-A>0 over finite real quantities."}
{"schema_version": "1.0", "item_id": "rcd2_3603d890199c57a2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B664293AE8AA", "name": "Etemadi's inequality", "statement_latex": "Pr(max_{1<=k<=n} |S_k| >= 3alpha) <= 3 max_{1<=k<=n} Pr(|S_k| >= alpha)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(Pr(max_{1≤k≤n} |S_k| ≥ 3alpha))-(3 max_{1≤k≤n} Pr(|S_k| ≥ alpha))\\}=0", "to_expression": "Pr(max_{1<=k<=n} |S_k| >= 3alpha) <= 3 max_{1<=k<=n} Pr(|S_k| >= alpha)", "justification": "positive part vanishes iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr(max_{1≤k≤n} |S_k| ≥ 3alpha) ≤ 3 max_{1≤k≤n} Pr(|S_k| ≥ alpha).\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_b707f2d18b1c7603", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "measure theory", "subfield": "real analysis", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8351934B4991", "name": "Egorov's theorem", "statement_latex": "\\mu(B) < \\varepsilon", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(\\mu(B))-(\\varepsilon)\\}=0\\;\\land\\;(\\mu(B))\\ne (\\varepsilon)", "to_expression": "\\mu(B) < \\varepsilon", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\mu(B) < \\\\varepsilon.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
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{"schema_version": "1.0", "item_id": "rcd2_7c3981b40a84f176", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "theoretical computer science", "subfield": "algebraic complexity theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d950260e0fe06168", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum mechanics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9078CE9B3CD4", "name": "Ehrenfest theorem", "statement_latex": "F = -V'(x)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((F),(-V'(x)))=0", "to_expression": "F = -V'(x)", "justification": "zero discrete distance iff equality", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as F = -V'(x).\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_bc0cd649fd2d1ea9", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "modular arithmetic", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_1b264a4e53d2e58e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "infinite-dimensional vector spaces", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B23BCA4AB254", "name": "Erdős–Kaplansky theorem", "statement_latex": "\\dim(E^*) = \\text{card}(\\mathbb{K}^I)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((\\dim(E^*)) \\in S \\land (\\text{card}(\\mathbb{K}^I)) \\notin S) \\lor ((\\text{card}(\\mathbb{K}^I)) \\in S \\land (\\dim(E^*)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\dim(E^*) = \\\\text{card}(\\\\mathbb{K}^I).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\dim(E^*) = \\text{card}(\\mathbb{K}^I)", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\dim(E^*) = \\\\text{card}(\\\\mathbb{K}^I).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
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{"schema_version": "1.0", "item_id": "rcd2_bda6f46a48027d50", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4273AD2C24D7", "name": "Bayes' theorem", "statement_latex": "P(A|B) = P(B|A)*P(A)/P(B)", "assumptions": "A and B are events with P(B)>0; conditional probabilities are defined."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((P(A|B)),(P(B|A)*P(A)/P(B)))=0", "to_expression": "P(A|B) = P(B|A)*P(A)/P(B)", "justification": "zero discrete distance iff equality", "preconditions": "A and B are events with P(B)>0; conditional probabilities are defined.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(A|B) = P(B|A)*P(A)/P(B).\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_14826c326bd44254", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "statistical mechanics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4D516E39AEB4", "name": "H-theorem", "statement_latex": "H(t) = ∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(H(t)) \\in {(∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover H(t) = ∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "H(t) = ∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE", "justification": "L in {R} iff L = R.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover H(t) = ∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_de30a881468d0dc2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "queueing theory", "subfield": "probability", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_91967E3D2C5F", "name": "Little's law", "statement_latex": "L = lambda*W", "assumptions": "The queueing system is stable and in steady state; L, lambda, and W are finite real quantities."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(L))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(lambda*W))", "to_expression": "L = lambda*W", "justification": "each argmin set is a singleton", "preconditions": "The queueing system is stable and in steady state; L, lambda, and W are finite real quantities.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as L = lambda*W.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_c115df5ca69d955e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "functional analysis", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_55A9511267AC", "name": "Bobkov's inequality", "statement_latex": "I(integral f d gamma^n) <= integral sqrt(I(f)^2 + |grad f|^2) d gamma^n", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (integral sqrt(I(f)^2 + |grad f|^2) d gamma^n)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (I(integral f d gamma^n))\\}", "to_expression": "I(integral f d gamma^n) <= integral sqrt(I(f)^2 + |grad f|^2) d gamma^n", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as I(integral f d gamma^n) ≤ integral sqrt(I(f)^2 + |grad f|^2) d gamma^n.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_f40743c39794bbb7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_94B20D8A1D74", "name": "Chebyshev's inequality", "statement_latex": "Pr(|X-mu| >= k*sigma) <= 1/k^2", "assumptions": "X has mean mu and finite nonzero standard deviation sigma; k>0."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((Pr(|X-mu| ≥ k*sigma))-(1/k^2))=0", "to_expression": "Pr(|X-mu| >= k*sigma) <= 1/k^2", "justification": "sup equals max(0,X-Y)", "preconditions": "X has mean mu and finite nonzero standard deviation sigma; k>0.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr(|X-mu| ≥ k*sigma) ≤ 1/k^2.\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
{"schema_version": "1.0", "item_id": "rcd2_45c0cdcdb1531f6f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "discrepancy theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E6FD55ACB598", "name": "Beck–Fiala theorem", "statement_latex": "|x(S_j)| <= 2t - 1", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((|x(S_j)|)-(2t - 1),(-\\infty,0])=0", "to_expression": "|x(S_j)| <= 2t - 1", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |x(S_j)| ≤ 2t - 1.\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
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{"schema_version": "1.0", "item_id": "rcd2_137bd49389530c79", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "geometry, probability theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3B9F2B85DC25", "name": "Gaussian isoperimetric inequality", "statement_latex": "liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)} >= φ(Φ^{-1}(γ^n(A)))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(φ(Φ^{-1}(γ^n(A))))}{\\exp(φ(Φ^{-1}(γ^n(A))))+\\exp(liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)})}\\le\\frac{1}{2}", "to_expression": "liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)} >= φ(Φ^{-1}(γ^n(A)))", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)} ≥ φ(Φ^{-1}(γ^n(A))).\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_ebfeea77f541eed5", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_617ACB7CFB1C", "name": "Specht's theorem", "statement_latex": "tr W(A, A*) = tr W(B, B*)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(tr W(A, A*)) \\in {(tr W(B, B*))}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover tr W(A, A*) = tr W(B, B*).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "tr W(A, A*) = tr W(B, B*)", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover tr W(A, A*) = tr W(B, B*).\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
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{"schema_version": "1.0", "item_id": "rcd2_d98da6e8928a5d71", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FE78D365CC0D", "name": "Weinstein–Aronszajn identity", "statement_latex": "det(I_m + AB) = det(I_n + BA)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(det(I_m + AB))\\Longleftrightarrow z=(det(I_n + BA)))", "to_expression": "det(I_m + AB) = det(I_n + BA)", "justification": "same equality fiber iff equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as det(I_m + AB) = det(I_n + BA).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_53f75062555041f8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_83A41482F4E9", "name": "Fekete's lemma", "statement_latex": "lim_{n→∞} (a_n / n) = inf_{n∈N} (a_n / n)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((lim_{n→∞} (a_n / n)),z):z\\in D\\}=\\{((inf_{n∈N} (a_n / n)),z):z\\in D\\}", "to_expression": "lim_{n→∞} (a_n / n) = inf_{n∈N} (a_n / n)", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{n→∞} (a_n / n) = inf_{n∈N} (a_n / n).\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
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{"schema_version": "1.0", "item_id": "rcd2_8fc1f7b79ad2b5d8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AE174D33AD68", "name": "Eaton's inequality", "statement_latex": "P(|sum_{i=1}^n a_i X_i| >= k) <= 2 inf_{0 <= c <= k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(2 inf_{0 ≤ c ≤ k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz)\\le t\\Rightarrow (P(|sum_{i=1}^n a_i X_i| ≥ k))\\le t", "to_expression": "P(|sum_{i=1}^n a_i X_i| >= k) <= 2 inf_{0 <= c <= k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz", "justification": "converse uses t=Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(|sum_{i=1}^n a_i X_i| ≥ k) ≤ 2 inf_{0 ≤ c ≤ k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz.\"]"}], "source_equivalence_identity": "converse uses t=Y"}
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{"schema_version": "1.0", "item_id": "rcd2_cc3a4daa4c10e727", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "approximation theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9336CBED6851", "name": "Bernstein's theorem (polynomials)", "statement_latex": "max_{|z|=1}|P'(z)| <= n max_{|z|=1}|P(z)|", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (max_{|z|=1}|P'(z)|) \\le x \\land x \\le (n max_{|z|=1}|P(z)|)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover max_{|z|=1}|P'(z)| <= n max_{|z|=1}|P(z)|.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover max_{|z|=1}|P'(z)| <= n max_{|z|=1}|P(z)|.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "max_{|z|=1}|P'(z)| <= n max_{|z|=1}|P(z)|", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover max_{|z|=1}|P'(z)| <= n max_{|z|=1}|P(z)|.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_2e15a57bf17f8747", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A6D639595426", "name": "Inverse function theorem", "statement_latex": "(f^{-1})'(b) = 1 / f'(a)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((f^{-1})'(b))=F((1 / f'(a)))", "to_expression": "(f^{-1})'(b) = 1 / f'(a)", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (f^{-1})'(b) = 1 / f'(a).\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
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{"schema_version": "1.0", "item_id": "rcd2_d182652f64d15377", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematical analysis", "subfield": "analytic number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F4E26A602D8B", "name": "Perron's formula", "statement_latex": "A(x) = (1 / (2*pi*i)) * integral_{c-i*infinity}^{c+i*infinity} g(z) * (x^z / z) dz", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(A(x)) \\in {((1 / (2*\\pi*i)) * integral_{c-i*\\infty}^{c+i*\\infty} g(z) * (x^z / z) dz)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover A(x) = (1 / (2*pi*i)) * integral_{c-i*infinity}^{c+i*infinity} g(z) * (x^z / z) dz.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "A(x) = (1 / (2*pi*i)) * integral_{c-i*infinity}^{c+i*infinity} g(z) * (x^z / z) dz", "justification": "L in {R} iff L = R.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover A(x) = (1 / (2*pi*i)) * integral_{c-i*infinity}^{c+i*infinity} g(z) * (x^z / z) dz.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
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{"schema_version": "1.0", "item_id": "rcd2_ebe1b10bcf0f7b5a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "combinatorics", "subfield": "combinatorial geometry", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_23E48784FFFD", "name": "Littlewood–Offord problem", "statement_latex": "P(S_n in A) <= binom(n,floor(n/2))/2^n", "assumptions": "S_n is the signed sum in the Littlewood-Offord setup; A is an interval or target set as specified by the theorem; n is positive."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(P(S_n in A))-(binom(n,floor(n/2))/2^n)\\}=0", "to_expression": "P(S_n in A) <= binom(n,floor(n/2))/2^n", "justification": "positive part vanishes iff X≤Y", "preconditions": "S_n is the signed sum in the Littlewood-Offord setup; A is an interval or target set as specified by the theorem; n is positive.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(S_n in A) ≤ binom(n,floor(n/2))/2^n.\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_7a0ab5268ae91117", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "polynomials", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_632ce6ae2a61542d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "coding theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_23b6ea248b4d1529", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "control theory", "subfield": "geometric control theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2ADEC5E217BE", "name": "Krener's theorem", "statement_latex": "\\dot{q}=f(q,u)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((\\dot{q}),z):z\\in D\\}=\\{((f(q,u)),z):z\\in D\\}", "to_expression": "\\dot{q}=f(q,u)", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\dot{q}=f(q,u).\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_2645634f4342ea29", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_692ce2fae1d2a99c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1C2EEE35D0F6", "name": "Fischer's inequality", "statement_latex": "det(M) <= det(A)*det(C)", "assumptions": "M is positive semidefinite and block-partitioned with principal blocks A and C."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(det(M))}{\\exp(det(M))+\\exp(det(A)*det(C))}\\le\\frac{1}{2}", "to_expression": "det(M) <= det(A)*det(C)", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "M is positive semidefinite and block-partitioned with principal blocks A and C.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as det(M) ≤ det(A)*det(C).\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_5c54e14841a65af5", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "orthogonal polynomials", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_223aaeb8e2cc29a5", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "analytic number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C97AADA78BC1", "name": "Vaughan's identity", "statement_latex": "-zeta'(s)/zeta(s) = F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(F(s) - z\\eta(s)F(s)G(s) - z\\eta'(s)G(s) + (-z\\eta'(s)/z\\eta(s) - F(s))(1 - z\\eta(s)G(s))) = (-z\\eta'(s)/z\\eta(s))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover -zeta'(s)/zeta(s) = F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s)).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "-zeta'(s)/zeta(s) = F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s))", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover -zeta'(s)/zeta(s) = F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s)).\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_0aae7d77a7f28cd1", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "representation theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_9a9f55fef332e85d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A75F5656C745", "name": "Grosswald–Schnitzer theorem", "statement_latex": "\\zeta(s) = \\prod_{n=1}^{\\infty} \\frac{1}{1-p_n^{-s}}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((\\zeta(s)),z):z\\in D\\}=\\{((\\prod_{n=1}^{\\infty} \\frac{1}{1-p_n^{-s}}),z):z\\in D\\}", "to_expression": "\\zeta(s) = \\prod_{n=1}^{\\infty} \\frac{1}{1-p_n^{-s}}", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\zeta(s) = \\\\prod_{n=1}^{\\\\infty} \\\\frac{1}{1-p_n^{-s}}.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_d0268e09a253ad50", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "Number theory", "subfield": "Elliptic functions", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_3e1606a8a05dfb32", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f721475edae8d8a7", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_538f1b3d64ba3d2e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "additive number theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_870FF97782B2", "name": "Goldbach's weak conjecture", "statement_latex": "n = p1 + p2 + p3", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((n),z)=((p1 + p2 + p3),z)", "to_expression": "n = p1 + p2 + p3", "justification": "pair equality forces first coordinates equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as n = p1 + p2 + p3.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_cd692272e550967d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_37DDB6F9F636", "name": "Duffin–Schaeffer theorem", "statement_latex": "\\sum_{q=1}^{\\infty}\\varphi(q)\\frac{f(q)}{q}=\\infty", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(\\sum_{q=1}^{\\infty}\\varphi(q)\\frac{f(q)}{q})\\Longleftrightarrow z=(\\infty))", "to_expression": "\\sum_{q=1}^{\\infty}\\varphi(q)\\frac{f(q)}{q}=\\infty", "justification": "same equality fiber iff equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\sum_{q=1}^{\\\\infty}\\\\varphi(q)\\\\frac{f(q)}{q}=\\\\infty.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_c38e1a9aa90cb743", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f6837ff58d88c0f4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "vector calculus", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_6CFB1B566265", "name": "Stokes' theorem", "statement_latex": "integral integral_{Sigma} (curl F) dot dSigma = integral_{partial Sigma} F dot dr", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(integral integral_{Sigma} (curl F) dot dSigma)\\Longleftrightarrow z=(integral_{partial Sigma} F dot dr))", "to_expression": "integral integral_{Sigma} (curl F) dot dSigma = integral_{partial Sigma} F dot dr", "justification": "same equality fiber iff equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as integral integral_{Sigma} (curl F) dot dSigma = integral_{partial Sigma} F dot dr.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_4d7c0c8e56be26ee", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "commutative algebra", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AB6EC963565B", "name": "Artin–Rees lemma", "statement_latex": "I^nM ∩ N = I^{n-k}(I^kM ∩ N)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(I^nM ∩ N)=z\\;\\land\\;z=(I^{n-k}(I^kM ∩ N))", "to_expression": "I^nM ∩ N = I^{n-k}(I^kM ∩ N)", "justification": "common witness iff equality", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as I^nM ∩ N = I^{n-k}(I^kM ∩ N).\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_580dfb9f389658fb", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_eb1cdafb46958471", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "representation theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D8B93C65D10E", "name": "McKay conjecture", "statement_latex": "|Irr_{p'}(G)|=|Irr_{p'}(N_{G}(P))|", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((|Irr_{p'}(G)|) \\in S \\land (|Irr_{p'}(N_{G}(P))|) \\notin S) \\lor ((|Irr_{p'}(N_{G}(P))|) \\in S \\land (|Irr_{p'}(G)|) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover |Irr_{p'}(G)|=|Irr_{p'}(N_{G}(P))|.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "|Irr_{p'}(G)|=|Irr_{p'}(N_{G}(P))|", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover |Irr_{p'}(G)|=|Irr_{p'}(N_{G}(P))|.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_54d0d54ab623cb4d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "combinatorics", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A5B93C0856B4", "name": "Handshaking lemma", "statement_latex": "Σ_{v∈V} deg(v) = 2|E|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((Σ_{v∈V} deg(v)),z):z\\in D\\}=\\{((2|E|),z):z\\in D\\}", "to_expression": "Σ_{v∈V} deg(v) = 2|E|", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Σ_{v∈V} deg(v) = 2|E|.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_4237db143118afc2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "analysis", "subfield": "approximation theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_538DF6DD562D", "name": "Favard's theorem", "statement_latex": "y_{n+1}=(x-c_{n})y_{n}-d_{n}y_{n-1}", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "((x-c_{n})y_{n}-d_{n}y_{n-1}) = (y_{n+1})", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover y_{n+1}=(x-c_{n})y_{n}-d_{n}y_{n-1}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "y_{n+1}=(x-c_{n})y_{n}-d_{n}y_{n-1}", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover y_{n+1}=(x-c_{n})y_{n}-d_{n}y_{n-1}.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_f94a6fd416e7841e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "queueing theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D0BC02092015", "name": "Bartlett's theorem", "statement_latex": "mu(t) = integral_{-infinity}^{t} A(s)p(s,t) ds", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((mu(t)) \\in S \\land (integral_{-\\infty}^{t} A(s)p(s,t) ds) \\notin S) \\lor ((integral_{-\\infty}^{t} A(s)p(s,t) ds) \\in S \\land (mu(t)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover mu(t) = integral_{-infinity}^{t} A(s)p(s,t) ds.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "mu(t) = integral_{-infinity}^{t} A(s)p(s,t) ds", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover mu(t) = integral_{-infinity}^{t} A(s)p(s,t) ds.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
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{"schema_version": "1.0", "item_id": "rcd2_e7c33840cee3d7e2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "scattering theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0C5D6466FC1D", "name": "Levinson's theorem", "statement_latex": "φ(+∞) − φ(0) = −π(1/2*n0 + nb)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(φ(+∞) − φ(0)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(−π(1/2*n0 + nb)))", "to_expression": "φ(+∞) − φ(0) = −π(1/2*n0 + nb)", "justification": "each argmin set is a singleton", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as φ(+∞) − φ(0) = −π(1/2*n0 + nb).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_0347f391dc47e27a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "statistical mechanics", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_63998BA24BBA", "name": "Fluctuation theorem", "statement_latex": "Pr(Σ_t=A) / Pr(Σ_t=-A) = e^(At)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((Pr(Σ_t=A) / Pr(Σ_t=-A)),z)=((e^(At)),z)", "to_expression": "Pr(Σ_t=A) / Pr(Σ_t=-A) = e^(At)", "justification": "pair equality forces first coordinates equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr(Σ_t=A) / Pr(Σ_t=-A) = e^(At).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
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{"schema_version": "1.0", "item_id": "rcd2_fb61a419b3fcebf7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4CEAA916D1E3", "name": "Riemann–Siegel formula", "statement_latex": "ζ(s) = Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(ζ(s)))+\\delta_D(z,(Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s))))=0", "to_expression": "ζ(s) = Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s)", "justification": "zero loss iff some z equals both", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ζ(s) = Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s).\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_6d8133b5fb95fcc3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "combinatorics", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_EE0D43BAAED5", "name": "Ahlswede–Daykin inequality", "statement_latex": "f_1(X)f_2(Y) ≤ f_3(X ∨ Y)f_4(X ∧ Y)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(f_3(X ∨ Y)f_4(X ∧ Y))\\le t\\Rightarrow (f_1(X)f_2(Y))\\le t", "to_expression": "f_1(X)f_2(Y) ≤ f_3(X ∨ Y)f_4(X ∧ Y)", "justification": "converse uses t=Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f_1(X)f_2(Y) ≤ f_3(X ∨ Y)f_4(X ∧ Y).\"]"}], "source_equivalence_identity": "converse uses t=Y"}
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{"schema_version": "1.0", "item_id": "rcd2_b1768ebca60d16ee", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "design theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_45A9C577BD36", "name": "Fisher's inequality", "statement_latex": "b >= v", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (v) \\le x \\land x \\le (b)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover b >= v.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover b >= v.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "b >= v", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover b >= v.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
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{"schema_version": "1.0", "item_id": "rcd2_f3af443a26ec7f27", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic functions", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FDD3E2DFBB65", "name": "Hermite's identity", "statement_latex": "sum_{k=0}^{n-1} floor(x+k/n) = floor(n*x)", "assumptions": "n is a positive integer; x is real."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((sum_{k=0}^{n-1} floor(x+k/n)),z)=((floor(n*x)),z)", "to_expression": "sum_{k=0}^{n-1} floor(x+k/n) = floor(n*x)", "justification": "pair equality forces first coordinates equal", "preconditions": "n is a positive integer; x is real.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{k=0}^{n-1} floor(x+k/n) = floor(n*x).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
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{"schema_version": "1.0", "item_id": "rcd2_78a5e934e382d75b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic geometry", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BB5E3AF7EFD7", "name": "Modularity theorem", "statement_latex": "L(E, s) = sum_{n=1}^∞ a_n / n^s", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((L(E, s)),(sum_{n=1}^∞ a_n / n^s))=0", "to_expression": "L(E, s) = sum_{n=1}^∞ a_n / n^s", "justification": "zero discrete distance iff equality", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as L(E, s) = sum_{n=1}^∞ a_n / n^s.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_b762a812f99a27f7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "field theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_ED1CB49A03EE", "name": "Steinitz's theorem", "statement_latex": "L = K(α)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(L))+\\delta_D(z,(K(α))))=0", "to_expression": "L = K(α)", "justification": "zero loss iff some z equals both", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as L = K(α).\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
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{"schema_version": "1.0", "item_id": "rcd2_760990843ea025a5", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_98220A4F883B", "name": "Siegel identity", "statement_latex": "\\frac{x_3-x_1}{x_2-x_1} + \\frac{x_2-x_3}{x_2-x_1} = 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\frac{x_3-x_1}{x_2-x_1} + \\frac{x_2-x_3}{x_2-x_1}))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(1))", "to_expression": "\\frac{x_3-x_1}{x_2-x_1} + \\frac{x_2-x_3}{x_2-x_1} = 1", "justification": "each argmin set is a singleton", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\frac{x_3-x_1}{x_2-x_1} + \\\\frac{x_2-x_3}{x_2-x_1} = 1.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
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{"schema_version": "1.0", "item_id": "rcd2_2e23884ea05c0362", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "coding theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F6DC10428DD6", "name": "Z-channel capacity", "statement_latex": "cap(Z) = H(p/(1+2^s))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((cap(Z)) \\in S \\land (H(p/(1+2^s))) \\notin S) \\lor ((H(p/(1+2^s))) \\in S \\land (cap(Z)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover cap(Z) = H(p/(1+2^s)).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "cap(Z) = H(p/(1+2^s))", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover cap(Z) = H(p/(1+2^s)).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
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{"schema_version": "1.0", "item_id": "rcd2_937c470731ce7876", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "statistical mechanics", "subfield": "non-equilibrium thermodynamics", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D06DBDBC4CC8", "name": "Crooks fluctuation theorem", "statement_latex": "P[σ(x(t))] / P[σ(x_tilde(t))] = e^σ[x(t)]", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(P[σ(x(t))] / P[σ(x_tilde(t))])\\}=\\{(e^σ[x(t)])\\}", "to_expression": "P[σ(x(t))] / P[σ(x_tilde(t))] = e^σ[x(t)]", "justification": "singletons equal iff elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P[σ(x(t))] / P[σ(x_tilde(t))] = e^σ[x(t)].\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
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{"schema_version": "1.0", "item_id": "rcd2_cb83f72f83e1e48d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2DC4FABA16C6", "name": "Kantorovich inequality", "statement_latex": "(Σ p_i x_i) * (Σ p_i / x_i) <= (a+b)^2 / (4ab)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(a+b)^2 / (4ab)\\le t\\Rightarrow (Σ p_i x_i) * (Σ p_i / x_i)\\le t", "to_expression": "(Σ p_i x_i) * (Σ p_i / x_i) <= (a+b)^2 / (4ab)", "justification": "converse uses t=Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (Σ p_i x_i) * (Σ p_i / x_i) ≤ (a+b)^2 / (4ab).\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_0d3aea59a89de18e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "combinatorics", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_51200980F031", "name": "Star of David theorem", "statement_latex": "gcd{(n-1 k-1), (n k+1), (n+1 k)} = gcd{(n-1 k), (n k-1), (n+1 k+1)}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((gcd{(n-1 k-1), (n k+1), (n+1 k)}),z)=((gcd{(n-1 k), (n k-1), (n+1 k+1)}),z)", "to_expression": "gcd{(n-1 k-1), (n k+1), (n+1 k)} = gcd{(n-1 k), (n k-1), (n+1 k+1)}", "justification": "pair equality forces first coordinates equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as gcd{(n-1 k-1), (n k+1), (n+1 k)} = gcd{(n-1 k), (n k-1), (n+1 k+1)}.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_6d309db7f1ec5026", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "combinatorial number theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3FBB7DDC427F", "name": "Erdős–Graham problem", "statement_latex": "sum_{n in S} (1/n) = 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(sum_{n in S} (1/n)))+\\delta_D(z,(1)))=0", "to_expression": "sum_{n in S} (1/n) = 1", "justification": "zero loss iff some z equals both", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{n in S} (1/n) = 1.\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_b3196fd766da7961", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum mechanics", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d428512277484fe4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "computer science", "subfield": "probabilistic combinatorics", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AA4BF27A6AAD", "name": "Algorithmic Lovász local lemma", "statement_latex": "Pr[A] <= x(A) * Product_{B in Gamma(A)} (1 - x(B))", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\max\\{0,(Pr[A])-(x(A) * Product_{B in Gamma(A)} (1 - x(B)))\\}=0", "to_expression": "Pr[A] <= x(A) * Product_{B in Gamma(A)} (1 - x(B))", "justification": "positive part vanishes iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr[A] ≤ x(A) * Product_{B in Gamma(A)} (1 - x(B)).\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_9dd923d1c1b44643", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_24003c96d7a2a2dc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "zeta and L-functions", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CEA3FEEC415F", "name": "Selberg's zeta function conjecture", "statement_latex": "N(T + H) - N(T) ≥ cH log T", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\le (cH log T)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (N(T + H) - N(T))\\}", "to_expression": "N(T + H) - N(T) ≥ cH log T", "justification": "lower-cut inclusion iff X≤Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as N(T + H) - N(T) ≥ cH log T.\"]"}], "source_equivalence_identity": "lower-cut inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_f5e0cd80f537f6d8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_94B20D8A1D74", "name": "Chebyshev's inequality", "statement_latex": "Pr(|X-mu| >= k*sigma) <= 1/k^2", "assumptions": "X has mean mu and finite nonzero standard deviation sigma; k>0."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(Pr(|X-mu| ≥ k*sigma))-(1/k^2)\\}=0", "to_expression": "Pr(|X-mu| >= k*sigma) <= 1/k^2", "justification": "positive part vanishes iff X≤Y", "preconditions": "X has mean mu and finite nonzero standard deviation sigma; k>0.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr(|X-mu| ≥ k*sigma) ≤ 1/k^2.\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_6544d70d1776ff60", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "symmetric functions", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2f5327c01596faa1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal set theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_03FC7B5F26C8", "name": "Erdős–Ko–Rado theorem", "statement_latex": "|A| <= binom(n-1,r-1)", "assumptions": "A is an intersecting family of r-subsets of an n-element set; n>=2r."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(|A|)}{\\exp(|A|)+\\exp(binom(n-1,r-1))}\\le\\frac{1}{2}", "to_expression": "|A| <= binom(n-1,r-1)", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "A is an intersecting family of r-subsets of an n-element set; n>=2r.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |A| ≤ binom(n-1,r-1).\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_2f7f66003e982af3", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "hypergraph theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_09166ca3c773844b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ab28979b838e1741", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "asymptotic geometric analysis", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c5ab0196ac8e3fc0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic geometry", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_49F57EB1100D", "name": "Faltings' theorem", "statement_latex": "g > 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(1)}{\\exp(1)+\\exp(g)}<\\frac{1}{2}", "to_expression": "g > 1", "justification": "equiv exp(X)<exp(Y)", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as g > 1.\"]"}], "source_equivalence_identity": "equiv exp(X)<exp(Y)"}
{"schema_version": "1.0", "item_id": "rcd2_0edebdbb3c41ca9c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "linear algebra", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_08D175804776", "name": "Cayley–Hamilton theorem", "statement_latex": "p_A(A) = 0", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(p_A(A))\\Longleftrightarrow z=(0))", "to_expression": "p_A(A) = 0", "justification": "same equality fiber iff equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p_A(A) = 0.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_9c2d7b83c8d37c21", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "large deviations theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_53869962DDDE", "name": "Laplace's principle", "statement_latex": "\\lim_{\\theta \\to \\infty}\\frac{1}{\\theta}\\log\\int_{A}e^{-\\theta\\varphi(x)}\\,dx=-\\mathop{\\mathrm{ess\\,inf}}_{x\\in A}\\varphi(x)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((\\lim_{\\theta → \\infty}\\frac{1}{\\theta}\\log\\int_{A}e^{-\\theta\\varphi(x)}\\,dx),z):z\\in D\\}=\\{((-\\mathop{\\mathrm{ess\\,inf}}_{x\\in A}\\varphi(x)),z):z\\in D\\}", "to_expression": "\\lim_{\\theta \\to \\infty}\\frac{1}{\\theta}\\log\\int_{A}e^{-\\theta\\varphi(x)}\\,dx=-\\mathop{\\mathrm{ess\\,inf}}_{x\\in A}\\varphi(x)", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\lim_{\\\\theta → \\\\infty}\\\\frac{1}{\\\\theta}\\\\log\\\\int_{A}e^{-\\\\theta\\\\varphi(x)}\\\\,dx=-\\\\mathop{\\\\mathrm{ess\\\\,inf}}_{x\\\\in A}\\\\varphi(x).\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_c681b1405e0173d2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "linear algebra", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FF8D78866125", "name": "Binet–Cauchy identity", "statement_latex": "(sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i<j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i)", "assumptions": "All sums are finite over the same index range; entries lie in R."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "-(j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i)) < -((sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "x -> -x is strictly decreasing, so A<B iff -B<-A.", "preconditions": "All sums are finite over the same index range; entries lie in R.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i < j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "x -> -x is strictly decreasing, so A<B iff -B<-A.", "preconditions": "All sums are finite over the same index range; entries lie in R.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i < j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "(sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i<j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i)", "justification": "x -> -x is strictly decreasing, so A<B iff -B<-A.", "preconditions": "All sums are finite over the same index range; entries lie in R.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i < j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i).\"]"}], "source_equivalence_identity": "x -> -x is strictly decreasing, so A<B iff -B<-A."}
{"schema_version": "1.0", "item_id": "rcd2_1dd8360b830c0df6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2DC4FABA16C6", "name": "Kantorovich inequality", "statement_latex": "(Σ p_i x_i) * (Σ p_i / x_i) <= (a+b)^2 / (4ab)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: ((Σ p_i x_i) * (Σ p_i / x_i)) \\le x \\land x \\le ((a+b)^2 / (4ab))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (Σ p_i x_i) * (Σ p_i / x_i) <= (a+b)^2 / (4ab).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (Σ p_i x_i) * (Σ p_i / x_i) <= (a+b)^2 / (4ab).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "(Σ p_i x_i) * (Σ p_i / x_i) <= (a+b)^2 / (4ab)", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (Σ p_i x_i) * (Σ p_i / x_i) <= (a+b)^2 / (4ab).\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_6e7fa6189d8c7212", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "extremal graph theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4D5DD853BD7", "name": "Szemerédi regularity lemma", "statement_latex": "|d(X,Y) - d(A,B)| <= ε", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((|d(X,Y) - d(A,B)|)-(ε),(-\\infty,0])=0", "to_expression": "|d(X,Y) - d(A,B)| <= ε", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |d(X,Y) - d(A,B)| ≤ ε.\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
{"schema_version": "1.0", "item_id": "rcd2_b9f7cd5e6e568da7", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "extremal graph theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_41e12b068a5358d2", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "combinatorics on words", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_9e7f6303a1cabcbb", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_97c8427a633aa7b1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "number theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9793AD078B54", "name": "Degen's eight-square identity", "statement_latex": "(∑_{i=1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2) = ∑_{i=1}^{8} c_i^2", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(0 = 0) \\Rightarrow ((∑_{i=1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2) = ∑_{i=1}^{8} c_i^2)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Since 0=0 is true, true=>Phi_T has the same truth value as Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "(∑_{i=1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2) = ∑_{i=1}^{8} c_i^2", "justification": "Since 0=0 is true, true=>Phi_T has the same truth value as Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "Since 0=0 is true, true=>Phi_T has the same truth value as Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_31566370fbd15d59", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "Ramsey theory", "subfield": "discrete mathematics", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d2f7bb4a50d88f2b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "number theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d1d1e7ae6a119084", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_41972AC1B752", "name": "Rothe–Hagen identity", "statement_latex": "\\sum _{k=0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k}={\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (\\sum _{k=0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k}={\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n}))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\sum _{k=0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k}={\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n}", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_71db181bd594c0a6", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "combinatorics", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b6b59c788eace49d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "statistics", "subfield": "statistical hypothesis testing", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_cccaa06a55c71a9c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "additive number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_06ee32a8f197ecfd", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_42828E58305E", "name": "Pascal's rule", "statement_latex": "\\binom{n-1}{k} + \\binom{n-1}{k-1} = \\binom{n}{k}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\binom{n-1}{k} + \\binom{n-1}{k-1}) \\in {(\\binom{n}{k})}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\binom{n-1}{k} + \\\\binom{n-1}{k-1} = \\\\binom{n}{k}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\binom{n-1}{k} + \\binom{n-1}{k-1} = \\binom{n}{k}", "justification": "L in {R} iff L = R.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\binom{n-1}{k} + \\\\binom{n-1}{k-1} = \\\\binom{n}{k}.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_2b56843269588bdf", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "functional analysis", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_55A9511267AC", "name": "Bobkov's inequality", "statement_latex": "I(integral f d gamma^n) <= integral sqrt(I(f)^2 + |grad f|^2) d gamma^n", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(integral sqrt(I(f)^2 + |grad f|^2) d gamma^n)\\le t\\Rightarrow (I(integral f d gamma^n))\\le t", "to_expression": "I(integral f d gamma^n) <= integral sqrt(I(f)^2 + |grad f|^2) d gamma^n", "justification": "converse uses t=Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as I(integral f d gamma^n) ≤ integral sqrt(I(f)^2 + |grad f|^2) d gamma^n.\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_e3c89872a1ec5fc2", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "geometry", "subfield": "Euclidean geometry", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_85f7214e4c61fdd5", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "combinatorics on words", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3E0E814BA7D9", "name": "Levi's lemma", "statement_latex": "uv = xy", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((uv),z)=((xy),z)", "to_expression": "uv = xy", "justification": "pair equality forces first coordinates equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as uv = xy.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_d1ffd58fe65a101e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "inequalities", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9822BD8B9BE6", "name": "Muirhead's inequality", "statement_latex": "[a] >= [b] for a majorizing b", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp([b] for a majorizing b)}{\\exp([b] for a majorizing b)+\\exp([a])}\\le\\frac{1}{2}", "to_expression": "[a] >= [b] for a majorizing b", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as [a] ≥ [b] for a majorizing b.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_70717d373fc802c4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "complex analysis", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_73CA77821F72", "name": "Rouché's theorem", "statement_latex": "|g(z)| < |f(z)|", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(|g(z)|)-(|f(z)|)\\}=0\\;\\land\\;(|g(z)|)\\ne (|f(z)|)", "to_expression": "|g(z)| < |f(z)|", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |g(z)| < |f(z)|.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
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{"schema_version": "1.0", "item_id": "rcd2_d237fab3e60e88d8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematical analysis", "subfield": "analytic number theory", "primary_mechanism": "distance", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F4E26A602D8B", "name": "Perron's formula", "statement_latex": "A(x) = (1 / (2*pi*i)) * integral_{c-i*infinity}^{c+i*infinity} g(z) * (x^z / z) dz", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists c \\in D: (A(x)) = c \\land ((1 / (2*\\pi*i)) * integral_{c-i*\\infty}^{c+i*\\infty} g(z) * (x^z / z) dz) = c", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A common c exists iff L and R are equal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover A(x) = (1 / (2*pi*i)) * integral_{c-i*infinity}^{c+i*infinity} g(z) * (x^z / z) dz.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "A(x) = (1 / (2*pi*i)) * integral_{c-i*infinity}^{c+i*infinity} g(z) * (x^z / z) dz", "justification": "A common c exists iff L and R are equal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover A(x) = (1 / (2*pi*i)) * integral_{c-i*infinity}^{c+i*infinity} g(z) * (x^z / z) dz.\"]"}], "source_equivalence_identity": "A common c exists iff L and R are equal."}
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{"schema_version": "1.0", "item_id": "rcd2_c7ec638fbb4e7bcb", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "complex analysis", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2C71326A0896", "name": "Area theorem", "statement_latex": "\\sum_{n=0}^{\\infty} n|a_n|^2 \\leq 1", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(1)\\le t\\Rightarrow (\\sum_{n=0}^{\\infty} n|a_n|^2)\\le t", "to_expression": "\\sum_{n=0}^{\\infty} n|a_n|^2 \\leq 1", "justification": "converse uses t=Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\sum_{n=0}^{\\\\infty} n|a_n|^2 ≤ 1.\"]"}], "source_equivalence_identity": "converse uses t=Y"}
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{"schema_version": "1.0", "item_id": "rcd2_f342e836b39f7f70", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "measure theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4450EEEA850", "name": "Borel–Cantelli lemma", "statement_latex": "limsup_{n->infinity} E_n = intersection_{n=1 to infinity} union_{k=n to infinity} E_k", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "-(limsup_{n-) < -(\\infty} E_n = intersection_{n=1 to \\infty} union_{k=n to \\infty} E_k)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "x -> -x is strictly decreasing, so A<B iff -B<-A.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover limsup_{n- > infinity} E_n = intersection_{n=1 to infinity} union_{k=n to infinity} E_k.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "x -> -x is strictly decreasing, so A<B iff -B<-A.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover limsup_{n- > infinity} E_n = intersection_{n=1 to infinity} union_{k=n to infinity} E_k.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "limsup_{n->infinity} E_n = intersection_{n=1 to infinity} union_{k=n to infinity} E_k", "justification": "x -> -x is strictly decreasing, so A<B iff -B<-A.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover limsup_{n- > infinity} E_n = intersection_{n=1 to infinity} union_{k=n to infinity} E_k.\"]"}], "source_equivalence_identity": "x -> -x is strictly decreasing, so A<B iff -B<-A."}
{"schema_version": "1.0", "item_id": "rcd2_a3f24e69be744f81", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_45A3C68C1B71", "name": "Linnik's theorem", "statement_latex": "p(a,d) < c * d^L", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(p(a,d))}{\\exp(p(a,d))+\\exp(c * d^L)}<\\frac{1}{2}", "to_expression": "p(a,d) < c * d^L", "justification": "equiv exp(X)<exp(Y)", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p(a,d) < c * d^L.\"]"}], "source_equivalence_identity": "equiv exp(X)<exp(Y)"}
{"schema_version": "1.0", "item_id": "rcd2_dacff1bab44377c7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_30AD154E7704", "name": "Cyclotomic identity", "statement_latex": "1/(1-alpha*z) = product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((1/(1-alpha*z)),(product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}))=0", "to_expression": "1/(1-alpha*z) = product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}", "justification": "zero discrete distance iff equality", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as 1/(1-alpha*z) = product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_a6bda8c5256b3428", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D236ADD607AB", "name": "Chernoff bound", "statement_latex": "P(X ≥ a) ≤ inf_{t>0} M(t)e^{-ta}", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((P(X ≥ a))-(inf_{t>0} M(t)e^{-ta}))=0", "to_expression": "P(X ≥ a) ≤ inf_{t>0} M(t)e^{-ta}", "justification": "sup equals max(0,X-Y)", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(X ≥ a) ≤ inf_{t>0} M(t)e^{-ta}.\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
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{"schema_version": "1.0", "item_id": "rcd2_284c4e36069f740d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal set theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3B6CADC238DF", "name": "Lubell–Yamamoto–Meshalkin inequality", "statement_latex": "sum_{k=0}^n a_k/binom(n,k) <= 1", "assumptions": "The family is Sperner; a_k counts chosen k-subsets of an n-element ground set."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (sum_{k=0}^n a_k/binom(n,k)) \\le x \\land x \\le (1)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "The family is Sperner; a_k counts chosen k-subsets of an n-element ground set.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover sum_{k=0}^n a_k/binom(n,k) <= 1.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "The family is Sperner; a_k counts chosen k-subsets of an n-element ground set.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover sum_{k=0}^n a_k/binom(n,k) <= 1.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "sum_{k=0}^n a_k/binom(n,k) <= 1", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "The family is Sperner; a_k counts chosen k-subsets of an n-element ground set.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover sum_{k=0}^n a_k/binom(n,k) <= 1.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
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{"schema_version": "1.0", "item_id": "rcd2_fbf68290c9316d3d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "graph theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_37489533E622", "name": "Dinitz theorem", "statement_latex": "chi'_l(K_{n,n}) = n", "assumptions": "K_{n,n} is the complete bipartite graph; chi_l prime denotes list chromatic index; n is a positive integer."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((chi'_l(K_{n,n})))=F((n))", "to_expression": "chi'_l(K_{n,n}) = n", "justification": "indicator of {X} separates unequal objects", "preconditions": "K_{n,n} is the complete bipartite graph; chi_l prime denotes list chromatic index; n is a positive integer.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as chi'_l(K_{n,n}) = n.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_d140174e28ee1376", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "geometry", "subfield": "discrete geometry", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_DCDC0E01EC8B", "name": "Covering problem of Rado", "statement_latex": "F(X) = inf_S sup_I (|I|/|S|)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(F(X))\\}=\\{(inf_S sup_I (|I|/|S|))\\}", "to_expression": "F(X) = inf_S sup_I (|I|/|S|)", "justification": "singletons equal iff elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as F(X) = inf_S sup_I (|I|/|S|).\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
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{"schema_version": "1.0", "item_id": "rcd2_b194512e976689ec", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "Ramsey theory", "subfield": "discrete mathematics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_7cca1f9d94753683", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "graph theory", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e5e1f8f74ab435f5", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "graph theory", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_07CDE34AC57A", "name": "Counting lemma", "statement_latex": "γ <= ε^h / 4h", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(γ)+s=(ε^h / 4h)", "to_expression": "γ <= ε^h / 4h", "justification": "nonnegative slack exists iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as γ ≤ ε^h / 4h.\"]"}], "source_equivalence_identity": "nonnegative slack exists iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_7d8931cab48db1ed", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "Ramsey theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A40EE0DC8F58", "name": "Van der Waerden's theorem", "statement_latex": "W(r,k) ≤ 2^{2^{r^{2^{2^{k+9}}}}}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (W(r,k)) \\le x \\land x \\le (2^{2^{r^{2^{2^{k+9}}}}})", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover W(r,k) <= 2^{2^{r^{2^{2^{k+9}}}}}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover W(r,k) <= 2^{2^{r^{2^{2^{k+9}}}}}.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "W(r,k) ≤ 2^{2^{r^{2^{2^{k+9}}}}}", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover W(r,k) <= 2^{2^{r^{2^{2^{k+9}}}}}.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_c523bdcb28fec188", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7922A15B98B4", "name": "Kolmogorov's inequality", "statement_latex": "Pr(max_{1<=k<=n}|S_{k}|>=λ) <= (1/λ^2) * Var[S_{n}]", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (Pr(max_{1\\lek\\len}|S_{k}|\\geλ) \\le (1/λ^2) * Var[S_{n}]))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "Pr(max_{1<=k<=n}|S_{k}|>=λ) <= (1/λ^2) * Var[S_{n}]", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
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{"schema_version": "1.0", "item_id": "rcd2_a477cacd1cf391fb", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "queueing theory", "subfield": "probability", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_91967E3D2C5F", "name": "Little's law", "statement_latex": "L = lambda*W", "assumptions": "The queueing system is stable and in steady state; L, lambda, and W are finite real quantities."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((L) \\in S \\land (lambda*W) \\notin S) \\lor ((lambda*W) \\in S \\land (L) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "The queueing system is stable and in steady state; L, lambda, and W are finite real quantities.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover L = lambda*W.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "L = lambda*W", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "The queueing system is stable and in steady state; L, lambda, and W are finite real quantities.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover L = lambda*W.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
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{"schema_version": "1.0", "item_id": "rcd2_56f76965713227a5", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "discrete mathematics", "subfield": "combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_fe29b83f296013a2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "combinatorial number theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3FBB7DDC427F", "name": "Erdős–Graham problem", "statement_latex": "sum_{n in S} (1/n) = 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((sum_{n in S} (1/n)))=F((1))", "to_expression": "sum_{n in S} (1/n) = 1", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{n in S} (1/n) = 1.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_d644a5555e82c43b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "design theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_863144497be80cef", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "metric geometry", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e9cfb5b4eec9cdb0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "time-reversal symmetry", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BACF4A4770A1", "name": "Kramers' theorem", "statement_latex": "T^2 = -1", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((T^2),(-1))=0", "to_expression": "T^2 = -1", "justification": "zero discrete distance iff equality", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as T^2 = -1.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_9bf0832f8f94f777", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "analysis", "subfield": "moment problem", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A82D2EB23B1F", "name": "Carleman's condition", "statement_latex": "\\sum _{n=1}^{\\infty }m_{2n}^{-{1/2n}}=+\\infty", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (\\sum _{n=1}^{\\infty }m_{2n}^{-{1/2n}}=+\\infty))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\sum _{n=1}^{\\infty }m_{2n}^{-{1/2n}}=+\\infty", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
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{"schema_version": "1.0", "item_id": "rcd2_140a0a59275c058e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "calculus of variations", "subfield": "Morse theory", "primary_mechanism": "distance", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5B2379DEDFD8", "name": "Morse–Palais lemma", "statement_latex": "f(x) = ⟨Aφ(x), φ(x)⟩", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists c \\in D: (f(x)) = c \\land (⟨Aφ(x), φ(x)⟩) = c", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A common c exists iff L and R are equal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(x) = ⟨Aφ(x), φ(x)⟩.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f(x) = ⟨Aφ(x), φ(x)⟩", "justification": "A common c exists iff L and R are equal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(x) = ⟨Aφ(x), φ(x)⟩.\"]"}], "source_equivalence_identity": "A common c exists iff L and R are equal."}
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{"schema_version": "1.0", "item_id": "rcd2_cfc82b4caabb831f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "symmetric functions", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4E5DD778CBB5", "name": "Newton's identities", "statement_latex": "k e_k = \\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (k e_k = \\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "k e_k = \\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
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{"schema_version": "1.0", "item_id": "rcd2_64e6c72608846363", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8CBF25034487", "name": "Heegner's lemma", "statement_latex": "y^{2}=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((y^{2}) \\in S \\land (a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}) \\notin S) \\lor ((a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}) \\in S \\land (y^{2}) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover y^{2}=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "y^{2}=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover y^{2}=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
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{"schema_version": "1.0", "item_id": "rcd2_0d8c0670d4f03b48", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "matching theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_69DD0D246726", "name": "Tutte–Berge formula", "statement_latex": "max_matching_size = 1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((max_matching_size))=F((1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)))", "to_expression": "max_matching_size = 1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)", "justification": "indicator of {X} separates unequal objects", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as max_matching_size = 1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|).\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
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{"schema_version": "1.0", "item_id": "rcd2_b24ebda69d5748ab", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "calculus", "subfield": "integral calculus", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_9168cf7e4ec22bfe", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "inner product spaces", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2EE2B752D03D", "name": "Polarization identity", "statement_latex": "\\operatorname{Re} \\langle x, y \\rangle = \\frac{1}{2} (\\|x + y\\|^2 - \\|x\\|^2 - \\|y\\|^2)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(\\operatorname{Re} \\langle x, y \\rangle)\\Longleftrightarrow z=(\\frac{1}{2} (\\|x + y\\|^2 - \\|x\\|^2 - \\|y\\|^2)))", "to_expression": "\\operatorname{Re} \\langle x, y \\rangle = \\frac{1}{2} (\\|x + y\\|^2 - \\|x\\|^2 - \\|y\\|^2)", "justification": "same equality fiber iff equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\operatorname{Re} \\\\langle x, y \\\\rangle = \\\\frac{1}{2} (\\\\|x + y\\\\|^2 - \\\\|x\\\\|^2 - \\\\|y\\\\|^2).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
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{"schema_version": "1.0", "item_id": "rcd2_7675c5800ff26d04", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic calculus", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_4c656adde11cce84", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "geometric probability", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F57E82952899", "name": "Wendel's theorem", "statement_latex": "p_{n,N}=2^{-N+1}\\sum _{k=0}^{n-1}{\\binom {N-1}{k}}", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((p_{n,N}),z)=((2^{-N+1}\\sum _{k=0}^{n-1}{\\binom {N-1}{k}}),z)", "to_expression": "p_{n,N}=2^{-N+1}\\sum _{k=0}^{n-1}{\\binom {N-1}{k}}", "justification": "pair equality forces first coordinates equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p_{n,N}=2^{-N+1}\\\\sum _{k=0}^{n-1}{\\\\binom {N-1}{k}}.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_73382c6671d7a7e4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "analysis", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A35964624017", "name": "AM–GM inequality", "statement_latex": "(x + y)/2 >= sqrt(x*y)", "assumptions": "x and y are nonnegative real numbers."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\le (sqrt(x*y))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le ((x + y)/2)\\}", "to_expression": "(x + y)/2 >= sqrt(x*y)", "justification": "lower-cut inclusion iff X≤Y", "preconditions": "x and y are nonnegative real numbers.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (x + y)/2 ≥ sqrt(x*y).\"]"}], "source_equivalence_identity": "lower-cut inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_7eedc912c750029b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_79DC6C43DD64", "name": "Perron–Frobenius theorem", "statement_latex": "Av = rv", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(rv) = (Av)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover Av = rv.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "Av = rv", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover Av = rv.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_b774d9f88a081ec4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "approximation theory", "subfield": "mathematical analysis", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_389D9173C8B7", "name": "Lebesgue's lemma", "statement_latex": "‖v - Pv‖ ≤ (1 + ‖P‖) inf_{u ∈ U} ‖v - u‖", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (‖v - Pv‖) \\le x \\land x \\le ((1 + ‖P‖) \\inf_{u ∈ U} ‖v - u‖)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ‖v - Pv‖ <= (1 + ‖P‖) inf_{u ∈ U} ‖v - u‖.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ‖v - Pv‖ <= (1 + ‖P‖) inf_{u ∈ U} ‖v - u‖.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "‖v - Pv‖ ≤ (1 + ‖P‖) inf_{u ∈ U} ‖v - u‖", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ‖v - Pv‖ <= (1 + ‖P‖) inf_{u ∈ U} ‖v - u‖.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_11e20257a0343a39", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "discrete geometry", "subfield": "geometry", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E02FAD9C277D", "name": "Erdős–Anning theorem", "statement_latex": "|d(A,X) - d(B,X)| <= d(A,B)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(d(A,B)) - (|d(A,X) - d(B,X)|) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |d(A,X) - d(B,X)| <= d(A,B).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |d(A,X) - d(B,X)| <= d(A,B).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "|d(A,X) - d(B,X)| <= d(A,B)", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |d(A,X) - d(B,X)| <= d(A,B).\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
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{"schema_version": "1.0", "item_id": "rcd2_42acb3e20cdb35e2", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "complex analysis", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f476df9c0195810f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "combinatorics on words", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_08AF905E8677", "name": "Fine and Wilf's theorem", "statement_latex": "a_n = a_{n+gcd(h,k)}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(a_n)\\Longleftrightarrow z=(a_{n+gcd(h,k)}))", "to_expression": "a_n = a_{n+gcd(h,k)}", "justification": "same equality fiber iff equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as a_n = a_{n+gcd(h,k)}.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_d36b73d914376f82", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d4919de59b28c6b8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "time-reversal symmetry", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BACF4A4770A1", "name": "Kramers' theorem", "statement_latex": "T^2 = -1", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((T^2),(-1))\\in\\{(z,z):z\\in D\\}", "to_expression": "T^2 = -1", "justification": "pair in diagonal iff X=Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as T^2 = -1.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_c34a3749e5249dea", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_936D29F3BF52", "name": "Hsu–Robbins–Erdős theorem", "statement_latex": "∑_{n>=1}P(|S_{n}|>εn)<∞", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "((∑_{n≥1}P(|S_{n}|>εn)),(∞))\\in\\{(u,v)\\in\\mathbb{R}^2:u\\le v\\}\\setminus\\{(u,u):u\\in\\mathbb{R}\\}", "to_expression": "∑_{n>=1}P(|S_{n}|>εn)<∞", "justification": "≤ plus not equal iff <", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∑_{n≥1}P(|S_{n}|>εn)<∞.\"]"}], "source_equivalence_identity": "≤ plus not equal iff <"}
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{"schema_version": "1.0", "item_id": "rcd2_1522c1604ce63b02", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal set theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_15d99cc07a09b062", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "vector calculus", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F8BCAAED7243", "name": "Green's theorem", "statement_latex": "∮_{C}(L dx + M dy) = ∬_{D} (∂M/∂x - ∂L/∂y) dA", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(∮_{C}(L dx + M dy)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(∬_{D} (∂M/∂x - ∂L/∂y) dA))", "to_expression": "∮_{C}(L dx + M dy) = ∬_{D} (∂M/∂x - ∂L/∂y) dA", "justification": "each argmin set is a singleton", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∮_{C}(L dx + M dy) = ∬_{D} (∂M/∂x - ∂L/∂y) dA.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_a81eaa15bf94d335", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "extremal graph theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_7525d4038685ff84", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "quadratic forms", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2db97c67f9117372", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "differential calculus", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F1B027EB50E5", "name": "Interior extremum theorem", "statement_latex": "f'(x_0) = 0", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(f'(x_0)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(0))", "to_expression": "f'(x_0) = 0", "justification": "each argmin set is a singleton", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f'(x_0) = 0.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_aa82f501ce247ee4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "statistics", "subfield": "statistical hypothesis testing", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_69B4EB18D461", "name": "Neyman–Pearson lemma", "statement_latex": "α = Pr_θ0(X ∈ R)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((α),z)=((Pr_θ0(X ∈ R)),z)", "to_expression": "α = Pr_θ0(X ∈ R)", "justification": "pair equality forces first coordinates equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as α = Pr_θ0(X ∈ R).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_1121d7f9c170e80f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F01A12C6F5E3", "name": "Taylor's theorem", "statement_latex": "f(x) = sum_{n=0}^k (f^(n)(a)/n!) * (x-a)^n + R_k(x)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(0 = 0) \\Rightarrow (f(x) = sum_{n=0}^k (f^(n)(a)/n!) * (x-a)^n + R_k(x))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Since 0=0 is true, true=>Phi_T has the same truth value as Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f(x) = sum_{n=0}^k (f^(n)(a)/n!) * (x-a)^n + R_k(x)", "justification": "Since 0=0 is true, true=>Phi_T has the same truth value as Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "Since 0=0 is true, true=>Phi_T has the same truth value as Phi_T."}
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{"schema_version": "1.0", "item_id": "rcd2_a0db399d494d6cc3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "physics", "subfield": "general relativity", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_737BDAA0F832", "name": "Stewart–Walker lemma", "statement_latex": "ΔδT = 0", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((ΔδT) \\in S \\land (0) \\notin S) \\lor ((0) \\in S \\land (ΔδT) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover ΔδT = 0.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "ΔδT = 0", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover ΔδT = 0.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
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{"schema_version": "1.0", "item_id": "rcd2_36f3c24413d9865a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "coding theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F6DC10428DD6", "name": "Z-channel capacity", "statement_latex": "cap(Z) = H(p/(1+2^s))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((cap(Z)),z)=((H(p/(1+2^s))),z)", "to_expression": "cap(Z) = H(p/(1+2^s))", "justification": "pair equality forces first coordinates equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as cap(Z) = H(p/(1+2^s)).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_bcb6b491db2cf325", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "geometry", "subfield": "Euclidean geometry", "primary_mechanism": "monotone", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F38D8FABD394", "name": "Ptolemy's inequality", "statement_latex": "AB * CD + BC * DA >= AC * BD", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\arctan(AC * BD)\\le\\arctan(AB * CD + BC * DA)", "to_expression": "AB * CD + BC * DA >= AC * BD", "justification": "arctan preserves and reflects order", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as AB * CD + BC * DA ≥ AC * BD.\"]"}], "source_equivalence_identity": "arctan preserves and reflects order"}
{"schema_version": "1.0", "item_id": "rcd2_fa4ff5680134fac4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7922A15B98B4", "name": "Kolmogorov's inequality", "statement_latex": "Pr(max_{1<=k<=n}|S_{k}|>=λ) <= (1/λ^2) * Var[S_{n}]", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge ((1/λ^2) * Var[S_{n}])\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (Pr(max_{1≤k≤n}|S_{k}|≥λ))\\}", "to_expression": "Pr(max_{1<=k<=n}|S_{k}|>=λ) <= (1/λ^2) * Var[S_{n}]", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr(max_{1≤k≤n}|S_{k}|≥λ) ≤ (1/λ^2) * Var[S_{n}].\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_329d353a6eb4c91b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FE78D365CC0D", "name": "Weinstein–Aronszajn identity", "statement_latex": "det(I_m + AB) = det(I_n + BA)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((det(I_m + AB)),z):z\\in D\\}=\\{((det(I_n + BA)),z):z\\in D\\}", "to_expression": "det(I_m + AB) = det(I_n + BA)", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as det(I_m + AB) = det(I_n + BA).\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_995e5cd574e03e9e", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "matrix theory", "subfield": "linear algebra", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_7b83d4c4ea8558cc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8DC5B9EC1AA5", "name": "Dimension theorem for vector spaces", "statement_latex": "|B_1| = |B_2|", "assumptions": "B_1 and B_2 are bases of the same finite-dimensional vector space."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((|B_1|),(|B_2|))\\in\\{(z,z):z\\in D\\}", "to_expression": "|B_1| = |B_2|", "justification": "pair in diagonal iff X=Y", "preconditions": "B_1 and B_2 are bases of the same finite-dimensional vector space.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |B_1| = |B_2|.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_0cf6fcecae3434be", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "modular arithmetic", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A1DCD328739C", "name": "Hensel's lemma", "statement_latex": "f(x) = 0 (mod p^n)", "assumptions": "All displayed arithmetic expressions are integers; the modulus is a positive integer."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(f(x)) \\in {(0 (mod p^n))}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed arithmetic expressions are integers; the modulus is a positive integer.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(x) = 0 (mod p^n).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f(x) = 0 (mod p^n)", "justification": "L in {R} iff L = R.", "preconditions": "All displayed arithmetic expressions are integers; the modulus is a positive integer.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(x) = 0 (mod p^n).\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_be4ff09239c0b1f5", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_9b8f21562034ceac", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum optics", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BFC6929833DF", "name": "Optical equivalence theorem", "statement_latex": "\\operatorname{tr}(\\hat{\\rho} \\cdot g_{\\Omega}(\\hat{a}, \\hat{a}^{\\dagger})) = \\int f_{\\bar{\\Omega}}(\\alpha, \\alpha^*) g_{\\Omega}(\\alpha, \\alpha^*) d^2\\alpha", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((\\operatorname{tr}(\\hat{\\rho} \\cdot g_{\\Omega}(\\hat{a}, \\hat{a}^{\\dagger}))),z):z\\in D\\}=\\{((\\int f_{\\bar{\\Omega}}(\\alpha, \\alpha^*) g_{\\Omega}(\\alpha, \\alpha^*) d^2\\alpha),z):z\\in D\\}", "to_expression": "\\operatorname{tr}(\\hat{\\rho} \\cdot g_{\\Omega}(\\hat{a}, \\hat{a}^{\\dagger})) = \\int f_{\\bar{\\Omega}}(\\alpha, \\alpha^*) g_{\\Omega}(\\alpha, \\alpha^*) d^2\\alpha", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\operatorname{tr}(\\\\hat{\\\\rho} \\\\cdot g_{\\\\Omega}(\\\\hat{a}, \\\\hat{a}^{\\\\dagger})) = \\\\int f_{\\\\bar{\\\\Omega}}(\\\\alpha, \\\\alpha^*) g_{\\\\Omega}(\\\\alpha, \\\\alpha^*) d^2\\\\alpha.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_233e1c22ab71f448", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "additive number theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_870FF97782B2", "name": "Goldbach's weak conjecture", "statement_latex": "n = p1 + p2 + p3", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((n),z):z\\in D\\}=\\{((p1 + p2 + p3),z):z\\in D\\}", "to_expression": "n = p1 + p2 + p3", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as n = p1 + p2 + p3.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_84c24e4318c53e53", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "field theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2412d6b170787f03", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "approximation theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_DC96C747092B", "name": "Müntz–Szász theorem", "statement_latex": "sum_{n in S} (1/n) = infinity", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((sum_{n in S} (1/n)),(infinity))=0", "to_expression": "sum_{n in S} (1/n) = infinity", "justification": "zero discrete distance iff equality", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{n in S} (1/n) = infinity.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_fc624f6c6eea273a", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_97d36d88d3b74c5d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic calculus", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_58173198312A", "name": "Boué–Dupuis formula", "statement_latex": "-log E[e^{-f(B)}] = inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)]", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((-log E[e^{-f(B)}]),z)=((inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)]),z)", "to_expression": "-log E[e^{-f(B)}] = inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)]", "justification": "pair equality forces first coordinates equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as -log E[e^{-f(B)}] = inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)].\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_f95d5f9eda29de92", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "convex geometry", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4F00365B64E5", "name": "Minkowski's first inequality for convex bodies", "statement_latex": "V_1(K, L) ≥ V(K)^((n-1)/n) V(L)^(1/n)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((V(K)^((n-1)/n) V(L)^(1/n))-(V_1(K, L)),(-\\infty,0])=0", "to_expression": "V_1(K, L) ≥ V(K)^((n-1)/n) V(L)^(1/n)", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as V_1(K, L) ≥ V(K)^((n-1)/n) V(L)^(1/n).\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
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{"schema_version": "1.0", "item_id": "rcd2_9c3eb0cff1973c09", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "zeta and L-functions", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0323683fa7cdb075", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0EDEF2DF9656", "name": "Fundamental theorem of arithmetic", "statement_latex": "n = p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(n)\\Longleftrightarrow z=(p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k}))", "to_expression": "n = p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k}", "justification": "same equality fiber iff equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as n = p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k}.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_f24d674cd9bb9c51", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "measure theory", "primary_mechanism": "set_membership", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4450EEEA850", "name": "Borel–Cantelli lemma", "statement_latex": "limsup_{n->infinity} E_n = intersection_{n=1 to infinity} union_{k=n to infinity} E_k", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists s \\in \\mathbb{R}_{>0}: (limsup_{n-) = (\\infty} E_n = intersection_{n=1 to \\infty} union_{k=n to \\infty} E_k) + s", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A<B iff s=B-A is positive and B=A+s.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover limsup_{n- > infinity} E_n = intersection_{n=1 to infinity} union_{k=n to infinity} E_k.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "A<B iff s=B-A is positive and B=A+s.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover limsup_{n- > infinity} E_n = intersection_{n=1 to infinity} union_{k=n to infinity} E_k.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "limsup_{n->infinity} E_n = intersection_{n=1 to infinity} union_{k=n to infinity} E_k", "justification": "A<B iff s=B-A is positive and B=A+s.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover limsup_{n- > infinity} E_n = intersection_{n=1 to infinity} union_{k=n to infinity} E_k.\"]"}], "source_equivalence_identity": "A<B iff s=B-A is positive and B=A+s."}
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{"schema_version": "1.0", "item_id": "rcd2_1e7e767a751b8627", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematical analysis", "subfield": "number theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_4c1c1dbd72e8b721", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal set theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4D5A3A69C1D", "name": "Sauer–Shelah lemma", "statement_latex": "|F| <= sum_{i=0}^{d} binom(n,i)", "assumptions": "F is a family of subsets of an n-element set with VC dimension d."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((|F|)-(sum_{i=0}^{d} binom(n,i)))=0", "to_expression": "|F| <= sum_{i=0}^{d} binom(n,i)", "justification": "sup equals max(0,X-Y)", "preconditions": "F is a family of subsets of an n-element set with VC dimension d.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |F| ≤ sum_{i=0}^{d} binom(n,i).\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
{"schema_version": "1.0", "item_id": "rcd2_e9bcdfb75e48fb4b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "measure theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3D75E63AA146", "name": "Vitale's random Brunn–Minkowski inequality", "statement_latex": "(vol_n(E[X]))^(1/n) >= E[vol_n(X)^(1/n)]", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(E[vol_n(X)^(1/n)])}{\\exp(E[vol_n(X)^(1/n)])+\\exp(vol_n(E[X]))^(1/n)}\\le\\frac{1}{2}", "to_expression": "(vol_n(E[X]))^(1/n) >= E[vol_n(X)^(1/n)]", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (vol_n(E[X]))^(1/n) ≥ E[vol_n(X)^(1/n)].\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_c131c588678e6c92", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "matrix analysis", "subfield": "linear algebra", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c79e627c250f825e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8CBF25034487", "name": "Heegner's lemma", "statement_latex": "y^{2}=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(y^{2}) \\in {(a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0})}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover y^{2}=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "y^{2}=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}", "justification": "L in {R} iff L = R.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover y^{2}=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_589cd8c72ec8ca6e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_936D29F3BF52", "name": "Hsu–Robbins–Erdős theorem", "statement_latex": "∑_{n>=1}P(|S_{n}|>εn)<∞", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{>0}\\;\\text{such that}\\;(∑_{n≥1}P(|S_{n}|>εn))+s=(∞)", "to_expression": "∑_{n>=1}P(|S_{n}|>εn)<∞", "justification": "positive slack iff X<Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∑_{n≥1}P(|S_{n}|>εn)<∞.\"]"}], "source_equivalence_identity": "positive slack iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_257c5e8f1b48073e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "enumerative combinatorics", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_30BF4DFA8E4C", "name": "MacMahon's master theorem", "statement_latex": "G(k1, ..., km) = [x1^k1 ... xm^km] Prod_{i=1 to m} (Sum_{j=1 to m} a_{ij} x_j)^k_i", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((G(k1, ..., km)),z):z\\in D\\}=\\{(([x1^k1 ... xm^km] Prod_{i=1 to m} (Sum_{j=1 to m} a_{ij} x_j)^k_i),z):z\\in D\\}", "to_expression": "G(k1, ..., km) = [x1^k1 ... xm^km] Prod_{i=1 to m} (Sum_{j=1 to m} a_{ij} x_j)^k_i", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as G(k1, ..., km) = [x1^k1 ... xm^km] Prod_{i=1 to m} (Sum_{j=1 to m} a_{ij} x_j)^k_i.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_679d534f7b2306bb", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_7f47e85457034d84", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FE78D365CC0D", "name": "Weinstein–Aronszajn identity", "statement_latex": "det(I_m + AB) = det(I_n + BA)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((det(I_m + AB)),z)=((det(I_n + BA)),z)", "to_expression": "det(I_m + AB) = det(I_n + BA)", "justification": "pair equality forces first coordinates equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as det(I_m + AB) = det(I_n + BA).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_f86e49aa033be464", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "complex analysis", "subfield": "complex geometry", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A0642B3B222A", "name": "Schwarz lemma", "statement_latex": "|f(z)| <= |z|", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (|f(z)|) \\le x \\land x \\le (|z|)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |f(z)| <= |z|.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |f(z)| <= |z|.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "|f(z)| <= |z|", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |f(z)| <= |z|.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
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{"schema_version": "1.0", "item_id": "rcd2_81151b04d69f1e22", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
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{"schema_version": "1.0", "item_id": "rcd2_7b9556f41f27f95c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c2b54d8c32350571", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "functional analysis", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1D2F074AF9A8", "name": "Spectral theorem", "statement_latex": "Ax = λx", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(Ax)=z\\;\\land\\;z=(λx)", "to_expression": "Ax = λx", "justification": "common witness iff equality", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Ax = λx.\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_5624d9cfc8f3057f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "physics", "subfield": "general relativity", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_354eefb0b3c95036", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "Mathematical analysis", "subfield": "Sequences and series", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F3C0470E95D9", "name": "Abel's inequality", "statement_latex": "|∑ a_k b_k| ≤ max|B_k|(|a_n| + |a_n - a_1|)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\operatorname{softplus}((|∑ a_k b_k|)-(max|B_k|(|a_n| + |a_n - a_1|)))\\le\\log 2", "to_expression": "|∑ a_k b_k| ≤ max|B_k|(|a_n| + |a_n - a_1|)", "justification": "softplus is increasing and softplus(0)=log2", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |∑ a_k b_k| ≤ max|B_k|(|a_n| + |a_n - a_1|).\"]"}], "source_equivalence_identity": "softplus is increasing and softplus(0)=log2"}
{"schema_version": "1.0", "item_id": "rcd2_9d582b6f48743811", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum information theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3795DDB6E155", "name": "NLTS conjecture", "statement_latex": "H = \\sum_{i=1}^{m} H_{i}", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(H)\\Longleftrightarrow z=(\\sum_{i=1}^{m} H_{i}))", "to_expression": "H = \\sum_{i=1}^{m} H_{i}", "justification": "same equality fiber iff equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as H = \\\\sum_{i=1}^{m} H_{i}.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_d9fa04ba3c46ed28", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "topology", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3919D9203C0B", "name": "Cantor's intersection theorem", "statement_latex": "\\bigcap _{k=0}^{\\infty }C_{k}\\neq \\emptyset", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(0}^{\\infty }C_{k}\\neq \\\\emptyset) = (\\bigcap _{k)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\bigcap _{k=0}^{\\\\infty }C_{k}\\\\neq \\\\emptyset.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\bigcap _{k=0}^{\\infty }C_{k}\\neq \\emptyset", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\bigcap _{k=0}^{\\\\infty }C_{k}\\\\neq \\\\emptyset.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_ac3cc683384f792e", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "inequalities", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a82b91c9cc25bbf4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "elementary number theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0F6BD7E91884", "name": "Bézout's identity", "statement_latex": "ax + by = gcd(a, b)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((ax + by),z):z\\in D\\}=\\{((gcd(a, b)),z):z\\in D\\}", "to_expression": "ax + by = gcd(a, b)", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ax + by = gcd(a, b).\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_86e266c9f6ffa02f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "approximation theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_DC96C747092B", "name": "Müntz–Szász theorem", "statement_latex": "sum_{n in S} (1/n) = infinity", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((sum_{n \\in S} (1/n)) \\in S \\land (\\infty) \\notin S) \\lor ((\\infty) \\in S \\land (sum_{n \\in S} (1/n)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover sum_{n in S} (1/n) = infinity.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "sum_{n in S} (1/n) = infinity", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover sum_{n in S} (1/n) = infinity.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_e3a5023d67bdabaa", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "polyhedral combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_687D9EB61DA6", "name": "Eberhard's theorem", "statement_latex": "sum (6-i)p_i = 12", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(12) = (sum (6-i)p_i)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover sum (6-i)p_i = 12.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "sum (6-i)p_i = 12", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover sum (6-i)p_i = 12.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
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{"schema_version": "1.0", "item_id": "rcd2_6158bc0d6a69616e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "calculus", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_819FFAB136D4", "name": "L'Hôpital's rule", "statement_latex": "lim_{x->c} f(x)/g(x) = lim_{x->c} f'(x)/g'(x)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((lim_{x→c} f(x)/g(x)),z)=((lim_{x→c} f'(x)/g'(x)),z)", "to_expression": "lim_{x->c} f(x)/g(x) = lim_{x->c} f'(x)/g'(x)", "justification": "pair equality forces first coordinates equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{x→c} f(x)/g(x) = lim_{x→c} f'(x)/g'(x).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_7db20c8f88fbd047", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "Ramsey theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A40EE0DC8F58", "name": "Van der Waerden's theorem", "statement_latex": "W(r,k) ≤ 2^{2^{r^{2^{2^{k+9}}}}}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (2^{2^{r^{2^{2^{k+9}}}}})\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (W(r,k))\\}", "to_expression": "W(r,k) ≤ 2^{2^{r^{2^{2^{k+9}}}}}", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as W(r,k) ≤ 2^{2^{r^{2^{2^{k+9}}}}}.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_0f4e90dc787a7e12", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7BB84422BC5F", "name": "Hadamard's inequality", "statement_latex": "|det(N)| <= prod_{i=1}^n ||v_i||", "assumptions": "N is a square matrix with row or column vectors v_i in an inner-product space; norms are Euclidean."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (|det(N)|) \\le x \\land x \\le (prod_{i=1}^n ||v_i||)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "N is a square matrix with row or column vectors v_i in an inner-product space; norms are Euclidean.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |det(N)| <= prod_{i=1}^n ||v_i||.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "N is a square matrix with row or column vectors v_i in an inner-product space; norms are Euclidean.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |det(N)| <= prod_{i=1}^n ||v_i||.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "|det(N)| <= prod_{i=1}^n ||v_i||", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "N is a square matrix with row or column vectors v_i in an inner-product space; norms are Euclidean.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |det(N)| <= prod_{i=1}^n ||v_i||.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
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{"schema_version": "1.0", "item_id": "rcd2_32d46eeb0976b08b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_20033DA37507", "name": "Kolmogorov–Arnold representation theorem", "statement_latex": "f(x_1, ..., x_n) = Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p))", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(f(x_1, ..., x_n)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p))))", "to_expression": "f(x_1, ..., x_n) = Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p))", "justification": "each argmin set is a singleton", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f(x_1, ..., x_n) = Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p)).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
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{"schema_version": "1.0", "item_id": "rcd2_80452960dd0ea28f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f8ab108aef8d07ca", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "enumerative combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E1416BDA0AD6", "name": "Pólya enumeration theorem", "statement_latex": "|Y^X/G| = (1/|G|) * Σ_{g∈G} m^{c(g)}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (|Y^X/G| = (1/|G|) * Σ_{g∈G} m^{c(g)}))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "|Y^X/G| = (1/|G|) * Σ_{g∈G} m^{c(g)}", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
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{"schema_version": "1.0", "item_id": "rcd2_64066911f7e0436b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "modular arithmetic", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_6B373783C03D", "name": "Thue's lemma", "statement_latex": "ay ≡ x (mod m)", "assumptions": "All displayed arithmetic expressions are integers; the modulus is a positive integer."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists k\\in\\mathbb{Z}:(ay)=(x)+k(m)", "to_expression": "ay ≡ x (mod m)", "justification": "same as congruence witness", "preconditions": "All displayed arithmetic expressions are integers; the modulus is a positive integer.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ay ≡ x (mod m).\"]"}], "source_equivalence_identity": "same as congruence witness"}
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{"schema_version": "1.0", "item_id": "rcd2_bb188a9c4a471096", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "probabilistic number theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a49f20f250c3a0bc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "prime number theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BFDCE2A9DBB2", "name": "Bertrand's postulate", "statement_latex": "exists prime p such that n < p < 2*n", "assumptions": "n is an integer greater than 1."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\operatorname{Id}_D[\\{prime p\\in D:n < p < 2*n\\}]\\ne\\emptyset", "to_expression": "exists prime p such that n < p < 2*n", "justification": "identity image equals S", "preconditions": "n is an integer greater than 1.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as exists prime p such that n < p < 2*n.\"]"}], "source_equivalence_identity": "identity image equals S"}
{"schema_version": "1.0", "item_id": "rcd2_aced321953bc42a3", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "measure theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_948d712cb0d53a3d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "intersection", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7C6C20BC69B4", "name": "Slepian's lemma", "statement_latex": "Pr[Intersection(X_i <= u_i)] <= Pr[Intersection(Y_i <= u_i)]", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\operatorname{softplus}((Pr[Intersection(X_i ≤ u_i)])-(Pr[Intersection(Y_i ≤ u_i)]))\\le\\log 2", "to_expression": "Pr[Intersection(X_i <= u_i)] <= Pr[Intersection(Y_i <= u_i)]", "justification": "softplus is increasing and softplus(0)=log2", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr[Intersection(X_i ≤ u_i)] ≤ Pr[Intersection(Y_i ≤ u_i)].\"]"}], "source_equivalence_identity": "softplus is increasing and softplus(0)=log2"}
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{"schema_version": "1.0", "item_id": "rcd2_5fece05c8bbb8f8c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "p-adic analysis", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3B924C720B22", "name": "Krasner's lemma", "statement_latex": "|alpha - beta| < |alpha - alpha_i|", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(|alpha - alpha_i|) - (|alpha - b\\eta|) > 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |alpha - beta| < |alpha - alpha_i|.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |alpha - beta| < |alpha - alpha_i|.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "|alpha - beta| < |alpha - alpha_i|", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |alpha - beta| < |alpha - alpha_i|.\"]"}], "source_equivalence_identity": "A<B iff B-A>0 over finite real quantities."}
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{"schema_version": "1.0", "item_id": "rcd2_c388a1ff72d4d5b2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4A7E73DE4982", "name": "Rayleigh's theorem", "statement_latex": "1/r + 1/s = 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(1) = (1/r + 1/s)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover 1/r + 1/s = 1.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "1/r + 1/s = 1", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover 1/r + 1/s = 1.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_8e617699330fc909", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "vector calculus", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F8BCAAED7243", "name": "Green's theorem", "statement_latex": "∮_{C}(L dx + M dy) = ∬_{D} (∂M/∂x - ∂L/∂y) dA", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(∮_{C}(L dx + M dy)) \\in {(∬_{D} (∂M/∂x - ∂L/∂y) dA)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover ∮_{C}(L dx + M dy) = ∬_{D} (∂M/∂x - ∂L/∂y) dA.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "∮_{C}(L dx + M dy) = ∬_{D} (∂M/∂x - ∂L/∂y) dA", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover ∮_{C}(L dx + M dy) = ∬_{D} (∂M/∂x - ∂L/∂y) dA.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_b371187124cbb364", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f21ca9f04335fec4", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_eaa87c740dbd8194", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_72498ba40bedf195", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "computer algebra", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F94B0806E6BB", "name": "Landau–Mignotte bound", "statement_latex": "||h||_1 <= 2^n * ||f||_1", "assumptions": "h is a factor of the integer polynomial f of degree n under the standard Landau-Mignotte hypotheses; l1 coefficient norms are finite."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(2^n * ||f||_1)\\le t\\Rightarrow (||h||_1)\\le t", "to_expression": "||h||_1 <= 2^n * ||f||_1", "justification": "converse uses t=Y", "preconditions": "h is a factor of the integer polynomial f of degree n under the standard Landau-Mignotte hypotheses; l1 coefficient norms are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ||h||_1 ≤ 2^n * ||f||_1.\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_d313640481cf5d51", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "intersection", "transformation_category": "optimization_projection", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ad1fd20c8971e7b4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "discrete mathematics", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2F917150CBA3", "name": "Pigeonhole principle", "statement_latex": "max_{1<=i<=m} |B_i| >= ceil(n/m)", "assumptions": "n objects are distributed among m nonempty boxes B_1,...,B_m; m>=1."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(ceil(n/m))+s=(max_{1≤i≤m} |B_i|)", "to_expression": "max_{1<=i<=m} |B_i| >= ceil(n/m)", "justification": "nonnegative slack exists iff X≤Y", "preconditions": "n objects are distributed among m nonempty boxes B_1,...,B_m; m>=1.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as max_{1≤i≤m} |B_i| ≥ ceil(n/m).\"]"}], "source_equivalence_identity": "nonnegative slack exists iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_fc2a1a7e18d01374", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "vector calculus", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d7639bc5c246cba8", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "combinatorics", "subfield": "combinatorial geometry", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6dfa49341e262190", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "decision theory", "subfield": "information theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D757C95A0326", "name": "Blackwell's informativeness theorem", "statement_latex": "V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((V(p)),(max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)))=0", "to_expression": "V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)", "justification": "zero discrete distance iff equality", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega).\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_52c36ead880e3490", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "combinatorial number theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3FBB7DDC427F", "name": "Erdős–Graham problem", "statement_latex": "sum_{n in S} (1/n) = 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{n in S} (1/n)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(1))", "to_expression": "sum_{n in S} (1/n) = 1", "justification": "each argmin set is a singleton", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{n in S} (1/n) = 1.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_8b269c82b3cf897e", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "graph theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_02e8447fd8f4924c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "quadratic forms", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_39adc236cdc88c2a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0FE477D29636", "name": "Hardy–Ramanujan theorem", "statement_latex": "|ω(n)−loglogn|<(loglogn)^(1/2+ε)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(|ω(n)−loglogn|) - ((loglogn)^(1/2+ε)) < 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A-B<0 iff A<B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |ω(n)−loglogn| < (loglogn)^(1/2+ε).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A-B<0 iff A<B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |ω(n)−loglogn| < (loglogn)^(1/2+ε).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "|ω(n)−loglogn|<(loglogn)^(1/2+ε)", "justification": "A-B<0 iff A<B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |ω(n)−loglogn| < (loglogn)^(1/2+ε).\"]"}], "source_equivalence_identity": "A-B<0 iff A<B."}
{"schema_version": "1.0", "item_id": "rcd2_92efe0eadb00bdec", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "Ramsey theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c28623cbe7005233", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "physics", "subfield": "general relativity", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_243e8a08abfee861", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "statistics", "subfield": "empirical Bayes methods", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2C7BEBF4ECBC", "name": "Robbins lemma", "statement_latex": "E(Xf(X-1)) = λE(f(X))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(E(Xf(X-1))) \\in {(λE(f(X)))}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E(Xf(X-1)) = λE(f(X)).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "E(Xf(X-1)) = λE(f(X))", "justification": "L in {R} iff L = R.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E(Xf(X-1)) = λE(f(X)).\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_c5045ac202e7b4f0", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "geometry of numbers", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2b95ee9f6ae21b22", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "statistical mechanics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_5caf95f0b8dbb1c8", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0ed49b8cd35c55cb", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a5f9ab5992454b61", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "algebra", "subfield": "combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2617f9a8d5fa0a60", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C3C168B97D27", "name": "Watson's quintuple product identity", "statement_latex": "prod_{n>=1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1})", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n \\\\in \\mathbb{Z}} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1})) \\le x \\land x \\le (prod_{n)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover prod_{n >= 1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1}).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover prod_{n >= 1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1}).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "prod_{n>=1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1})", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover prod_{n >= 1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1}).\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_c2c03f1e05799b1b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "topology", "subfield": "algebraic topology", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_48CD393CFABB", "name": "Borsuk–Ulam theorem", "statement_latex": "f(-x) = f(x)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((f(-x)))=F((f(x)))", "to_expression": "f(-x) = f(x)", "justification": "indicator of {X} separates unequal objects", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f(-x) = f(x).\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_4fbced180603254d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "probability theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8F3F3785BDA2", "name": "Gibbs' inequality", "statement_latex": "-Σ p_i log p_i <= -Σ p_i log q_i", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (-Σ p_i log p_i) \\le x \\land x \\le (-Σ p_i log q_i)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover -Σ p_i log p_i <= -Σ p_i log q_i.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover -Σ p_i log p_i <= -Σ p_i log q_i.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "-Σ p_i log p_i <= -Σ p_i log q_i", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover -Σ p_i log p_i <= -Σ p_i log q_i.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
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{"schema_version": "1.0", "item_id": "rcd2_9e146cb929aed98a", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "topology", "primary_mechanism": "distance", "transformation_category": "residual_nonnegativity", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_42dc19e39f12a5e2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "matroid theory", "subfield": "combinatorics", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_09298FD96036", "name": "Ingleton's inequality", "statement_latex": "ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4) ≤ ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;t\\le (ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4))\\Rightarrow t\\le (ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4))", "to_expression": "ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4) ≤ ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4)", "justification": "converse uses t=X", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4) ≤ ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4).\"]"}], "source_equivalence_identity": "converse uses t=X"}
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{"schema_version": "1.0", "item_id": "rcd2_3ce3b0ab5b92b143", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "additive combinatorics", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e7c6e537eeffd4cd", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_7f62c7aba88ac23b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "physics", "subfield": "general relativity", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_47756d4f4013d84d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "quadratic residues", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A18C6C225F06", "name": "Zolotarev's lemma", "statement_latex": "(a/p) = epsilon(pi_a)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\varepsilon(pi_a)) = ((a/p))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover (a/p) = epsilon(pi_a).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "(a/p) = epsilon(pi_a)", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover (a/p) = epsilon(pi_a).\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_c4e984d9b3650490", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "harmonic analysis", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_dc406facfa87c6f1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "metric geometry", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BDB71158A694", "name": "Besicovitch inequality", "statement_latex": "prod_{i} d_{i} <= Vol(M)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(prod_{i} d_{i})+s=(Vol(M))", "to_expression": "prod_{i} d_{i} <= Vol(M)", "justification": "nonnegative slack exists iff X≤Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as prod_{i} d_{i} ≤ Vol(M).\"]"}], "source_equivalence_identity": "nonnegative slack exists iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_d3626908e6c7a4e2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "representation theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_13CEC210D728", "name": "Wigner–Eckart theorem", "statement_latex": "⟨j m | T_q^(k) | j' m'⟩ = ⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((⟨j m | T_q^(k) | j' m'⟩),z)=((⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩),z)", "to_expression": "⟨j m | T_q^(k) | j' m'⟩ = ⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩", "justification": "pair equality forces first coordinates equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ⟨j m | T_q^(k) | j' m'⟩ = ⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_7968c1d56820f3e3", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "symplectic geometry", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_097eac3dbe50fedd", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "functional analysis", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1D2F074AF9A8", "name": "Spectral theorem", "statement_latex": "Ax = λx", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((Ax),z)=((λx),z)", "to_expression": "Ax = λx", "justification": "pair equality forces first coordinates equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Ax = λx.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_ccb8f9546dfb5b59", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_58a4ecc1bf09f353", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "field theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4E2C0DED1A31", "name": "Joubert's theorem", "statement_latex": "p(t) = t^6 + c_4t^4 + c_2t^2 + c_1t + c_0", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(p(t))\\}=\\{(t^6 + c_4t^4 + c_2t^2 + c_1t + c_0)\\}", "to_expression": "p(t) = t^6 + c_4t^4 + c_2t^2 + c_1t + c_0", "justification": "singletons equal iff elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p(t) = t^6 + c_4t^4 + c_2t^2 + c_1t + c_0.\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_87b8609d7fac92a2", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "convex geometry", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_5d9a970557c9818a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0C6AE95E0EE0", "name": "Hoeffding's inequality", "statement_latex": "P(S_n - E[S_n] >= t) <= exp(-2*t^2 / sum_i (b_i-a_i)^2)", "assumptions": "The summands are independent and bounded with X_i in [a_i,b_i]; t>=0; denominator is positive."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (P(S_n - E[S_n] \\ge t) \\le \\exp(-2*t^2 / sum_i (b_i-a_i)^2)))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "The summands are independent and bounded with X_i in [a_i,b_i]; t>=0; denominator is positive.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "P(S_n - E[S_n] >= t) <= exp(-2*t^2 / sum_i (b_i-a_i)^2)", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "The summands are independent and bounded with X_i in [a_i,b_i]; t>=0; denominator is positive.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_7fa2f111773629ca", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "topology", "subfield": "general topology", "primary_mechanism": "quantifier", "transformation_category": "counting_combinatorial", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_00c7886d15ed0922", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "modular arithmetic", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_cb9df7b73dd393f3", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "inclusion-exclusion principle", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_88248e992df8669a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "statistical mechanics", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4D516E39AEB4", "name": "H-theorem", "statement_latex": "H(t) = ∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((H(t)),z)=((∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE),z)", "to_expression": "H(t) = ∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE", "justification": "pair equality forces first coordinates equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as H(t) = ∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_64e0fe4d84fd62c3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "probability theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_27AD75237F2D", "name": "Entropy power inequality", "statement_latex": "N(X+Y) >= N(X) + N(Y)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(N(X) + N(Y)) - (N(X+Y)) \\le 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A-B <= 0 iff A <= B by adding B to both sides.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover N(X+Y) >= N(X) + N(Y).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A-B <= 0 iff A <= B by adding B to both sides.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover N(X+Y) >= N(X) + N(Y).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "N(X+Y) >= N(X) + N(Y)", "justification": "A-B <= 0 iff A <= B by adding B to both sides.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover N(X+Y) >= N(X) + N(Y).\"]"}], "source_equivalence_identity": "A-B <= 0 iff A <= B by adding B to both sides."}
{"schema_version": "1.0", "item_id": "rcd2_f2326a292c9193da", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "design theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_3951853aba3dc64d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "statistical mechanics", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4D516E39AEB4", "name": "H-theorem", "statement_latex": "H(t) = ∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((H(t)),z):z\\in D\\}=\\{((∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE),z):z\\in D\\}", "to_expression": "H(t) = ∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as H(t) = ∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_2e71e8324783bfbd", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "vector calculus", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_6CFB1B566265", "name": "Stokes' theorem", "statement_latex": "integral integral_{Sigma} (curl F) dot dSigma = integral_{partial Sigma} F dot dr", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((integral integral_{Sigma} (curl F) dot dSigma),(integral_{partial Sigma} F dot dr))=0", "to_expression": "integral integral_{Sigma} (curl F) dot dSigma = integral_{partial Sigma} F dot dr", "justification": "zero discrete distance iff equality", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as integral integral_{Sigma} (curl F) dot dSigma = integral_{partial Sigma} F dot dr.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_3c6baff5744619de", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "inequalities", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9822BD8B9BE6", "name": "Muirhead's inequality", "statement_latex": "[a] >= [b] for a majorizing b", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: ([b] for a majorizing b) \\le x \\land x \\le ([a])", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover [a] >= [b] for a majorizing b.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover [a] >= [b] for a majorizing b.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "[a] >= [b] for a majorizing b", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover [a] >= [b] for a majorizing b.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_f635bc37077cc843", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8DC5B9EC1AA5", "name": "Dimension theorem for vector spaces", "statement_latex": "|B_1| = |B_2|", "assumptions": "B_1 and B_2 are bases of the same finite-dimensional vector space."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(|B_1|)=z\\;\\land\\;z=(|B_2|)", "to_expression": "|B_1| = |B_2|", "justification": "common witness iff equality", "preconditions": "B_1 and B_2 are bases of the same finite-dimensional vector space.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |B_1| = |B_2|.\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_74a3cc29c414f4c9", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "zeta and L-functions", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_cf2dbc3b52d43dc7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_99B889CC4005", "name": "Hoeffding's lemma", "statement_latex": "E[exp(lambda*(X-E[X]))] <= exp(lambda^2*(b-a)^2/8)", "assumptions": "X is almost surely in [a,b]; lambda is real; expectations are finite."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\exp(lambda^2*(b-a)^2/8)) - (E[\\exp(lambda*(X-E[X]))]) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "X is almost surely in [a,b]; lambda is real; expectations are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover E[exp(lambda*(X-E[X]))] <= exp(lambda^2*(b-a)^2/8).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "X is almost surely in [a,b]; lambda is real; expectations are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover E[exp(lambda*(X-E[X]))] <= exp(lambda^2*(b-a)^2/8).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "E[exp(lambda*(X-E[X]))] <= exp(lambda^2*(b-a)^2/8)", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "X is almost surely in [a,b]; lambda is real; expectations are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover E[exp(lambda*(X-E[X]))] <= exp(lambda^2*(b-a)^2/8).\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
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{"schema_version": "1.0", "item_id": "rcd2_859dbdc76e7d133f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "functional analysis", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_462346AD2251", "name": "Ehrling's lemma", "statement_latex": "||x||_Y <= ε||x||_X + C(ε)||x||_Z", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(||x||_Y)-(ε||x||_X + C(ε)||x||_Z)\\}=0", "to_expression": "||x||_Y <= ε||x||_X + C(ε)||x||_Z", "justification": "positive part vanishes iff X≤Y", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ||x||_Y ≤ ε||x||_X + C(ε)||x||_Z.\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_966e3224f42df48d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "field theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_ED1CB49A03EE", "name": "Steinitz's theorem", "statement_latex": "L = K(α)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "(\\{(L)\\}\\times\\{(K(α))\\})\\cap\\{(z,z):z\\in D\\}=\\{((L),(K(α)))\\}", "to_expression": "L = K(α)", "justification": "intersection is singleton iff pair diagonal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as L = K(α).\"]"}], "source_equivalence_identity": "intersection is singleton iff pair diagonal"}
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{"schema_version": "1.0", "item_id": "rcd2_5b4554633aa4fdf6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_EB73116410D1", "name": "Hurwitz's theorem", "statement_latex": "|\\xi - m/n| < 1 / (\\sqrt{5} * n^2)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(|\\xi - m/n|)-(1 / (\\sqrt{5} * n^2))\\}=0\\;\\land\\;(|\\xi - m/n|)\\ne (1 / (\\sqrt{5} * n^2))", "to_expression": "|\\xi - m/n| < 1 / (\\sqrt{5} * n^2)", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |\\\\xi - m/n| < 1 / (\\\\sqrt{5} * n^2).\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
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{"schema_version": "1.0", "item_id": "rcd2_7354341cda00c486", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "convex geometry", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
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{"schema_version": "1.0", "item_id": "rcd2_b8111c563d3a2bb0", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "matroid theory", "subfield": "combinatorics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2453181529f15c7a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "Ramsey theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8E938CA76307", "name": "Milliken's tree theorem", "statement_latex": "S_T^n = C_1 ∪ ... ∪ C_r", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(C_1 ∪ ... ∪ C_r) = (S_T^n)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover S_T^n = C_1 ∪ ... ∪ C_r.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "S_T^n = C_1 ∪ ... ∪ C_r", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover S_T^n = C_1 ∪ ... ∪ C_r.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_03ec65b64f40a088", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "numerical analysis, finite element method", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7C7FF749E01A", "name": "Bramble–Hilbert lemma", "statement_latex": "inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)} <= C |u|_{W^{m,p}(\\Omega)}", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)})}{\\exp(inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)})+\\exp(C |u|_{W^{m,p}(\\Omega)})}\\le\\frac{1}{2}", "to_expression": "inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)} <= C |u|_{W^{m,p}(\\Omega)}", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\\\Omega)} ≤ C |u|_{W^{m,p}(\\\\Omega)}.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_851ddeb1ace23c04", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum information theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_cc0b3fbb52957df9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D737B48D674F", "name": "Borell–TIS inequality", "statement_latex": "P(‖f‖_T > E(‖f‖_T) + u) ≤ exp(-u^2 / (2σ_T^2))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(P(‖f‖_T > E(‖f‖_T) + u))-(exp(-u^2 / (2σ_T^2)))\\}=0", "to_expression": "P(‖f‖_T > E(‖f‖_T) + u) ≤ exp(-u^2 / (2σ_T^2))", "justification": "positive part vanishes iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(‖f‖_T > E(‖f‖_T) + u) ≤ exp(-u^2 / (2σ_T^2)).\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_b31fc8e3197da93a", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "elementary number theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b86800624469e903", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_368EF3E8339F", "name": "Summation by parts", "statement_latex": "sum_{k=m}^{n} f_k Delta g_k = (f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} Delta f_k", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(sum_{k=m}^{n} f_k Delta g_k))+\\delta_D(z,((f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} Delta f_k)))=0", "to_expression": "sum_{k=m}^{n} f_k Delta g_k = (f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} Delta f_k", "justification": "zero loss iff some z equals both", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{k=m}^{n} f_k Delta g_k = (f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} Delta f_k.\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_d663cbf2c7b6adfd", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "measure theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3D75E63AA146", "name": "Vitale's random Brunn–Minkowski inequality", "statement_latex": "(vol_n(E[X]))^(1/n) >= E[vol_n(X)^(1/n)]", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (vol_n(E[X]))^(1/n)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (E[vol_n(X)^(1/n)])\\}", "to_expression": "(vol_n(E[X]))^(1/n) >= E[vol_n(X)^(1/n)]", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (vol_n(E[X]))^(1/n) ≥ E[vol_n(X)^(1/n)].\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_2f43ee36bf2710e0", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "martingale theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_634754a1eb4b7c75", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_98220A4F883B", "name": "Siegel identity", "statement_latex": "\\frac{x_3-x_1}{x_2-x_1} + \\frac{x_2-x_3}{x_2-x_1} = 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(\\frac{x_3-x_1}{x_2-x_1} + \\frac{x_2-x_3}{x_2-x_1})=z\\;\\land\\;z=(1)", "to_expression": "\\frac{x_3-x_1}{x_2-x_1} + \\frac{x_2-x_3}{x_2-x_1} = 1", "justification": "common witness iff equality", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\frac{x_3-x_1}{x_2-x_1} + \\\\frac{x_2-x_3}{x_2-x_1} = 1.\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_02cc30dd6ea0028e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_30AD154E7704", "name": "Cyclotomic identity", "statement_latex": "1/(1-alpha*z) = product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((1/(1-alpha*z)))=F((product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}))", "to_expression": "1/(1-alpha*z) = product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as 1/(1-alpha*z) = product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_dd323150feaf36a3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "approximation theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9336CBED6851", "name": "Bernstein's theorem (polynomials)", "statement_latex": "max_{|z|=1}|P'(z)| <= n max_{|z|=1}|P(z)|", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((max_{|z|=1}|P'(z)|)-(n max_{|z|=1}|P(z)|))=0", "to_expression": "max_{|z|=1}|P'(z)| <= n max_{|z|=1}|P(z)|", "justification": "sup equals max(0,X-Y)", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as max_{|z|=1}|P'(z)| ≤ n max_{|z|=1}|P(z)|.\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
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{"schema_version": "1.0", "item_id": "rcd2_94b3835d843a9bd2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_99B889CC4005", "name": "Hoeffding's lemma", "statement_latex": "E[exp(lambda*(X-E[X]))] <= exp(lambda^2*(b-a)^2/8)", "assumptions": "X is almost surely in [a,b]; lambda is real; expectations are finite."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((E[exp(lambda*(X-E[X]))])-(exp(lambda^2*(b-a)^2/8)))=0", "to_expression": "E[exp(lambda*(X-E[X]))] <= exp(lambda^2*(b-a)^2/8)", "justification": "sup equals max(0,X-Y)", "preconditions": "X is almost surely in [a,b]; lambda is real; expectations are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as E[exp(lambda*(X-E[X]))] ≤ exp(lambda^2*(b-a)^2/8).\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
{"schema_version": "1.0", "item_id": "rcd2_6e854aa38de2f006", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9DBE3057770F", "name": "Kneser's theorem", "statement_latex": "|A+B|=|A+H|+|B+H|-|H|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(|A+H|+|B+H|-|H|) = (|A+B|)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover |A+B|=|A+H|+|B+H|-|H|.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "|A+B|=|A+H|+|B+H|-|H|", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover |A+B|=|A+H|+|B+H|-|H|.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_4765990d914e8410", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A75F5656C745", "name": "Grosswald–Schnitzer theorem", "statement_latex": "\\zeta(s) = \\prod_{n=1}^{\\infty} \\frac{1}{1-p_n^{-s}}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((\\zeta(s)),(\\prod_{n=1}^{\\infty} \\frac{1}{1-p_n^{-s}}))\\in\\{(z,z):z\\in D\\}", "to_expression": "\\zeta(s) = \\prod_{n=1}^{\\infty} \\frac{1}{1-p_n^{-s}}", "justification": "pair in diagonal iff X=Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\zeta(s) = \\\\prod_{n=1}^{\\\\infty} \\\\frac{1}{1-p_n^{-s}}.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
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{"schema_version": "1.0", "item_id": "rcd2_419a52fb6906f864", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "algebra", "subfield": "combinatorics", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E24E6480323C", "name": "Multinomial theorem", "statement_latex": "(x_{1}+x_{2}+\\cdots +x_{m})^{n}=\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;((x_{1}+x_{2}+\\cdots +x_{m})^{n})=z\\;\\land\\;z=(\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}})", "to_expression": "(x_{1}+x_{2}+\\cdots +x_{m})^{n}=\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}}", "justification": "common witness iff equality", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (x_{1}+x_{2}+\\\\cdots +x_{m})^{n}=\\\\sum _{k_{1}+k_{2}+\\\\cdots +k_{m}=n} \\\\binom{n}{k_1, k_2, \\\\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\\\cdots x_{m}^{k_{m}}.\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_a91d2b6ab8a35bd7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "elementary number theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_677EBF92D0D7", "name": "Lifting-the-exponent lemma", "statement_latex": "nu_p(x^n - y^n) = nu_p(x - y) + nu_p(n)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(nu_p(x^n - y^n))\\Longleftrightarrow z=(nu_p(x - y) + nu_p(n)))", "to_expression": "nu_p(x^n - y^n) = nu_p(x - y) + nu_p(n)", "justification": "same equality fiber iff equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as nu_p(x^n - y^n) = nu_p(x - y) + nu_p(n).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
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{"schema_version": "1.0", "item_id": "rcd2_fbfcfa6f1d48cdaf", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "physics", "subfield": "quantum field theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5C335B571C48", "name": "Gell-Mann and Low theorem", "statement_latex": "H = H_0 + gV", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((H),z)=((H_0 + gV),z)", "to_expression": "H = H_0 + gV", "justification": "pair equality forces first coordinates equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as H = H_0 + gV.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_ec6b675d0918f946", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "quadratic forms", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_1a61b1350c6f72a6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "infinite-dimensional vector spaces", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B23BCA4AB254", "name": "Erdős–Kaplansky theorem", "statement_latex": "\\dim(E^*) = \\text{card}(\\mathbb{K}^I)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((\\dim(E^*)),(\\text{card}(\\mathbb{K}^I)))\\in\\{(z,z):z\\in D\\}", "to_expression": "\\dim(E^*) = \\text{card}(\\mathbb{K}^I)", "justification": "pair in diagonal iff X=Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\dim(E^*) = \\\\text{card}(\\\\mathbb{K}^I).\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_75f2e9c2113da37a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8CBF25034487", "name": "Heegner's lemma", "statement_latex": "y^{2}=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((y^{2}),(a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}))\\in\\{(z,z):z\\in D\\}", "to_expression": "y^{2}=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}", "justification": "pair in diagonal iff X=Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as y^{2}=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
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{"schema_version": "1.0", "item_id": "rcd2_0890c184756195c3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "polyhedral combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_687D9EB61DA6", "name": "Eberhard's theorem", "statement_latex": "sum (6-i)p_i = 12", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((sum (6-i)p_i))=F((12))", "to_expression": "sum (6-i)p_i = 12", "justification": "indicator of {X} separates unequal objects", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum (6-i)p_i = 12.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_b8e12304e3be51fc", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "asymptotic geometric analysis", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_1880bdb7bf1bdccf", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "inequalities", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ae9f3d2582ae81be", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "computer algebra", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F94B0806E6BB", "name": "Landau–Mignotte bound", "statement_latex": "||h||_1 <= 2^n * ||f||_1", "assumptions": "h is a factor of the integer polynomial f of degree n under the standard Landau-Mignotte hypotheses; l1 coefficient norms are finite."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (2^n * ||f||_1)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (||h||_1)\\}", "to_expression": "||h||_1 <= 2^n * ||f||_1", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "h is a factor of the integer polynomial f of degree n under the standard Landau-Mignotte hypotheses; l1 coefficient norms are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ||h||_1 ≤ 2^n * ||f||_1.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_052024d51ce6cdc8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "calculus of variations", "subfield": "Morse theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5B2379DEDFD8", "name": "Morse–Palais lemma", "statement_latex": "f(x) = ⟨Aφ(x), φ(x)⟩", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(⟨Aφ(x), φ(x)⟩) = (f(x))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(x) = ⟨Aφ(x), φ(x)⟩.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f(x) = ⟨Aφ(x), φ(x)⟩", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(x) = ⟨Aφ(x), φ(x)⟩.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_d681e3d9b8f77b08", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "approximation theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9336CBED6851", "name": "Bernstein's theorem (polynomials)", "statement_latex": "max_{|z|=1}|P'(z)| <= n max_{|z|=1}|P(z)|", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((max_{|z|=1}|P'(z)|)-(n max_{|z|=1}|P(z)|),(-\\infty,0])=0", "to_expression": "max_{|z|=1}|P'(z)| <= n max_{|z|=1}|P(z)|", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as max_{|z|=1}|P'(z)| ≤ n max_{|z|=1}|P(z)|.\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
{"schema_version": "1.0", "item_id": "rcd2_dc3871ff051016d8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "measure theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3D75E63AA146", "name": "Vitale's random Brunn–Minkowski inequality", "statement_latex": "(vol_n(E[X]))^(1/n) >= E[vol_n(X)^(1/n)]", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "((vol_n(E[X]))^(1/n)) - (E[vol_n(X)^(1/n)]) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (vol_n(E[X]))^(1/n) >= E[vol_n(X)^(1/n)].\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (vol_n(E[X]))^(1/n) >= E[vol_n(X)^(1/n)].\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "(vol_n(E[X]))^(1/n) >= E[vol_n(X)^(1/n)]", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (vol_n(E[X]))^(1/n) >= E[vol_n(X)^(1/n)].\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_96384db79cc21de9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "symbolic dynamics", "subfield": "cellular automata", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_03C6BB7FE5A2", "name": "Curtis–Hedlund–Lyndon theorem", "statement_latex": "f(s(x)) = s(f(x))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(f(s(x))) \\in {(s(f(x)))}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(s(x)) = s(f(x)).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f(s(x)) = s(f(x))", "justification": "L in {R} iff L = R.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(s(x)) = s(f(x)).\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_2cf597607883193e", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal combinatorics", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c263c3123478521c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "Ramsey theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_05F0F9A888B8", "name": "Erdős–Szekeres theorem", "statement_latex": "n = (r - 1)(s - 1) + 1", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "(\\{(n)\\}\\times\\{((r - 1)(s - 1) + 1)\\})\\cap\\{(z,z):z\\in D\\}=\\{((n),((r - 1)(s - 1) + 1))\\}", "to_expression": "n = (r - 1)(s - 1) + 1", "justification": "intersection is singleton iff pair diagonal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as n = (r - 1)(s - 1) + 1.\"]"}], "source_equivalence_identity": "intersection is singleton iff pair diagonal"}
{"schema_version": "1.0", "item_id": "rcd2_f1c66880ecd94478", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "physics", "subfield": "general relativity", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FAE91EE049E6", "name": "Lovelock's theorem", "statement_latex": "A^{\\mu \\nu} = aG^{\\mu \\nu} + bg^{\\mu \\nu}", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(A^{\\mu \\nu})=z\\;\\land\\;z=(aG^{\\mu \\nu} + bg^{\\mu \\nu})", "to_expression": "A^{\\mu \\nu} = aG^{\\mu \\nu} + bg^{\\mu \\nu}", "justification": "common witness iff equality", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as A^{\\\\mu \\\\nu} = aG^{\\\\mu \\\\nu} + bg^{\\\\mu \\\\nu}.\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_3008bad5f97eefe7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "symplectic geometry", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C62ED9627235", "name": "Williamson theorem", "statement_latex": "SMS^T = D ⊕ D", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((SMS^T))=F((D ⊕ D))", "to_expression": "SMS^T = D ⊕ D", "justification": "indicator of {X} separates unequal objects", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as SMS^T = D ⊕ D.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_72cee9e25a352c10", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "statistical mechanics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_fe9c08768ee3e835", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "combinatorics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_EE0D43BAAED5", "name": "Ahlswede–Daykin inequality", "statement_latex": "f_1(X)f_2(Y) ≤ f_3(X ∨ Y)f_4(X ∧ Y)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (f_1(X)f_2(Y)) \\le x \\land x \\le (f_3(X ∨ Y)f_4(X ∧ Y))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover f_1(X)f_2(Y) <= f_3(X ∨ Y)f_4(X ∧ Y).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover f_1(X)f_2(Y) <= f_3(X ∨ Y)f_4(X ∧ Y).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "f_1(X)f_2(Y) ≤ f_3(X ∨ Y)f_4(X ∧ Y)", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover f_1(X)f_2(Y) <= f_3(X ∨ Y)f_4(X ∧ Y).\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_cee3925e826249e1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "polynomial ring theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BF300C15A52D", "name": "Gauss's lemma (polynomials)", "statement_latex": "F(X) = a_0 + a_1X + ... + a_nX^n", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(F(X))\\Longleftrightarrow z=(a_0 + a_1X + ... + a_nX^n))", "to_expression": "F(X) = a_0 + a_1X + ... + a_nX^n", "justification": "same equality fiber iff equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as F(X) = a_0 + a_1X + ... + a_nX^n.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_57b2e6571f37e1d1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "representation theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D8B93C65D10E", "name": "McKay conjecture", "statement_latex": "|Irr_{p'}(G)|=|Irr_{p'}(N_{G}(P))|", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((|Irr_{p'}(G)|),(|Irr_{p'}(N_{G}(P))|))\\in\\{(z,z):z\\in D\\}", "to_expression": "|Irr_{p'}(G)|=|Irr_{p'}(N_{G}(P))|", "justification": "pair in diagonal iff X=Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |Irr_{p'}(G)|=|Irr_{p'}(N_{G}(P))|.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_6c5de4f87f02d035", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "combinatorics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ee1a42b84d296e37", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c019550e70373d8f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "analysis", "subfield": "moment problem", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A82D2EB23B1F", "name": "Carleman's condition", "statement_latex": "\\sum _{n=1}^{\\infty }m_{2n}^{-{1/2n}}=+\\infty", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\sum _{n=1}^{\\infty }m_{2n}^{-{1/2n}}=+\\infty) \\land (0 = 0)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\sum _{n=1}^{\\infty }m_{2n}^{-{1/2n}}=+\\infty", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "0=0 is true, so Phi_T and true iff Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_fb0aac224f3c9401", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "inner product spaces", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a8d059727b1a87ff", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "orthogonal polynomials", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AEDF9C91417B", "name": "Turán's inequalities", "statement_latex": "P_n(x)^2 > P_{n-1}(x)P_{n+1}(x)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(P_{n-1}(x)P_{n+1}(x))-(P_n(x)^2)\\}=0\\;\\land\\;(P_{n-1}(x)P_{n+1}(x))\\ne (P_n(x)^2)", "to_expression": "P_n(x)^2 > P_{n-1}(x)P_{n+1}(x)", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P_n(x)^2 > P_{n-1}(x)P_{n+1}(x).\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_25b7aafb33caa4ae", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "geometric probability", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F57E82952899", "name": "Wendel's theorem", "statement_latex": "p_{n,N}=2^{-N+1}\\sum _{k=0}^{n-1}{\\binom {N-1}{k}}", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(p_{n,N})\\Longleftrightarrow z=(2^{-N+1}\\sum _{k=0}^{n-1}{\\binom {N-1}{k}}))", "to_expression": "p_{n,N}=2^{-N+1}\\sum _{k=0}^{n-1}{\\binom {N-1}{k}}", "justification": "same equality fiber iff equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p_{n,N}=2^{-N+1}\\\\sum _{k=0}^{n-1}{\\\\binom {N-1}{k}}.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_e6324a202700bbff", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "inequalities", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_907e8e62a9b3cc93", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_94B20D8A1D74", "name": "Chebyshev's inequality", "statement_latex": "Pr(|X-mu| >= k*sigma) <= 1/k^2", "assumptions": "X has mean mu and finite nonzero standard deviation sigma; k>0."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(Pr(|X-mu| ≥ k*sigma))+s=(1/k^2)", "to_expression": "Pr(|X-mu| >= k*sigma) <= 1/k^2", "justification": "nonnegative slack exists iff X≤Y", "preconditions": "X has mean mu and finite nonzero standard deviation sigma; k>0.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr(|X-mu| ≥ k*sigma) ≤ 1/k^2.\"]"}], "source_equivalence_identity": "nonnegative slack exists iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_e76865528cf245a2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "decision theory", "subfield": "information theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D757C95A0326", "name": "Blackwell's informativeness theorem", "statement_latex": "V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((V(p)),z)=((max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)),z)", "to_expression": "V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)", "justification": "pair equality forces first coordinates equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_1b37b61b755b08c5", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal combinatorics", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_670a6515d5028282", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "vector calculus", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F8BCAAED7243", "name": "Green's theorem", "statement_latex": "∮_{C}(L dx + M dy) = ∬_{D} (∂M/∂x - ∂L/∂y) dA", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(∬_{D} (∂M/∂x - ∂L/∂y) dA) = (∮_{C}(L dx + M dy))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover ∮_{C}(L dx + M dy) = ∬_{D} (∂M/∂x - ∂L/∂y) dA.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "∮_{C}(L dx + M dy) = ∬_{D} (∂M/∂x - ∂L/∂y) dA", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover ∮_{C}(L dx + M dy) = ∬_{D} (∂M/∂x - ∂L/∂y) dA.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_af510d626a5e6451", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4A7E73DE4982", "name": "Rayleigh's theorem", "statement_latex": "1/r + 1/s = 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(1/r + 1/s) \\in {(1)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover 1/r + 1/s = 1.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "1/r + 1/s = 1", "justification": "L in {R} iff L = R.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover 1/r + 1/s = 1.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_be08db64fdbf8be9", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematical analysis", "subfield": "analytic number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_21ebc5c300b2d092", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c675fb3bedf7e775", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "polynomial factorization", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_299359E3C599", "name": "Sophie Germain's identity", "statement_latex": "x^{4}+4y^{4}=(x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2})", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((x^{4}+4y^{4}) \\in S \\land ((x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2})) \\notin S) \\lor (((x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2})) \\in S \\land (x^{4}+4y^{4}) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover x^{4}+4y^{4}=(x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2}).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "x^{4}+4y^{4}=(x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2})", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover x^{4}+4y^{4}=(x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2}).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_3ba74ac1204ebda6", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "differential calculus", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_1f1a4a2aecedcaec", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f7dd10464b664bba", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "theoretical computer science", "subfield": "algebraic complexity theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_46E2A92077B9", "name": "Schwartz–Zippel lemma", "statement_latex": "Pr[P(r1, ..., rn) = 0] ≤ d/|S|", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(Pr[P(r1, ..., rn) = 0])}{\\exp(Pr[P(r1, ..., rn) = 0])+\\exp(d/|S|)}\\le\\frac{1}{2}", "to_expression": "Pr[P(r1, ..., rn) = 0] ≤ d/|S|", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr[P(r1, ..., rn) = 0] ≤ d/|S|.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_d26fba0b5e7bf73c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "cryptography", "subfield": "cryptanalysis", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E35608C3DE5E", "name": "Piling-up lemma", "statement_latex": "ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn) = 2^{n-1}∏_{i=1}^{n}ε(Xi)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn) = 2^{n-1}∏_{i=1}^{n}ε(Xi)) \\land (0 = 0)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn) = 2^{n-1}∏_{i=1}^{n}ε(Xi)", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "0=0 is true, so Phi_T and true iff Phi_T."}
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{"schema_version": "1.0", "item_id": "rcd2_f3f214ae12bb31b9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F881715F08CD", "name": "Sylvester's determinant identity", "statement_latex": "det(A) = det(A_v^u) * det(A_tilde_v^u)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "(\\{(det(A))\\}\\times\\{(det(A_v^u) * det(A_tilde_v^u))\\})\\cap\\{(z,z):z\\in D\\}=\\{((det(A)),(det(A_v^u) * det(A_tilde_v^u)))\\}", "to_expression": "det(A) = det(A_v^u) * det(A_tilde_v^u)", "justification": "intersection is singleton iff pair diagonal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as det(A) = det(A_v^u) * det(A_tilde_v^u).\"]"}], "source_equivalence_identity": "intersection is singleton iff pair diagonal"}
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{"schema_version": "1.0", "item_id": "rcd2_64daec978b365b77", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "multilinear algebra", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_702761e49087d166", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "modular arithmetic", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_6B373783C03D", "name": "Thue's lemma", "statement_latex": "ay ≡ x (mod m)", "assumptions": "All displayed arithmetic expressions are integers; the modulus is a positive integer."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists k\\in\\mathbb{Z}:(ay)-(x)=k(m)", "to_expression": "ay ≡ x (mod m)", "justification": "difference is integer multiple of M", "preconditions": "All displayed arithmetic expressions are integers; the modulus is a positive integer.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ay ≡ x (mod m).\"]"}], "source_equivalence_identity": "difference is integer multiple of M"}
{"schema_version": "1.0", "item_id": "rcd2_1a9a1c87d82570ea", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_42ED352AD252", "name": "Bernstein inequalities", "statement_latex": "P(|(1/n) * sum_{i=1}^n X_i| > epsilon) <= 2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3)))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(P(|(1/n) * sum_{i=1}^n X_i| > epsilon))+s=(2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3))))", "to_expression": "P(|(1/n) * sum_{i=1}^n X_i| > epsilon) <= 2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3)))", "justification": "nonnegative slack exists iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(|(1/n) * sum_{i=1}^n X_i| > epsilon) ≤ 2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3))).\"]"}], "source_equivalence_identity": "nonnegative slack exists iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_d0ed9f734b94c771", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_93F7C5A7A33A", "name": "Sun's curious identity", "statement_latex": "(x+m+1)\\sum _{i=0}^{m}(-1)^{i}{\\dbinom {x+y+i}{m-i}}{\\dbinom {y+2i}{i}}-\\sum _{i=0}^{m}{\\dbinom {x+i}{m-i}}(-4)^{i}=(x-m){\\dbinom {x}{m}}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=((x+m+1)\\sum _{i=0}^{m}(-1)^{i}{\\dbinom {x+y+i}{m-i}}{\\dbinom {y+2i}{i}}-\\sum _{i=0}^{m}{\\dbinom {x+i}{m-i}}(-4)^{i})\\Longleftrightarrow z=((x-m){\\dbinom {x}{m}}))", "to_expression": "(x+m+1)\\sum _{i=0}^{m}(-1)^{i}{\\dbinom {x+y+i}{m-i}}{\\dbinom {y+2i}{i}}-\\sum _{i=0}^{m}{\\dbinom {x+i}{m-i}}(-4)^{i}=(x-m){\\dbinom {x}{m}}", "justification": "same equality fiber iff equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (x+m+1)\\\\sum _{i=0}^{m}(-1)^{i}{\\\\dbinom {x+y+i}{m-i}}{\\\\dbinom {y+2i}{i}}-\\\\sum _{i=0}^{m}{\\\\dbinom {x+i}{m-i}}(-4)^{i}=(x-m){\\\\dbinom {x}{m}}.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
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{"schema_version": "1.0", "item_id": "rcd2_cf36bad801bb7d69", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "numerical linear algebra", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_139164e5f15d8d58", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2754949756464731", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "percolation theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4AB97BAC2783", "name": "Van den Berg–Kesten inequality", "statement_latex": "P(A square B) <= P(A)P(B)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(P(A square B))-(P(A)P(B))\\}=0", "to_expression": "P(A square B) <= P(A)P(B)", "justification": "positive part vanishes iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(A square B) ≤ P(A)P(B).\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_0083c364bac5c646", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "elementary algebra", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D16BD70DAEA0", "name": "Difference of two squares", "statement_latex": "a^2 - b^2 = (a + b)(a - b)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(a^2 - b^2)\\Longleftrightarrow z=(a + b)(a - b))", "to_expression": "a^2 - b^2 = (a + b)(a - b)", "justification": "same equality fiber iff equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as a^2 - b^2 = (a + b)(a - b).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_38df1f41c4b8bdb4", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "measure theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_00576ae0a7ad4edd", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "geometry, probability theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3B9F2B85DC25", "name": "Gaussian isoperimetric inequality", "statement_latex": "liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)} >= φ(Φ^{-1}(γ^n(A)))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)})\\le t\\Rightarrow (φ(Φ^{-1}(γ^n(A))))\\le t", "to_expression": "liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)} >= φ(Φ^{-1}(γ^n(A)))", "justification": "converse uses t=Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)} ≥ φ(Φ^{-1}(γ^n(A))).\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_b07850bf1721e4a4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "extremal graph theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F0F061CBC3F5", "name": "Blow-up lemma", "statement_latex": "e(X,Y) > delta |X| |Y|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;t\\le (delta |X| |Y|)\\Rightarrow t<(e(X,Y))", "to_expression": "e(X,Y) > delta |X| |Y|", "justification": "converse uses t=X", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as e(X,Y) > delta |X| |Y|.\"]"}], "source_equivalence_identity": "converse uses t=X"}
{"schema_version": "1.0", "item_id": "rcd2_9140ab99617daa3d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal set theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3B6CADC238DF", "name": "Lubell–Yamamoto–Meshalkin inequality", "statement_latex": "sum_{k=0}^n a_k/binom(n,k) <= 1", "assumptions": "The family is Sperner; a_k counts chosen k-subsets of an n-element ground set."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (1)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (sum_{k=0}^n a_k/binom(n,k))\\}", "to_expression": "sum_{k=0}^n a_k/binom(n,k) <= 1", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "The family is Sperner; a_k counts chosen k-subsets of an n-element ground set.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{k=0}^n a_k/binom(n,k) ≤ 1.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_e8b182682bc94b8b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "Sobolev spaces", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_35353A67F17E", "name": "Pólya–Szegő inequality", "statement_latex": "∫ |∇u*|^p dx <= ∫ |∇u|^p dx", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(∫ |∇u*|^p dx)+s=(∫ |∇u|^p dx)", "to_expression": "∫ |∇u*|^p dx <= ∫ |∇u|^p dx", "justification": "nonnegative slack exists iff X≤Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∫ |∇u*|^p dx ≤ ∫ |∇u|^p dx.\"]"}], "source_equivalence_identity": "nonnegative slack exists iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_b71fc318aa688e24", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "multilinear algebra", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F538F79B0750", "name": "Lagrange's identity", "statement_latex": "(\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2 = \\sum_{i<j} (a_i b_j - a_j b_i)^2", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "(((\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2),(\\sum_{i<j} (a_i b_j - a_j b_i)^2))\\in\\{(z,z):z\\in D\\}", "to_expression": "(\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2 = \\sum_{i<j} (a_i b_j - a_j b_i)^2", "justification": "pair in diagonal iff X=Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (\\\\sum a_k^2)(\\\\sum b_k^2) - (\\\\sum a_k b_k)^2 = \\\\sum_{i<j} (a_i b_j - a_j b_i)^2.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
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{"schema_version": "1.0", "item_id": "rcd2_6e735f17e4cb2579", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "inequalities", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B1E33965D95E", "name": "Nesbitt's inequality", "statement_latex": "a/(b+c) + b/(a+c) + c/(a+b) >= 3/2", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (3/2) \\le x \\land x \\le (a/(b+c) + b/(a+c) + c/(a+b))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover a/(b+c) + b/(a+c) + c/(a+b) >= 3/2.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover a/(b+c) + b/(a+c) + c/(a+b) >= 3/2.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "a/(b+c) + b/(a+c) + c/(a+b) >= 3/2", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover a/(b+c) + b/(a+c) + c/(a+b) >= 3/2.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
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{"schema_version": "1.0", "item_id": "rcd2_bc1719c4880987a0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5795F6B3279F", "name": "Chain rule (probability)", "statement_latex": "P(A ∩ B) = P(B|A)*P(A)", "assumptions": "A and B are events with P(A)>0 for the displayed conditional form."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((P(A ∩ B)),(P(B|A)*P(A)))=0", "to_expression": "P(A ∩ B) = P(B|A)*P(A)", "justification": "zero discrete distance iff equality", "preconditions": "A and B are events with P(A)>0 for the displayed conditional form.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(A ∩ B) = P(B|A)*P(A).\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_221e6e4a3a8f8912", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "automorphic forms", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_64D8463F3766", "name": "Fundamental lemma", "statement_latex": "SO_{\\gamma _{H}}(1_{K_{H}})=\\Delta (\\gamma _{H},\\gamma _{G})O_{\\gamma _{G}}^{\\kappa }(1_{K_{G}})", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((SO_{\\gamma _{H}}(1_{K_{H}})),z)=((\\Delta (\\gamma _{H},\\gamma _{G})O_{\\gamma _{G}}^{\\kappa }(1_{K_{G}})),z)", "to_expression": "SO_{\\gamma _{H}}(1_{K_{H}})=\\Delta (\\gamma _{H},\\gamma _{G})O_{\\gamma _{G}}^{\\kappa }(1_{K_{G}})", "justification": "pair equality forces first coordinates equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as SO_{\\\\gamma _{H}}(1_{K_{H}})=\\\\Delta (\\\\gamma _{H},\\\\gamma _{G})O_{\\\\gamma _{G}}^{\\\\kappa }(1_{K_{G}}).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
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{"schema_version": "1.0", "item_id": "rcd2_2c50e3f33967f970", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_EAE642263ABE", "name": "Catalan's conjecture", "statement_latex": "x^a - y^b = 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(x^a - y^b) \\in {(1)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover x^a - y^b = 1.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "x^a - y^b = 1", "justification": "L in {R} iff L = R.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover x^a - y^b = 1.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
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{"schema_version": "1.0", "item_id": "rcd2_62a13f9073cccc13", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "differential equations", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_891C2603FAE3", "name": "Grönwall's inequality", "statement_latex": "u(t) <= u(a) * exp(integral from a to t of beta(s) ds)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(u(t))-(u(a) * exp(integral from a to t of beta(s) ds))\\}=0", "to_expression": "u(t) <= u(a) * exp(integral from a to t of beta(s) ds)", "justification": "positive part vanishes iff X≤Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as u(t) ≤ u(a) * exp(integral from a to t of beta(s) ds).\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_44a7c50d61d6d32a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "additive combinatorics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_081265713252", "name": "Erdős–Szemerédi theorem", "statement_latex": "max(|A + A|, |A * A|) >= c|A|^{1 + ε}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\le (c|A|^{1 + ε})\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (max(|A + A|, |A * A|))\\}", "to_expression": "max(|A + A|, |A * A|) >= c|A|^{1 + ε}", "justification": "lower-cut inclusion iff X≤Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as max(|A + A|, |A * A|) ≥ c|A|^{1 + ε}.\"]"}], "source_equivalence_identity": "lower-cut inclusion iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_a0adb7309e16abf0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "enumerative combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E1416BDA0AD6", "name": "Pólya enumeration theorem", "statement_latex": "|Y^X/G| = (1/|G|) * Σ_{g∈G} m^{c(g)}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((|Y^X/G|))=F(((1/|G|) * Σ_{g∈G} m^{c(g)}))", "to_expression": "|Y^X/G| = (1/|G|) * Σ_{g∈G} m^{c(g)}", "justification": "indicator of {X} separates unequal objects", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |Y^X/G| = (1/|G|) * Σ_{g∈G} m^{c(g)}.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_523a092009c1b1d7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_DB1D004A0930", "name": "Cramer's rule", "statement_latex": "x_i = \\det(A_i) / \\det(A)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "(\\{(x_i)\\}\\times\\{(\\det(A_i) / \\det(A))\\})\\cap\\{(z,z):z\\in D\\}=\\{((x_i),(\\det(A_i) / \\det(A)))\\}", "to_expression": "x_i = \\det(A_i) / \\det(A)", "justification": "intersection is singleton iff pair diagonal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as x_i = \\\\det(A_i) / \\\\det(A).\"]"}], "source_equivalence_identity": "intersection is singleton iff pair diagonal"}
{"schema_version": "1.0", "item_id": "rcd2_cf34bfd0ced7a71e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D236ADD607AB", "name": "Chernoff bound", "statement_latex": "P(X ≥ a) ≤ inf_{t>0} M(t)e^{-ta}", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (inf_{t>0} M(t)e^{-ta})\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (P(X ≥ a))\\}", "to_expression": "P(X ≥ a) ≤ inf_{t>0} M(t)e^{-ta}", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(X ≥ a) ≤ inf_{t>0} M(t)e^{-ta}.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_36f3579ab6197755", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "complex analysis", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_1053f371d5bbc04f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum information theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B8493CAFC989", "name": "No-deleting theorem", "statement_latex": "U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C = |\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(|\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C) = (U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover U|\\\\psi \\\\rangle_A |\\\\psi \\\\rangle_B |A \\\\rangle_C = |\\\\psi \\\\rangle_A |0 \\\\rangle_B |A' \\\\rangle_C.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C = |\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover U|\\\\psi \\\\rangle_A |\\\\psi \\\\rangle_B |A \\\\rangle_C = |\\\\psi \\\\rangle_A |0 \\\\rangle_B |A' \\\\rangle_C.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_7015e8794026b826", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "additive number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E3E323366217", "name": "Lagrange's four-square theorem", "statement_latex": "p = a^2 + b^2 + c^2 + d^2", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((p),(a^2 + b^2 + c^2 + d^2))=0", "to_expression": "p = a^2 + b^2 + c^2 + d^2", "justification": "zero discrete distance iff equality", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p = a^2 + b^2 + c^2 + d^2.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_f1d9202c410fe858", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "Algebra", "subfield": "Lie algebras", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AE35A2B1281A", "name": "Jacobi identity", "statement_latex": "x * (y * z) + y * (z * x) + z * (x * y) = 0", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(x * (y * z) + y * (z * x) + z * (x * y)) \\in {(0)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover x * (y * z) + y * (z * x) + z * (x * y) = 0.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "x * (y * z) + y * (z * x) + z * (x * y) = 0", "justification": "L in {R} iff L = R.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover x * (y * z) + y * (z * x) + z * (x * y) = 0.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_ee5a6149c394ff89", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "computer algebra", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F94B0806E6BB", "name": "Landau–Mignotte bound", "statement_latex": "||h||_1 <= 2^n * ||f||_1", "assumptions": "h is a factor of the integer polynomial f of degree n under the standard Landau-Mignotte hypotheses; l1 coefficient norms are finite."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (||h||_1) \\le x \\land x \\le (2^n * ||f||_1)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "h is a factor of the integer polynomial f of degree n under the standard Landau-Mignotte hypotheses; l1 coefficient norms are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ||h||_1 <= 2^n * ||f||_1.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "h is a factor of the integer polynomial f of degree n under the standard Landau-Mignotte hypotheses; l1 coefficient norms are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ||h||_1 <= 2^n * ||f||_1.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "||h||_1 <= 2^n * ||f||_1", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "h is a factor of the integer polynomial f of degree n under the standard Landau-Mignotte hypotheses; l1 coefficient norms are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ||h||_1 <= 2^n * ||f||_1.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
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{"schema_version": "1.0", "item_id": "rcd2_0774fd29a9444691", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "algebra", "subfield": "combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_3f02ce8647f5d9b3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "calculus", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_88C089C7703F", "name": "Leibniz integral rule", "statement_latex": "d/dx integral_{a(x)}^{b(x)} f(x,t) dt = f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(d/dx integral_{a(x)}^{b(x)} f(x,t) dt) \\in {(f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover d/dx integral_{a(x)}^{b(x)} f(x,t) dt = f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "d/dx integral_{a(x)}^{b(x)} f(x,t) dt = f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover d/dx integral_{a(x)}^{b(x)} f(x,t) dt = f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
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{"schema_version": "1.0", "item_id": "rcd2_4f084de370643632", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "algebraic graph theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_8fc01340d78a2598", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0C6AE95E0EE0", "name": "Hoeffding's inequality", "statement_latex": "P(S_n - E[S_n] >= t) <= exp(-2*t^2 / sum_i (b_i-a_i)^2)", "assumptions": "The summands are independent and bounded with X_i in [a_i,b_i]; t>=0; denominator is positive."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(exp(-2*t^2 / sum_i (b_i-a_i)^2))\\le t\\Rightarrow (P(S_n - E[S_n] ≥ t))\\le t", "to_expression": "P(S_n - E[S_n] >= t) <= exp(-2*t^2 / sum_i (b_i-a_i)^2)", "justification": "converse uses t=Y", "preconditions": "The summands are independent and bounded with X_i in [a_i,b_i]; t>=0; denominator is positive.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(S_n - E[S_n] ≥ t) ≤ exp(-2*t^2 / sum_i (b_i-a_i)^2).\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_220c81d8b362f866", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "polyhedral combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b0d6efa3d983b13b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal set theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3B6CADC238DF", "name": "Lubell–Yamamoto–Meshalkin inequality", "statement_latex": "sum_{k=0}^n a_k/binom(n,k) <= 1", "assumptions": "The family is Sperner; a_k counts chosen k-subsets of an n-element ground set."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(sum_{k=0}^n a_k/binom(n,k))}{\\exp(sum_{k=0}^n a_k/binom(n,k))+\\exp(1)}\\le\\frac{1}{2}", "to_expression": "sum_{k=0}^n a_k/binom(n,k) <= 1", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "The family is Sperner; a_k counts chosen k-subsets of an n-element ground set.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{k=0}^n a_k/binom(n,k) ≤ 1.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_3283df5361d881cf", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A5B93C0856B4", "name": "Handshaking lemma", "statement_latex": "Σ_{v∈V} deg(v) = 2|E|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((Σ_{v∈V} deg(v)))=F((2|E|))", "to_expression": "Σ_{v∈V} deg(v) = 2|E|", "justification": "indicator of {X} separates unequal objects", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Σ_{v∈V} deg(v) = 2|E|.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
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{"schema_version": "1.0", "item_id": "rcd2_312c7c894ebb0841", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AD101C6FC364", "name": "Adiabatic theorem", "statement_latex": "E = nhv", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(E)\\Longleftrightarrow z=(nhv))", "to_expression": "E = nhv", "justification": "same equality fiber iff equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as E = nhv.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
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{"schema_version": "1.0", "item_id": "rcd2_67b20158f5f3c59d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "algebraic combinatorics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B8C05F31E817", "name": "Dyson conjecture", "statement_latex": "constant_term(prod_{1<=i!=j<=n} (1-t_i/t_j)^{a_i}) = (sum a_i)! / prod(a_i!)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((constant_term(prod_{1≤i≠j≤n} (1-t_i/t_j)^{a_i})),(sum a_i)! / prod(a_i!))\\in\\{(z,z):z\\in D\\}", "to_expression": "constant_term(prod_{1<=i!=j<=n} (1-t_i/t_j)^{a_i}) = (sum a_i)! / prod(a_i!)", "justification": "pair in diagonal iff X=Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as constant_term(prod_{1≤i≠j≤n} (1-t_i/t_j)^{a_i}) = (sum a_i)! / prod(a_i!).\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_43498570b9f5ed2b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic calculus", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_58173198312A", "name": "Boué–Dupuis formula", "statement_latex": "-log E[e^{-f(B)}] = inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)]", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)]) = (-log E[e^{-f(B)}])", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover -log E[e^{-f(B)}] = inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)].\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "-log E[e^{-f(B)}] = inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)]", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover -log E[e^{-f(B)}] = inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)].\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
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{"schema_version": "1.0", "item_id": "rcd2_f23638ce87b5236e", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "point processes", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_871e60af784b6cfc", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "Number theory", "subfield": "Elliptic functions", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_244848d2006756c6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "additive number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E3E323366217", "name": "Lagrange's four-square theorem", "statement_latex": "p = a^2 + b^2 + c^2 + d^2", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(a^2 + b^2 + c^2 + d^2) = (p)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover p = a^2 + b^2 + c^2 + d^2.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "p = a^2 + b^2 + c^2 + d^2", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover p = a^2 + b^2 + c^2 + d^2.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_a8591160976e2bbc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "analysis", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A35964624017", "name": "AM–GM inequality", "statement_latex": "(x + y)/2 >= sqrt(x*y)", "assumptions": "x and y are nonnegative real numbers."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;((x + y)/2)\\le t\\Rightarrow (sqrt(x*y))\\le t", "to_expression": "(x + y)/2 >= sqrt(x*y)", "justification": "converse uses t=Y", "preconditions": "x and y are nonnegative real numbers.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (x + y)/2 ≥ sqrt(x*y).\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_17b28f42d0daff3e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "combinatorics", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_684A9634E4F6", "name": "Hypergeometric identity", "statement_latex": "sum_{i=0}^n binom(n,i) = 2^n", "assumptions": "n is a nonnegative integer."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(sum_{i=0}^n binom(n,i))=z\\;\\land\\;z=(2^n)", "to_expression": "sum_{i=0}^n binom(n,i) = 2^n", "justification": "common witness iff equality", "preconditions": "n is a nonnegative integer.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{i=0}^n binom(n,i) = 2^n.\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_cf6131a88c5702dd", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "number theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0e258b6bb675d4a0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "physics", "subfield": "quantum field theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5C335B571C48", "name": "Gell-Mann and Low theorem", "statement_latex": "H = H_0 + gV", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((H),z):z\\in D\\}=\\{((H_0 + gV),z):z\\in D\\}", "to_expression": "H = H_0 + gV", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as H = H_0 + gV.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
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{"schema_version": "1.0", "item_id": "rcd2_d6256fe8ae9c0301", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "Number theory", "subfield": "Additive number theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_472A65DC1391", "name": "Romanov's theorem", "statement_latex": "d_ > 0", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(0)-(d_)\\}=0\\;\\land\\;(0)\\ne (d_)", "to_expression": "d_ > 0", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as d_ > 0.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
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{"schema_version": "1.0", "item_id": "rcd2_9dae063dc56fd3ba", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "Ramsey theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_546d95a02dfd916b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic geometry", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BB5E3AF7EFD7", "name": "Modularity theorem", "statement_latex": "L(E, s) = sum_{n=1}^∞ a_n / n^s", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (L(E, s) = sum_{n=1}^∞ a_n / n^s))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "L(E, s) = sum_{n=1}^∞ a_n / n^s", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_3f79c4fbaea8ee80", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "inequalities", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A3A584B25561", "name": "Ky Fan inequality", "statement_latex": "(∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n} ≤ (∑ x_i) / (∑ (1-x_i))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (∑ x_i) / (∑ (1-x_i))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge ((∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n})\\}", "to_expression": "(∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n} ≤ (∑ x_i) / (∑ (1-x_i))", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n} ≤ (∑ x_i) / (∑ (1-x_i)).\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_6f34f8d0cf0d2b15", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "complex analysis", "subfield": "complex geometry", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1CFF4C1B7932", "name": "Andreotti–Grauert theorem", "statement_latex": "dim_C H^i(X, F) < infinity", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,\\infty)}\\lambda((infinity)-(dim_C H^i(X, F)))=+\\infty", "to_expression": "dim_C H^i(X, F) < infinity", "justification": "unbounded iff Y-X>0", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as dim_C H^i(X, F) < infinity.\"]"}], "source_equivalence_identity": "unbounded iff Y-X>0"}
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{"schema_version": "1.0", "item_id": "rcd2_a16554ed1c7d80f6", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "matching theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d3403565b00e5b14", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_98220A4F883B", "name": "Siegel identity", "statement_latex": "\\frac{x_3-x_1}{x_2-x_1} + \\frac{x_2-x_3}{x_2-x_1} = 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(\\frac{x_3-x_1}{x_2-x_1} + \\frac{x_2-x_3}{x_2-x_1})\\}=\\{(1)\\}", "to_expression": "\\frac{x_3-x_1}{x_2-x_1} + \\frac{x_2-x_3}{x_2-x_1} = 1", "justification": "singletons equal iff elements equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\frac{x_3-x_1}{x_2-x_1} + \\\\frac{x_2-x_3}{x_2-x_1} = 1.\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_68bc150e83fe54ad", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "number theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C3C168B97D27", "name": "Watson's quintuple product identity", "statement_latex": "prod_{n>=1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1})", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(prod_{n≥1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2})))+\\delta_D(z,(sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1}))))=0", "to_expression": "prod_{n>=1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1})", "justification": "zero loss iff some z equals both", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as prod_{n≥1} (1-s^n)(1-s^nt)(1-s^{n-1}t^{-1})(1-s^{2n-1}t^2)(1-s^{2n-1}t^{-2}) = sum_{n in Z} s^{(3n^2+n)/2} (t^{3n} - t^{-3n-1}).\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_3efeebf990d21724", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0506260C92DB", "name": "Bennett's inequality", "statement_latex": "Pr(S_n > t) <= exp(-(sigma^2 / a^2) * h(at / sigma^2))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((Pr(S_n > t))-(exp(-(sigma^2 / a^2) * h(at / sigma^2))),(-\\infty,0])=0", "to_expression": "Pr(S_n > t) <= exp(-(sigma^2 / a^2) * h(at / sigma^2))", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr(S_n > t) ≤ exp(-(sigma^2 / a^2) * h(at / sigma^2)).\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
{"schema_version": "1.0", "item_id": "rcd2_23596090158edf6c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "decision theory", "subfield": "information theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D757C95A0326", "name": "Blackwell's informativeness theorem", "statement_latex": "V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((V(p)),z):z\\in D\\}=\\{((max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)),z):z\\in D\\}", "to_expression": "V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega)", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as V(p) = max_{a in A} sum_{omega in Omega} u(a, omega) p(omega).\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_693fbd3b8e1b67ae", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_39638a357deb4ba3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum mechanics", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9078CE9B3CD4", "name": "Ehrenfest theorem", "statement_latex": "F = -V'(x)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(F))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(-V'(x)))", "to_expression": "F = -V'(x)", "justification": "each argmin set is a singleton", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as F = -V'(x).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_f7a8e363ba445803", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "foundations of probability", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F79EB6FC7340", "name": "Cox's theorem", "statement_latex": "w(AB|X) = w(A|X)w(B|AX)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((w(AB|X)),(w(A|X)w(B|AX)))=0", "to_expression": "w(AB|X) = w(A|X)w(B|AX)", "justification": "zero discrete distance iff equality", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as w(AB|X) = w(A|X)w(B|AX).\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_a2b19b2a9a3d9ee1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "computer science", "subfield": "theoretical computer science", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_30A4875B9AAE", "name": "Isolation lemma", "statement_latex": "w(S) = sum_{x in S} w(x)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(w(S)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{x in S} w(x)))", "to_expression": "w(S) = sum_{x in S} w(x)", "justification": "each argmin set is a singleton", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as w(S) = sum_{x in S} w(x).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_31643e59601b7728", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "commutative algebra", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a88b04ab08c0568b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "combinatorics", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_51200980F031", "name": "Star of David theorem", "statement_latex": "gcd{(n-1 k-1), (n k+1), (n+1 k)} = gcd{(n-1 k), (n k-1), (n+1 k+1)}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(gcd{(n-1 k-1), (n k+1), (n+1 k)}))+\\delta_D(z,(gcd{(n-1 k), (n k-1), (n+1 k+1)})))=0", "to_expression": "gcd{(n-1 k-1), (n k+1), (n+1 k)} = gcd{(n-1 k), (n k-1), (n+1 k+1)}", "justification": "zero loss iff some z equals both", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as gcd{(n-1 k-1), (n k+1), (n+1 k)} = gcd{(n-1 k), (n k-1), (n+1 k+1)}.\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_b51fee6fe29de607", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "linear algebra", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FF8D78866125", "name": "Binet–Cauchy identity", "statement_latex": "(sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i<j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i)", "assumptions": "All sums are finite over the same index range; entries lie in R."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "((sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i) - (j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i)) < 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A-B<0 iff A<B.", "preconditions": "All sums are finite over the same index range; entries lie in R.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i < j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A-B<0 iff A<B.", "preconditions": "All sums are finite over the same index range; entries lie in R.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i < j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "(sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i<j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i)", "justification": "A-B<0 iff A<B.", "preconditions": "All sums are finite over the same index range; entries lie in R.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i < j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i).\"]"}], "source_equivalence_identity": "A-B<0 iff A<B."}
{"schema_version": "1.0", "item_id": "rcd2_3929fec4ad3eb4ad", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "recreational mathematics", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_92d22c7368982b1d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "analysis", "subfield": "approximation theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6d765398ff87bf0f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6e9fc170b9b780ef", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "approximation theory", "subfield": "mathematical analysis", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_389D9173C8B7", "name": "Lebesgue's lemma", "statement_latex": "‖v - Pv‖ ≤ (1 + ‖P‖) inf_{u ∈ U} ‖v - u‖", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(‖v - Pv‖)}{\\exp(‖v - Pv‖)+\\exp((1 + ‖P‖) inf_{u ∈ U} ‖v - u‖)}\\le\\frac{1}{2}", "to_expression": "‖v - Pv‖ ≤ (1 + ‖P‖) inf_{u ∈ U} ‖v - u‖", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ‖v - Pv‖ ≤ (1 + ‖P‖) inf_{u ∈ U} ‖v - u‖.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_fd8e09e4be656699", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "incidence geometry", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_cec1e2ba840ba517", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D737B48D674F", "name": "Borell–TIS inequality", "statement_latex": "P(‖f‖_T > E(‖f‖_T) + u) ≤ exp(-u^2 / (2σ_T^2))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\exp(-u^2 / (2σ_T^2))) - (P(‖f‖_T > E(‖f‖_T) + u)) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover P(‖f‖_T > E(‖f‖_T) + u) <= exp(-u^2 / (2σ_T^2)).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover P(‖f‖_T > E(‖f‖_T) + u) <= exp(-u^2 / (2σ_T^2)).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "P(‖f‖_T > E(‖f‖_T) + u) ≤ exp(-u^2 / (2σ_T^2))", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover P(‖f‖_T > E(‖f‖_T) + u) <= exp(-u^2 / (2σ_T^2)).\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_95985190ecca4d66", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "theoretical computer science", "subfield": "algebraic complexity theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_46E2A92077B9", "name": "Schwartz–Zippel lemma", "statement_latex": "Pr[P(r1, ..., rn) = 0] ≤ d/|S|", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (d/|S|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (Pr[P(r1, ..., rn) = 0])\\}", "to_expression": "Pr[P(r1, ..., rn) = 0] ≤ d/|S|", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr[P(r1, ..., rn) = 0] ≤ d/|S|.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_bcee159fc31934c2", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_cfc80f04c5aa4555", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "calculus of variations", "subfield": "Morse theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_190ca1d899cdf4ae", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C95D5999DB6B", "name": "Nullity theorem", "statement_latex": "nullity A = nullity H", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((nullity A) \\in S \\land (nullity H) \\notin S) \\lor ((nullity H) \\in S \\land (nullity A) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover nullity A = nullity H.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "nullity A = nullity H", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover nullity A = nullity H.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_5a8d8a8c0dcdad95", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_793aceeafe82e861", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "large deviations theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_944433ba6c355b91", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_ABB949903DF7", "name": "Vandermonde's identity", "statement_latex": "{m+n \\choose r}=\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "(\\{({m+n \\choose r})\\}\\times\\{(\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k})\\})\\cap\\{(z,z):z\\in D\\}=\\{(({m+n \\choose r}),(\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k}))\\}", "to_expression": "{m+n \\choose r}=\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k}", "justification": "intersection is singleton iff pair diagonal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as {m+n \\\\choose r}=\\\\sum _{k=0}^{r}{m \\\\choose k}{n \\\\choose r-k}.\"]"}], "source_equivalence_identity": "intersection is singleton iff pair diagonal"}
{"schema_version": "1.0", "item_id": "rcd2_af9a177f5cbd6c05", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A00EEAE41A77", "name": "Intermediate value theorem", "statement_latex": "f(x) = s", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(f(x)) \\in {(s)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(x) = s.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f(x) = s", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(x) = s.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_341c71eedf0090c9", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_dbc514a2f24ac528", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "combinatorics", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_30AD154E7704", "name": "Cyclotomic identity", "statement_latex": "1/(1-alpha*z) = product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(1/(1-alpha*z)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}))", "to_expression": "1/(1-alpha*z) = product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}", "justification": "each argmin set is a singleton", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as 1/(1-alpha*z) = product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_60030e5b63e3af9f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "calculus", "subfield": "integral calculus", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F8E7F707512F", "name": "Bioche's rules", "statement_latex": "ω(-t) = ω(t)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((ω(-t)))=F((ω(t)))", "to_expression": "ω(-t) = ω(t)", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ω(-t) = ω(t).\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_b650746152392497", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "probabilistic number theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9D5ECA10E501", "name": "Turán–Kubilius inequality", "statement_latex": "(1/x) * sum_{n<=x} |f(n) - A(x)|^2 <= (2 + epsilon(x)) * B(x)^2", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge ((2 + epsilon(x)) * B(x)^2)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge ((1/x) * sum_{n≤x} |f(n) - A(x)|^2)\\}", "to_expression": "(1/x) * sum_{n<=x} |f(n) - A(x)|^2 <= (2 + epsilon(x)) * B(x)^2", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (1/x) * sum_{n≤x} |f(n) - A(x)|^2 ≤ (2 + epsilon(x)) * B(x)^2.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_767846360d8c4333", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "measure theory", "subfield": "probability theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_35D604A69AA1", "name": "Kakutani's theorem", "statement_latex": "prod_{n=1}^\\infty \\int \\sqrt{d\\mu_n d\\nu_n} > 0", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(0)-(prod_{n=1}^\\infty \\int \\sqrt{d\\mu_n d\\nu_n})\\}=0\\;\\land\\;(0)\\ne (prod_{n=1}^\\infty \\int \\sqrt{d\\mu_n d\\nu_n})", "to_expression": "prod_{n=1}^\\infty \\int \\sqrt{d\\mu_n d\\nu_n} > 0", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as prod_{n=1}^\\\\infty \\\\int \\\\sqrt{d\\\\mu_n d\\\\nu_n} > 0.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
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{"schema_version": "1.0", "item_id": "rcd2_fd222614e7fb28cc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "design theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_45A9C577BD36", "name": "Fisher's inequality", "statement_latex": "b >= v", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((v)-(b),(-\\infty,0])=0", "to_expression": "b >= v", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as b ≥ v.\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
{"schema_version": "1.0", "item_id": "rcd2_eebd6b7162958bd9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "numerical analysis, finite element method", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7C7FF749E01A", "name": "Bramble–Hilbert lemma", "statement_latex": "inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)} <= C |u|_{W^{m,p}(\\Omega)}", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (\\inf_{v \\in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)}) \\le x \\land x \\le (C |u|_{W^{m,p}(\\Omega)})", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\\\Omega)} <= C |u|_{W^{m,p}(\\\\Omega)}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\\\Omega)} <= C |u|_{W^{m,p}(\\\\Omega)}.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)} <= C |u|_{W^{m,p}(\\Omega)}", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\\\Omega)} <= C |u|_{W^{m,p}(\\\\Omega)}.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_d2cfca90c5462668", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "inequalities", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A3A584B25561", "name": "Ky Fan inequality", "statement_latex": "(∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n} ≤ (∑ x_i) / (∑ (1-x_i))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: ((∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n}) \\le x \\land x \\le ((∑ x_i) / (∑ (1-x_i)))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n} <= (∑ x_i) / (∑ (1-x_i)).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n} <= (∑ x_i) / (∑ (1-x_i)).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "(∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n} ≤ (∑ x_i) / (∑ (1-x_i))", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n} <= (∑ x_i) / (∑ (1-x_i)).\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
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{"schema_version": "1.0", "item_id": "rcd2_cfbb1aa385189c64", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_ABB949903DF7", "name": "Vandermonde's identity", "statement_latex": "{m+n \\choose r}=\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=({m+n \\choose r})\\Longleftrightarrow z=(\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k}))", "to_expression": "{m+n \\choose r}=\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k}", "justification": "same equality fiber iff equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as {m+n \\\\choose r}=\\\\sum _{k=0}^{r}{m \\\\choose k}{n \\\\choose r-k}.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_a35351e4016be346", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematical economics", "subfield": "linear algebra", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1A4ADB6F0297", "name": "Hawkins–Simon condition", "statement_latex": "[I - A]x = d", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D(([I - A]x),(d))=0", "to_expression": "[I - A]x = d", "justification": "zero discrete distance iff equality", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as [I - A]x = d.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_8d88e26efcbfd644", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "functional analysis", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1D2F074AF9A8", "name": "Spectral theorem", "statement_latex": "Ax = λx", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(Ax)\\}=\\{(λx)\\}", "to_expression": "Ax = λx", "justification": "singletons equal iff elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Ax = λx.\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_88d072b751a637f3", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "symbolic dynamics", "subfield": "cellular automata", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_647ef1a497eb0ec3", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "physics", "subfield": "general relativity", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_7bf8e183d5ebd752", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "algebra", "subfield": "combinatorics", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_24005F6E3998", "name": "Abel's binomial theorem", "statement_latex": "Σ [k=0 to m] (m choose k) * (w + m - k)^(m - k - 1) * (z + k)^k = w^(-1) * (z + w + m)^m", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((Σ [k=0 to m] (m choose k) * (w + m - k)^(m - k - 1) * (z + k)^k),z)=((w^(-1) * (z + w + m)^m),z)", "to_expression": "Σ [k=0 to m] (m choose k) * (w + m - k)^(m - k - 1) * (z + k)^k = w^(-1) * (z + w + m)^m", "justification": "pair equality forces first coordinates equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Σ [k=0 to m] (m choose k) * (w + m - k)^(m - k - 1) * (z + k)^k = w^(-1) * (z + w + m)^m.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
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{"schema_version": "1.0", "item_id": "rcd2_fb7c634e8d3fc268", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "statistical mechanics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_bb73da85addec97f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "number theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9793AD078B54", "name": "Degen's eight-square identity", "statement_latex": "(∑_{i=1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2) = ∑_{i=1}^{8} c_i^2", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "((∑_{i=1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2) = ∑_{i=1}^{8} c_i^2) \\lor (0 = 1)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "0=1 is false, so Phi_T or false iff Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "(∑_{i=1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2) = ∑_{i=1}^{8} c_i^2", "justification": "0=1 is false, so Phi_T or false iff Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "0=1 is false, so Phi_T or false iff Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_90f78859c14cbbbe", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum field theory", "subfield": "axiomatic quantum field theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FEF97AED413B", "name": "Cluster decomposition property", "statement_latex": "S_βα → S_β1α1 * S_β2α2", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall r\\in\\mathbb{Q}_{>0}\\exists N\\forall n\\ge N:d((S_βα),(S_β1α1 * S_β2α2))<r", "to_expression": "S_βα → S_β1α1 * S_β2α2", "justification": "positive rationals cofinal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as S_βα → S_β1α1 * S_β2α2.\"]"}], "source_equivalence_identity": "positive rationals cofinal"}
{"schema_version": "1.0", "item_id": "rcd2_068008c354e81834", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ec68cf907459bc64", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum mechanics", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d7dd1072b819ebcb", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "computer science", "subfield": "theoretical computer science", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_30A4875B9AAE", "name": "Isolation lemma", "statement_latex": "w(S) = sum_{x in S} w(x)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((w(S)),z)=((sum_{x in S} w(x)),z)", "to_expression": "w(S) = sum_{x in S} w(x)", "justification": "pair equality forces first coordinates equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as w(S) = sum_{x in S} w(x).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_ed2a28d23fa12b58", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8DC5B9EC1AA5", "name": "Dimension theorem for vector spaces", "statement_latex": "|B_1| = |B_2|", "assumptions": "B_1 and B_2 are bases of the same finite-dimensional vector space."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(|B_2|) = (|B_1|)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "B_1 and B_2 are bases of the same finite-dimensional vector space.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover |B_1| = |B_2|.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "|B_1| = |B_2|", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "B_1 and B_2 are bases of the same finite-dimensional vector space.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover |B_1| = |B_2|.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_890d38a74ff86e8b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5A3D942AED87", "name": "Markov's inequality", "statement_latex": "P(X >= a) <= E[X] / a", "assumptions": "X is a nonnegative random variable; a>0; E[X] is finite."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (P(X \\ge a) \\le E[X] / a))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "X is a nonnegative random variable; a>0; E[X] is finite.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "P(X >= a) <= E[X] / a", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "X is a nonnegative random variable; a>0; E[X] is finite.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_ca164c71d3ee8faa", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "topology", "subfield": "algebraic topology", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_3609065936e58407", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "calculus", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_88C089C7703F", "name": "Leibniz integral rule", "statement_latex": "d/dx integral_{a(x)}^{b(x)} f(x,t) dt = f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(d/dx integral_{a(x)}^{b(x)} f(x,t) dt))+\\delta_D(z,(f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt)))=0", "to_expression": "d/dx integral_{a(x)}^{b(x)} f(x,t) dt = f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt", "justification": "zero loss iff some z equals both", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as d/dx integral_{a(x)}^{b(x)} f(x,t) dt = f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt.\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_eb88865c72e145e2", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "inner product spaces", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b35e3ba8cba73746", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_051344808DC5", "name": "Gauss congruence", "statement_latex": "sum_{d|n} mu(d)*a_{n/d} ≡ 0 (mod n)", "assumptions": "n is a positive integer; the displayed sum is an integer; mu is the Möbius function."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "(sum_{d|n} mu(d)*a_{n/d})-(0)\\in (n)\\mathbb{Z}", "to_expression": "sum_{d|n} mu(d)*a_{n/d} ≡ 0 (mod n)", "justification": "membership in MZ is divisibility", "preconditions": "n is a positive integer; the displayed sum is an integer; mu is the Möbius function.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{d|n} mu(d)*a_{n/d} ≡ 0 (mod n).\"]"}], "source_equivalence_identity": "membership in MZ is divisibility"}
{"schema_version": "1.0", "item_id": "rcd2_2517333cb46b1e86", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "functional analysis", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_462346AD2251", "name": "Ehrling's lemma", "statement_latex": "||x||_Y <= ε||x||_X + C(ε)||x||_Z", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (||x||_Y) \\le x \\land x \\le (ε||x||_X + C(ε)||x||_Z)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ||x||_Y <= ε||x||_X + C(ε)||x||_Z.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ||x||_Y <= ε||x||_X + C(ε)||x||_Z.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "||x||_Y <= ε||x||_X + C(ε)||x||_Z", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ||x||_Y <= ε||x||_X + C(ε)||x||_Z.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_e0ec56edf1b74a98", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "matroid theory", "subfield": "combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_09298FD96036", "name": "Ingleton's inequality", "statement_latex": "ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4) ≤ ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4)) - (ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4)) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4) <= ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4) <= ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4) ≤ ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4)", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4) <= ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4).\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_0a1259f665e3ec95", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "physics", "subfield": "general relativity", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_616d5b69320ed85b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "zeta and L-functions", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_cbfbbf21f7af9351", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "additive combinatorics", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_081265713252", "name": "Erdős–Szemerédi theorem", "statement_latex": "max(|A + A|, |A * A|) >= c|A|^{1 + ε}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(c|A|^{1 + ε})-(max(|A + A|, |A * A|))\\}=0", "to_expression": "max(|A + A|, |A * A|) >= c|A|^{1 + ε}", "justification": "positive part vanishes iff X≤Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as max(|A + A|, |A * A|) ≥ c|A|^{1 + ε}.\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_55e852bfa2da2fce", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "complex analysis", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_411151b88a3e8ef4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "complex analysis", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D3ABA1826867", "name": "Fundamental theorem of algebra", "statement_latex": "exists z in C such that p(z)=0", "assumptions": "p in C[x] is nonconstant."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (\\exists z \\in C such that p(z)=0))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "p in C[x] is nonconstant.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "exists z in C such that p(z)=0", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "p in C[x] is nonconstant.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_b31b2faf9eca9380", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_20033DA37507", "name": "Kolmogorov–Arnold representation theorem", "statement_latex": "f(x_1, ..., x_n) = Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p))", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(f(x_1, ..., x_n)))+\\delta_D(z,(Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p)))))=0", "to_expression": "f(x_1, ..., x_n) = Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p))", "justification": "zero loss iff some z equals both", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f(x_1, ..., x_n) = Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p)).\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_f50e75abbb78d90b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis, measure theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4A564A0B747B", "name": "Monotone convergence theorem", "statement_latex": "lim_{n -> infinity} a_n = sup {a_n}", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\infty} a_n = sup {a_n}) - (lim_{n -) < 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A-B<0 iff A<B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover lim_{n - > infinity} a_n = sup {a_n}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A-B<0 iff A<B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover lim_{n - > infinity} a_n = sup {a_n}.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "lim_{n -> infinity} a_n = sup {a_n}", "justification": "A-B<0 iff A<B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover lim_{n - > infinity} a_n = sup {a_n}.\"]"}], "source_equivalence_identity": "A-B<0 iff A<B."}
{"schema_version": "1.0", "item_id": "rcd2_4dfc63745a1758f9", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "percolation theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d312f99ac6b693d9", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "numerical linear algebra", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c74ce8e8d4b43025", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "Combinatorics", "subfield": "Order theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_5e29fcf19657a4dc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "quadratic forms", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D2FCDF28ABB7", "name": "Pfister's sixteen-square identity", "statement_latex": "(x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2) = z_1^2 + z_2^2 + ... + z_{16}^2", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(z_1^2 + z_2^2 + ... + z_{16}^2) = ((x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover (x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2) = z_1^2 + z_2^2 + ... + z_{16}^2.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "(x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2) = z_1^2 + z_2^2 + ... + z_{16}^2", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover (x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2) = z_1^2 + z_2^2 + ... + z_{16}^2.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_5b96dc0b0510033d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CCCCB5088787", "name": "Mean value theorem", "statement_latex": "f'(c) = (f(b)-f(a))/(b-a)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "((f(b)-f(a))/(b-a)) = (f'(c))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f'(c) = (f(b)-f(a))/(b-a).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f'(c) = (f(b)-f(a))/(b-a)", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f'(c) = (f(b)-f(a))/(b-a).\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_77281c02683a32a2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "geometric probability", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F57E82952899", "name": "Wendel's theorem", "statement_latex": "p_{n,N}=2^{-N+1}\\sum _{k=0}^{n-1}{\\binom {N-1}{k}}", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((p_{n,N}),z):z\\in D\\}=\\{((2^{-N+1}\\sum _{k=0}^{n-1}{\\binom {N-1}{k}}),z):z\\in D\\}", "to_expression": "p_{n,N}=2^{-N+1}\\sum _{k=0}^{n-1}{\\binom {N-1}{k}}", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p_{n,N}=2^{-N+1}\\\\sum _{k=0}^{n-1}{\\\\binom {N-1}{k}}.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
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{"schema_version": "1.0", "item_id": "rcd2_d72b30310d558329", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "functional analysis", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AB221FDAC285", "name": "Lumer–Phillips theorem", "statement_latex": "D(A) is dense in X, A is dissipative, and A − λ0I is surjective for some λ0 > 0", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(0)-(D(A) is dense in X, A is dissipative, and A − λ0I is surjective for some λ0)\\}=0\\;\\land\\;(0)\\ne (D(A) is dense in X, A is dissipative, and A − λ0I is surjective for some λ0)", "to_expression": "D(A) is dense in X, A is dissipative, and A − λ0I is surjective for some λ0 > 0", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as D(A) is dense in X, A is dissipative, and A − λ0I is surjective for some λ0 > 0.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_c165c87dc83b14de", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "matrix analysis", "subfield": "linear algebra", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D76BA21A085C", "name": "Crouzeix's conjecture", "statement_latex": "||f(A)|| <= 2 sup_{z in W(A)} |f(z)|", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;t\\le (||f(A)||)\\Rightarrow t\\le (2 sup_{z in W(A)} |f(z)|)", "to_expression": "||f(A)|| <= 2 sup_{z in W(A)} |f(z)|", "justification": "converse uses t=X", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ||f(A)|| ≤ 2 sup_{z in W(A)} |f(z)|.\"]"}], "source_equivalence_identity": "converse uses t=X"}
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{"schema_version": "1.0", "item_id": "rcd2_2a197999b9591aa6", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "approximation theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2c8461ee86eddb03", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "asymptotic geometric analysis", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0B6D1767746C", "name": "Milman's reverse Brunn–Minkowski inequality", "statement_latex": "vol(s φ K + t ψ L)^(1/n) ≤ C (s vol(φ K)^(1/n) + t vol(ψ L)^(1/n))", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(vol(s φ K + t ψ L)^(1/n))-(C (s vol(φ K)^(1/n) + t vol(ψ L)^(1/n)))\\}=0", "to_expression": "vol(s φ K + t ψ L)^(1/n) ≤ C (s vol(φ K)^(1/n) + t vol(ψ L)^(1/n))", "justification": "positive part vanishes iff X≤Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as vol(s φ K + t ψ L)^(1/n) ≤ C (s vol(φ K)^(1/n) + t vol(ψ L)^(1/n)).\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_517c42360016a53d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "geometry, probability theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3B9F2B85DC25", "name": "Gaussian isoperimetric inequality", "statement_latex": "liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)} >= φ(Φ^{-1}(γ^n(A)))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (φ(Φ^{-1}(γ^n(A)))) \\le x \\land x \\le (liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)})", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)} >= φ(Φ^{-1}(γ^n(A))).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)} >= φ(Φ^{-1}(γ^n(A))).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)} >= φ(Φ^{-1}(γ^n(A)))", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)} >= φ(Φ^{-1}(γ^n(A))).\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
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{"schema_version": "1.0", "item_id": "rcd2_7587757d52c29b6c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "hypergeometric series", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_94538B67A269", "name": "Dixon's identity", "statement_latex": "sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3 = (3a)! / (a!)^3", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3),((3a)! / (a!)^3))=0", "to_expression": "sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3 = (3a)! / (a!)^3", "justification": "zero discrete distance iff equality", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3 = (3a)! / (a!)^3.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_55a44ddb5f6fabde", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematical analysis", "subfield": "analytic number theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F4E26A602D8B", "name": "Perron's formula", "statement_latex": "A(x) = (1 / (2*pi*i)) * integral_{c-i*infinity}^{c+i*infinity} g(z) * (x^z / z) dz", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((A(x)),z):z\\in D\\}=\\{(((1 / (2*pi*i)) * integral_{c-i*infinity}^{c+i*infinity} g(z) * (x^z / z) dz),z):z\\in D\\}", "to_expression": "A(x) = (1 / (2*pi*i)) * integral_{c-i*infinity}^{c+i*infinity} g(z) * (x^z / z) dz", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as A(x) = (1 / (2*pi*i)) * integral_{c-i*infinity}^{c+i*infinity} g(z) * (x^z / z) dz.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
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{"schema_version": "1.0", "item_id": "rcd2_ee3e90cff8968545", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "enumerative combinatorics", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7EDAF21632BF", "name": "Labelled enumeration theorem", "statement_latex": "F(z,t)=e^{g(z)t}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(F(z,t))\\}=\\{(e^{g(z)t})\\}", "to_expression": "F(z,t)=e^{g(z)t}", "justification": "singletons equal iff elements equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as F(z,t)=e^{g(z)t}.\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_52ac27e141145b01", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "queueing theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D0BC02092015", "name": "Bartlett's theorem", "statement_latex": "mu(t) = integral_{-infinity}^{t} A(s)p(s,t) ds", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((mu(t)))=F((integral_{-infinity}^{t} A(s)p(s,t) ds))", "to_expression": "mu(t) = integral_{-infinity}^{t} A(s)p(s,t) ds", "justification": "indicator of {X} separates unequal objects", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as mu(t) = integral_{-infinity}^{t} A(s)p(s,t) ds.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
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{"schema_version": "1.0", "item_id": "rcd2_bb69aa8cc62c4a2e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "Combinatorics", "subfield": "Order theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B5F5A5581E5D", "name": "Kruskal's tree theorem", "statement_latex": "T1 <= T2", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(T2) - (T1) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover T1 <= T2.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover T1 <= T2.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "T1 <= T2", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover T1 <= T2.\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_19c1225e47d56d54", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "calculus", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3967E74F7B6F", "name": "Increment theorem", "statement_latex": "Δy = f'(x)Δx + εΔx", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(Δy)\\Longleftrightarrow z=(f'(x)Δx + εΔx))", "to_expression": "Δy = f'(x)Δx + εΔx", "justification": "same equality fiber iff equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Δy = f'(x)Δx + εΔx.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
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{"schema_version": "1.0", "item_id": "rcd2_123f14a7f2633b9e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "number theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_051344808DC5", "name": "Gauss congruence", "statement_latex": "sum_{d|n} mu(d)*a_{n/d} ≡ 0 (mod n)", "assumptions": "n is a positive integer; the displayed sum is an integer; mu is the Möbius function."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\pi_{(n)}((sum_{d|n} mu(d)*a_{n/d}))=\\pi_{(n)}((0))", "to_expression": "sum_{d|n} mu(d)*a_{n/d} ≡ 0 (mod n)", "justification": "quotient map identifies congruent integers", "preconditions": "n is a positive integer; the displayed sum is an integer; mu is the Möbius function.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{d|n} mu(d)*a_{n/d} ≡ 0 (mod n).\"]"}], "source_equivalence_identity": "quotient map identifies congruent integers"}
{"schema_version": "1.0", "item_id": "rcd2_04d14a2d515e1840", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "polyhedral combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_687D9EB61DA6", "name": "Eberhard's theorem", "statement_latex": "sum (6-i)p_i = 12", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((sum (6-i)p_i),(12))=0", "to_expression": "sum (6-i)p_i = 12", "justification": "zero discrete distance iff equality", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum (6-i)p_i = 12.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_84dfbbc36744f196", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "symplectic geometry", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C62ED9627235", "name": "Williamson theorem", "statement_latex": "SMS^T = D ⊕ D", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(SMS^T)=z\\;\\land\\;z=(D ⊕ D)", "to_expression": "SMS^T = D ⊕ D", "justification": "common witness iff equality", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as SMS^T = D ⊕ D.\"]"}], "source_equivalence_identity": "common witness iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_10978a235b412221", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "numerical linear algebra", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FBD08849D0F3", "name": "Kreiss matrix theorem", "statement_latex": "K(A) = sup_{|z|>1} (|z|-1) ||(z-A)^-1||", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(K(A) = sup_{|z|) - (1} (|z|-1) ||(z-A)^-1||) > 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover K(A) = sup_{|z| > 1} (|z|-1) ||(z-A)^-1||.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover K(A) = sup_{|z| > 1} (|z|-1) ||(z-A)^-1||.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "K(A) = sup_{|z|>1} (|z|-1) ||(z-A)^-1||", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover K(A) = sup_{|z| > 1} (|z|-1) ||(z-A)^-1||.\"]"}], "source_equivalence_identity": "A<B iff B-A>0 over finite real quantities."}
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{"schema_version": "1.0", "item_id": "rcd2_b60dec5e2d42ce2f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_525542abde309078", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic geometry", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_49F57EB1100D", "name": "Faltings' theorem", "statement_latex": "g > 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;t\\le (1)\\Rightarrow t<(g)", "to_expression": "g > 1", "justification": "converse uses t=X", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as g > 1.\"]"}], "source_equivalence_identity": "converse uses t=X"}
{"schema_version": "1.0", "item_id": "rcd2_cbd0ea899c7ec55d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f0127f6912998dda", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "computer science", "subfield": "formal language theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_37D96EA4CFB7", "name": "Pumping lemma for regular languages", "statement_latex": "w = xyz", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(w) \\in {(xyz)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover w = xyz.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "w = xyz", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover w = xyz.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_a64017c2b02a4ea3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum mechanics", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9078CE9B3CD4", "name": "Ehrenfest theorem", "statement_latex": "F = -V'(x)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(F))+\\delta_D(z,(-V'(x))))=0", "to_expression": "F = -V'(x)", "justification": "zero loss iff some z equals both", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as F = -V'(x).\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_ac3af4352833ad00", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_98543c2a9525bff1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "combinatorial number theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3FBB7DDC427F", "name": "Erdős–Graham problem", "statement_latex": "sum_{n in S} (1/n) = 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((sum_{n \\in S} (1/n)) \\in S \\land (1) \\notin S) \\lor ((1) \\in S \\land (sum_{n \\in S} (1/n)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover sum_{n in S} (1/n) = 1.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "sum_{n in S} (1/n) = 1", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover sum_{n in S} (1/n) = 1.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_d45f1486dfb2f3f9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "probabilistic number theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9D5ECA10E501", "name": "Turán–Kubilius inequality", "statement_latex": "(1/x) * sum_{n<=x} |f(n) - A(x)|^2 <= (2 + epsilon(x)) * B(x)^2", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "((1/x) * sum_{n\\lex} |f(n) - A(x)|^2 \\le (2 + \\varepsilon(x)) * B(x)^2) \\land (0 = 0)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "(1/x) * sum_{n<=x} |f(n) - A(x)|^2 <= (2 + epsilon(x)) * B(x)^2", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "0=0 is true, so Phi_T and true iff Phi_T."}
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{"schema_version": "1.0", "item_id": "rcd2_1b197f365cd76033", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "combinatorics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_30AD154E7704", "name": "Cyclotomic identity", "statement_latex": "1/(1-alpha*z) = product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((1/(1-alpha*z)),(product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}))\\in\\{(z,z):z\\in D\\}", "to_expression": "1/(1-alpha*z) = product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}", "justification": "pair in diagonal iff X=Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as 1/(1-alpha*z) = product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_a2101a693d97c462", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "metric geometry", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e30669bda4940bea", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7C6C20BC69B4", "name": "Slepian's lemma", "statement_latex": "Pr[Intersection(X_i <= u_i)] <= Pr[Intersection(Y_i <= u_i)]", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\arctan(Pr[Intersection(X_i ≤ u_i)])\\le\\arctan(Pr[Intersection(Y_i ≤ u_i)])", "to_expression": "Pr[Intersection(X_i <= u_i)] <= Pr[Intersection(Y_i <= u_i)]", "justification": "arctan preserves and reflects order", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr[Intersection(X_i ≤ u_i)] ≤ Pr[Intersection(Y_i ≤ u_i)].\"]"}], "source_equivalence_identity": "arctan preserves and reflects order"}
{"schema_version": "1.0", "item_id": "rcd2_d5df18460cbc8020", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A6D639595426", "name": "Inverse function theorem", "statement_latex": "(f^{-1})'(b) = 1 / f'(a)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: ((((f^{-1})'(b)) \\in S \\land (1 / f'(a)) \\notin S) \\lor ((1 / f'(a)) \\in S \\land ((f^{-1})'(b)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover (f^{-1})'(b) = 1 / f'(a).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "(f^{-1})'(b) = 1 / f'(a)", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover (f^{-1})'(b) = 1 / f'(a).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_a5bdc561039802de", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A75F5656C745", "name": "Grosswald–Schnitzer theorem", "statement_latex": "\\zeta(s) = \\prod_{n=1}^{\\infty} \\frac{1}{1-p_n^{-s}}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(\\zeta(s))\\}=\\{(\\prod_{n=1}^{\\infty} \\frac{1}{1-p_n^{-s}})\\}", "to_expression": "\\zeta(s) = \\prod_{n=1}^{\\infty} \\frac{1}{1-p_n^{-s}}", "justification": "singletons equal iff elements equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\zeta(s) = \\\\prod_{n=1}^{\\\\infty} \\\\frac{1}{1-p_n^{-s}}.\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_f2f6b2975524b2f8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "quadratic forms", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D2FCDF28ABB7", "name": "Pfister's sixteen-square identity", "statement_latex": "(x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2) = z_1^2 + z_2^2 + ... + z_{16}^2", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2))=F((z_1^2 + z_2^2 + ... + z_{16}^2))", "to_expression": "(x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2) = z_1^2 + z_2^2 + ... + z_{16}^2", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2) = z_1^2 + z_2^2 + ... + z_{16}^2.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_223e2d28caa08d61", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "symplectic geometry", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0e8b5934018610a2", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic calculus", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_9c3b1ae695c4e7f3", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a3218b641672cd96", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_cb1e9e8b1f3c2814", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis, measure theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_356b21caa34d6606", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "differential topology", "subfield": "geometry", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CC0CC8596114", "name": "Morse theory", "statement_latex": "M^a = {x in M | f(x) <= a}", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (M^a = {x \\in M | f(x)) \\le x \\land x \\le (a})", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover M^a = {x in M | f(x) <= a}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover M^a = {x in M | f(x) <= a}.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "M^a = {x in M | f(x) <= a}", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover M^a = {x in M | f(x) <= a}.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
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{"schema_version": "1.0", "item_id": "rcd2_d94dca54eb3f0710", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "modular arithmetic", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A1DCD328739C", "name": "Hensel's lemma", "statement_latex": "f(x) = 0 (mod p^n)", "assumptions": "All displayed arithmetic expressions are integers; the modulus is a positive integer."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(f(x)))+\\delta_D(z,(0 (mod p^n))))=0", "to_expression": "f(x) = 0 (mod p^n)", "justification": "zero loss iff some z equals both", "preconditions": "All displayed arithmetic expressions are integers; the modulus is a positive integer.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f(x) = 0 (mod p^n).\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_388ac31b6d540511", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7BB84422BC5F", "name": "Hadamard's inequality", "statement_latex": "|det(N)| <= prod_{i=1}^n ||v_i||", "assumptions": "N is a square matrix with row or column vectors v_i in an inner-product space; norms are Euclidean."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (prod_{i=1}^n ||v_i||)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|det(N)|)\\}", "to_expression": "|det(N)| <= prod_{i=1}^n ||v_i||", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "N is a square matrix with row or column vectors v_i in an inner-product space; norms are Euclidean.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |det(N)| ≤ prod_{i=1}^n ||v_i||.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_6350289756401f51", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "discrete geometry", "subfield": "geometry", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E02FAD9C277D", "name": "Erdős–Anning theorem", "statement_latex": "|d(A,X) - d(B,X)| <= d(A,B)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\le (|d(A,X) - d(B,X)|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (d(A,B))\\}", "to_expression": "|d(A,X) - d(B,X)| <= d(A,B)", "justification": "lower-cut inclusion iff X≤Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |d(A,X) - d(B,X)| ≤ d(A,B).\"]"}], "source_equivalence_identity": "lower-cut inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_d1eaa9c524bf4dcf", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "Algebra", "subfield": "Lie algebras", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a80cf1081bd6ca47", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "geometry of numbers", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_638EC6734424", "name": "Minkowski–Hlawka theorem", "statement_latex": "Δ ≥ ζ(n) / 2^(n-1)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (Δ)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (ζ(n) / 2^(n-1))\\}", "to_expression": "Δ ≥ ζ(n) / 2^(n-1)", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Δ ≥ ζ(n) / 2^(n-1).\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_139791a8e845a3aa", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_672c640d21c42c2e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "topology", "subfield": "differential topology", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2151776292B6", "name": "Thom's second isotopy lemma", "statement_latex": "ker(d(f|_Y)_y) subset tau", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(ker(d(f|_Y)_y) \\subseteq tau) \\land (0 = 0)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "ker(d(f|_Y)_y) subset tau", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "0=0 is true, so Phi_T and true iff Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_81562b21b537047b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E6F93BEA2A51", "name": "Cassini and Catalan identities", "statement_latex": "F_{n-1}*F_{n+1} - F_n^2 = (-1)^n", "assumptions": "F_n denotes the Fibonacci sequence with the indexing convention used by the theorem; n is an integer in the stated range."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((F_{n-1}*F_{n+1} - F_n^2),((-1)^n))=0", "to_expression": "F_{n-1}*F_{n+1} - F_n^2 = (-1)^n", "justification": "zero discrete distance iff equality", "preconditions": "F_n denotes the Fibonacci sequence with the indexing convention used by the theorem; n is an integer in the stated range.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as F_{n-1}*F_{n+1} - F_n^2 = (-1)^n.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_83b3698f24359686", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "numerical analysis", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2045edbbecbd8a44", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "field theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_6E9382B37B92", "name": "Primitive element theorem", "statement_latex": "E = F(α)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((E),z)=((F(α)),z)", "to_expression": "E = F(α)", "justification": "pair equality forces first coordinates equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as E = F(α).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_05f8a13a9a09953b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d0f4d369fa758b42", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "analysis", "subfield": "approximation theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_538DF6DD562D", "name": "Favard's theorem", "statement_latex": "y_{n+1}=(x-c_{n})y_{n}-d_{n}y_{n-1}", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((y_{n+1}))=F(((x-c_{n})y_{n}-d_{n}y_{n-1}))", "to_expression": "y_{n+1}=(x-c_{n})y_{n}-d_{n}y_{n-1}", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as y_{n+1}=(x-c_{n})y_{n}-d_{n}y_{n-1}.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_c4e498a54d518de7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "analytic number theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C97AADA78BC1", "name": "Vaughan's identity", "statement_latex": "-zeta'(s)/zeta(s) = F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(-zeta'(s)/zeta(s))\\}=\\{(F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s)))\\}", "to_expression": "-zeta'(s)/zeta(s) = F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s))", "justification": "singletons equal iff elements equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as -zeta'(s)/zeta(s) = F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s)).\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_55c3c4a8494aedf8", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "algebra", "subfield": "combinatorics", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d643e1a82c86e5e5", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8DC5B9EC1AA5", "name": "Dimension theorem for vector spaces", "statement_latex": "|B_1| = |B_2|", "assumptions": "B_1 and B_2 are bases of the same finite-dimensional vector space."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((|B_1|),(|B_2|))=0", "to_expression": "|B_1| = |B_2|", "justification": "zero discrete distance iff equality", "preconditions": "B_1 and B_2 are bases of the same finite-dimensional vector space.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |B_1| = |B_2|.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_f75ec3554dd84866", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B63439ADF46F", "name": "Rising sun lemma", "statement_latex": "E = \\bigcup_{k}(a_{k},b_{k})", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((E) \\in S \\land (\\bigcup_{k}(a_{k},b_{k})) \\notin S) \\lor ((\\bigcup_{k}(a_{k},b_{k})) \\in S \\land (E) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E = \\\\bigcup_{k}(a_{k},b_{k}).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "E = \\bigcup_{k}(a_{k},b_{k})", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E = \\\\bigcup_{k}(a_{k},b_{k}).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_93df160222180fd3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_93F7C5A7A33A", "name": "Sun's curious identity", "statement_latex": "(x+m+1)\\sum _{i=0}^{m}(-1)^{i}{\\dbinom {x+y+i}{m-i}}{\\dbinom {y+2i}{i}}-\\sum _{i=0}^{m}{\\dbinom {x+i}{m-i}}(-4)^{i}=(x-m){\\dbinom {x}{m}}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;(((x+m+1)\\sum _{i=0}^{m}(-1)^{i}{\\dbinom {x+y+i}{m-i}}{\\dbinom {y+2i}{i}}-\\sum _{i=0}^{m}{\\dbinom {x+i}{m-i}}(-4)^{i}),z)=(((x-m){\\dbinom {x}{m}}),z)", "to_expression": "(x+m+1)\\sum _{i=0}^{m}(-1)^{i}{\\dbinom {x+y+i}{m-i}}{\\dbinom {y+2i}{i}}-\\sum _{i=0}^{m}{\\dbinom {x+i}{m-i}}(-4)^{i}=(x-m){\\dbinom {x}{m}}", "justification": "pair equality forces first coordinates equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (x+m+1)\\\\sum _{i=0}^{m}(-1)^{i}{\\\\dbinom {x+y+i}{m-i}}{\\\\dbinom {y+2i}{i}}-\\\\sum _{i=0}^{m}{\\\\dbinom {x+i}{m-i}}(-4)^{i}=(x-m){\\\\dbinom {x}{m}}.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
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{"schema_version": "1.0", "item_id": "rcd2_ee583f3e101490b0", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "numerical analysis", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b4a1afbad71560d0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "symbolic dynamics", "subfield": "cellular automata", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_03C6BB7FE5A2", "name": "Curtis–Hedlund–Lyndon theorem", "statement_latex": "f(s(x)) = s(f(x))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "{(f(s(x)))} = {(s(f(x)))}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "{L} = {R} iff L = R by extensionality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(s(x)) = s(f(x)).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f(s(x)) = s(f(x))", "justification": "{L} = {R} iff L = R by extensionality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(s(x)) = s(f(x)).\"]"}], "source_equivalence_identity": "{L} = {R} iff L = R by extensionality."}
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{"schema_version": "1.0", "item_id": "rcd2_5be8fa486a3df452", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "probability theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8F3F3785BDA2", "name": "Gibbs' inequality", "statement_latex": "-Σ p_i log p_i <= -Σ p_i log q_i", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (-Σ p_i log q_i)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (-Σ p_i log p_i)\\}", "to_expression": "-Σ p_i log p_i <= -Σ p_i log q_i", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as -Σ p_i log p_i ≤ -Σ p_i log q_i.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_2ee622389ec0b8fb", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_7e0320fae30d262d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "Number theory", "subfield": "Elliptic functions", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_78B717A820D4", "name": "Jacobi triple product", "statement_latex": "product_{m=1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2}) = sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\{(product_{m=1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2}))\\}=\\{(sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n})\\}", "to_expression": "product_{m=1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2}) = sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n}", "justification": "singletons equal iff elements equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as product_{m=1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2}) = sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n}.\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
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{"schema_version": "1.0", "item_id": "rcd2_ec6f4c2dd6493552", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "Ramsey theory", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6373106f31e15a01", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e3cb509ebf9ff516", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "convex geometry", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4F00365B64E5", "name": "Minkowski's first inequality for convex bodies", "statement_latex": "V_1(K, L) ≥ V(K)^((n-1)/n) V(L)^(1/n)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(V(K)^((n-1)/n) V(L)^(1/n))+s=(V_1(K, L))", "to_expression": "V_1(K, L) ≥ V(K)^((n-1)/n) V(L)^(1/n)", "justification": "nonnegative slack exists iff X≤Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as V_1(K, L) ≥ V(K)^((n-1)/n) V(L)^(1/n).\"]"}], "source_equivalence_identity": "nonnegative slack exists iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_faf9a4052cf78f49", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "elementary number theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_9d286672ddfae1b6", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_3ea278a9b6aad7a0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "quadratic forms", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D2FCDF28ABB7", "name": "Pfister's sixteen-square identity", "statement_latex": "(x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2) = z_1^2 + z_2^2 + ... + z_{16}^2", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2),(z_1^2 + z_2^2 + ... + z_{16}^2))=0", "to_expression": "(x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2) = z_1^2 + z_2^2 + ... + z_{16}^2", "justification": "zero discrete distance iff equality", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2) = z_1^2 + z_2^2 + ... + z_{16}^2.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_ead79d6c07914ca6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal set theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3B6CADC238DF", "name": "Lubell–Yamamoto–Meshalkin inequality", "statement_latex": "sum_{k=0}^n a_k/binom(n,k) <= 1", "assumptions": "The family is Sperner; a_k counts chosen k-subsets of an n-element ground set."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(1) - (sum_{k=0}^n a_k/binom(n,k)) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "The family is Sperner; a_k counts chosen k-subsets of an n-element ground set.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover sum_{k=0}^n a_k/binom(n,k) <= 1.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "The family is Sperner; a_k counts chosen k-subsets of an n-element ground set.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover sum_{k=0}^n a_k/binom(n,k) <= 1.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "sum_{k=0}^n a_k/binom(n,k) <= 1", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "The family is Sperner; a_k counts chosen k-subsets of an n-element ground set.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover sum_{k=0}^n a_k/binom(n,k) <= 1.\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_66b3a30f40c0cb4c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "multilinear algebra", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F538F79B0750", "name": "Lagrange's identity", "statement_latex": "(\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2 = \\sum_{i<j} (a_i b_j - a_j b_i)^2", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D(((\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2),(\\sum_{i<j} (a_i b_j - a_j b_i)^2))=0", "to_expression": "(\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2 = \\sum_{i<j} (a_i b_j - a_j b_i)^2", "justification": "zero discrete distance iff equality", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (\\\\sum a_k^2)(\\\\sum b_k^2) - (\\\\sum a_k b_k)^2 = \\\\sum_{i<j} (a_i b_j - a_j b_i)^2.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_07e1ecb9b17e135b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum information theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_42507491714c8865", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "theoretical computer science", "subfield": "algebraic complexity theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_46E2A92077B9", "name": "Schwartz–Zippel lemma", "statement_latex": "Pr[P(r1, ..., rn) = 0] ≤ d/|S|", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(d/|S|)\\le t\\Rightarrow (Pr[P(r1, ..., rn) = 0])\\le t", "to_expression": "Pr[P(r1, ..., rn) = 0] ≤ d/|S|", "justification": "converse uses t=Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr[P(r1, ..., rn) = 0] ≤ d/|S|.\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_ecf5252812140704", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "prime numbers", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_71AA6DBCD731", "name": "Bonse's inequality", "statement_latex": "p_n# > p_{n+1}^2", "assumptions": "n>=4; p_n# is the product of the first n primes; p_{n+1} is the next prime."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (p_{n+1}^2) < x \\land x < (p_n#)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "n>=4; p_n# is the product of the first n primes; p_{n+1} is the next prime.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover p_n# > p_{n+1}^2.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "n>=4; p_n# is the product of the first n primes; p_{n+1} is the next prime.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover p_n# > p_{n+1}^2.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "p_n# > p_{n+1}^2", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "n>=4; p_n# is the product of the first n primes; p_{n+1} is the next prime.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover p_n# > p_{n+1}^2.\"]"}], "source_equivalence_identity": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B."}
{"schema_version": "1.0", "item_id": "rcd2_19c07cb5e2729996", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "scattering theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0C5D6466FC1D", "name": "Levinson's theorem", "statement_latex": "φ(+∞) − φ(0) = −π(1/2*n0 + nb)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(φ(+∞) − φ(0))\\Longleftrightarrow z=(−π(1/2*n0 + nb)))", "to_expression": "φ(+∞) − φ(0) = −π(1/2*n0 + nb)", "justification": "same equality fiber iff equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as φ(+∞) − φ(0) = −π(1/2*n0 + nb).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_513ee5d5fa6e80ca", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "polynomial factorization", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_4ec79913cc3a6f71", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0C6AE95E0EE0", "name": "Hoeffding's inequality", "statement_latex": "P(S_n - E[S_n] >= t) <= exp(-2*t^2 / sum_i (b_i-a_i)^2)", "assumptions": "The summands are independent and bounded with X_i in [a_i,b_i]; t>=0; denominator is positive."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(P(S_n - E[S_n] ≥ t))}{\\exp(P(S_n - E[S_n] ≥ t))+\\exp(exp(-2*t^2 / sum_i (b_i-a_i)^2))}\\le\\frac{1}{2}", "to_expression": "P(S_n - E[S_n] >= t) <= exp(-2*t^2 / sum_i (b_i-a_i)^2)", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "The summands are independent and bounded with X_i in [a_i,b_i]; t>=0; denominator is positive.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(S_n - E[S_n] ≥ t) ≤ exp(-2*t^2 / sum_i (b_i-a_i)^2).\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_545672e45bb5b356", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "inequalities", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B1E33965D95E", "name": "Nesbitt's inequality", "statement_latex": "a/(b+c) + b/(a+c) + c/(a+b) >= 3/2", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(a/(b+c) + b/(a+c) + c/(a+b)) - (3/2) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover a/(b+c) + b/(a+c) + c/(a+b) >= 3/2.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover a/(b+c) + b/(a+c) + c/(a+b) >= 3/2.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "a/(b+c) + b/(a+c) + c/(a+b) >= 3/2", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover a/(b+c) + b/(a+c) + c/(a+b) >= 3/2.\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_390c1c6d2995b6a5", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_89D2662A27BE", "name": "Freiman's theorem", "statement_latex": "|A+A| <= K|A|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(|A+A|)}{\\exp(|A+A|)+\\exp(K|A|)}\\le\\frac{1}{2}", "to_expression": "|A+A| <= K|A|", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |A+A| ≤ K|A|.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_426b341ddb8f6280", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "combinatorics", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_059ff7760fca7545", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "combinatorics on words", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_08AF905E8677", "name": "Fine and Wilf's theorem", "statement_latex": "a_n = a_{n+gcd(h,k)}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(a_n)=z\\;\\land\\;z=(a_{n+gcd(h,k)})", "to_expression": "a_n = a_{n+gcd(h,k)}", "justification": "common witness iff equality", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as a_n = a_{n+gcd(h,k)}.\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_f8e4a5f75863c9b8", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "computer science", "subfield": "theoretical computer science", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_43edff8786f78d9b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_8bcdae22d06e8581", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "sieve theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5DC5BE909937", "name": "Fundamental lemma of sieve theory", "statement_latex": "|A_d| = \\frac{w(d)}{d}X + R_d", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((|A_d|) \\in S \\land (\\frac{w(d)}{d}X + R_d) \\notin S) \\lor ((\\frac{w(d)}{d}X + R_d) \\in S \\land (|A_d|) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover |A_d| = \\\\frac{w(d)}{d}X + R_d.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "|A_d| = \\frac{w(d)}{d}X + R_d", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover |A_d| = \\\\frac{w(d)}{d}X + R_d.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_1892c77e0386bf97", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9EF5E48E9DAF", "name": "Rolle's theorem", "statement_latex": "f'(c) = 0", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(f'(c)) \\in {(0)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f'(c) = 0.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f'(c) = 0", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f'(c) = 0.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_ad2870e899701403", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "martingale theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1E094F8EB368", "name": "Azuma's inequality", "statement_latex": "P(X_N - X_0 >= epsilon) <= exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;t\\le (P(X_N - X_0 ≥ epsilon))\\Rightarrow t\\le (exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2)))", "to_expression": "P(X_N - X_0 >= epsilon) <= exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2))", "justification": "converse uses t=X", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(X_N - X_0 ≥ epsilon) ≤ exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2)).\"]"}], "source_equivalence_identity": "converse uses t=X"}
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{"schema_version": "1.0", "item_id": "rcd2_7d912f7d77dce5b3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FE78D365CC0D", "name": "Weinstein–Aronszajn identity", "statement_latex": "det(I_m + AB) = det(I_n + BA)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(det(I_m + AB)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(det(I_n + BA)))", "to_expression": "det(I_m + AB) = det(I_n + BA)", "justification": "each argmin set is a singleton", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as det(I_m + AB) = det(I_n + BA).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
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{"schema_version": "1.0", "item_id": "rcd2_f12633f809bb3f38", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "percolation theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4AB97BAC2783", "name": "Van den Berg–Kesten inequality", "statement_latex": "P(A square B) <= P(A)P(B)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(P(A square B))}{\\exp(P(A square B))+\\exp(P(A)P(B))}\\le\\frac{1}{2}", "to_expression": "P(A square B) <= P(A)P(B)", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(A square B) ≤ P(A)P(B).\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_f9b9a92afa64d97f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "computer science", "subfield": "probabilistic combinatorics", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AA4BF27A6AAD", "name": "Algorithmic Lovász local lemma", "statement_latex": "Pr[A] <= x(A) * Product_{B in Gamma(A)} (1 - x(B))", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(Pr[A])}{\\exp(Pr[A])+\\exp(x(A) * Product_{B in Gamma(A)} (1 - x(B)))}\\le\\frac{1}{2}", "to_expression": "Pr[A] <= x(A) * Product_{B in Gamma(A)} (1 - x(B))", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr[A] ≤ x(A) * Product_{B in Gamma(A)} (1 - x(B)).\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_77e089cdb878a33c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4273AD2C24D7", "name": "Bayes' theorem", "statement_latex": "P(A|B) = P(B|A)*P(A)/P(B)", "assumptions": "A and B are events with P(B)>0; conditional probabilities are defined."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(P(B|A)*P(A)/P(B)) = (P(A|B))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "A and B are events with P(B)>0; conditional probabilities are defined.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover P(A|B) = P(B|A)*P(A)/P(B).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "P(A|B) = P(B|A)*P(A)/P(B)", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "A and B are events with P(B)>0; conditional probabilities are defined.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover P(A|B) = P(B|A)*P(A)/P(B).\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_adde373de73699ee", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_93F7C5A7A33A", "name": "Sun's curious identity", "statement_latex": "(x+m+1)\\sum _{i=0}^{m}(-1)^{i}{\\dbinom {x+y+i}{m-i}}{\\dbinom {y+2i}{i}}-\\sum _{i=0}^{m}{\\dbinom {x+i}{m-i}}(-4)^{i}=(x-m){\\dbinom {x}{m}}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg ((x+m+1)\\sum _{i=0}^{m}(-1)^{i}{\\dbinom {x+y+i}{m-i}}{\\dbinom {y+2i}{i}}-\\sum _{i=0}^{m}{\\dbinom {x+i}{m-i}}(-4)^{i}=(x-m){\\dbinom {x}{m}}))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "(x+m+1)\\sum _{i=0}^{m}(-1)^{i}{\\dbinom {x+y+i}{m-i}}{\\dbinom {y+2i}{i}}-\\sum _{i=0}^{m}{\\dbinom {x+i}{m-i}}(-4)^{i}=(x-m){\\dbinom {x}{m}}", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
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{"schema_version": "1.0", "item_id": "rcd2_bcae77716a3be1e4", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "discrete geometry", "subfield": "geometry", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e6e9fb695b581585", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c4d79778b15ea76e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "algebraic graph theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0ABE50B0E12D", "name": "Expander mixing lemma", "statement_latex": "|e(S,T) - (d/n)|S||T|| <= λ√|S||T|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((|e(S,T) - (d/n)|S||T||)-(λ√|S||T|))=0", "to_expression": "|e(S,T) - (d/n)|S||T|| <= λ√|S||T|", "justification": "sup equals max(0,X-Y)", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |e(S,T) - (d/n)|S||T|| ≤ λ√|S||T|.\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
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{"schema_version": "1.0", "item_id": "rcd2_00f7292d85e84eaa", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "calculus", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
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{"schema_version": "1.0", "item_id": "rcd2_448b410649f3402c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "control theory", "subfield": "geometric control theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2ADEC5E217BE", "name": "Krener's theorem", "statement_latex": "\\dot{q}=f(q,u)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((\\dot{q}) \\in S \\land (f(q,u)) \\notin S) \\lor ((f(q,u)) \\in S \\land (\\dot{q}) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\dot{q}=f(q,u).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\dot{q}=f(q,u)", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\dot{q}=f(q,u).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_87f6a8843be67c9b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "computer science", "subfield": "theoretical computer science", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_30A4875B9AAE", "name": "Isolation lemma", "statement_latex": "w(S) = sum_{x in S} w(x)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((w(S)) \\in S \\land (sum_{x \\in S} w(x)) \\notin S) \\lor ((sum_{x \\in S} w(x)) \\in S \\land (w(S)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover w(S) = sum_{x in S} w(x).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "w(S) = sum_{x in S} w(x)", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover w(S) = sum_{x in S} w(x).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_9678baf8091d35df", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "approximation theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_DC96C747092B", "name": "Müntz–Szász theorem", "statement_latex": "sum_{n in S} (1/n) = infinity", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\infty) = (sum_{n \\in S} (1/n))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover sum_{n in S} (1/n) = infinity.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "sum_{n in S} (1/n) = infinity", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover sum_{n in S} (1/n) = infinity.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_9b4ec8d2c442e68c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic calculus", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4D5944961D8", "name": "Itô's lemma", "statement_latex": "dX_t = mu_t dt + sigma_t dB_t", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(dX_t))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(mu_t dt + sigma_t dB_t))", "to_expression": "dX_t = mu_t dt + sigma_t dB_t", "justification": "each argmin set is a singleton", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as dX_t = mu_t dt + sigma_t dB_t.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_bf24cb9649814781", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_abfc580d0ff9b97c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "hypergraph theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0E38B7F67353", "name": "Baranyai's theorem", "statement_latex": "\\binom{k}{r} \\frac{r}{k} = \\binom{k-1}{r-1}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\binom{k}{r} \\frac{r}{k}) \\in {(\\binom{k-1}{r-1})}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\binom{k}{r} \\\\frac{r}{k} = \\\\binom{k-1}{r-1}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\binom{k}{r} \\frac{r}{k} = \\binom{k-1}{r-1}", "justification": "L in {R} iff L = R.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\binom{k}{r} \\\\frac{r}{k} = \\\\binom{k-1}{r-1}.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_b4084de9493facb4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "recreational mathematics", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_96F33F66821C", "name": "Tupper's self-referential formula", "statement_latex": "1/2 < floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2))", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(1/2)}{\\exp(1/2)+\\exp(floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2)))}<\\frac{1}{2}", "to_expression": "1/2 < floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2))", "justification": "equiv exp(X)<exp(Y)", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as 1/2 < floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2)).\"]"}], "source_equivalence_identity": "equiv exp(X)<exp(Y)"}
{"schema_version": "1.0", "item_id": "rcd2_3ab01ef82febac18", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "modular arithmetic", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_292bbc9ad9a22281", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "classical mechanics", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BC55A9432F80", "name": "Helmholtz theorem (classical mechanics)", "statement_latex": "dS = (dE + PdV) / T", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(dS)\\Longleftrightarrow z=((dE + PdV) / T))", "to_expression": "dS = (dE + PdV) / T", "justification": "same equality fiber iff equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as dS = (dE + PdV) / T.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_678db6a4a076b3e3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "approximation theory", "subfield": "mathematical analysis", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_389D9173C8B7", "name": "Lebesgue's lemma", "statement_latex": "‖v - Pv‖ ≤ (1 + ‖P‖) inf_{u ∈ U} ‖v - u‖", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "((1 + ‖P‖) \\inf_{u ∈ U} ‖v - u‖) - (‖v - Pv‖) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ‖v - Pv‖ <= (1 + ‖P‖) inf_{u ∈ U} ‖v - u‖.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ‖v - Pv‖ <= (1 + ‖P‖) inf_{u ∈ U} ‖v - u‖.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "‖v - Pv‖ ≤ (1 + ‖P‖) inf_{u ∈ U} ‖v - u‖", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ‖v - Pv‖ <= (1 + ‖P‖) inf_{u ∈ U} ‖v - u‖.\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_6af3100ebe91b319", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "discrete geometry", "subfield": "geometry", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E02FAD9C277D", "name": "Erdős–Anning theorem", "statement_latex": "|d(A,X) - d(B,X)| <= d(A,B)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(|d(A,X) - d(B,X)|)}{\\exp(|d(A,X) - d(B,X)|)+\\exp(d(A,B))}\\le\\frac{1}{2}", "to_expression": "|d(A,X) - d(B,X)| <= d(A,B)", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |d(A,X) - d(B,X)| ≤ d(A,B).\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_b8a1adbd0c47a076", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "combinatorics", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_229d3de8d637192c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "representation theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a57cc9249e9eaafb", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "additive combinatorics", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_28783c94838eedbb", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "elementary algebra", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_8b58ccee5ab5f1e8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_51200980F031", "name": "Star of David theorem", "statement_latex": "gcd{(n-1 k-1), (n k+1), (n+1 k)} = gcd{(n-1 k), (n k-1), (n+1 k+1)}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((gcd{(n-1 k-1), (n k+1), (n+1 k)}))=F((gcd{(n-1 k), (n k-1), (n+1 k+1)}))", "to_expression": "gcd{(n-1 k-1), (n k+1), (n+1 k)} = gcd{(n-1 k), (n k-1), (n+1 k+1)}", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as gcd{(n-1 k-1), (n k+1), (n+1 k)} = gcd{(n-1 k), (n k-1), (n+1 k+1)}.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_66e2c109960877e1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0506260C92DB", "name": "Bennett's inequality", "statement_latex": "Pr(S_n > t) <= exp(-(sigma^2 / a^2) * h(at / sigma^2))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(Pr(S_n > t))-(exp(-(sigma^2 / a^2) * h(at / sigma^2)))\\}=0", "to_expression": "Pr(S_n > t) <= exp(-(sigma^2 / a^2) * h(at / sigma^2))", "justification": "positive part vanishes iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr(S_n > t) ≤ exp(-(sigma^2 / a^2) * h(at / sigma^2)).\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_5aa698f378cd6601", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "combinatorics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E6F93BEA2A51", "name": "Cassini and Catalan identities", "statement_latex": "F_{n-1}*F_{n+1} - F_n^2 = (-1)^n", "assumptions": "F_n denotes the Fibonacci sequence with the indexing convention used by the theorem; n is an integer in the stated range."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((F_{n-1}*F_{n+1} - F_n^2) \\in S \\land ((-1)^n) \\notin S) \\lor (((-1)^n) \\in S \\land (F_{n-1}*F_{n+1} - F_n^2) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "F_n denotes the Fibonacci sequence with the indexing convention used by the theorem; n is an integer in the stated range.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover F_{n-1}*F_{n+1} - F_n^2 = (-1)^n.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "F_{n-1}*F_{n+1} - F_n^2 = (-1)^n", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "F_n denotes the Fibonacci sequence with the indexing convention used by the theorem; n is an integer in the stated range.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover F_{n-1}*F_{n+1} - F_n^2 = (-1)^n.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
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{"schema_version": "1.0", "item_id": "rcd2_cc8f4ec765740b24", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F01A12C6F5E3", "name": "Taylor's theorem", "statement_latex": "f(x) = sum_{n=0}^k (f^(n)(a)/n!) * (x-a)^n + R_k(x)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(f(x) = sum_{n=0}^k (f^(n)(a)/n!) * (x-a)^n + R_k(x)) \\land (0 = 0)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f(x) = sum_{n=0}^k (f^(n)(a)/n!) * (x-a)^n + R_k(x)", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "0=0 is true, so Phi_T and true iff Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_bc815f2e7e777c70", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "permutation patterns", "primary_mechanism": "quantifier", "transformation_category": "counting_combinatorial", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8A6780273F9B", "name": "Stanley–Wilf conjecture", "statement_latex": "lim_{n→∞} |S_n(β)|^(1/n)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall k\\in\\mathbb{N}_{>0}\\exists N\\forall n\\ge N:d((lim_{n),(∞} |S_n(β)|^(1/n)))<1/k", "to_expression": "lim_{n→∞} |S_n(β)|^(1/n)", "justification": "1/k base is cofinal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{n→∞} |S_n(β)|^(1/n).\"]"}], "source_equivalence_identity": "1/k base is cofinal"}
{"schema_version": "1.0", "item_id": "rcd2_a33d2b8ea9a34098", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FE78D365CC0D", "name": "Weinstein–Aronszajn identity", "statement_latex": "det(I_m + AB) = det(I_n + BA)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "(\\{(det(I_m + AB))\\}\\times\\{(det(I_n + BA))\\})\\cap\\{(z,z):z\\in D\\}=\\{((det(I_m + AB)),(det(I_n + BA)))\\}", "to_expression": "det(I_m + AB) = det(I_n + BA)", "justification": "intersection is singleton iff pair diagonal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as det(I_m + AB) = det(I_n + BA).\"]"}], "source_equivalence_identity": "intersection is singleton iff pair diagonal"}
{"schema_version": "1.0", "item_id": "rcd2_95a59689f07826cd", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "field theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_ED1CB49A03EE", "name": "Steinitz's theorem", "statement_latex": "L = K(α)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(L))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(K(α)))", "to_expression": "L = K(α)", "justification": "each argmin set is a singleton", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as L = K(α).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_ab54464c52621203", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic functions", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FDD3E2DFBB65", "name": "Hermite's identity", "statement_latex": "sum_{k=0}^{n-1} floor(x+k/n) = floor(n*x)", "assumptions": "n is a positive integer; x is real."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{k=0}^{n-1} floor(x+k/n)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(floor(n*x)))", "to_expression": "sum_{k=0}^{n-1} floor(x+k/n) = floor(n*x)", "justification": "each argmin set is a singleton", "preconditions": "n is a positive integer; x is real.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{k=0}^{n-1} floor(x+k/n) = floor(n*x).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
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{"schema_version": "1.0", "item_id": "rcd2_d69e63e6f0f560e2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "physics", "subfield": "general relativity", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_737BDAA0F832", "name": "Stewart–Walker lemma", "statement_latex": "ΔδT = 0", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(ΔδT))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(0))", "to_expression": "ΔδT = 0", "justification": "each argmin set is a singleton", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ΔδT = 0.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_1eca4e8561fe55cb", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "queueing theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D0BC02092015", "name": "Bartlett's theorem", "statement_latex": "mu(t) = integral_{-infinity}^{t} A(s)p(s,t) ds", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(mu(t))\\Longleftrightarrow z=(integral_{-infinity}^{t} A(s)p(s,t) ds))", "to_expression": "mu(t) = integral_{-infinity}^{t} A(s)p(s,t) ds", "justification": "same equality fiber iff equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as mu(t) = integral_{-infinity}^{t} A(s)p(s,t) ds.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
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{"schema_version": "1.0", "item_id": "rcd2_c62238b8d614cb54", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "analytic number theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C97AADA78BC1", "name": "Vaughan's identity", "statement_latex": "-zeta'(s)/zeta(s) = F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((-z\\eta'(s)/z\\eta(s)) \\in S \\land (F(s) - z\\eta(s)F(s)G(s) - z\\eta'(s)G(s) + (-z\\eta'(s)/z\\eta(s) - F(s))(1 - z\\eta(s)G(s))) \\notin S) \\lor ((F(s) - z\\eta(s)F(s)G(s) - z\\eta'(s)G(s) + (-z\\eta'(s)/z\\eta(s) - F(s))(1 - z\\eta(s)G(s))) \\in S \\land (-z\\eta'(s)/z\\eta(s)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover -zeta'(s)/zeta(s) = F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s)).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "-zeta'(s)/zeta(s) = F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s))", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover -zeta'(s)/zeta(s) = F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s)).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_a70539358d6b8895", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "quadratic forms", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D2FCDF28ABB7", "name": "Pfister's sixteen-square identity", "statement_latex": "(x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2) = z_1^2 + z_2^2 + ... + z_{16}^2", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: ((((x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2)) \\in S \\land (z_1^2 + z_2^2 + ... + z_{16}^2) \\notin S) \\lor ((z_1^2 + z_2^2 + ... + z_{16}^2) \\in S \\land ((x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover (x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2) = z_1^2 + z_2^2 + ... + z_{16}^2.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "(x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2) = z_1^2 + z_2^2 + ... + z_{16}^2", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover (x_1^2 + x_2^2 + ... + x_{16}^2)(y_1^2 + y_2^2 + ... + y_{16}^2) = z_1^2 + z_2^2 + ... + z_{16}^2.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_6ab45faaab7baed8", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "elementary algebra", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_fe52ce3c0d962c16", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "quadratic residues", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A18C6C225F06", "name": "Zolotarev's lemma", "statement_latex": "(a/p) = epsilon(pi_a)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((a/p))=F((epsilon(pi_a)))", "to_expression": "(a/p) = epsilon(pi_a)", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (a/p) = epsilon(pi_a).\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_de24c01ecd7aeb77", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "finite fields", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_24697C70978F", "name": "Chevalley–Warning theorem", "statement_latex": "n > sum_{j=1}^r d_j", "assumptions": "n is the number of variables; d_j are degrees of the polynomial equations over a finite field. This row encodes the standard degree-sum hypothesis."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\operatorname{softplus}((n)-(sum_{j=1}^r d_j))>\\log 2", "to_expression": "n > sum_{j=1}^r d_j", "justification": "softplus(Y-X)>log2 iff Y-X>0", "preconditions": "n is the number of variables; d_j are degrees of the polynomial equations over a finite field. This row encodes the standard degree-sum hypothesis.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as n > sum_{j=1}^r d_j.\"]"}], "source_equivalence_identity": "softplus(Y-X)>log2 iff Y-X>0"}
{"schema_version": "1.0", "item_id": "rcd2_fbd185b52fde8bbe", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "infinite-dimensional vector spaces", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_9a270db8379099c1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "graph theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9A3C13F73177", "name": "Burr–Erdős conjecture", "statement_latex": "r(G) <= cp * n", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(cp * n)\\le t\\Rightarrow (r(G))\\le t", "to_expression": "r(G) <= cp * n", "justification": "converse uses t=Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as r(G) ≤ cp * n.\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_4412091170965e01", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "number theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9793AD078B54", "name": "Degen's eight-square identity", "statement_latex": "(∑_{i=1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2) = ∑_{i=1}^{8} c_i^2", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "((∑_{i=1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2) = ∑_{i=1}^{8} c_i^2) \\land (0 = 0)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "(∑_{i=1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2) = ∑_{i=1}^{8} c_i^2", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "0=0 is true, so Phi_T and true iff Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_c6974771d988f042", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal set theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_005d0644e5e46849", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "extremal graph theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2dc0e04ae38e143b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "statistics", "subfield": "statistical hypothesis testing", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_69B4EB18D461", "name": "Neyman–Pearson lemma", "statement_latex": "α = Pr_θ0(X ∈ R)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(α))+\\delta_D(z,(Pr_θ0(X ∈ R))))=0", "to_expression": "α = Pr_θ0(X ∈ R)", "justification": "zero loss iff some z equals both", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as α = Pr_θ0(X ∈ R).\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_16710332b864dbbc", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_dc4bfb7fac90b623", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A75F5656C745", "name": "Grosswald–Schnitzer theorem", "statement_latex": "\\zeta(s) = \\prod_{n=1}^{\\infty} \\frac{1}{1-p_n^{-s}}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((\\zeta(s)),(\\prod_{n=1}^{\\infty} \\frac{1}{1-p_n^{-s}}))=0", "to_expression": "\\zeta(s) = \\prod_{n=1}^{\\infty} \\frac{1}{1-p_n^{-s}}", "justification": "zero discrete distance iff equality", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\zeta(s) = \\\\prod_{n=1}^{\\\\infty} \\\\frac{1}{1-p_n^{-s}}.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_28ec816d4ab8603c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "graph theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9A3C13F73177", "name": "Burr–Erdős conjecture", "statement_latex": "r(G) <= cp * n", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(r(G))}{\\exp(r(G))+\\exp(cp * n)}\\le\\frac{1}{2}", "to_expression": "r(G) <= cp * n", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as r(G) ≤ cp * n.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_80483b838cd774f6", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "sieve theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a654532661c8c35e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "prime number theory", "primary_mechanism": "characterization", "transformation_category": "probabilistic_distributional", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BFDCE2A9DBB2", "name": "Bertrand's postulate", "statement_latex": "exists prime p such that n < p < 2*n", "assumptions": "n is an integer greater than 1."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\sup_{x\\in D}\\mathbf{1}_{\\{prime p\\in D:n < p < 2*n\\}}(x)=1", "to_expression": "exists prime p such that n < p < 2*n", "justification": "sup one iff member exists", "preconditions": "n is an integer greater than 1.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as exists prime p such that n < p < 2*n.\"]"}], "source_equivalence_identity": "sup one iff member exists"}
{"schema_version": "1.0", "item_id": "rcd2_f538a7a3b898fd69", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "geometry", "subfield": "Euclidean geometry", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0f3a2138109240b9", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "representation theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
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{"schema_version": "1.0", "item_id": "rcd2_f9789c4f4a4f53c0", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "field theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_1d7cf339c224f272", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "number theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9793AD078B54", "name": "Degen's eight-square identity", "statement_latex": "(∑_{i=1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2) = ∑_{i=1}^{8} c_i^2", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(∑_{i=1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2)\\Longleftrightarrow z=(∑_{i=1}^{8} c_i^2))", "to_expression": "(∑_{i=1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2) = ∑_{i=1}^{8} c_i^2", "justification": "same equality fiber iff equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (∑_{i=1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2) = ∑_{i=1}^{8} c_i^2.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
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{"schema_version": "1.0", "item_id": "rcd2_9521c9bd4bd9f179", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "commutative algebra", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0EBFF1A397C6", "name": "Local criterion for flatness", "statement_latex": "Tor_1^A(C, M) = 0", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((Tor_1^A(C, M)) \\in S \\land (0) \\notin S) \\lor ((0) \\in S \\land (Tor_1^A(C, M)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover Tor_1^A(C, M) = 0.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "Tor_1^A(C, M) = 0", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover Tor_1^A(C, M) = 0.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_850c547721551c95", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "classical mechanics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BC55A9432F80", "name": "Helmholtz theorem (classical mechanics)", "statement_latex": "dS = (dE + PdV) / T", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "((dE + PdV) / T) = (dS)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover dS = (dE + PdV) / T.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "dS = (dE + PdV) / T", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover dS = (dE + PdV) / T.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_275de04eea35c9fd", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "Sobolev spaces", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_35353A67F17E", "name": "Pólya–Szegő inequality", "statement_latex": "∫ |∇u*|^p dx <= ∫ |∇u|^p dx", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((∫ |∇u*|^p dx)-(∫ |∇u|^p dx))=0", "to_expression": "∫ |∇u*|^p dx <= ∫ |∇u|^p dx", "justification": "sup equals max(0,X-Y)", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∫ |∇u*|^p dx ≤ ∫ |∇u|^p dx.\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
{"schema_version": "1.0", "item_id": "rcd2_7c66fe04e18bf4c4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "large deviations theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_53869962DDDE", "name": "Laplace's principle", "statement_latex": "\\lim_{\\theta \\to \\infty}\\frac{1}{\\theta}\\log\\int_{A}e^{-\\theta\\varphi(x)}\\,dx=-\\mathop{\\mathrm{ess\\,inf}}_{x\\in A}\\varphi(x)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((\\lim_{\\theta → \\infty}\\frac{1}{\\theta}\\log\\int_{A}e^{-\\theta\\varphi(x)}\\,dx))=F((-\\mathop{\\mathrm{ess\\,inf}}_{x\\in A}\\varphi(x)))", "to_expression": "\\lim_{\\theta \\to \\infty}\\frac{1}{\\theta}\\log\\int_{A}e^{-\\theta\\varphi(x)}\\,dx=-\\mathop{\\mathrm{ess\\,inf}}_{x\\in A}\\varphi(x)", "justification": "indicator of {X} separates unequal objects", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\lim_{\\\\theta → \\\\infty}\\\\frac{1}{\\\\theta}\\\\log\\\\int_{A}e^{-\\\\theta\\\\varphi(x)}\\\\,dx=-\\\\mathop{\\\\mathrm{ess\\\\,inf}}_{x\\\\in A}\\\\varphi(x).\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_691cc755f1a93d5a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum information theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B8493CAFC989", "name": "No-deleting theorem", "statement_latex": "U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C = |\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C),z):z\\in D\\}=\\{((|\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C),z):z\\in D\\}", "to_expression": "U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C = |\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as U|\\\\psi \\\\rangle_A |\\\\psi \\\\rangle_B |A \\\\rangle_C = |\\\\psi \\\\rangle_A |0 \\\\rangle_B |A' \\\\rangle_C.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_0eb4892b9fc19f7e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "cryptography", "subfield": "cryptanalysis", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E35608C3DE5E", "name": "Piling-up lemma", "statement_latex": "ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn) = 2^{n-1}∏_{i=1}^{n}ε(Xi)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn)))=F((2^{n-1}∏_{i=1}^{n}ε(Xi)))", "to_expression": "ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn) = 2^{n-1}∏_{i=1}^{n}ε(Xi)", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn) = 2^{n-1}∏_{i=1}^{n}ε(Xi).\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
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{"schema_version": "1.0", "item_id": "rcd2_af0b099bca616ed3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "topology", "subfield": "differential topology", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2151776292B6", "name": "Thom's second isotopy lemma", "statement_latex": "ker(d(f|_Y)_y) subset tau", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "(ker(d(f|_Y)_y))\\setminus (tau)=\\emptyset", "to_expression": "ker(d(f|_Y)_y) subset tau", "justification": "X\\Y empty iff subset", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ker(d(f|_Y)_y) ⊆ tau.\"]"}], "source_equivalence_identity": "X\\Y empty iff subset"}
{"schema_version": "1.0", "item_id": "rcd2_a66f422a07b2c53c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "matrix analysis", "subfield": "linear algebra", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_4fcfe661fdc3aead", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_93F7C5A7A33A", "name": "Sun's curious identity", "statement_latex": "(x+m+1)\\sum _{i=0}^{m}(-1)^{i}{\\dbinom {x+y+i}{m-i}}{\\dbinom {y+2i}{i}}-\\sum _{i=0}^{m}{\\dbinom {x+i}{m-i}}(-4)^{i}=(x-m){\\dbinom {x}{m}}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F(((x+m+1)\\sum _{i=0}^{m}(-1)^{i}{\\dbinom {x+y+i}{m-i}}{\\dbinom {y+2i}{i}}-\\sum _{i=0}^{m}{\\dbinom {x+i}{m-i}}(-4)^{i}))=F(((x-m){\\dbinom {x}{m}}))", "to_expression": "(x+m+1)\\sum _{i=0}^{m}(-1)^{i}{\\dbinom {x+y+i}{m-i}}{\\dbinom {y+2i}{i}}-\\sum _{i=0}^{m}{\\dbinom {x+i}{m-i}}(-4)^{i}=(x-m){\\dbinom {x}{m}}", "justification": "indicator of {X} separates unequal objects", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (x+m+1)\\\\sum _{i=0}^{m}(-1)^{i}{\\\\dbinom {x+y+i}{m-i}}{\\\\dbinom {y+2i}{i}}-\\\\sum _{i=0}^{m}{\\\\dbinom {x+i}{m-i}}(-4)^{i}=(x-m){\\\\dbinom {x}{m}}.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_7dd5d847575ed667", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "complex analysis", "subfield": "complex geometry", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b008c0d460b0c578", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "inner product spaces", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2EE2B752D03D", "name": "Polarization identity", "statement_latex": "\\operatorname{Re} \\langle x, y \\rangle = \\frac{1}{2} (\\|x + y\\|^2 - \\|x\\|^2 - \\|y\\|^2)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(\\operatorname{Re} \\langle x, y \\rangle)=z\\;\\land\\;z=(\\frac{1}{2} (\\|x + y\\|^2 - \\|x\\|^2 - \\|y\\|^2))", "to_expression": "\\operatorname{Re} \\langle x, y \\rangle = \\frac{1}{2} (\\|x + y\\|^2 - \\|x\\|^2 - \\|y\\|^2)", "justification": "common witness iff equality", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\operatorname{Re} \\\\langle x, y \\\\rangle = \\\\frac{1}{2} (\\\\|x + y\\\\|^2 - \\\\|x\\\\|^2 - \\\\|y\\\\|^2).\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_e2aa4bf92abc3fdc", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e5c2ac94a3ec22da", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "enumerative combinatorics", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7EDAF21632BF", "name": "Labelled enumeration theorem", "statement_latex": "F(z,t)=e^{g(z)t}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((F(z,t)),z):z\\in D\\}=\\{((e^{g(z)t}),z):z\\in D\\}", "to_expression": "F(z,t)=e^{g(z)t}", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as F(z,t)=e^{g(z)t}.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_7cc0939406122ebc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_DAE7CEB541EE", "name": "Lusin's theorem", "statement_latex": "μ(E) > b - a - ε", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(b - a - ε)-(μ(E))\\}=0\\;\\land\\;(b - a - ε)\\ne (μ(E))", "to_expression": "μ(E) > b - a - ε", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as μ(E) > b - a - ε.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_dda6064ebb91af30", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "algebra", "subfield": "algebraic combinatorics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_396E0535E013", "name": "n! conjecture", "statement_latex": "dim(H_μ) = n!", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((dim(H_μ)),(n!))\\in\\{(z,z):z\\in D\\}", "to_expression": "dim(H_μ) = n!", "justification": "pair in diagonal iff X=Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as dim(H_μ) = n!.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_494838f73995c0c1", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_fffa2c2f5d8cf148", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "Number theory", "subfield": "Elliptic functions", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_78B717A820D4", "name": "Jacobi triple product", "statement_latex": "product_{m=1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2}) = sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((product_{m=1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2})),(sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n}))\\in\\{(z,z):z\\in D\\}", "to_expression": "product_{m=1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2}) = sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n}", "justification": "pair in diagonal iff X=Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as product_{m=1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2}) = sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n}.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
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{"schema_version": "1.0", "item_id": "rcd2_cade97c75e131eb2", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_7b0732caa42abd70", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C95D5999DB6B", "name": "Nullity theorem", "statement_latex": "nullity A = nullity H", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(nullity A))+\\delta_D(z,(nullity H)))=0", "to_expression": "nullity A = nullity H", "justification": "zero loss iff some z equals both", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as nullity A = nullity H.\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_6808157837ffdef0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8CBF25034487", "name": "Heegner's lemma", "statement_latex": "y^{2}=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}) = (y^{2})", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover y^{2}=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "y^{2}=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover y^{2}=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_a070b3a2af167276", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "polynomial ring theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BF300C15A52D", "name": "Gauss's lemma (polynomials)", "statement_latex": "F(X) = a_0 + a_1X + ... + a_nX^n", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(a_0 + a_1X + ... + a_nX^n) = (F(X))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover F(X) = a_0 + a_1X + ... + a_nX^n.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "F(X) = a_0 + a_1X + ... + a_nX^n", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover F(X) = a_0 + a_1X + ... + a_nX^n.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_9e2e3da6d990a813", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "approximation theory", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F6B5A5657F99", "name": "Fejér's theorem", "statement_latex": "lim_{n→∞} σ_n(f,x) = f(x)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(lim_{n→∞} σ_n(f,x))=z\\;\\land\\;z=(f(x))", "to_expression": "lim_{n→∞} σ_n(f,x) = f(x)", "justification": "common witness iff equality", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{n→∞} σ_n(f,x) = f(x).\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_de5b3e7d4334c211", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "Sobolev spaces", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_35353A67F17E", "name": "Pólya–Szegő inequality", "statement_latex": "∫ |∇u*|^p dx <= ∫ |∇u|^p dx", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(∫ |∇u|^p dx)\\le t\\Rightarrow (∫ |∇u*|^p dx)\\le t", "to_expression": "∫ |∇u*|^p dx <= ∫ |∇u|^p dx", "justification": "converse uses t=Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∫ |∇u*|^p dx ≤ ∫ |∇u|^p dx.\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_13e6cf76e18a8f90", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "physics", "subfield": "quantum field theory", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_3750474a51d76f54", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "design theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_41032C4E68AC", "name": "Bruck–Ryser–Chowla theorem", "statement_latex": "x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2} = 0", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "(\\{(x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2})\\}\\times\\{(0)\\})\\cap\\{(z,z):z\\in D\\}=\\{((x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2}),(0))\\}", "to_expression": "x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2} = 0", "justification": "intersection is singleton iff pair diagonal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as x^{2} - (k - \\\\lambda)y^{2} - (-1)^{(v-1)/2} \\\\lambda z^{2} = 0.\"]"}], "source_equivalence_identity": "intersection is singleton iff pair diagonal"}
{"schema_version": "1.0", "item_id": "rcd2_404492ef3627b02a", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis, measure theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2ccc162e9253b2c5", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "measure theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e1fec499524060e9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_45A3C68C1B71", "name": "Linnik's theorem", "statement_latex": "p(a,d) < c * d^L", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (p(a,d)) < x \\land x < (c * d^L)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover p(a,d) < c * d^L.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover p(a,d) < c * d^L.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "p(a,d) < c * d^L", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover p(a,d) < c * d^L.\"]"}], "source_equivalence_identity": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B."}
{"schema_version": "1.0", "item_id": "rcd2_0c929cee094cea58", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "approximation theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F6B5A5657F99", "name": "Fejér's theorem", "statement_latex": "lim_{n→∞} σ_n(f,x) = f(x)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(lim_{n→∞} σ_n(f,x))\\}=\\{(f(x))\\}", "to_expression": "lim_{n→∞} σ_n(f,x) = f(x)", "justification": "singletons equal iff elements equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{n→∞} σ_n(f,x) = f(x).\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_6415491885836691", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A00EEAE41A77", "name": "Intermediate value theorem", "statement_latex": "f(x) = s", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((f(x)),(s))=0", "to_expression": "f(x) = s", "justification": "zero discrete distance iff equality", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f(x) = s.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_bb07a0d78b828d25", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "additive number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_af3a3c04798ee679", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "elementary number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_677EBF92D0D7", "name": "Lifting-the-exponent lemma", "statement_latex": "nu_p(x^n - y^n) = nu_p(x - y) + nu_p(n)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((nu_p(x^n - y^n)),(nu_p(x - y) + nu_p(n)))=0", "to_expression": "nu_p(x^n - y^n) = nu_p(x - y) + nu_p(n)", "justification": "zero discrete distance iff equality", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as nu_p(x^n - y^n) = nu_p(x - y) + nu_p(n).\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_4d200d2b3fdbde4f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic geometry", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c975bcb15daabdbe", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F902B76211EA", "name": "Cantelli's inequality", "statement_latex": "Pr(X - E[X] ≥ λ) ≤ σ^2 / (σ^2 + λ^2)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(Pr(X - E[X] ≥ λ) ≤ σ^2 / (σ^2 + λ^2)) \\lor (0 = 1)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "0=1 is false, so Phi_T or false iff Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "Pr(X - E[X] ≥ λ) ≤ σ^2 / (σ^2 + λ^2)", "justification": "0=1 is false, so Phi_T or false iff Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "0=1 is false, so Phi_T or false iff Phi_T."}
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{"schema_version": "1.0", "item_id": "rcd2_adc8358cfd0a8e84", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "discrete mathematics", "subfield": "combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AE455D38D038", "name": "Bregman–Minc inequality", "statement_latex": "per(A) <= prod_{i=1}^n (r_i!)^(1/r_i)", "assumptions": "A is an n by n 0-1 matrix; r_i is the sum of row i; terms with r_i=0 are handled by the standard zero-permanent convention."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(prod_{i=1}^n (r_i!)^(1/r_i)) - (per(A)) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "A is an n by n 0-1 matrix; r_i is the sum of row i; terms with r_i=0 are handled by the standard zero-permanent convention.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover per(A) <= prod_{i=1}^n (r_i!)^(1/r_i).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "A is an n by n 0-1 matrix; r_i is the sum of row i; terms with r_i=0 are handled by the standard zero-permanent convention.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover per(A) <= prod_{i=1}^n (r_i!)^(1/r_i).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "per(A) <= prod_{i=1}^n (r_i!)^(1/r_i)", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "A is an n by n 0-1 matrix; r_i is the sum of row i; terms with r_i=0 are handled by the standard zero-permanent convention.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover per(A) <= prod_{i=1}^n (r_i!)^(1/r_i).\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_d4673509932cb4e0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "p-adic analysis", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3B924C720B22", "name": "Krasner's lemma", "statement_latex": "|alpha - beta| < |alpha - alpha_i|", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;t\\le (|alpha - beta|)\\Rightarrow t<(|alpha - alpha_i|)", "to_expression": "|alpha - beta| < |alpha - alpha_i|", "justification": "converse uses t=X", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |alpha - beta| < |alpha - alpha_i|.\"]"}], "source_equivalence_identity": "converse uses t=X"}
{"schema_version": "1.0", "item_id": "rcd2_3269b9002160d6f7", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e3f8312cec75081f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_95775DC75EC2", "name": "Oppenheim conjecture", "statement_latex": "|Q(x)| < \\epsilon", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;t\\le (|Q(x)|)\\Rightarrow t<(\\epsilon)", "to_expression": "|Q(x)| < \\epsilon", "justification": "converse uses t=X", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |Q(x)| < \\\\epsilon.\"]"}], "source_equivalence_identity": "converse uses t=X"}
{"schema_version": "1.0", "item_id": "rcd2_e273e50a5e56f4d5", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "functional analysis", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1D2F074AF9A8", "name": "Spectral theorem", "statement_latex": "Ax = λx", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(Ax))+\\delta_D(z,(λx)))=0", "to_expression": "Ax = λx", "justification": "zero loss iff some z equals both", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Ax = λx.\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_3a61226e04116575", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "statistical mechanics", "subfield": "non-equilibrium thermodynamics", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_84998867e4a3d538", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal combinatorics", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_768aac7be8f4c363", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AD101C6FC364", "name": "Adiabatic theorem", "statement_latex": "E = nhv", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((E),(nhv))\\in\\{(z,z):z\\in D\\}", "to_expression": "E = nhv", "justification": "pair in diagonal iff X=Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as E = nhv.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_a796f3c6ecab9ac6", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "approximation theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_bda5136915527a43", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "vector calculus", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2E42699EEB3D", "name": "Gradient theorem", "statement_latex": "∫_{γ} ∇φ(r) · dr = φ(q) - φ(p)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(∫_{γ} ∇φ(r) · dr)\\Longleftrightarrow z=(φ(q) - φ(p)))", "to_expression": "∫_{γ} ∇φ(r) · dr = φ(q) - φ(p)", "justification": "same equality fiber iff equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∫_{γ} ∇φ(r) · dr = φ(q) - φ(p).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_27e07fcd1cdc9d62", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "discrete mathematics", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_bdae8ff794ea0d25", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "martingale theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_1dd11f73b8117925", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "measure theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_16dc5d43d3c5ec9e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "permutation patterns", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8A6780273F9B", "name": "Stanley–Wilf conjecture", "statement_latex": "lim_{n→∞} |S_n(β)|^(1/n)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\varepsilon>0}\\inf_N\\mathbf{1}_{\\{\\forall n\\ge N:d((lim_{n),(∞} |S_n(β)|^(1/n)))<\\varepsilon\\}}=1", "to_expression": "lim_{n→∞} |S_n(β)|^(1/n)", "justification": "value one iff every epsilon has tail", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{n→∞} |S_n(β)|^(1/n).\"]"}], "source_equivalence_identity": "value one iff every epsilon has tail"}
{"schema_version": "1.0", "item_id": "rcd2_99560e698f57ab3a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "inequalities", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A3A584B25561", "name": "Ky Fan inequality", "statement_latex": "(∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n} ≤ (∑ x_i) / (∑ (1-x_i))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(∑ x_i) / (∑ (1-x_i))\\le t\\Rightarrow ((∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n})\\le t", "to_expression": "(∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n} ≤ (∑ x_i) / (∑ (1-x_i))", "justification": "converse uses t=Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n} ≤ (∑ x_i) / (∑ (1-x_i)).\"]"}], "source_equivalence_identity": "converse uses t=Y"}
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{"schema_version": "1.0", "item_id": "rcd2_b3880f3d52b86c85", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "computer science", "subfield": "probabilistic combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AA4BF27A6AAD", "name": "Algorithmic Lovász local lemma", "statement_latex": "Pr[A] <= x(A) * Product_{B in Gamma(A)} (1 - x(B))", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(x(A) * Product_{B \\in Gamma(A)} (1 - x(B))) - (Pr[A]) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Pr[A] <= x(A) * Product_{B in Gamma(A)} (1 - x(B)).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Pr[A] <= x(A) * Product_{B in Gamma(A)} (1 - x(B)).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "Pr[A] <= x(A) * Product_{B in Gamma(A)} (1 - x(B))", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Pr[A] <= x(A) * Product_{B in Gamma(A)} (1 - x(B)).\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_f533324a6de5e222", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "Kleinian groups", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C0AAA35160EF", "name": "Jørgensen's inequality", "statement_latex": "|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2| >= 1", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(1)}{\\exp(1)+\\exp(|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2|)}\\le\\frac{1}{2}", "to_expression": "|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2| >= 1", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2| ≥ 1.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_c3fc3f6bd5b96e11", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "physics", "subfield": "quantum field theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5C335B571C48", "name": "Gell-Mann and Low theorem", "statement_latex": "H = H_0 + gV", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((H) \\in S \\land (H_0 + gV) \\notin S) \\lor ((H_0 + gV) \\in S \\land (H) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover H = H_0 + gV.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "H = H_0 + gV", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover H = H_0 + gV.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_b30a476e59a213e1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B63439ADF46F", "name": "Rising sun lemma", "statement_latex": "E = \\bigcup_{k}(a_{k},b_{k})", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(E))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\bigcup_{k}(a_{k},b_{k})))", "to_expression": "E = \\bigcup_{k}(a_{k},b_{k})", "justification": "each argmin set is a singleton", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as E = \\\\bigcup_{k}(a_{k},b_{k}).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_98ecc5ef9c776bd2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "combinatorics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_51200980F031", "name": "Star of David theorem", "statement_latex": "gcd{(n-1 k-1), (n k+1), (n+1 k)} = gcd{(n-1 k), (n k-1), (n+1 k+1)}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((gcd{(n-1 k-1), (n k+1), (n+1 k)}) \\in S \\land (gcd{(n-1 k), (n k-1), (n+1 k+1)}) \\notin S) \\lor ((gcd{(n-1 k), (n k-1), (n+1 k+1)}) \\in S \\land (gcd{(n-1 k-1), (n k+1), (n+1 k)}) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover gcd{(n-1 k-1), (n k+1), (n+1 k)} = gcd{(n-1 k), (n k-1), (n+1 k+1)}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "gcd{(n-1 k-1), (n k+1), (n+1 k)} = gcd{(n-1 k), (n k-1), (n+1 k+1)}", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover gcd{(n-1 k-1), (n k+1), (n+1 k)} = gcd{(n-1 k), (n k-1), (n+1 k+1)}.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_282451c1b462ee09", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "representation theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e173c5769f0bed8d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "inclusion-exclusion principle", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_58BBE6141124", "name": "Schuette–Nesbitt formula", "statement_latex": "sum_{k=0}^m c_k 1_{N=k} = sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{k=0}^m c_k 1_{N=k}))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k))", "to_expression": "sum_{k=0}^m c_k 1_{N=k} = sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k", "justification": "each argmin set is a singleton", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{k=0}^m c_k 1_{N=k} = sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_1fe19d54e14a0898", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ceeabe31a7bce07a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "computer science", "subfield": "formal language theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_37D96EA4CFB7", "name": "Pumping lemma for regular languages", "statement_latex": "w = xyz", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((w))=F((xyz))", "to_expression": "w = xyz", "justification": "indicator of {X} separates unequal objects", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as w = xyz.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
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{"schema_version": "1.0", "item_id": "rcd2_7603b65c7e27dce9", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis, measure theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_5a220b7cfa7fa281", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "statistics", "subfield": "empirical Bayes methods", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_38e0732ea820b800", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "measure theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4450EEEA850", "name": "Borel–Cantelli lemma", "statement_latex": "limsup_{n->infinity} E_n = intersection_{n=1 to infinity} union_{k=n to infinity} E_k", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\infty} E_n = intersection_{n=1 to \\infty} union_{k=n to \\infty} E_k) - (limsup_{n-) < 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A-B<0 iff A<B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover limsup_{n- > infinity} E_n = intersection_{n=1 to infinity} union_{k=n to infinity} E_k.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "A-B<0 iff A<B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover limsup_{n- > infinity} E_n = intersection_{n=1 to infinity} union_{k=n to infinity} E_k.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "limsup_{n->infinity} E_n = intersection_{n=1 to infinity} union_{k=n to infinity} E_k", "justification": "A-B<0 iff A<B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover limsup_{n- > infinity} E_n = intersection_{n=1 to infinity} union_{k=n to infinity} E_k.\"]"}], "source_equivalence_identity": "A-B<0 iff A<B."}
{"schema_version": "1.0", "item_id": "rcd2_a88011a8227e86d3", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "combinatorics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_55cc7a0478b4614a", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum field theory", "subfield": "axiomatic quantum field theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_01a5ec8a733c6baf", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_41972AC1B752", "name": "Rothe–Hagen identity", "statement_latex": "\\sum _{k=0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k}={\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\sum _{k=0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k}))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,({\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n}))", "to_expression": "\\sum _{k=0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k}={\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n}", "justification": "each argmin set is a singleton", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\sum _{k=0}^{n}{\\\\frac {x}{x+kz}}{x+kz \\\\choose k}{\\\\frac {y}{y+(n-k)z}}{y+(n-k)z \\\\choose n-k}={\\\\frac {x+y}{x+y+nz}}{x+y+nz \\\\choose n}.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_49bfa0ea95f38b85", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "orthogonal polynomials", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_85f2b7fb846a1e3f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AE174D33AD68", "name": "Eaton's inequality", "statement_latex": "P(|sum_{i=1}^n a_i X_i| >= k) <= 2 inf_{0 <= c <= k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(P(|sum_{i=1}^n a_i X_i| ≥ k))-(2 inf_{0 ≤ c ≤ k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz)\\}=0", "to_expression": "P(|sum_{i=1}^n a_i X_i| >= k) <= 2 inf_{0 <= c <= k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz", "justification": "positive part vanishes iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(|sum_{i=1}^n a_i X_i| ≥ k) ≤ 2 inf_{0 ≤ c ≤ k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz.\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_a52b6da84ee4e45d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "functional analysis", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_391d8faac86ec551", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_42ED352AD252", "name": "Bernstein inequalities", "statement_latex": "P(|(1/n) * sum_{i=1}^n X_i| > epsilon) <= 2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3)))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3))))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (P(|(1/n) * sum_{i=1}^n X_i| > epsilon))\\}", "to_expression": "P(|(1/n) * sum_{i=1}^n X_i| > epsilon) <= 2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3)))", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(|(1/n) * sum_{i=1}^n X_i| > epsilon) ≤ 2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3))).\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_f4958c1685c71da9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "monotone", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1C2EEE35D0F6", "name": "Fischer's inequality", "statement_latex": "det(M) <= det(A)*det(C)", "assumptions": "M is positive semidefinite and block-partitioned with principal blocks A and C."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\arctan(det(M))\\le\\arctan(det(A)*det(C))", "to_expression": "det(M) <= det(A)*det(C)", "justification": "arctan preserves and reflects order", "preconditions": "M is positive semidefinite and block-partitioned with principal blocks A and C.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as det(M) ≤ det(A)*det(C).\"]"}], "source_equivalence_identity": "arctan preserves and reflects order"}
{"schema_version": "1.0", "item_id": "rcd2_b0dd3b915de94d37", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "linear algebra", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0038d302c8389325", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a224e8b53c19b83a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "algebraic K-theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_582614A6491A", "name": "Whitehead's lemma", "statement_latex": "[[u, 0], [0, u^-1]] = e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{(([[u, 0], [0, u^-1]]),z):z\\in D\\}=\\{((e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)),z):z\\in D\\}", "to_expression": "[[u, 0], [0, u^-1]] = e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as [[u, 0], [0, u^-1]] = e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1).\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_c2f396de0c53428d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_bc4e74f9d8a7e259", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "algebraic combinatorics", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_91e8e8ff50e8990b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_23d9eadad982bcc8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "polynomial ring theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BF300C15A52D", "name": "Gauss's lemma (polynomials)", "statement_latex": "F(X) = a_0 + a_1X + ... + a_nX^n", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((F(X)) \\in S \\land (a_0 + a_1X + ... + a_nX^n) \\notin S) \\lor ((a_0 + a_1X + ... + a_nX^n) \\in S \\land (F(X)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover F(X) = a_0 + a_1X + ... + a_nX^n.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "F(X) = a_0 + a_1X + ... + a_nX^n", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover F(X) = a_0 + a_1X + ... + a_nX^n.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_dcf7226473eb7275", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_3d052674afea3d0c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "Sobolev spaces", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_7f75e07c198829ec", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "combinatorics", "primary_mechanism": "distance", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_843AE41690C1", "name": "q-Vandermonde identity", "statement_latex": "binom(m+n, k)_q = sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists c \\in D: (binom(m+n, k)_q) = c \\land (sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)}) = c", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A common c exists iff L and R are equal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover binom(m+n, k)_q = sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "binom(m+n, k)_q = sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)}", "justification": "A common c exists iff L and R are equal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover binom(m+n, k)_q = sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)}.\"]"}], "source_equivalence_identity": "A common c exists iff L and R are equal."}
{"schema_version": "1.0", "item_id": "rcd2_bf54e7e1851c563b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_90d48bcb975e08ad", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "measure theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c99f5649989aada9", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum field theory", "subfield": "lattice field theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_4249c824540c2276", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "polyhedral combinatorics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_687D9EB61DA6", "name": "Eberhard's theorem", "statement_latex": "sum (6-i)p_i = 12", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((sum (6-i)p_i) \\in S \\land (12) \\notin S) \\lor ((12) \\in S \\land (sum (6-i)p_i) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover sum (6-i)p_i = 12.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "sum (6-i)p_i = 12", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover sum (6-i)p_i = 12.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_113bcf81c3648a06", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "recreational mathematics", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_96F33F66821C", "name": "Tupper's self-referential formula", "statement_latex": "1/2 < floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2))", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(1/2)-(floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2)))\\}=0\\;\\land\\;(1/2)\\ne (floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2)))", "to_expression": "1/2 < floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2))", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as 1/2 < floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2)).\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_e7235ceb938ed6dc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "statistics", "subfield": "statistical hypothesis testing", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_69B4EB18D461", "name": "Neyman–Pearson lemma", "statement_latex": "α = Pr_θ0(X ∈ R)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(α = Pr_θ0(X ∈ R)) \\land (0 = 0)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "α = Pr_θ0(X ∈ R)", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "0=0 is true, so Phi_T and true iff Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_5938d37836ca37f5", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_99B889CC4005", "name": "Hoeffding's lemma", "statement_latex": "E[exp(lambda*(X-E[X]))] <= exp(lambda^2*(b-a)^2/8)", "assumptions": "X is almost surely in [a,b]; lambda is real; expectations are finite."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (E[\\exp(lambda*(X-E[X]))]) \\le x \\land x \\le (\\exp(lambda^2*(b-a)^2/8))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "X is almost surely in [a,b]; lambda is real; expectations are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover E[exp(lambda*(X-E[X]))] <= exp(lambda^2*(b-a)^2/8).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "X is almost surely in [a,b]; lambda is real; expectations are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover E[exp(lambda*(X-E[X]))] <= exp(lambda^2*(b-a)^2/8).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "E[exp(lambda*(X-E[X]))] <= exp(lambda^2*(b-a)^2/8)", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "X is almost surely in [a,b]; lambda is real; expectations are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover E[exp(lambda*(X-E[X]))] <= exp(lambda^2*(b-a)^2/8).\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_af47569c9f42aeb9", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal set theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6e8e2e3717c1d9d9", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "statistics", "subfield": "empirical Bayes methods", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_de6198d4ab81cc34", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "point processes", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C81FC5D06BF2", "name": "Campbell's theorem", "statement_latex": "E[sum_{x in N} f(x)] = integral f(x) dLambda(x)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(integral f(x) dLambda(x)) = (E[sum_{x \\in N} f(x)])", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E[sum_{x in N} f(x)] = integral f(x) dLambda(x).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "E[sum_{x in N} f(x)] = integral f(x) dLambda(x)", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E[sum_{x in N} f(x)] = integral f(x) dLambda(x).\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_a294ca1d5b1a95e9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_37DDB6F9F636", "name": "Duffin–Schaeffer theorem", "statement_latex": "\\sum_{q=1}^{\\infty}\\varphi(q)\\frac{f(q)}{q}=\\infty", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((\\sum_{q=1}^{\\infty}\\varphi(q)\\frac{f(q)}{q}),z)=((\\infty),z)", "to_expression": "\\sum_{q=1}^{\\infty}\\varphi(q)\\frac{f(q)}{q}=\\infty", "justification": "pair equality forces first coordinates equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\sum_{q=1}^{\\\\infty}\\\\varphi(q)\\\\frac{f(q)}{q}=\\\\infty.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_aa68eaee28e257c4", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "geometry", "subfield": "discrete geometry", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f9a6d103ff491438", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "design theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_45A9C577BD36", "name": "Fisher's inequality", "statement_latex": "b >= v", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(v)}{\\exp(v)+\\exp(b)}\\le\\frac{1}{2}", "to_expression": "b >= v", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as b ≥ v.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_480f1394ca8c8975", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "numerical analysis", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_DC75D2014574", "name": "Céa's lemma", "statement_latex": "||u - u_h|| <= (gamma / alpha) * ||u - v||", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(||u - u_h||)}{\\exp(||u - u_h||)+\\exp((gamma / alpha) * ||u - v||)}\\le\\frac{1}{2}", "to_expression": "||u - u_h|| <= (gamma / alpha) * ||u - v||", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ||u - u_h|| ≤ (gamma / alpha) * ||u - v||.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_039661508b85f17a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "combinatorics", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_684A9634E4F6", "name": "Hypergeometric identity", "statement_latex": "sum_{i=0}^n binom(n,i) = 2^n", "assumptions": "n is a nonnegative integer."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((sum_{i=0}^n binom(n,i)),z):z\\in D\\}=\\{((2^n),z):z\\in D\\}", "to_expression": "sum_{i=0}^n binom(n,i) = 2^n", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "n is a nonnegative integer.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{i=0}^n binom(n,i) = 2^n.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_f6bb5195e6ddc2a2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "graph theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_07CDE34AC57A", "name": "Counting lemma", "statement_latex": "γ <= ε^h / 4h", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (γ) \\le x \\land x \\le (ε^h / 4h)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover γ <= ε^h / 4h.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover γ <= ε^h / 4h.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "γ <= ε^h / 4h", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover γ <= ε^h / 4h.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_ebb4abf3d4b401c8", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_550231221e61d08d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4CEAA916D1E3", "name": "Riemann–Siegel formula", "statement_latex": "ζ(s) = Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(ζ(s)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s)))", "to_expression": "ζ(s) = Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s)", "justification": "each argmin set is a singleton", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ζ(s) = Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_1dba66bcc7de73a9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2DC4FABA16C6", "name": "Kantorovich inequality", "statement_latex": "(Σ p_i x_i) * (Σ p_i / x_i) <= (a+b)^2 / (4ab)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(Σ p_i x_i) * (Σ p_i / x_i)}{\\exp(Σ p_i x_i) * (Σ p_i / x_i)+\\exp(a+b)^2 / (4ab)}\\le\\frac{1}{2}", "to_expression": "(Σ p_i x_i) * (Σ p_i / x_i) <= (a+b)^2 / (4ab)", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (Σ p_i x_i) * (Σ p_i / x_i) ≤ (a+b)^2 / (4ab).\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_1169e45cfc979067", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "classical mechanics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BC55A9432F80", "name": "Helmholtz theorem (classical mechanics)", "statement_latex": "dS = (dE + PdV) / T", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((dS) \\in S \\land ((dE + PdV) / T) \\notin S) \\lor (((dE + PdV) / T) \\in S \\land (dS) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover dS = (dE + PdV) / T.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "dS = (dE + PdV) / T", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover dS = (dE + PdV) / T.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_2638ad9acc58fc85", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "design theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_45A9C577BD36", "name": "Fisher's inequality", "statement_latex": "b >= v", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(b) - (v) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover b >= v.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover b >= v.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "b >= v", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover b >= v.\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_54027e68dd6beac6", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematical analysis", "subfield": "analytic number theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_5942feebfe8ee762", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "functional analysis", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_8d65e076da662b72", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "discrepancy theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_656862810de393d1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "Kleinian groups", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C0AAA35160EF", "name": "Jørgensen's inequality", "statement_latex": "|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2| >= 1", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2|)\\le t\\Rightarrow (1)\\le t", "to_expression": "|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2| >= 1", "justification": "converse uses t=Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2| ≥ 1.\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_1d82a62747df7f85", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "complex analysis", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_62a87da7b06b464b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_14ba6b9bc13226a4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "measure theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_448CEF326F89", "name": "Doob–Dynkin lemma", "statement_latex": "f = g ∘ T", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(g ∘ T) = (f)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f = g ∘ T.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f = g ∘ T", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f = g ∘ T.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_a017934a6be66995", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "vector calculus", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_6CFB1B566265", "name": "Stokes' theorem", "statement_latex": "integral integral_{Sigma} (curl F) dot dSigma = integral_{partial Sigma} F dot dr", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((integral integral_{Sigma} (curl F) dot dSigma),(integral_{partial Sigma} F dot dr))\\in\\{(z,z):z\\in D\\}", "to_expression": "integral integral_{Sigma} (curl F) dot dSigma = integral_{partial Sigma} F dot dr", "justification": "pair in diagonal iff X=Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as integral integral_{Sigma} (curl F) dot dSigma = integral_{partial Sigma} F dot dr.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_6591819bbda0c8e2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "combinatorics", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A5B93C0856B4", "name": "Handshaking lemma", "statement_latex": "Σ_{v∈V} deg(v) = 2|E|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((Σ_{v∈V} deg(v)),z)=((2|E|),z)", "to_expression": "Σ_{v∈V} deg(v) = 2|E|", "justification": "pair equality forces first coordinates equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Σ_{v∈V} deg(v) = 2|E|.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_95f5b93ac91cd531", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_89D2662A27BE", "name": "Freiman's theorem", "statement_latex": "|A+A| <= K|A|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(K|A|) - (|A+A|) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |A+A| <= K|A|.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |A+A| <= K|A|.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "|A+A| <= K|A|", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |A+A| <= K|A|.\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_1aff73574afd9168", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9EF5E48E9DAF", "name": "Rolle's theorem", "statement_latex": "f'(c) = 0", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((f'(c)),z):z\\in D\\}=\\{((0),z):z\\in D\\}", "to_expression": "f'(c) = 0", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f'(c) = 0.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
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{"schema_version": "1.0", "item_id": "rcd2_d0d584207fee9601", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "elementary number theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_677EBF92D0D7", "name": "Lifting-the-exponent lemma", "statement_latex": "nu_p(x^n - y^n) = nu_p(x - y) + nu_p(n)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(nu_p(x^n - y^n))\\}=\\{(nu_p(x - y) + nu_p(n))\\}", "to_expression": "nu_p(x^n - y^n) = nu_p(x - y) + nu_p(n)", "justification": "singletons equal iff elements equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as nu_p(x^n - y^n) = nu_p(x - y) + nu_p(n).\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_a7adc8be532ca95a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "functional analysis", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1D2F074AF9A8", "name": "Spectral theorem", "statement_latex": "Ax = λx", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((Ax),z):z\\in D\\}=\\{((λx),z):z\\in D\\}", "to_expression": "Ax = λx", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Ax = λx.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_c8c8b5cd5edab309", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0FE477D29636", "name": "Hardy–Ramanujan theorem", "statement_latex": "|ω(n)−loglogn|<(loglogn)^(1/2+ε)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(|ω(n)−loglogn|)-(loglogn)^(1/2+ε)\\}=0\\;\\land\\;(|ω(n)−loglogn|)\\ne (loglogn)^(1/2+ε)", "to_expression": "|ω(n)−loglogn|<(loglogn)^(1/2+ε)", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |ω(n)−loglogn|<(loglogn)^(1/2+ε).\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
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{"schema_version": "1.0", "item_id": "rcd2_f935bf2164e83505", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "complex analysis", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_3b5e403ccbd3af35", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "enumerative combinatorics", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E1416BDA0AD6", "name": "Pólya enumeration theorem", "statement_latex": "|Y^X/G| = (1/|G|) * Σ_{g∈G} m^{c(g)}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((|Y^X/G|),z)=(((1/|G|) * Σ_{g∈G} m^{c(g)}),z)", "to_expression": "|Y^X/G| = (1/|G|) * Σ_{g∈G} m^{c(g)}", "justification": "pair equality forces first coordinates equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |Y^X/G| = (1/|G|) * Σ_{g∈G} m^{c(g)}.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_6448ba8ef14df525", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "graph theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_7eb2d5df4f9c574b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "differential equations", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_891C2603FAE3", "name": "Grönwall's inequality", "statement_latex": "u(t) <= u(a) * exp(integral from a to t of beta(s) ds)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(u(a) * exp(integral from a to t of beta(s) ds))\\le t\\Rightarrow (u(t))\\le t", "to_expression": "u(t) <= u(a) * exp(integral from a to t of beta(s) ds)", "justification": "converse uses t=Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as u(t) ≤ u(a) * exp(integral from a to t of beta(s) ds).\"]"}], "source_equivalence_identity": "converse uses t=Y"}
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{"schema_version": "1.0", "item_id": "rcd2_0d75835b742e97e7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "geometry", "subfield": "Euclidean geometry", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F38D8FABD394", "name": "Ptolemy's inequality", "statement_latex": "AB * CD + BC * DA >= AC * BD", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(AC * BD)}{\\exp(AC * BD)+\\exp(AB * CD + BC * DA)}\\le\\frac{1}{2}", "to_expression": "AB * CD + BC * DA >= AC * BD", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as AB * CD + BC * DA ≥ AC * BD.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_6697df450bab002f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "topology", "subfield": "algebraic topology", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_48CD393CFABB", "name": "Borsuk–Ulam theorem", "statement_latex": "f(-x) = f(x)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((f(-x)),(f(x)))=0", "to_expression": "f(-x) = f(x)", "justification": "zero discrete distance iff equality", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f(-x) = f(x).\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_facc11731b089854", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "numerical linear algebra", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FBD08849D0F3", "name": "Kreiss matrix theorem", "statement_latex": "K(A) = sup_{|z|>1} (|z|-1) ||(z-A)^-1||", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (1} (|z|-1) ||(z-A)^-1||) < x \\land x < (K(A) = sup_{|z|)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover K(A) = sup_{|z| > 1} (|z|-1) ||(z-A)^-1||.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover K(A) = sup_{|z| > 1} (|z|-1) ||(z-A)^-1||.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "K(A) = sup_{|z|>1} (|z|-1) ||(z-A)^-1||", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover K(A) = sup_{|z| > 1} (|z|-1) ||(z-A)^-1||.\"]"}], "source_equivalence_identity": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B."}
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{"schema_version": "1.0", "item_id": "rcd2_da6e0955c4d2c33a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "modular arithmetic", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_6B373783C03D", "name": "Thue's lemma", "statement_latex": "ay ≡ x (mod m)", "assumptions": "All displayed arithmetic expressions are integers; the modulus is a positive integer."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "(m)\\mid((ay)-(x))", "to_expression": "ay ≡ x (mod m)", "justification": "A≡B mod M iff M divides A-B", "preconditions": "All displayed arithmetic expressions are integers; the modulus is a positive integer.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ay ≡ x (mod m).\"]"}], "source_equivalence_identity": "A≡B mod M iff M divides A-B"}
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{"schema_version": "1.0", "item_id": "rcd2_56c7bd2ac3fdc710", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "inequalities", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A3A584B25561", "name": "Ky Fan inequality", "statement_latex": "(∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n} ≤ (∑ x_i) / (∑ (1-x_i))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\le ((∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n})\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (∑ x_i) / (∑ (1-x_i))\\}", "to_expression": "(∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n} ≤ (∑ x_i) / (∑ (1-x_i))", "justification": "lower-cut inclusion iff X≤Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (∏ x_i)^{1/n} / (∏ (1-x_i))^{1/n} ≤ (∑ x_i) / (∑ (1-x_i)).\"]"}], "source_equivalence_identity": "lower-cut inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_56258ee2d948cb5c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_736807BC5C8F", "name": "Brun–Titchmarsh theorem", "statement_latex": "pi(x;q,a) <= 2x / (phi(q) log(x/q))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(2x / (phi(q) log(x/q)))\\le t\\Rightarrow (pi(x;q,a))\\le t", "to_expression": "pi(x;q,a) <= 2x / (phi(q) log(x/q))", "justification": "converse uses t=Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as pi(x;q,a) ≤ 2x / (phi(q) log(x/q)).\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_ce7fb3e2aeb2ba7a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "infinite-dimensional vector spaces", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B23BCA4AB254", "name": "Erdős–Kaplansky theorem", "statement_latex": "\\dim(E^*) = \\text{card}(\\mathbb{K}^I)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((\\dim(E^*)))=F((\\text{card}(\\mathbb{K}^I)))", "to_expression": "\\dim(E^*) = \\text{card}(\\mathbb{K}^I)", "justification": "indicator of {X} separates unequal objects", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\dim(E^*) = \\\\text{card}(\\\\mathbb{K}^I).\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_b2382b12ae637256", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "combinatorics on words", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3E0E814BA7D9", "name": "Levi's lemma", "statement_latex": "uv = xy", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((uv),z):z\\in D\\}=\\{((xy),z):z\\in D\\}", "to_expression": "uv = xy", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as uv = xy.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_37a962c23fc3058e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "Ramsey theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8E938CA76307", "name": "Milliken's tree theorem", "statement_latex": "S_T^n = C_1 ∪ ... ∪ C_r", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((S_T^n),(C_1 ∪ ... ∪ C_r))=0", "to_expression": "S_T^n = C_1 ∪ ... ∪ C_r", "justification": "zero discrete distance iff equality", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as S_T^n = C_1 ∪ ... ∪ C_r.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_1d6a497a1f20600d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_45A3C68C1B71", "name": "Linnik's theorem", "statement_latex": "p(a,d) < c * d^L", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;t\\le (p(a,d))\\Rightarrow t<(c * d^L)", "to_expression": "p(a,d) < c * d^L", "justification": "converse uses t=X", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p(a,d) < c * d^L.\"]"}], "source_equivalence_identity": "converse uses t=X"}
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{"schema_version": "1.0", "item_id": "rcd2_44e2d7aade481194", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "calculus", "subfield": "vector calculus / electrostatics", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D5081E95CCA1", "name": "Uniqueness theorem for Poisson's equation", "statement_latex": "∇^2φ = -ρ_f/ε_0", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(∇^2φ))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(-ρ_f/ε_0))", "to_expression": "∇^2φ = -ρ_f/ε_0", "justification": "each argmin set is a singleton", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∇^2φ = -ρ_f/ε_0.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
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{"schema_version": "1.0", "item_id": "rcd2_272310ea22e8e7b6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "Number theory", "subfield": "Elliptic functions", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_78B717A820D4", "name": "Jacobi triple product", "statement_latex": "product_{m=1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2}) = sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((product_{m=1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2})))=F((sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n}))", "to_expression": "product_{m=1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2}) = sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n}", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as product_{m=1}^{infinity} (1-x^{2m})(1+x^{2m-1}y^{2})(1+x^{2m-1}/y^{2}) = sum_{n=-infinity}^{infinity} x^{n^{2}}y^{2n}.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
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{"schema_version": "1.0", "item_id": "rcd2_a62d47290fda9323", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B63439ADF46F", "name": "Rising sun lemma", "statement_latex": "E = \\bigcup_{k}(a_{k},b_{k})", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((E),z)=((\\bigcup_{k}(a_{k},b_{k})),z)", "to_expression": "E = \\bigcup_{k}(a_{k},b_{k})", "justification": "pair equality forces first coordinates equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as E = \\\\bigcup_{k}(a_{k},b_{k}).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_cf3c6c030f638cdc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_773C0118E21F", "name": "Darboux's theorem", "statement_latex": "f'(x) = y", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((f'(x)))=F((y))", "to_expression": "f'(x) = y", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f'(x) = y.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
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{"schema_version": "1.0", "item_id": "rcd2_8d25a2c6a866c114", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "field theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_ED1CB49A03EE", "name": "Steinitz's theorem", "statement_latex": "L = K(α)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(L)\\Longleftrightarrow z=(K(α)))", "to_expression": "L = K(α)", "justification": "same equality fiber iff equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as L = K(α).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
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{"schema_version": "1.0", "item_id": "rcd2_f4008dd1c874b265", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "number theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A54D9D349D9E", "name": "Candido's identity", "statement_latex": "[x^2 + y^2 + (x+y)^2]^2 = 2[x^4 + y^4 + (x+y)^4]", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{(([x^2 + y^2 + (x+y)^2]^2),z):z\\in D\\}=\\{((2[x^4 + y^4 + (x+y)^4]),z):z\\in D\\}", "to_expression": "[x^2 + y^2 + (x+y)^2]^2 = 2[x^4 + y^4 + (x+y)^4]", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as [x^2 + y^2 + (x+y)^2]^2 = 2[x^4 + y^4 + (x+y)^4].\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_7954be269f3b5ee4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "geometry of numbers", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_638EC6734424", "name": "Minkowski–Hlawka theorem", "statement_latex": "Δ ≥ ζ(n) / 2^(n-1)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(Δ) - (ζ(n) / 2^(n-1)) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Δ >= ζ(n) / 2^(n-1).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Δ >= ζ(n) / 2^(n-1).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "Δ ≥ ζ(n) / 2^(n-1)", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Δ >= ζ(n) / 2^(n-1).\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_bbe29c857fb13bc5", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "numerical analysis", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_DC75D2014574", "name": "Céa's lemma", "statement_latex": "||u - u_h|| <= (gamma / alpha) * ||u - v||", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((||u - u_h||)-((gamma / alpha) * ||u - v||),(-\\infty,0])=0", "to_expression": "||u - u_h|| <= (gamma / alpha) * ||u - v||", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ||u - u_h|| ≤ (gamma / alpha) * ||u - v||.\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
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{"schema_version": "1.0", "item_id": "rcd2_0592dc6b1209164c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "matrix analysis", "subfield": "linear algebra", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_90db8f4b8cd03441", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_42ED352AD252", "name": "Bernstein inequalities", "statement_latex": "P(|(1/n) * sum_{i=1}^n X_i| > epsilon) <= 2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3)))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(P(|(1/n) * sum_{i=1}^n X_i| > epsilon))}{\\exp(P(|(1/n) * sum_{i=1}^n X_i| > epsilon))+\\exp(2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3))))}\\le\\frac{1}{2}", "to_expression": "P(|(1/n) * sum_{i=1}^n X_i| > epsilon) <= 2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3)))", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(|(1/n) * sum_{i=1}^n X_i| > epsilon) ≤ 2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3))).\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_e1f62a8836c2fd88", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "calculus", "subfield": "vector calculus / electrostatics", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D5081E95CCA1", "name": "Uniqueness theorem for Poisson's equation", "statement_latex": "∇^2φ = -ρ_f/ε_0", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(∇^2φ)\\Longleftrightarrow z=(-ρ_f/ε_0))", "to_expression": "∇^2φ = -ρ_f/ε_0", "justification": "same equality fiber iff equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∇^2φ = -ρ_f/ε_0.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_752d5512b51d29d8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematical analysis", "subfield": "analytic number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F4E26A602D8B", "name": "Perron's formula", "statement_latex": "A(x) = (1 / (2*pi*i)) * integral_{c-i*infinity}^{c+i*infinity} g(z) * (x^z / z) dz", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "((1 / (2*\\pi*i)) * integral_{c-i*\\infty}^{c+i*\\infty} g(z) * (x^z / z) dz) = (A(x))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover A(x) = (1 / (2*pi*i)) * integral_{c-i*infinity}^{c+i*infinity} g(z) * (x^z / z) dz.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "A(x) = (1 / (2*pi*i)) * integral_{c-i*infinity}^{c+i*infinity} g(z) * (x^z / z) dz", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover A(x) = (1 / (2*pi*i)) * integral_{c-i*infinity}^{c+i*infinity} g(z) * (x^z / z) dz.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
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{"schema_version": "1.0", "item_id": "rcd2_49468bf94f62a944", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "analysis", "subfield": "moment problem", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A82D2EB23B1F", "name": "Carleman's condition", "statement_latex": "\\sum _{n=1}^{\\infty }m_{2n}^{-{1/2n}}=+\\infty", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\neg (\\sum _{n=1}^{\\infty }m_{2n}^{-{1/2n}}=+\\infty)) \\Rightarrow (0 = 1)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "not Phi_T => false is not(not Phi_T), equivalent to Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\sum _{n=1}^{\\infty }m_{2n}^{-{1/2n}}=+\\infty", "justification": "not Phi_T => false is not(not Phi_T), equivalent to Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "not Phi_T => false is not(not Phi_T), equivalent to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_4524b4325ae105a1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "combinatorics", "subfield": "combinatorial geometry", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_23E48784FFFD", "name": "Littlewood–Offord problem", "statement_latex": "P(S_n in A) <= binom(n,floor(n/2))/2^n", "assumptions": "S_n is the signed sum in the Littlewood-Offord setup; A is an interval or target set as specified by the theorem; n is positive."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((P(S_n in A))-(binom(n,floor(n/2))/2^n),(-\\infty,0])=0", "to_expression": "P(S_n in A) <= binom(n,floor(n/2))/2^n", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "S_n is the signed sum in the Littlewood-Offord setup; A is an interval or target set as specified by the theorem; n is positive.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(S_n in A) ≤ binom(n,floor(n/2))/2^n.\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
{"schema_version": "1.0", "item_id": "rcd2_e72cee521c6b5b9e", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_8cd62835f45e2344", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "complex analysis", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2C71326A0896", "name": "Area theorem", "statement_latex": "\\sum_{n=0}^{\\infty} n|a_n|^2 \\leq 1", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (\\sum_{n=0}^{\\infty} n|a_n|^2) \\le x \\land x \\le (1)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover \\\\sum_{n=0}^{\\\\infty} n|a_n|^2 <= 1.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover \\\\sum_{n=0}^{\\\\infty} n|a_n|^2 <= 1.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "\\sum_{n=0}^{\\infty} n|a_n|^2 \\leq 1", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover \\\\sum_{n=0}^{\\\\infty} n|a_n|^2 <= 1.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_11008a8b84388ff6", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "graph theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d1f826006e82e7f1", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e5ccbbee5e1ae3bc", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "hypergraph theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_42eebf24fead3cea", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "Ramsey theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_05F0F9A888B8", "name": "Erdős–Szekeres theorem", "statement_latex": "n = (r - 1)(s - 1) + 1", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(n)\\Longleftrightarrow z=((r - 1)(s - 1) + 1))", "to_expression": "n = (r - 1)(s - 1) + 1", "justification": "same equality fiber iff equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as n = (r - 1)(s - 1) + 1.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_771da698e4d61f4c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "analysis", "subfield": "moment problem", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b2045d1659499f37", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "prime number theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_04257ed938782d63", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "modular arithmetic", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_30416e9cf4098f7c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_49710848aa230a7b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "additive number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_282ddd610c4121be", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "probability theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_27AD75237F2D", "name": "Entropy power inequality", "statement_latex": "N(X+Y) >= N(X) + N(Y)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((N(X) + N(Y))-(N(X+Y)))=0", "to_expression": "N(X+Y) >= N(X) + N(Y)", "justification": "sup equals max(0,X-Y)", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as N(X+Y) ≥ N(X) + N(Y).\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
{"schema_version": "1.0", "item_id": "rcd2_cc1d2f851135aec2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum field theory", "subfield": "lattice field theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7FD1BE25E27D", "name": "Nielsen–Ninomiya theorem", "statement_latex": "H = ∑_{x,y} ψ^†(x) F(x, y) ψ(y)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((H))=F((∑_{x,y} ψ^†(x) F(x, y) ψ(y)))", "to_expression": "H = ∑_{x,y} ψ^†(x) F(x, y) ψ(y)", "justification": "indicator of {X} separates unequal objects", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as H = ∑_{x,y} ψ^†(x) F(x, y) ψ(y).\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_1749c9a5a5fc0ce3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "diophantine approximation", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_023EE81526D0", "name": "Subspace theorem", "statement_latex": "|L_{1}(x)\\cdots L_{n}(x)|<|x|^{-\\epsilon }", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(|L_{1}(x)\\cdots L_{n}(x)|)-(|x|^{-\\epsilon })\\}=0\\;\\land\\;(|L_{1}(x)\\cdots L_{n}(x)|)\\ne (|x|^{-\\epsilon })", "to_expression": "|L_{1}(x)\\cdots L_{n}(x)|<|x|^{-\\epsilon }", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |L_{1}(x)\\\\cdots L_{n}(x)|<|x|^{-\\\\epsilon }.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_967048845364e1f4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3BD55A296126", "name": "Rank–nullity theorem", "statement_latex": "rank(T) + nullity(T) = dim(V)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "(\\{(rank(T) + nullity(T))\\}\\times\\{(dim(V))\\})\\cap\\{(z,z):z\\in D\\}=\\{((rank(T) + nullity(T)),(dim(V)))\\}", "to_expression": "rank(T) + nullity(T) = dim(V)", "justification": "intersection is singleton iff pair diagonal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as rank(T) + nullity(T) = dim(V).\"]"}], "source_equivalence_identity": "intersection is singleton iff pair diagonal"}
{"schema_version": "1.0", "item_id": "rcd2_c92c6d0ba29d8250", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "scattering theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0C5D6466FC1D", "name": "Levinson's theorem", "statement_latex": "φ(+∞) − φ(0) = −π(1/2*n0 + nb)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((φ(+∞) − φ(0)),z):z\\in D\\}=\\{((−π(1/2*n0 + nb)),z):z\\in D\\}", "to_expression": "φ(+∞) − φ(0) = −π(1/2*n0 + nb)", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as φ(+∞) − φ(0) = −π(1/2*n0 + nb).\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_11227f6cb3e3a4a3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "commutative algebra", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_666997CDC49C", "name": "Noether normalization lemma", "statement_latex": "A = k[y_1, ..., y_m]", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((A),(k[y_1, ..., y_m]))\\in\\{(z,z):z\\in D\\}", "to_expression": "A = k[y_1, ..., y_m]", "justification": "pair in diagonal iff X=Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as A = k[y_1, ..., y_m].\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_fdf7cd6192a674af", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "incidence geometry", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_1119025fad34acfb", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_736807BC5C8F", "name": "Brun–Titchmarsh theorem", "statement_latex": "pi(x;q,a) <= 2x / (phi(q) log(x/q))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(2x / (phi(q) log(x/q))) - (\\pi(x;q,a)) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover pi(x;q,a) <= 2x / (phi(q) log(x/q)).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover pi(x;q,a) <= 2x / (phi(q) log(x/q)).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "pi(x;q,a) <= 2x / (phi(q) log(x/q))", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover pi(x;q,a) <= 2x / (phi(q) log(x/q)).\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_34f76b32030fb837", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "elementary algebra", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_40DB1B003904", "name": "Binomial theorem", "statement_latex": "(x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg ((x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "(x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_83f1064588e7fe70", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "sieve theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5DC5BE909937", "name": "Fundamental lemma of sieve theory", "statement_latex": "|A_d| = \\frac{w(d)}{d}X + R_d", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(|A_d|))+\\delta_D(z,(\\frac{w(d)}{d}X + R_d)))=0", "to_expression": "|A_d| = \\frac{w(d)}{d}X + R_d", "justification": "zero loss iff some z equals both", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |A_d| = \\\\frac{w(d)}{d}X + R_d.\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_cb1990578508996c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "queueing theory", "subfield": "probability", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_91967E3D2C5F", "name": "Little's law", "statement_latex": "L = lambda*W", "assumptions": "The queueing system is stable and in steady state; L, lambda, and W are finite real quantities."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((L),z):z\\in D\\}=\\{((lambda*W),z):z\\in D\\}", "to_expression": "L = lambda*W", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "The queueing system is stable and in steady state; L, lambda, and W are finite real quantities.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as L = lambda*W.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_4d738a419cfcaf0d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ff62dfd60b68c406", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "percolation theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4AB97BAC2783", "name": "Van den Berg–Kesten inequality", "statement_latex": "P(A square B) <= P(A)P(B)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((P(A square B))-(P(A)P(B)))=0", "to_expression": "P(A square B) <= P(A)P(B)", "justification": "sup equals max(0,X-Y)", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(A square B) ≤ P(A)P(B).\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
{"schema_version": "1.0", "item_id": "rcd2_22d826af043cd6aa", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "large deviations theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_53869962DDDE", "name": "Laplace's principle", "statement_latex": "\\lim_{\\theta \\to \\infty}\\frac{1}{\\theta}\\log\\int_{A}e^{-\\theta\\varphi(x)}\\,dx=-\\mathop{\\mathrm{ess\\,inf}}_{x\\in A}\\varphi(x)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(\\lim_{\\theta → \\infty}\\frac{1}{\\theta}\\log\\int_{A}e^{-\\theta\\varphi(x)}\\,dx)\\Longleftrightarrow z=(-\\mathop{\\mathrm{ess\\,inf}}_{x\\in A}\\varphi(x)))", "to_expression": "\\lim_{\\theta \\to \\infty}\\frac{1}{\\theta}\\log\\int_{A}e^{-\\theta\\varphi(x)}\\,dx=-\\mathop{\\mathrm{ess\\,inf}}_{x\\in A}\\varphi(x)", "justification": "same equality fiber iff equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\lim_{\\\\theta → \\\\infty}\\\\frac{1}{\\\\theta}\\\\log\\\\int_{A}e^{-\\\\theta\\\\varphi(x)}\\\\,dx=-\\\\mathop{\\\\mathrm{ess\\\\,inf}}_{x\\\\in A}\\\\varphi(x).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_fb0e739aec0f1b3b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "algebra", "subfield": "polynomials", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_235D1B5AD510", "name": "Routh–Hurwitz theorem", "statement_latex": "p - q = (1/π) Δ arg f(iy)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((p - q) \\in S \\land ((1/π) Δ arg f(iy)) \\notin S) \\lor (((1/π) Δ arg f(iy)) \\in S \\land (p - q) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover p - q = (1/π) Δ arg f(iy).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "p - q = (1/π) Δ arg f(iy)", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover p - q = (1/π) Δ arg f(iy).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_a4f8ef602ee41c34", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2aa0f17770de0b8f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "functional analysis", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_55A9511267AC", "name": "Bobkov's inequality", "statement_latex": "I(integral f d gamma^n) <= integral sqrt(I(f)^2 + |grad f|^2) d gamma^n", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (I(integral f d gamma^n)) \\le x \\land x \\le (integral sqrt(I(f)^2 + |grad f|^2) d gamma^n)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover I(integral f d gamma^n) <= integral sqrt(I(f)^2 + |grad f|^2) d gamma^n.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover I(integral f d gamma^n) <= integral sqrt(I(f)^2 + |grad f|^2) d gamma^n.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "I(integral f d gamma^n) <= integral sqrt(I(f)^2 + |grad f|^2) d gamma^n", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover I(integral f d gamma^n) <= integral sqrt(I(f)^2 + |grad f|^2) d gamma^n.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_6cac0d09b1e6be9d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "hypergeometric series", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_94538B67A269", "name": "Dixon's identity", "statement_latex": "sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3 = (3a)! / (a!)^3", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3)\\}=\\{((3a)! / (a!)^3)\\}", "to_expression": "sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3 = (3a)! / (a!)^3", "justification": "singletons equal iff elements equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3 = (3a)! / (a!)^3.\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_3a266c82bb77d354", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_20033DA37507", "name": "Kolmogorov–Arnold representation theorem", "statement_latex": "f(x_1, ..., x_n) = Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p))", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (f(x_1, ..., x_n) = Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p))))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f(x_1, ..., x_n) = Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p))", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_87729d1c4f987337", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "topology", "subfield": "differential topology", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f01b64f6b85e8879", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "convex geometry", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4F00365B64E5", "name": "Minkowski's first inequality for convex bodies", "statement_latex": "V_1(K, L) ≥ V(K)^((n-1)/n) V(L)^(1/n)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (V(K)^((n-1)/n) V(L)^(1/n)) \\le x \\land x \\le (V_1(K, L))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover V_1(K, L) >= V(K)^((n-1)/n) V(L)^(1/n).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover V_1(K, L) >= V(K)^((n-1)/n) V(L)^(1/n).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "V_1(K, L) ≥ V(K)^((n-1)/n) V(L)^(1/n)", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover V_1(K, L) >= V(K)^((n-1)/n) V(L)^(1/n).\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_acbb74a2743d0fe9", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "statistical mechanics", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_3c57acbe566c734c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic geometry", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_8f18f8f70c8a6538", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "geometry, probability theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3B9F2B85DC25", "name": "Gaussian isoperimetric inequality", "statement_latex": "liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)} >= φ(Φ^{-1}(γ^n(A)))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)}) - (φ(Φ^{-1}(γ^n(A)))) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)} >= φ(Φ^{-1}(γ^n(A))).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)} >= φ(Φ^{-1}(γ^n(A))).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)} >= φ(Φ^{-1}(γ^n(A)))", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)} >= φ(Φ^{-1}(γ^n(A))).\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_b194a0c844836aba", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "functional analysis", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_55A9511267AC", "name": "Bobkov's inequality", "statement_latex": "I(integral f d gamma^n) <= integral sqrt(I(f)^2 + |grad f|^2) d gamma^n", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((I(integral f d gamma^n))-(integral sqrt(I(f)^2 + |grad f|^2) d gamma^n))=0", "to_expression": "I(integral f d gamma^n) <= integral sqrt(I(f)^2 + |grad f|^2) d gamma^n", "justification": "sup equals max(0,X-Y)", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as I(integral f d gamma^n) ≤ integral sqrt(I(f)^2 + |grad f|^2) d gamma^n.\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
{"schema_version": "1.0", "item_id": "rcd2_04955883f41f0790", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "topology", "subfield": "combinatorial topology", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_fca508cdb44f6ed2", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_89ccdeb4cc84c939", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2477ce38cb345f21", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "discrete geometry", "subfield": "geometry", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6145c73e83bb9d2a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "graph theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9A3C13F73177", "name": "Burr–Erdős conjecture", "statement_latex": "r(G) <= cp * n", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (cp * n)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (r(G))\\}", "to_expression": "r(G) <= cp * n", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as r(G) ≤ cp * n.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_a43d3e2960679347", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematical economics", "subfield": "linear algebra", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_49ed1307e114a1a7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "orthogonal polynomials", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AEDF9C91417B", "name": "Turán's inequalities", "statement_latex": "P_n(x)^2 > P_{n-1}(x)P_{n+1}(x)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "((P_{n-1}(x)P_{n+1}(x)),(P_n(x)^2))\\in\\{(u,v)\\in\\mathbb{R}^2:u\\le v\\}\\setminus\\{(u,u):u\\in\\mathbb{R}\\}", "to_expression": "P_n(x)^2 > P_{n-1}(x)P_{n+1}(x)", "justification": "≤ plus not equal iff <", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P_n(x)^2 > P_{n-1}(x)P_{n+1}(x).\"]"}], "source_equivalence_identity": "≤ plus not equal iff <"}
{"schema_version": "1.0", "item_id": "rcd2_130542eb76165550", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "matroid theory", "subfield": "combinatorics", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_09298FD96036", "name": "Ingleton's inequality", "statement_latex": "ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4) ≤ ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4))\\le t\\Rightarrow (ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4))\\le t", "to_expression": "ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4) ≤ ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4)", "justification": "converse uses t=Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4) ≤ ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4).\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_4403b0a1e2aa43c9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum information theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3795DDB6E155", "name": "NLTS conjecture", "statement_latex": "H = \\sum_{i=1}^{m} H_{i}", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((H),z)=((\\sum_{i=1}^{m} H_{i}),z)", "to_expression": "H = \\sum_{i=1}^{m} H_{i}", "justification": "pair equality forces first coordinates equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as H = \\\\sum_{i=1}^{m} H_{i}.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_d98966094f29c3d3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_617ACB7CFB1C", "name": "Specht's theorem", "statement_latex": "tr W(A, A*) = tr W(B, B*)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((tr W(A, A*)),(tr W(B, B*)))\\in\\{(z,z):z\\in D\\}", "to_expression": "tr W(A, A*) = tr W(B, B*)", "justification": "pair in diagonal iff X=Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as tr W(A, A*) = tr W(B, B*).\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_e4ceaba2256aef50", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "vector calculus", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F8BCAAED7243", "name": "Green's theorem", "statement_latex": "∮_{C}(L dx + M dy) = ∬_{D} (∂M/∂x - ∂L/∂y) dA", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((∮_{C}(L dx + M dy)),(∬_{D} (∂M/∂x - ∂L/∂y) dA))=0", "to_expression": "∮_{C}(L dx + M dy) = ∬_{D} (∂M/∂x - ∂L/∂y) dA", "justification": "zero discrete distance iff equality", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∮_{C}(L dx + M dy) = ∬_{D} (∂M/∂x - ∂L/∂y) dA.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_fc2e20b8841fb29e", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d2cebb775cb8f266", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "graph theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_07CDE34AC57A", "name": "Counting lemma", "statement_latex": "γ <= ε^h / 4h", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((γ)-(ε^h / 4h))=0", "to_expression": "γ <= ε^h / 4h", "justification": "sup equals max(0,X-Y)", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as γ ≤ ε^h / 4h.\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
{"schema_version": "1.0", "item_id": "rcd2_ba801b2f3442053c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b5c9876b8132e43b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "Sobolev spaces", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e4b6b18ccf1f6957", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8CBF25034487", "name": "Heegner's lemma", "statement_latex": "y^{2}=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((y^{2}))=F((a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}))", "to_expression": "y^{2}=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as y^{2}=a_{4}x^{4}+a_{3}x^{3}+a_{2}x^{2}+a_{1}x+a_{0}.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_c52e61133b7bdc26", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "control theory", "subfield": "geometric control theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a4378380f6665536", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic functions", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FDD3E2DFBB65", "name": "Hermite's identity", "statement_latex": "sum_{k=0}^{n-1} floor(x+k/n) = floor(n*x)", "assumptions": "n is a positive integer; x is real."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((sum_{k=0}^{n-1} floor(x+k/n)),z):z\\in D\\}=\\{((floor(n*x)),z):z\\in D\\}", "to_expression": "sum_{k=0}^{n-1} floor(x+k/n) = floor(n*x)", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "n is a positive integer; x is real.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{k=0}^{n-1} floor(x+k/n) = floor(n*x).\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_169fad19f02a8db3", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "functional analysis", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c6b2c4975ab65e63", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b41bac78c2cc3496", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_6883BDE34C4C", "name": "Le Cam's theorem", "statement_latex": "Σ|Pr(Sn=k) - (λ^k * e^-λ / k!)| < 2 * Σ pi^2", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(Σ|Pr(Sn=k) - (λ^k * e^-λ / k!)|)-(2 * Σ pi^2)\\}=0\\;\\land\\;(Σ|Pr(Sn=k) - (λ^k * e^-λ / k!)|)\\ne (2 * Σ pi^2)", "to_expression": "Σ|Pr(Sn=k) - (λ^k * e^-λ / k!)| < 2 * Σ pi^2", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Σ|Pr(Sn=k) - (λ^k * e^-λ / k!)| < 2 * Σ pi^2.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_5d59ed9b08d70933", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "modular arithmetic", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_92b7dde6099c23a7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "probabilistic method", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8F1DAFA4A16E", "name": "Lovász local lemma", "statement_latex": "e*p*(d+1) <= 1", "assumptions": "Symmetric Lovasz local lemma setting; each bad event has probability at most p and dependency degree at most d."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (1)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (e*p*(d+1))\\}", "to_expression": "e*p*(d+1) <= 1", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "Symmetric Lovasz local lemma setting; each bad event has probability at most p and dependency degree at most d.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as e*p*(d+1) ≤ 1.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_8d39ec158210289b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_68687FF65AAB", "name": "Landsberg–Schaar relation", "statement_latex": "1/sqrt(p) * sum_{n=0}^{p-1} exp(2*pi*i*n^2*q/p) = e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((1/sqrt(p) * sum_{n=0}^{p-1} exp(2*pi*i*n^2*q/p)),(e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q))))\\in\\{(z,z):z\\in D\\}", "to_expression": "1/sqrt(p) * sum_{n=0}^{p-1} exp(2*pi*i*n^2*q/p) = e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q))", "justification": "pair in diagonal iff X=Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as 1/sqrt(p) * sum_{n=0}^{p-1} exp(2*pi*i*n^2*q/p) = e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q)).\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_82077ad3a8147ab8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic geometry", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_49F57EB1100D", "name": "Faltings' theorem", "statement_latex": "g > 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(g) - (1) > 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover g > 1.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover g > 1.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "g > 1", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover g > 1.\"]"}], "source_equivalence_identity": "A<B iff B-A>0 over finite real quantities."}
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{"schema_version": "1.0", "item_id": "rcd2_080a1c22a3066b46", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematical analysis", "subfield": "number theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0AAEED4B3B51", "name": "Goldbach–Euler theorem", "statement_latex": "sum_{p=2}^{\\infty} 1/(p-1) = 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{p=2}^{\\infty} 1/(p-1)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(1))", "to_expression": "sum_{p=2}^{\\infty} 1/(p-1) = 1", "justification": "each argmin set is a singleton", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{p=2}^{\\\\infty} 1/(p-1) = 1.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_cb40bd57454601b2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "geometry", "subfield": "discrete geometry", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_DCDC0E01EC8B", "name": "Covering problem of Rado", "statement_latex": "F(X) = inf_S sup_I (|I|/|S|)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(F(X)))+\\delta_D(z,(inf_S sup_I (|I|/|S|))))=0", "to_expression": "F(X) = inf_S sup_I (|I|/|S|)", "justification": "zero loss iff some z equals both", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as F(X) = inf_S sup_I (|I|/|S|).\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_845be9aaf1ea86da", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis, measure theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4A564A0B747B", "name": "Monotone convergence theorem", "statement_latex": "lim_{n -> infinity} a_n = sup {a_n}", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (\\infty} a_n = sup {a_n}) < x \\land x < (lim_{n -)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover lim_{n - > infinity} a_n = sup {a_n}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover lim_{n - > infinity} a_n = sup {a_n}.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "lim_{n -> infinity} a_n = sup {a_n}", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover lim_{n - > infinity} a_n = sup {a_n}.\"]"}], "source_equivalence_identity": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B."}
{"schema_version": "1.0", "item_id": "rcd2_e45efcfff5cad74e", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "approximation theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f576b49501370506", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "martingale theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1E094F8EB368", "name": "Azuma's inequality", "statement_latex": "P(X_N - X_0 >= epsilon) <= exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(P(X_N - X_0 ≥ epsilon))}{\\exp(P(X_N - X_0 ≥ epsilon))+\\exp(exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2)))}\\le\\frac{1}{2}", "to_expression": "P(X_N - X_0 >= epsilon) <= exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2))", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(X_N - X_0 ≥ epsilon) ≤ exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2)).\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_54a974f51abc18b6", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "polyhedral combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_bf6837a7b3dc9073", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "combinatorics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_43eda8f7af5f750f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "Ramsey theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_4fd8cc11b5e6952b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "matrix theory", "subfield": "linear algebra", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_6095A64AB713", "name": "Golden–Thompson inequality", "statement_latex": "tr(exp(A+B)) <= tr(exp(A)*exp(B))", "assumptions": "A and B are Hermitian matrices of the same finite size."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(tr(exp(A)*exp(B)))\\le t\\Rightarrow (tr(exp(A+B)))\\le t", "to_expression": "tr(exp(A+B)) <= tr(exp(A)*exp(B))", "justification": "converse uses t=Y", "preconditions": "A and B are Hermitian matrices of the same finite size.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as tr(exp(A+B)) ≤ tr(exp(A)*exp(B)).\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_9f126c713db902da", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "prime numbers", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_71AA6DBCD731", "name": "Bonse's inequality", "statement_latex": "p_n# > p_{n+1}^2", "assumptions": "n>=4; p_n# is the product of the first n primes; p_{n+1} is the next prime."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{>0}\\;\\text{such that}\\;(p_{n+1}^2)+s=(p_n#)", "to_expression": "p_n# > p_{n+1}^2", "justification": "positive slack iff X<Y", "preconditions": "n>=4; p_n# is the product of the first n primes; p_{n+1} is the next prime.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p_n# > p_{n+1}^2.\"]"}], "source_equivalence_identity": "positive slack iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_6b40e1b591439f69", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal set theory", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a33f6fedecc84855", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "Ramsey theory", "subfield": "discrete mathematics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4B3C8E83990", "name": "Schur's theorem", "statement_latex": "x + y = z", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(x + y) \\in {(z)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover x + y = z.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "x + y = z", "justification": "L in {R} iff L = R.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover x + y = z.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_7b2e0814cc088dd1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1C2EEE35D0F6", "name": "Fischer's inequality", "statement_latex": "det(M) <= det(A)*det(C)", "assumptions": "M is positive semidefinite and block-partitioned with principal blocks A and C."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (det(M)) \\le x \\land x \\le (det(A)*det(C))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "M is positive semidefinite and block-partitioned with principal blocks A and C.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover det(M) <= det(A)*det(C).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "M is positive semidefinite and block-partitioned with principal blocks A and C.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover det(M) <= det(A)*det(C).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "det(M) <= det(A)*det(C)", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "M is positive semidefinite and block-partitioned with principal blocks A and C.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover det(M) <= det(A)*det(C).\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_8bc23fed7b4478fa", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "graph theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_57B1E9BA3579", "name": "Crossing number inequality", "statement_latex": "cr(G) >= e^3 / (29n^2)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(cr(G))\\le t\\Rightarrow (e^3 / (29n^2))\\le t", "to_expression": "cr(G) >= e^3 / (29n^2)", "justification": "converse uses t=Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as cr(G) ≥ e^3 / (29n^2).\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_e70818ebc226a210", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "graph theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9A3C13F73177", "name": "Burr–Erdős conjecture", "statement_latex": "r(G) <= cp * n", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(cp * n) - (r(G)) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover r(G) <= cp * n.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover r(G) <= cp * n.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "r(G) <= cp * n", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover r(G) <= cp * n.\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_355d004c8d60e40f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "discrete mathematics", "subfield": "combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AE455D38D038", "name": "Bregman–Minc inequality", "statement_latex": "per(A) <= prod_{i=1}^n (r_i!)^(1/r_i)", "assumptions": "A is an n by n 0-1 matrix; r_i is the sum of row i; terms with r_i=0 are handled by the standard zero-permanent convention."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(per(A)) - (prod_{i=1}^n (r_i!)^(1/r_i)) \\le 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A-B <= 0 iff A <= B by adding B to both sides.", "preconditions": "A is an n by n 0-1 matrix; r_i is the sum of row i; terms with r_i=0 are handled by the standard zero-permanent convention.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover per(A) <= prod_{i=1}^n (r_i!)^(1/r_i).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A-B <= 0 iff A <= B by adding B to both sides.", "preconditions": "A is an n by n 0-1 matrix; r_i is the sum of row i; terms with r_i=0 are handled by the standard zero-permanent convention.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover per(A) <= prod_{i=1}^n (r_i!)^(1/r_i).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "per(A) <= prod_{i=1}^n (r_i!)^(1/r_i)", "justification": "A-B <= 0 iff A <= B by adding B to both sides.", "preconditions": "A is an n by n 0-1 matrix; r_i is the sum of row i; terms with r_i=0 are handled by the standard zero-permanent convention.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover per(A) <= prod_{i=1}^n (r_i!)^(1/r_i).\"]"}], "source_equivalence_identity": "A-B <= 0 iff A <= B by adding B to both sides."}
{"schema_version": "1.0", "item_id": "rcd2_58fb15c0517e0365", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_95775DC75EC2", "name": "Oppenheim conjecture", "statement_latex": "|Q(x)| < \\epsilon", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(|Q(x)|)}{\\exp(|Q(x)|)+\\exp(\\epsilon)}<\\frac{1}{2}", "to_expression": "|Q(x)| < \\epsilon", "justification": "equiv exp(X)<exp(Y)", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |Q(x)| < \\\\epsilon.\"]"}], "source_equivalence_identity": "equiv exp(X)<exp(Y)"}
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{"schema_version": "1.0", "item_id": "rcd2_10d850f97e0066a8", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "Mathematical analysis", "subfield": "Sequences and series", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_9ceaf0b4ec130520", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_33e1c332ed9b118d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "vector calculus", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_24234F1E6547", "name": "Divergence theorem", "statement_latex": "\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S} = \\int_{V} (\\nabla \\cdot \\mathbf{F}) dV", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S})\\Longleftrightarrow z=(\\int_{V} (\\nabla \\cdot \\mathbf{F}) dV))", "to_expression": "\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S} = \\int_{V} (\\nabla \\cdot \\mathbf{F}) dV", "justification": "same equality fiber iff equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\oint_{S} \\\\mathbf{F} \\\\cdot d\\\\mathbf{S} = \\\\int_{V} (\\\\nabla \\\\cdot \\\\mathbf{F}) dV.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_95a01f841579162d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_773C0118E21F", "name": "Darboux's theorem", "statement_latex": "f'(x) = y", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(f'(x)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(y))", "to_expression": "f'(x) = y", "justification": "each argmin set is a singleton", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f'(x) = y.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_3c86dcbbe005e857", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_3bdd2ba9fb10a26a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "topology", "subfield": "algebraic topology", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_48CD393CFABB", "name": "Borsuk–Ulam theorem", "statement_latex": "f(-x) = f(x)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(f(x)) = (f(-x))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(-x) = f(x).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f(-x) = f(x)", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(-x) = f(x).\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_6a3cbdc5fe37db32", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "approximation theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a002b4305929b955", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "sieve theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_1f6e18036eca8262", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "quadratic forms", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2009d7d6444c3f9d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic calculus", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2fb8d0510250ad14", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "additive number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E3E323366217", "name": "Lagrange's four-square theorem", "statement_latex": "p = a^2 + b^2 + c^2 + d^2", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(p) \\in {(a^2 + b^2 + c^2 + d^2)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover p = a^2 + b^2 + c^2 + d^2.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "p = a^2 + b^2 + c^2 + d^2", "justification": "L in {R} iff L = R.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover p = a^2 + b^2 + c^2 + d^2.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_099ebc6bfb394ff4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "topology", "subfield": "combinatorial topology", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CB3E317B00C5", "name": "Tucker's lemma", "statement_latex": "L(-v) = -L(v)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(-L(v)) = (L(-v))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover L(-v) = -L(v).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "L(-v) = -L(v)", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover L(-v) = -L(v).\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_ba7258ee7f351a8a", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematical economics", "subfield": "linear algebra", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_05db1eec20fd3815", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "Kleinian groups", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C0AAA35160EF", "name": "Jørgensen's inequality", "statement_latex": "|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2| >= 1", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (1) \\le x \\land x \\le (|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2|)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2| >= 1.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2| >= 1.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "|Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2| >= 1", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |Tr(A)^2 - 4| + |Tr(ABA^-1B^-1) - 2| >= 1.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_18704e558d6f72a0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_98220A4F883B", "name": "Siegel identity", "statement_latex": "\\frac{x_3-x_1}{x_2-x_1} + \\frac{x_2-x_3}{x_2-x_1} = 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((\\frac{x_3-x_1}{x_2-x_1} + \\frac{x_2-x_3}{x_2-x_1}),z):z\\in D\\}=\\{((1),z):z\\in D\\}", "to_expression": "\\frac{x_3-x_1}{x_2-x_1} + \\frac{x_2-x_3}{x_2-x_1} = 1", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\frac{x_3-x_1}{x_2-x_1} + \\\\frac{x_2-x_3}{x_2-x_1} = 1.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_e76eb95395bfaa9a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "matrix analysis", "subfield": "linear algebra", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D76BA21A085C", "name": "Crouzeix's conjecture", "statement_latex": "||f(A)|| <= 2 sup_{z in W(A)} |f(z)|", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(||f(A)||)}{\\exp(||f(A)||)+\\exp(2 sup_{z in W(A)} |f(z)|)}\\le\\frac{1}{2}", "to_expression": "||f(A)|| <= 2 sup_{z in W(A)} |f(z)|", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ||f(A)|| ≤ 2 sup_{z in W(A)} |f(z)|.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_8e225d951e57d279", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "geometry of numbers", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_965ebde59d450e3f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_59658ff6baea6c1d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "discrete geometry", "subfield": "geometry", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_fef0264c6c40dc7c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "design theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_41032C4E68AC", "name": "Bruck–Ryser–Chowla theorem", "statement_latex": "x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2} = 0", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2}) \\in {(0)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover x^{2} - (k - \\\\lambda)y^{2} - (-1)^{(v-1)/2} \\\\lambda z^{2} = 0.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2} = 0", "justification": "L in {R} iff L = R.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover x^{2} - (k - \\\\lambda)y^{2} - (-1)^{(v-1)/2} \\\\lambda z^{2} = 0.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_e7408d3202c34d43", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_fa480fa8d5ee7540", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "statistics", "subfield": "statistical hypothesis testing", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_69B4EB18D461", "name": "Neyman–Pearson lemma", "statement_latex": "α = Pr_θ0(X ∈ R)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(α)\\Longleftrightarrow z=(Pr_θ0(X ∈ R)))", "to_expression": "α = Pr_θ0(X ∈ R)", "justification": "same equality fiber iff equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as α = Pr_θ0(X ∈ R).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_cea93580048cb44f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e7eb3b3a77498584", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "probabilistic method", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_24b1ed83b1c0e544", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7C6C20BC69B4", "name": "Slepian's lemma", "statement_latex": "Pr[Intersection(X_i <= u_i)] <= Pr[Intersection(Y_i <= u_i)]", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(Pr[Intersection(X_i ≤ u_i)])-(Pr[Intersection(Y_i ≤ u_i)])\\}=0", "to_expression": "Pr[Intersection(X_i <= u_i)] <= Pr[Intersection(Y_i <= u_i)]", "justification": "positive part vanishes iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr[Intersection(X_i ≤ u_i)] ≤ Pr[Intersection(Y_i ≤ u_i)].\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_c4b17c696c3ed646", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f31f1a01e3f7a4d6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D236ADD607AB", "name": "Chernoff bound", "statement_latex": "P(X ≥ a) ≤ inf_{t>0} M(t)e^{-ta}", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (P(X ≥ a) ≤ \\inf_{t>0} M(t)e^{-ta}))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "P(X ≥ a) ≤ inf_{t>0} M(t)e^{-ta}", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_3ec4def4724de4f8", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_5f3ef3a979003a96", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "polynomials", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e3a3fb3da1381a04", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "time-reversal symmetry", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b57d2c5ddd051acf", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_bdb6e042d5211797", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "hypergeometric series", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_94538B67A269", "name": "Dixon's identity", "statement_latex": "sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3 = (3a)! / (a!)^3", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3),z):z\\in D\\}=\\{(((3a)! / (a!)^3),z):z\\in D\\}", "to_expression": "sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3 = (3a)! / (a!)^3", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3 = (3a)! / (a!)^3.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_5f12f245474d1776", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B664293AE8AA", "name": "Etemadi's inequality", "statement_latex": "Pr(max_{1<=k<=n} |S_k| >= 3alpha) <= 3 max_{1<=k<=n} Pr(|S_k| >= alpha)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (Pr(max_{1\\lek\\len} |S_k| \\ge 3alpha) \\le 3 max_{1\\lek\\len} Pr(|S_k| \\ge alpha)))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "Pr(max_{1<=k<=n} |S_k| >= 3alpha) <= 3 max_{1<=k<=n} Pr(|S_k| >= alpha)", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_31d083bf12326a85", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_ABB949903DF7", "name": "Vandermonde's identity", "statement_latex": "{m+n \\choose r}=\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,({m+n \\choose r}))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k}))", "to_expression": "{m+n \\choose r}=\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k}", "justification": "each argmin set is a singleton", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as {m+n \\\\choose r}=\\\\sum _{k=0}^{r}{m \\\\choose k}{n \\\\choose r-k}.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_11a570a33330040f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "field theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_6E9382B37B92", "name": "Primitive element theorem", "statement_latex": "E = F(α)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(E)\\}=\\{(F(α))\\}", "to_expression": "E = F(α)", "justification": "singletons equal iff elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as E = F(α).\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_18420ffd27816a6a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "coding theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8CDA9CA5B14E", "name": "Noisy-channel coding theorem", "statement_latex": "R < C", "assumptions": "R is the coding rate and C is the channel capacity, both finite real values. This row encodes the achievability-side rate condition."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(R)-(C)\\}=0\\;\\land\\;(R)\\ne (C)", "to_expression": "R < C", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "R is the coding rate and C is the channel capacity, both finite real values. This row encodes the achievability-side rate condition.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as R < C.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_40056d2cdd27b1af", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "intersection", "transformation_category": "optimization_projection", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a97e2b0f603d1a97", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "number theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_de892c5278040ba5", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "calculus", "subfield": "vector calculus / electrostatics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D5081E95CCA1", "name": "Uniqueness theorem for Poisson's equation", "statement_latex": "∇^2φ = -ρ_f/ε_0", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(∇^2φ) \\in {(-ρ_f/ε_0)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover ∇^2φ = -ρ_f/ε_0.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "∇^2φ = -ρ_f/ε_0", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover ∇^2φ = -ρ_f/ε_0.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_1fd441b632a16712", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AE174D33AD68", "name": "Eaton's inequality", "statement_latex": "P(|sum_{i=1}^n a_i X_i| >= k) <= 2 inf_{0 <= c <= k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(P(|sum_{i=1}^n a_i X_i| ≥ k))}{\\exp(P(|sum_{i=1}^n a_i X_i| ≥ k))+\\exp(2 inf_{0 ≤ c ≤ k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz)}\\le\\frac{1}{2}", "to_expression": "P(|sum_{i=1}^n a_i X_i| >= k) <= 2 inf_{0 <= c <= k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(|sum_{i=1}^n a_i X_i| ≥ k) ≤ 2 inf_{0 ≤ c ≤ k} integral_{c}^inf ((z-c)/(k-c))^3 phi(z) dz.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_5b8871f88e409e7c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "differential topology", "subfield": "geometry", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CC0CC8596114", "name": "Morse theory", "statement_latex": "M^a = {x in M | f(x) <= a}", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(M^a = {x \\in M | f(x)) - (a}) \\le 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A-B <= 0 iff A <= B by adding B to both sides.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover M^a = {x in M | f(x) <= a}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A-B <= 0 iff A <= B by adding B to both sides.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover M^a = {x in M | f(x) <= a}.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "M^a = {x in M | f(x) <= a}", "justification": "A-B <= 0 iff A <= B by adding B to both sides.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover M^a = {x in M | f(x) <= a}.\"]"}], "source_equivalence_identity": "A-B <= 0 iff A <= B by adding B to both sides."}
{"schema_version": "1.0", "item_id": "rcd2_ef3cec19cbb31dfa", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "matrix analysis", "subfield": "linear algebra", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e8048ba6ea38d6d8", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "elementary number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_954a0a325b6c0fab", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "Ramsey theory", "subfield": "discrete mathematics", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3F2BB2E376A3", "name": "Rado's theorem", "statement_latex": "Ax = 0", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(Ax))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(0))", "to_expression": "Ax = 0", "justification": "each argmin set is a singleton", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Ax = 0.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_535473af68767ad4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "analysis", "subfield": "moment problem", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A82D2EB23B1F", "name": "Carleman's condition", "statement_latex": "\\sum _{n=1}^{\\infty }m_{2n}^{-{1/2n}}=+\\infty", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\sum _{n=1}^{\\infty }m_{2n}^{-{1/2n}}=+\\infty) \\lor (0 = 1)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "0=1 is false, so Phi_T or false iff Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\sum _{n=1}^{\\infty }m_{2n}^{-{1/2n}}=+\\infty", "justification": "0=1 is false, so Phi_T or false iff Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "0=1 is false, so Phi_T or false iff Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_bc43fec177079186", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "martingale theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1E094F8EB368", "name": "Azuma's inequality", "statement_latex": "P(X_N - X_0 >= epsilon) <= exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\operatorname{softplus}((P(X_N - X_0 ≥ epsilon))-(exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2))))\\le\\log 2", "to_expression": "P(X_N - X_0 >= epsilon) <= exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2))", "justification": "softplus is increasing and softplus(0)=log2", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(X_N - X_0 ≥ epsilon) ≤ exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2)).\"]"}], "source_equivalence_identity": "softplus is increasing and softplus(0)=log2"}
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{"schema_version": "1.0", "item_id": "rcd2_f2e44ddc4fc48fd2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_37DDB6F9F636", "name": "Duffin–Schaeffer theorem", "statement_latex": "\\sum_{q=1}^{\\infty}\\varphi(q)\\frac{f(q)}{q}=\\infty", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(\\sum_{q=1}^{\\infty}\\varphi(q)\\frac{f(q)}{q})\\}=\\{(\\infty)\\}", "to_expression": "\\sum_{q=1}^{\\infty}\\varphi(q)\\frac{f(q)}{q}=\\infty", "justification": "singletons equal iff elements equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\sum_{q=1}^{\\\\infty}\\\\varphi(q)\\\\frac{f(q)}{q}=\\\\infty.\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_0645d74f886b867e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "probability theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8F3F3785BDA2", "name": "Gibbs' inequality", "statement_latex": "-Σ p_i log p_i <= -Σ p_i log q_i", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((-Σ p_i log p_i)-(-Σ p_i log q_i),(-\\infty,0])=0", "to_expression": "-Σ p_i log p_i <= -Σ p_i log q_i", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as -Σ p_i log p_i ≤ -Σ p_i log q_i.\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
{"schema_version": "1.0", "item_id": "rcd2_3b74cdf1e78d09ff", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "hypergraph theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0E38B7F67353", "name": "Baranyai's theorem", "statement_latex": "\\binom{k}{r} \\frac{r}{k} = \\binom{k-1}{r-1}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\binom{k}{r} \\frac{r}{k}))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\binom{k-1}{r-1}))", "to_expression": "\\binom{k}{r} \\frac{r}{k} = \\binom{k-1}{r-1}", "justification": "each argmin set is a singleton", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\binom{k}{r} \\\\frac{r}{k} = \\\\binom{k-1}{r-1}.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
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{"schema_version": "1.0", "item_id": "rcd2_9cc4d30bb8a537e5", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "approximation theory", "subfield": "mathematical analysis", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_389D9173C8B7", "name": "Lebesgue's lemma", "statement_latex": "‖v - Pv‖ ≤ (1 + ‖P‖) inf_{u ∈ U} ‖v - u‖", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((‖v - Pv‖)-((1 + ‖P‖) inf_{u ∈ U} ‖v - u‖),(-\\infty,0])=0", "to_expression": "‖v - Pv‖ ≤ (1 + ‖P‖) inf_{u ∈ U} ‖v - u‖", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ‖v - Pv‖ ≤ (1 + ‖P‖) inf_{u ∈ U} ‖v - u‖.\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
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{"schema_version": "1.0", "item_id": "rcd2_25457e44bff2e48e", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "cryptography", "subfield": "cryptanalysis", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
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{"schema_version": "1.0", "item_id": "rcd2_dc394818607c617b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6315221e1d97f578", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A5B93C0856B4", "name": "Handshaking lemma", "statement_latex": "Σ_{v∈V} deg(v) = 2|E|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (Σ_{v∈V} deg(v) = 2|E|))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "Σ_{v∈V} deg(v) = 2|E|", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_4fe53f7165c7551e", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "algebra", "subfield": "polynomials", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_53cbe730fe1eeff7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "inner product spaces", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2EE2B752D03D", "name": "Polarization identity", "statement_latex": "\\operatorname{Re} \\langle x, y \\rangle = \\frac{1}{2} (\\|x + y\\|^2 - \\|x\\|^2 - \\|y\\|^2)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\operatorname{Re} \\langle x, y \\rangle) \\in {(\\frac{1}{2} (\\|x + y\\|^2 - \\|x\\|^2 - \\|y\\|^2))}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\operatorname{Re} \\\\langle x, y \\\\rangle = \\\\frac{1}{2} (\\\\|x + y\\\\|^2 - \\\\|x\\\\|^2 - \\\\|y\\\\|^2).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\operatorname{Re} \\langle x, y \\rangle = \\frac{1}{2} (\\|x + y\\|^2 - \\|x\\|^2 - \\|y\\|^2)", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\operatorname{Re} \\\\langle x, y \\\\rangle = \\\\frac{1}{2} (\\\\|x + y\\\\|^2 - \\\\|x\\\\|^2 - \\\\|y\\\\|^2).\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_76f1337a81cca68d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_42828E58305E", "name": "Pascal's rule", "statement_latex": "\\binom{n-1}{k} + \\binom{n-1}{k-1} = \\binom{n}{k}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(\\binom{n-1}{k} + \\binom{n-1}{k-1})\\Longleftrightarrow z=(\\binom{n}{k}))", "to_expression": "\\binom{n-1}{k} + \\binom{n-1}{k-1} = \\binom{n}{k}", "justification": "same equality fiber iff equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\binom{n-1}{k} + \\\\binom{n-1}{k-1} = \\\\binom{n}{k}.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_e3a9a72c7c69707d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "differential topology", "subfield": "geometry", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_89e4f5212d5f52b7", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "inequalities", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_628ca4d862b3a7d3", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_27f747fe39062872", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum information theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3795DDB6E155", "name": "NLTS conjecture", "statement_latex": "H = \\sum_{i=1}^{m} H_{i}", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((H),z):z\\in D\\}=\\{((\\sum_{i=1}^{m} H_{i}),z):z\\in D\\}", "to_expression": "H = \\sum_{i=1}^{m} H_{i}", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as H = \\\\sum_{i=1}^{m} H_{i}.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
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{"schema_version": "1.0", "item_id": "rcd2_dff80eca668d6807", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "hypergeometric series", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_94538B67A269", "name": "Dixon's identity", "statement_latex": "sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3 = (3a)! / (a!)^3", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3),z)=(((3a)! / (a!)^3),z)", "to_expression": "sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3 = (3a)! / (a!)^3", "justification": "pair equality forces first coordinates equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3 = (3a)! / (a!)^3.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
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{"schema_version": "1.0", "item_id": "rcd2_a844a9812f1b6e84", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A371695EA43E", "name": "Davenport–Schmidt theorem", "statement_latex": "|\\xi - \\alpha | < CH(\\alpha )^{-3}\\max(1, \\xi ^{2})", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(|\\xi - \\alpha |)-(CH(\\alpha )^{-3}\\max(1, \\xi ^{2}))\\}=0\\;\\land\\;(|\\xi - \\alpha |)\\ne (CH(\\alpha )^{-3}\\max(1, \\xi ^{2}))", "to_expression": "|\\xi - \\alpha | < CH(\\alpha )^{-3}\\max(1, \\xi ^{2})", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |\\\\xi - \\\\alpha | < CH(\\\\alpha )^{-3}\\\\max(1, \\\\xi ^{2}).\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
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{"schema_version": "1.0", "item_id": "rcd2_8d08172b8d642901", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_695c009e0cc9c41d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_763D96A1B9F8", "name": "Rouché–Capelli theorem", "statement_latex": "rank(A) = rank([A|b])", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(rank(A))=z\\;\\land\\;z=(rank([A|b]))", "to_expression": "rank(A) = rank([A|b])", "justification": "common witness iff equality", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as rank(A) = rank([A|b]).\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_3cf5db7a0efa4712", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_45A3C68C1B71", "name": "Linnik's theorem", "statement_latex": "p(a,d) < c * d^L", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(c * d^L) - (p(a,d)) > 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover p(a,d) < c * d^L.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover p(a,d) < c * d^L.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "p(a,d) < c * d^L", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover p(a,d) < c * d^L.\"]"}], "source_equivalence_identity": "A<B iff B-A>0 over finite real quantities."}
{"schema_version": "1.0", "item_id": "rcd2_33a36e40fb580f2e", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "graph theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_66f17ffc8d6beec8", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "hypergeometric series", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_07e0209488f83e97", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "approximation theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_55fddcf3da6b761b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_42ED352AD252", "name": "Bernstein inequalities", "statement_latex": "P(|(1/n) * sum_{i=1}^n X_i| > epsilon) <= 2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3)))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(2 * \\exp(-n * \\varepsilon^2 / (2 * (1 + \\varepsilon/3)))) - (P(|(1/n) * sum_{i=1}^n X_i| > \\varepsilon)) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover P(|(1/n) * sum_{i=1}^n X_i| > epsilon) <= 2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3))).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover P(|(1/n) * sum_{i=1}^n X_i| > epsilon) <= 2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3))).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "P(|(1/n) * sum_{i=1}^n X_i| > epsilon) <= 2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3)))", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover P(|(1/n) * sum_{i=1}^n X_i| > epsilon) <= 2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3))).\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_fb412d7b06a0652c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "calculus", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3967E74F7B6F", "name": "Increment theorem", "statement_latex": "Δy = f'(x)Δx + εΔx", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((Δy))=F((f'(x)Δx + εΔx))", "to_expression": "Δy = f'(x)Δx + εΔx", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Δy = f'(x)Δx + εΔx.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_dae4b3c7432ca41f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "control theory", "subfield": "geometric control theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2ADEC5E217BE", "name": "Krener's theorem", "statement_latex": "\\dot{q}=f(q,u)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(f(q,u)) = (\\dot{q})", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\dot{q}=f(q,u).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\dot{q}=f(q,u)", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\dot{q}=f(q,u).\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_6ebdd7266840b928", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "zeta and L-functions", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CEA3FEEC415F", "name": "Selberg's zeta function conjecture", "statement_latex": "N(T + H) - N(T) ≥ cH log T", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(N(T + H) - N(T))\\le t\\Rightarrow (cH log T)\\le t", "to_expression": "N(T + H) - N(T) ≥ cH log T", "justification": "converse uses t=Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as N(T + H) - N(T) ≥ cH log T.\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_471d4653b2ca566b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AD101C6FC364", "name": "Adiabatic theorem", "statement_latex": "E = nhv", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((E),(nhv))=0", "to_expression": "E = nhv", "justification": "zero discrete distance iff equality", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as E = nhv.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_ba4d6a89d699b6b0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "point processes", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C81FC5D06BF2", "name": "Campbell's theorem", "statement_latex": "E[sum_{x in N} f(x)] = integral f(x) dLambda(x)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((E[sum_{x \\in N} f(x)]) \\in S \\land (integral f(x) dLambda(x)) \\notin S) \\lor ((integral f(x) dLambda(x)) \\in S \\land (E[sum_{x \\in N} f(x)]) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E[sum_{x in N} f(x)] = integral f(x) dLambda(x).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "E[sum_{x in N} f(x)] = integral f(x) dLambda(x)", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E[sum_{x in N} f(x)] = integral f(x) dLambda(x).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_8f8f6366aba160d2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "differential topology", "subfield": "geometry", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CC0CC8596114", "name": "Morse theory", "statement_latex": "M^a = {x in M | f(x) <= a}", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(a}) \\in (M^a = {x \\in M | f(x)) + [0,\\infty)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover M^a = {x in M | f(x) <= a}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover M^a = {x in M | f(x) <= a}.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "M^a = {x in M | f(x) <= a}", "justification": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover M^a = {x in M | f(x) <= a}.\"]"}], "source_equivalence_identity": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_c7df54e27e7a2cfb", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "additive number theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_870FF97782B2", "name": "Goldbach's weak conjecture", "statement_latex": "n = p1 + p2 + p3", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(n))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(p1 + p2 + p3))", "to_expression": "n = p1 + p2 + p3", "justification": "each argmin set is a singleton", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as n = p1 + p2 + p3.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_84e521ed634c6baa", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_58f3137d5d64e47d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_be3bee22515c77e5", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "analysis", "subfield": "complex analysis", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_065364B1C6FB", "name": "Arakelyan's theorem", "statement_latex": "|g − f| < ε", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(|g − f|)-(ε)\\}=0\\;\\land\\;(|g − f|)\\ne (ε)", "to_expression": "|g − f| < ε", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |g − f| < ε.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_809eb1606b4abc2b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "computer science", "subfield": "formal language theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_37D96EA4CFB7", "name": "Pumping lemma for regular languages", "statement_latex": "w = xyz", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((w),(xyz))=0", "to_expression": "w = xyz", "justification": "zero discrete distance iff equality", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as w = xyz.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_cc55aa98c64e83f2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B664293AE8AA", "name": "Etemadi's inequality", "statement_latex": "Pr(max_{1<=k<=n} |S_k| >= 3alpha) <= 3 max_{1<=k<=n} Pr(|S_k| >= alpha)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(Pr(max_{1≤k≤n} |S_k| ≥ 3alpha))+s=(3 max_{1≤k≤n} Pr(|S_k| ≥ alpha))", "to_expression": "Pr(max_{1<=k<=n} |S_k| >= 3alpha) <= 3 max_{1<=k<=n} Pr(|S_k| >= alpha)", "justification": "nonnegative slack exists iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr(max_{1≤k≤n} |S_k| ≥ 3alpha) ≤ 3 max_{1≤k≤n} Pr(|S_k| ≥ alpha).\"]"}], "source_equivalence_identity": "nonnegative slack exists iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_835af8e3c13d0f44", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "calculus", "subfield": "integral calculus", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F8E7F707512F", "name": "Bioche's rules", "statement_latex": "ω(-t) = ω(t)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(ω(-t)) \\in {(ω(t))}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover ω(-t) = ω(t).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "ω(-t) = ω(t)", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover ω(-t) = ω(t).\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_f2fc3a110c25bb79", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D737B48D674F", "name": "Borell–TIS inequality", "statement_latex": "P(‖f‖_T > E(‖f‖_T) + u) ≤ exp(-u^2 / (2σ_T^2))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (P(‖f‖_T > E(‖f‖_T) + u)) \\le x \\land x \\le (\\exp(-u^2 / (2σ_T^2)))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover P(‖f‖_T > E(‖f‖_T) + u) <= exp(-u^2 / (2σ_T^2)).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover P(‖f‖_T > E(‖f‖_T) + u) <= exp(-u^2 / (2σ_T^2)).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "P(‖f‖_T > E(‖f‖_T) + u) ≤ exp(-u^2 / (2σ_T^2))", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover P(‖f‖_T > E(‖f‖_T) + u) <= exp(-u^2 / (2σ_T^2)).\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_07e2a9f9a49de255", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "combinatorics", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_51200980F031", "name": "Star of David theorem", "statement_latex": "gcd{(n-1 k-1), (n k+1), (n+1 k)} = gcd{(n-1 k), (n k-1), (n+1 k+1)}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((gcd{(n-1 k-1), (n k+1), (n+1 k)}),z):z\\in D\\}=\\{((gcd{(n-1 k), (n k-1), (n+1 k+1)}),z):z\\in D\\}", "to_expression": "gcd{(n-1 k-1), (n k+1), (n+1 k)} = gcd{(n-1 k), (n k-1), (n+1 k+1)}", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as gcd{(n-1 k-1), (n k+1), (n+1 k)} = gcd{(n-1 k), (n k-1), (n+1 k+1)}.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
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{"schema_version": "1.0", "item_id": "rcd2_0d6ffd8895ca1ecc", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "calculus", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_70869e1b784557e1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "Ramsey theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A40EE0DC8F58", "name": "Van der Waerden's theorem", "statement_latex": "W(r,k) ≤ 2^{2^{r^{2^{2^{k+9}}}}}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((W(r,k))-(2^{2^{r^{2^{2^{k+9}}}}}),(-\\infty,0])=0", "to_expression": "W(r,k) ≤ 2^{2^{r^{2^{2^{k+9}}}}}", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as W(r,k) ≤ 2^{2^{r^{2^{2^{k+9}}}}}.\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
{"schema_version": "1.0", "item_id": "rcd2_67ffb5b5c755da8b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F881715F08CD", "name": "Sylvester's determinant identity", "statement_latex": "det(A) = det(A_v^u) * det(A_tilde_v^u)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(det(A))\\Longleftrightarrow z=(det(A_v^u) * det(A_tilde_v^u)))", "to_expression": "det(A) = det(A_v^u) * det(A_tilde_v^u)", "justification": "same equality fiber iff equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as det(A) = det(A_v^u) * det(A_tilde_v^u).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_0ce6baf8a0914e53", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_21c97b5599a059bd", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_588ea1712adf0a19", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AFD3A111ECE4", "name": "Boole's inequality", "statement_latex": "P(∪_{i=1}^∞ A_i) ≤ ∑_{i=1}^∞ P(A_i)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(P(∪_{i=1}^∞ A_i))}{\\exp(P(∪_{i=1}^∞ A_i))+\\exp(∑_{i=1}^∞ P(A_i))}\\le\\frac{1}{2}", "to_expression": "P(∪_{i=1}^∞ A_i) ≤ ∑_{i=1}^∞ P(A_i)", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(∪_{i=1}^∞ A_i) ≤ ∑_{i=1}^∞ P(A_i).\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_20c6b8b64298e7d9", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "probability theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0dee5bee8ffd71a4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "statistical mechanics", "subfield": "non-equilibrium thermodynamics", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D06DBDBC4CC8", "name": "Crooks fluctuation theorem", "statement_latex": "P[σ(x(t))] / P[σ(x_tilde(t))] = e^σ[x(t)]", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((P[σ(x(t))] / P[σ(x_tilde(t))]),z)=((e^σ[x(t)]),z)", "to_expression": "P[σ(x(t))] / P[σ(x_tilde(t))] = e^σ[x(t)]", "justification": "pair equality forces first coordinates equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P[σ(x(t))] / P[σ(x_tilde(t))] = e^σ[x(t)].\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_4fe38d3d526d8349", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "inequalities", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9822BD8B9BE6", "name": "Muirhead's inequality", "statement_latex": "[a] >= [b] for a majorizing b", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge ([a])\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge ([b] for a majorizing b)\\}", "to_expression": "[a] >= [b] for a majorizing b", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as [a] ≥ [b] for a majorizing b.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_10b22d6f2547a3e9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "elementary algebra", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D16BD70DAEA0", "name": "Difference of two squares", "statement_latex": "a^2 - b^2 = (a + b)(a - b)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(a^2 - b^2)\\}=\\{(a + b)(a - b)\\}", "to_expression": "a^2 - b^2 = (a + b)(a - b)", "justification": "singletons equal iff elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as a^2 - b^2 = (a + b)(a - b).\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_6896bec9df54ff65", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "measure theory", "subfield": "stochastic analysis", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b7602ba0d388bfa4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "extremal graph theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F0F061CBC3F5", "name": "Blow-up lemma", "statement_latex": "e(X,Y) > delta |X| |Y|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(delta |X| |Y|)}{\\exp(delta |X| |Y|)+\\exp(e(X,Y))}<\\frac{1}{2}", "to_expression": "e(X,Y) > delta |X| |Y|", "justification": "equiv exp(X)<exp(Y)", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as e(X,Y) > delta |X| |Y|.\"]"}], "source_equivalence_identity": "equiv exp(X)<exp(Y)"}
{"schema_version": "1.0", "item_id": "rcd2_5da7ef165ff800e3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "mathematical analysis", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_25FCC5C4BBD9", "name": "Cauchy's convergence test", "statement_latex": "|a_{n+1} + a_{n+2} + \\cdots + a_{n+p}| < \\varepsilon", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(|a_{n+1} + a_{n+2} + \\cdots + a_{n+p}|)-(\\varepsilon)\\}=0\\;\\land\\;(|a_{n+1} + a_{n+2} + \\cdots + a_{n+p}|)\\ne (\\varepsilon)", "to_expression": "|a_{n+1} + a_{n+2} + \\cdots + a_{n+p}| < \\varepsilon", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |a_{n+1} + a_{n+2} + \\\\cdots + a_{n+p}| < \\\\varepsilon.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
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{"schema_version": "1.0", "item_id": "rcd2_b0f2bc0ce3495ac6", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "modular arithmetic", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a78b46b58ac8a9bb", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "large deviations theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_53869962DDDE", "name": "Laplace's principle", "statement_latex": "\\lim_{\\theta \\to \\infty}\\frac{1}{\\theta}\\log\\int_{A}e^{-\\theta\\varphi(x)}\\,dx=-\\mathop{\\mathrm{ess\\,inf}}_{x\\in A}\\varphi(x)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((\\lim_{\\th\\eta \\to \\infty}\\frac{1}{\\th\\eta}\\log\\int_{A}e^{-\\th\\eta\\varphi(x)}\\,dx) \\in S \\land (-\\mathop{\\mathrm{ess\\,inf}}_{x\\\\in A}\\varphi(x)) \\notin S) \\lor ((-\\mathop{\\mathrm{ess\\,inf}}_{x\\\\in A}\\varphi(x)) \\in S \\land (\\lim_{\\th\\eta \\to \\infty}\\frac{1}{\\th\\eta}\\log\\int_{A}e^{-\\th\\eta\\varphi(x)}\\,dx) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\lim_{\\\\theta \\\\to \\\\infty}\\\\frac{1}{\\\\theta}\\\\log\\\\int_{A}e^{-\\\\theta\\\\varphi(x)}\\\\,dx=-\\\\mathop{\\\\mathrm{ess\\\\,inf}}_{x\\\\in A}\\\\varphi(x).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\lim_{\\theta \\to \\infty}\\frac{1}{\\theta}\\log\\int_{A}e^{-\\theta\\varphi(x)}\\,dx=-\\mathop{\\mathrm{ess\\,inf}}_{x\\in A}\\varphi(x)", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\lim_{\\\\theta \\\\to \\\\infty}\\\\frac{1}{\\\\theta}\\\\log\\\\int_{A}e^{-\\\\theta\\\\varphi(x)}\\\\,dx=-\\\\mathop{\\\\mathrm{ess\\\\,inf}}_{x\\\\in A}\\\\varphi(x).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_5832c6c1ab5c5f03", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D737B48D674F", "name": "Borell–TIS inequality", "statement_latex": "P(‖f‖_T > E(‖f‖_T) + u) ≤ exp(-u^2 / (2σ_T^2))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (exp(-u^2 / (2σ_T^2)))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (P(‖f‖_T > E(‖f‖_T) + u))\\}", "to_expression": "P(‖f‖_T > E(‖f‖_T) + u) ≤ exp(-u^2 / (2σ_T^2))", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(‖f‖_T > E(‖f‖_T) + u) ≤ exp(-u^2 / (2σ_T^2)).\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_5a99deca5d584e2a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "matching theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_69DD0D246726", "name": "Tutte–Berge formula", "statement_latex": "max_matching_size = 1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((max_matching_size),z):z\\in D\\}=\\{((1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)),z):z\\in D\\}", "to_expression": "max_matching_size = 1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as max_matching_size = 1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|).\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_c57a5c5226a20234", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "combinatorics", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_684A9634E4F6", "name": "Hypergeometric identity", "statement_latex": "sum_{i=0}^n binom(n,i) = 2^n", "assumptions": "n is a nonnegative integer."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(sum_{i=0}^n binom(n,i)))+\\delta_D(z,(2^n)))=0", "to_expression": "sum_{i=0}^n binom(n,i) = 2^n", "justification": "zero loss iff some z equals both", "preconditions": "n is a nonnegative integer.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{i=0}^n binom(n,i) = 2^n.\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_722af4bdc94e6510", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4273AD2C24D7", "name": "Bayes' theorem", "statement_latex": "P(A|B) = P(B|A)*P(A)/P(B)", "assumptions": "A and B are events with P(B)>0; conditional probabilities are defined."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(P(A|B)) \\in {(P(B|A)*P(A)/P(B))}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "A and B are events with P(B)>0; conditional probabilities are defined.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover P(A|B) = P(B|A)*P(A)/P(B).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "P(A|B) = P(B|A)*P(A)/P(B)", "justification": "L in {R} iff L = R.", "preconditions": "A and B are events with P(B)>0; conditional probabilities are defined.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover P(A|B) = P(B|A)*P(A)/P(B).\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_0ca051fa53581be8", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "elementary algebra", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_67aee65f62f83258", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4CEAA916D1E3", "name": "Riemann–Siegel formula", "statement_latex": "ζ(s) = Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(ζ(s))\\Longleftrightarrow z=(Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s)))", "to_expression": "ζ(s) = Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s)", "justification": "same equality fiber iff equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ζ(s) = Σ[n=1,N] n^{-s} + γ(1-s) Σ[n=1,M] n^{s-1} + R(s).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_b2a4d6719c742303", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "combinatorics", "subfield": "combinatorial geometry", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_23E48784FFFD", "name": "Littlewood–Offord problem", "statement_latex": "P(S_n in A) <= binom(n,floor(n/2))/2^n", "assumptions": "S_n is the signed sum in the Littlewood-Offord setup; A is an interval or target set as specified by the theorem; n is positive."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((P(S_n in A))-(binom(n,floor(n/2))/2^n))=0", "to_expression": "P(S_n in A) <= binom(n,floor(n/2))/2^n", "justification": "sup equals max(0,X-Y)", "preconditions": "S_n is the signed sum in the Littlewood-Offord setup; A is an interval or target set as specified by the theorem; n is positive.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(S_n in A) ≤ binom(n,floor(n/2))/2^n.\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
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{"schema_version": "1.0", "item_id": "rcd2_9e868fe41c2c3ae3", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "Kleinian groups", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e15660d9614163eb", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_79DC6C43DD64", "name": "Perron–Frobenius theorem", "statement_latex": "Av = rv", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((Av),(rv))\\in\\{(z,z):z\\in D\\}", "to_expression": "Av = rv", "justification": "pair in diagonal iff X=Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Av = rv.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
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{"schema_version": "1.0", "item_id": "rcd2_8e5e3a43d1299b16", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "algebra", "subfield": "combinatorics", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_24005F6E3998", "name": "Abel's binomial theorem", "statement_latex": "Σ [k=0 to m] (m choose k) * (w + m - k)^(m - k - 1) * (z + k)^k = w^(-1) * (z + w + m)^m", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(Σ [k=0 to m] (m choose k) * (w + m - k)^(m - k - 1) * (z + k)^k))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(w^(-1) * (z + w + m)^m))", "to_expression": "Σ [k=0 to m] (m choose k) * (w + m - k)^(m - k - 1) * (z + k)^k = w^(-1) * (z + w + m)^m", "justification": "each argmin set is a singleton", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Σ [k=0 to m] (m choose k) * (w + m - k)^(m - k - 1) * (z + k)^k = w^(-1) * (z + w + m)^m.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_7941eae54953ab9a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B664293AE8AA", "name": "Etemadi's inequality", "statement_latex": "Pr(max_{1<=k<=n} |S_k| >= 3alpha) <= 3 max_{1<=k<=n} Pr(|S_k| >= alpha)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (3 max_{1≤k≤n} Pr(|S_k| ≥ alpha))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (Pr(max_{1≤k≤n} |S_k| ≥ 3alpha))\\}", "to_expression": "Pr(max_{1<=k<=n} |S_k| >= 3alpha) <= 3 max_{1<=k<=n} Pr(|S_k| >= alpha)", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr(max_{1≤k≤n} |S_k| ≥ 3alpha) ≤ 3 max_{1≤k≤n} Pr(|S_k| ≥ alpha).\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_42028074736de97b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_773C0118E21F", "name": "Darboux's theorem", "statement_latex": "f'(x) = y", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((f'(x)),(y))=0", "to_expression": "f'(x) = y", "justification": "zero discrete distance iff equality", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f'(x) = y.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_7332a36585688e05", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "polynomial factorization", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_299359E3C599", "name": "Sophie Germain's identity", "statement_latex": "x^{4}+4y^{4}=(x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2})", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((x^{4}+4y^{4}),z)=((x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2}),z)", "to_expression": "x^{4}+4y^{4}=(x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2})", "justification": "pair equality forces first coordinates equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as x^{4}+4y^{4}=(x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2}).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_5a1f91ae720f585a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_20033DA37507", "name": "Kolmogorov–Arnold representation theorem", "statement_latex": "f(x_1, ..., x_n) = Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p))", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((f(x_1, ..., x_n)))=F((Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p))))", "to_expression": "f(x_1, ..., x_n) = Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p))", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f(x_1, ..., x_n) = Σ_{q=0}^{2n} Φ_q(Σ_{p=1}^n φ_{q,p}(x_p)).\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_bde311b499ebdb06", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_736807BC5C8F", "name": "Brun–Titchmarsh theorem", "statement_latex": "pi(x;q,a) <= 2x / (phi(q) log(x/q))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\pi(x;q,a)) - (2x / (phi(q) log(x/q))) \\le 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A-B <= 0 iff A <= B by adding B to both sides.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover pi(x;q,a) <= 2x / (phi(q) log(x/q)).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A-B <= 0 iff A <= B by adding B to both sides.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover pi(x;q,a) <= 2x / (phi(q) log(x/q)).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "pi(x;q,a) <= 2x / (phi(q) log(x/q))", "justification": "A-B <= 0 iff A <= B by adding B to both sides.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover pi(x;q,a) <= 2x / (phi(q) log(x/q)).\"]"}], "source_equivalence_identity": "A-B <= 0 iff A <= B by adding B to both sides."}
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{"schema_version": "1.0", "item_id": "rcd2_1a60aab885387a94", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "incidence geometry", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B3E12B4FAA4C", "name": "Loomis–Whitney inequality", "statement_latex": "|| \\prod_{j=1}^{d} g_j \\circ \\pi_j ||_{L^1} \\le \\prod_{j=1}^{d} ||g_j||_{L^{d-1}}", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (|| \\prod_{j=1}^{d} g_j \\circ \\pi_j ||_{L^1}) \\le x \\land x \\le (\\prod_{j=1}^{d} ||g_j||_{L^{d-1}})", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover || \\\\prod_{j=1}^{d} g_j \\\\circ \\\\pi_j ||_{L^1} <= \\\\prod_{j=1}^{d} ||g_j||_{L^{d-1}}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover || \\\\prod_{j=1}^{d} g_j \\\\circ \\\\pi_j ||_{L^1} <= \\\\prod_{j=1}^{d} ||g_j||_{L^{d-1}}.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "|| \\prod_{j=1}^{d} g_j \\circ \\pi_j ||_{L^1} \\le \\prod_{j=1}^{d} ||g_j||_{L^{d-1}}", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover || \\\\prod_{j=1}^{d} g_j \\\\circ \\\\pi_j ||_{L^1} <= \\\\prod_{j=1}^{d} ||g_j||_{L^{d-1}}.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
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{"schema_version": "1.0", "item_id": "rcd2_9aab336cfffd8fc6", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_9e23199ba2775020", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_ABB949903DF7", "name": "Vandermonde's identity", "statement_latex": "{m+n \\choose r}=\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;({m+n \\choose r})=z\\;\\land\\;z=(\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k})", "to_expression": "{m+n \\choose r}=\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k}", "justification": "common witness iff equality", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as {m+n \\\\choose r}=\\\\sum _{k=0}^{r}{m \\\\choose k}{n \\\\choose r-k}.\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_0f7af6115a1c1441", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "combinatorics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_EE0D43BAAED5", "name": "Ahlswede–Daykin inequality", "statement_latex": "f_1(X)f_2(Y) ≤ f_3(X ∨ Y)f_4(X ∧ Y)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (f_3(X ∨ Y)f_4(X ∧ Y))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (f_1(X)f_2(Y))\\}", "to_expression": "f_1(X)f_2(Y) ≤ f_3(X ∨ Y)f_4(X ∧ Y)", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f_1(X)f_2(Y) ≤ f_3(X ∨ Y)f_4(X ∧ Y).\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_fbca91cac2968b7b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "incidence geometry", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B3E12B4FAA4C", "name": "Loomis–Whitney inequality", "statement_latex": "|| \\prod_{j=1}^{d} g_j \\circ \\pi_j ||_{L^1} \\le \\prod_{j=1}^{d} ||g_j||_{L^{d-1}}", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(|| \\prod_{j=1}^{d} g_j \\circ \\pi_j ||_{L^1})-(\\prod_{j=1}^{d} ||g_j||_{L^{d-1}})\\}=0", "to_expression": "|| \\prod_{j=1}^{d} g_j \\circ \\pi_j ||_{L^1} \\le \\prod_{j=1}^{d} ||g_j||_{L^{d-1}}", "justification": "positive part vanishes iff X≤Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as || \\\\prod_{j=1}^{d} g_j \\\\circ \\\\pi_j ||_{L^1} ≤ \\\\prod_{j=1}^{d} ||g_j||_{L^{d-1}}.\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_6274bbeb00902284", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "classical mechanics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BC55A9432F80", "name": "Helmholtz theorem (classical mechanics)", "statement_latex": "dS = (dE + PdV) / T", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((dS))=F(((dE + PdV) / T))", "to_expression": "dS = (dE + PdV) / T", "justification": "indicator of {X} separates unequal objects", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as dS = (dE + PdV) / T.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
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{"schema_version": "1.0", "item_id": "rcd2_8751337729d00930", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "Ramsey theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8E938CA76307", "name": "Milliken's tree theorem", "statement_latex": "S_T^n = C_1 ∪ ... ∪ C_r", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((S_T^n))=F((C_1 ∪ ... ∪ C_r))", "to_expression": "S_T^n = C_1 ∪ ... ∪ C_r", "justification": "indicator of {X} separates unequal objects", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as S_T^n = C_1 ∪ ... ∪ C_r.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
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{"schema_version": "1.0", "item_id": "rcd2_1b34fc9834c8b80e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "complex analysis", "subfield": "complex geometry", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A0642B3B222A", "name": "Schwarz lemma", "statement_latex": "|f(z)| <= |z|", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(|z|) - (|f(z)|) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |f(z)| <= |z|.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |f(z)| <= |z|.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "|f(z)| <= |z|", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |f(z)| <= |z|.\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
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{"schema_version": "1.0", "item_id": "rcd2_e065b4f130aa96a4", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c77701d429e1834b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "probability theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_27AD75237F2D", "name": "Entropy power inequality", "statement_latex": "N(X+Y) >= N(X) + N(Y)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(N(X+Y)) \\in (N(X) + N(Y)) + [0,\\infty)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover N(X+Y) >= N(X) + N(Y).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover N(X+Y) >= N(X) + N(Y).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "N(X+Y) >= N(X) + N(Y)", "justification": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover N(X+Y) >= N(X) + N(Y).\"]"}], "source_equivalence_identity": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_dcc77a81881dc9fd", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum field theory", "subfield": "axiomatic quantum field theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FEF97AED413B", "name": "Cluster decomposition property", "statement_latex": "S_βα → S_β1α1 * S_β2α2", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\varepsilon>0}\\inf_N\\mathbf{1}_{\\{\\forall n\\ge N:d((S_βα),(S_β1α1 * S_β2α2))<\\varepsilon\\}}=1", "to_expression": "S_βα → S_β1α1 * S_β2α2", "justification": "value one iff every epsilon has tail", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as S_βα → S_β1α1 * S_β2α2.\"]"}], "source_equivalence_identity": "value one iff every epsilon has tail"}
{"schema_version": "1.0", "item_id": "rcd2_0e7544fa0308cb6f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "representation theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D8B93C65D10E", "name": "McKay conjecture", "statement_latex": "|Irr_{p'}(G)|=|Irr_{p'}(N_{G}(P))|", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(|Irr_{p'}(G)|) \\in {(|Irr_{p'}(N_{G}(P))|)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover |Irr_{p'}(G)|=|Irr_{p'}(N_{G}(P))|.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "|Irr_{p'}(G)|=|Irr_{p'}(N_{G}(P))|", "justification": "L in {R} iff L = R.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover |Irr_{p'}(G)|=|Irr_{p'}(N_{G}(P))|.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_eab103abb55017b1", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "calculus", "subfield": "vector calculus / electrostatics", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_46f36bc88d1d6aed", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_24859b870420ccf5", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0EDEF2DF9656", "name": "Fundamental theorem of arithmetic", "statement_latex": "n = p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(n)=z\\;\\land\\;z=(p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k})", "to_expression": "n = p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k}", "justification": "common witness iff equality", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as n = p_1^{a_1} * p_2^{a_2} * ... * p_k^{a_k}.\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_18ea570ac1e86a55", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "symmetric functions", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4E5DD778CBB5", "name": "Newton's identities", "statement_latex": "k e_k = \\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((k e_k),z):z\\in D\\}=\\{((\\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i),z):z\\in D\\}", "to_expression": "k e_k = \\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as k e_k = \\\\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_f7c3b41a213805f3", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "number theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e58874c65a5f734e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "number theory", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_051344808DC5", "name": "Gauss congruence", "statement_latex": "sum_{d|n} mu(d)*a_{n/d} ≡ 0 (mod n)", "assumptions": "n is a positive integer; the displayed sum is an integer; mu is the Möbius function."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists k\\in\\mathbb{Z}:(sum_{d|n} mu(d)*a_{n/d})-(0)=k(n)", "to_expression": "sum_{d|n} mu(d)*a_{n/d} ≡ 0 (mod n)", "justification": "difference is integer multiple of M", "preconditions": "n is a positive integer; the displayed sum is an integer; mu is the Möbius function.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{d|n} mu(d)*a_{n/d} ≡ 0 (mod n).\"]"}], "source_equivalence_identity": "difference is integer multiple of M"}
{"schema_version": "1.0", "item_id": "rcd2_3347097ed4f0632e", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "cryptography", "subfield": "cryptanalysis", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2036e8f41d0bdae8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_93F7C5A7A33A", "name": "Sun's curious identity", "statement_latex": "(x+m+1)\\sum _{i=0}^{m}(-1)^{i}{\\dbinom {x+y+i}{m-i}}{\\dbinom {y+2i}{i}}-\\sum _{i=0}^{m}{\\dbinom {x+i}{m-i}}(-4)^{i}=(x-m){\\dbinom {x}{m}}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D(((x+m+1)\\sum _{i=0}^{m}(-1)^{i}{\\dbinom {x+y+i}{m-i}}{\\dbinom {y+2i}{i}}-\\sum _{i=0}^{m}{\\dbinom {x+i}{m-i}}(-4)^{i}),((x-m){\\dbinom {x}{m}}))=0", "to_expression": "(x+m+1)\\sum _{i=0}^{m}(-1)^{i}{\\dbinom {x+y+i}{m-i}}{\\dbinom {y+2i}{i}}-\\sum _{i=0}^{m}{\\dbinom {x+i}{m-i}}(-4)^{i}=(x-m){\\dbinom {x}{m}}", "justification": "zero discrete distance iff equality", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (x+m+1)\\\\sum _{i=0}^{m}(-1)^{i}{\\\\dbinom {x+y+i}{m-i}}{\\\\dbinom {y+2i}{i}}-\\\\sum _{i=0}^{m}{\\\\dbinom {x+i}{m-i}}(-4)^{i}=(x-m){\\\\dbinom {x}{m}}.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_f8b3b6fbbee2c75a", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6bbd506b88f42bcc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "functional analysis", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_462346AD2251", "name": "Ehrling's lemma", "statement_latex": "||x||_Y <= ε||x||_X + C(ε)||x||_Z", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((||x||_Y)-(ε||x||_X + C(ε)||x||_Z),(-\\infty,0])=0", "to_expression": "||x||_Y <= ε||x||_X + C(ε)||x||_Z", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ||x||_Y ≤ ε||x||_X + C(ε)||x||_Z.\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
{"schema_version": "1.0", "item_id": "rcd2_a110d3eda998e413", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "multilinear algebra", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d490cefb0bb1130e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "modular arithmetic", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A1DCD328739C", "name": "Hensel's lemma", "statement_latex": "f(x) = 0 (mod p^n)", "assumptions": "All displayed arithmetic expressions are integers; the modulus is a positive integer."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(f(x)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(0 (mod p^n)))", "to_expression": "f(x) = 0 (mod p^n)", "justification": "each argmin set is a singleton", "preconditions": "All displayed arithmetic expressions are integers; the modulus is a positive integer.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f(x) = 0 (mod p^n).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_0eac8de4b3aec383", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_7869af97cf76c9a6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "multilinear algebra", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F538F79B0750", "name": "Lagrange's identity", "statement_latex": "(\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2 = \\sum_{i<j} (a_i b_j - a_j b_i)^2", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: ((\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2 = \\sum_{i) < x \\land x < (j} (a_i b_j - a_j b_i)^2)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (\\\\sum a_k^2)(\\\\sum b_k^2) - (\\\\sum a_k b_k)^2 = \\\\sum_{i < j} (a_i b_j - a_j b_i)^2.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (\\\\sum a_k^2)(\\\\sum b_k^2) - (\\\\sum a_k b_k)^2 = \\\\sum_{i < j} (a_i b_j - a_j b_i)^2.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "(\\sum a_k^2)(\\sum b_k^2) - (\\sum a_k b_k)^2 = \\sum_{i<j} (a_i b_j - a_j b_i)^2", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (\\\\sum a_k^2)(\\\\sum b_k^2) - (\\\\sum a_k b_k)^2 = \\\\sum_{i < j} (a_i b_j - a_j b_i)^2.\"]"}], "source_equivalence_identity": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B."}
{"schema_version": "1.0", "item_id": "rcd2_618b2ea269c90a5a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_37DDB6F9F636", "name": "Duffin–Schaeffer theorem", "statement_latex": "\\sum_{q=1}^{\\infty}\\varphi(q)\\frac{f(q)}{q}=\\infty", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(\\sum_{q=1}^{\\infty}\\varphi(q)\\frac{f(q)}{q})=z\\;\\land\\;z=(\\infty)", "to_expression": "\\sum_{q=1}^{\\infty}\\varphi(q)\\frac{f(q)}{q}=\\infty", "justification": "common witness iff equality", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\sum_{q=1}^{\\\\infty}\\\\varphi(q)\\\\frac{f(q)}{q}=\\\\infty.\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_cf9715e5d821143a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "coding theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F6DC10428DD6", "name": "Z-channel capacity", "statement_latex": "cap(Z) = H(p/(1+2^s))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(cap(Z))\\Longleftrightarrow z=(H(p/(1+2^s))))", "to_expression": "cap(Z) = H(p/(1+2^s))", "justification": "same equality fiber iff equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as cap(Z) = H(p/(1+2^s)).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_7ed208b2b3419ecf", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "statistical mechanics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4D516E39AEB4", "name": "H-theorem", "statement_latex": "H(t) = ∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE) = (H(t))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover H(t) = ∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "H(t) = ∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover H(t) = ∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_46ea84216709210f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "design theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_41032C4E68AC", "name": "Bruck–Ryser–Chowla theorem", "statement_latex": "x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2} = 0", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2})\\Longleftrightarrow z=(0))", "to_expression": "x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2} = 0", "justification": "same equality fiber iff equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as x^{2} - (k - \\\\lambda)y^{2} - (-1)^{(v-1)/2} \\\\lambda z^{2} = 0.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_81d0947b2c828d85", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "hypergeometric series", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_94538B67A269", "name": "Dixon's identity", "statement_latex": "sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3 = (3a)! / (a!)^3", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,((3a)! / (a!)^3))", "to_expression": "sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3 = (3a)! / (a!)^3", "justification": "each argmin set is a singleton", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{k=-a}^{a} (-1)^k * (2a choose k+a)^3 = (3a)! / (a!)^3.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_53dae4d7ba6c0d2c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "calculus", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e492ef52e7947e98", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "field theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_9bc532fdf747294c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_68687FF65AAB", "name": "Landsberg–Schaar relation", "statement_latex": "1/sqrt(p) * sum_{n=0}^{p-1} exp(2*pi*i*n^2*q/p) = e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((1/sqrt(p) * sum_{n=0}^{p-1} exp(2*pi*i*n^2*q/p)),(e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q))))=0", "to_expression": "1/sqrt(p) * sum_{n=0}^{p-1} exp(2*pi*i*n^2*q/p) = e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q))", "justification": "zero discrete distance iff equality", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as 1/sqrt(p) * sum_{n=0}^{p-1} exp(2*pi*i*n^2*q/p) = e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q)).\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_c01b7117f9afbc8c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "functional analysis", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_462346AD2251", "name": "Ehrling's lemma", "statement_latex": "||x||_Y <= ε||x||_X + C(ε)||x||_Z", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(ε||x||_X + C(ε)||x||_Z) - (||x||_Y) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ||x||_Y <= ε||x||_X + C(ε)||x||_Z.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ||x||_Y <= ε||x||_X + C(ε)||x||_Z.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "||x||_Y <= ε||x||_X + C(ε)||x||_Z", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover ||x||_Y <= ε||x||_X + C(ε)||x||_Z.\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_cbbf9f35eabe4fb3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0FE477D29636", "name": "Hardy–Ramanujan theorem", "statement_latex": "|ω(n)−loglogn|<(loglogn)^(1/2+ε)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;t\\le (|ω(n)−loglogn|)\\Rightarrow t<(loglogn)^(1/2+ε)", "to_expression": "|ω(n)−loglogn|<(loglogn)^(1/2+ε)", "justification": "converse uses t=X", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |ω(n)−loglogn|<(loglogn)^(1/2+ε).\"]"}], "source_equivalence_identity": "converse uses t=X"}
{"schema_version": "1.0", "item_id": "rcd2_14175dd4112e6df2", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "number theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_82095829f3422637", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
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{"schema_version": "1.0", "item_id": "rcd2_fdad63a87b9df775", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5A3D942AED87", "name": "Markov's inequality", "statement_latex": "P(X >= a) <= E[X] / a", "assumptions": "X is a nonnegative random variable; a>0; E[X] is finite."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((P(X ≥ a))-(E[X] / a))=0", "to_expression": "P(X >= a) <= E[X] / a", "justification": "sup equals max(0,X-Y)", "preconditions": "X is a nonnegative random variable; a>0; E[X] is finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(X ≥ a) ≤ E[X] / a.\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
{"schema_version": "1.0", "item_id": "rcd2_757ea1cc449befad", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6c287e51690c6850", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "algebraic combinatorics", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_95918AB9C155", "name": "Kruskal–Katona theorem", "statement_latex": "N^{(i-1)}=\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(N^{(i-1)})=z\\;\\land\\;z=(\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1})", "to_expression": "N^{(i-1)}=\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1}", "justification": "common witness iff equality", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as N^{(i-1)}=\\\\binom{n_i}{i-1}+\\\\binom{n_{i-1}}{i-2}+\\\\dots+\\\\binom{n_j}{j-1}.\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_68c6251dac3ab253", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "number theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_893C1D2D0065", "name": "Fermat's Last Theorem", "statement_latex": "a^n + b^n = c^n", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(a^n + b^n))+\\delta_D(z,(c^n)))=0", "to_expression": "a^n + b^n = c^n", "justification": "zero loss iff some z equals both", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as a^n + b^n = c^n.\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
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{"schema_version": "1.0", "item_id": "rcd2_aa55396e7a68b228", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C321BE7114E8", "name": "Ahlswede–Khachatrian theorem", "statement_latex": "|F| <= max_{r: t+2r <= n} |F_{n,k,t,r}|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (|F| \\le max_{r: t+2r \\le n} |F_{n,k,t,r}|))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "|F| <= max_{r: t+2r <= n} |F_{n,k,t,r}|", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
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{"schema_version": "1.0", "item_id": "rcd2_f2ac2affe38d22f4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "martingale theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1E094F8EB368", "name": "Azuma's inequality", "statement_latex": "P(X_N - X_0 >= epsilon) <= exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2)))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (P(X_N - X_0 ≥ epsilon))\\}", "to_expression": "P(X_N - X_0 >= epsilon) <= exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2))", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(X_N - X_0 ≥ epsilon) ≤ exp(-epsilon^2 / (2 * sum_{k=1}^N c_k^2)).\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_2b0e43b626063df8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "special functions", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CA3873AC4ACD", "name": "Ramanujan's master theorem", "statement_latex": "∫[0,∞] x^{s-1} f(x) dx = Γ(s) φ(-s)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(∫[0,∞] x^{s-1} f(x) dx)\\}=\\{(Γ(s) φ(-s))\\}", "to_expression": "∫[0,∞] x^{s-1} f(x) dx = Γ(s) φ(-s)", "justification": "singletons equal iff elements equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∫[0,∞] x^{s-1} f(x) dx = Γ(s) φ(-s).\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_49c591676dc9cefa", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "matrix theory", "subfield": "linear algebra", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_6095A64AB713", "name": "Golden–Thompson inequality", "statement_latex": "tr(exp(A+B)) <= tr(exp(A)*exp(B))", "assumptions": "A and B are Hermitian matrices of the same finite size."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (tr(exp(A)*exp(B)))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (tr(exp(A+B)))\\}", "to_expression": "tr(exp(A+B)) <= tr(exp(A)*exp(B))", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "A and B are Hermitian matrices of the same finite size.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as tr(exp(A+B)) ≤ tr(exp(A)*exp(B)).\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_8c8636fee8205870", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "statistics", "subfield": "empirical Bayes methods", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_71fa4d0a88309b7b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3BD55A296126", "name": "Rank–nullity theorem", "statement_latex": "rank(T) + nullity(T) = dim(V)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(rank(T) + nullity(T))\\}=\\{(dim(V))\\}", "to_expression": "rank(T) + nullity(T) = dim(V)", "justification": "singletons equal iff elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as rank(T) + nullity(T) = dim(V).\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_2fa263315a3976f3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal combinatorics", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C321BE7114E8", "name": "Ahlswede–Khachatrian theorem", "statement_latex": "|F| <= max_{r: t+2r <= n} |F_{n,k,t,r}|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((|F|)-(max_{r: t+2r ≤ n} |F_{n,k,t,r}|))=0", "to_expression": "|F| <= max_{r: t+2r <= n} |F_{n,k,t,r}|", "justification": "sup equals max(0,X-Y)", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |F| ≤ max_{r: t+2r ≤ n} |F_{n,k,t,r}|.\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
{"schema_version": "1.0", "item_id": "rcd2_decc735c497f3351", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "infinite-dimensional vector spaces", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B23BCA4AB254", "name": "Erdős–Kaplansky theorem", "statement_latex": "\\dim(E^*) = \\text{card}(\\mathbb{K}^I)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\dim(E^*)) \\in {(\\text{card}(\\mathbb{K}^I))}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\dim(E^*) = \\\\text{card}(\\\\mathbb{K}^I).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\dim(E^*) = \\text{card}(\\mathbb{K}^I)", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\dim(E^*) = \\\\text{card}(\\\\mathbb{K}^I).\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
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{"schema_version": "1.0", "item_id": "rcd2_1ec90c2cc2eeb89f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9103F62922E0", "name": "Multidimensional Chebyshev's inequality", "statement_latex": "Pr(sqrt((X-μ)^T * V^-1 * (X-μ)) > t) <= N / t^2", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(N / t^2) - (Pr(sqrt((X-μ)^T * V^-1 * (X-μ)) > t)) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Pr(sqrt((X-μ)^T * V^-1 * (X-μ)) > t) <= N / t^2.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Pr(sqrt((X-μ)^T * V^-1 * (X-μ)) > t) <= N / t^2.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "Pr(sqrt((X-μ)^T * V^-1 * (X-μ)) > t) <= N / t^2", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Pr(sqrt((X-μ)^T * V^-1 * (X-μ)) > t) <= N / t^2.\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
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{"schema_version": "1.0", "item_id": "rcd2_e107d9746c6b878e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "vector calculus", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_24234F1E6547", "name": "Divergence theorem", "statement_latex": "\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S} = \\int_{V} (\\nabla \\cdot \\mathbf{F}) dV", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S}))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\int_{V} (\\nabla \\cdot \\mathbf{F}) dV))", "to_expression": "\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S} = \\int_{V} (\\nabla \\cdot \\mathbf{F}) dV", "justification": "each argmin set is a singleton", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\oint_{S} \\\\mathbf{F} \\\\cdot d\\\\mathbf{S} = \\\\int_{V} (\\\\nabla \\\\cdot \\\\mathbf{F}) dV.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_0b4084ae550b26dc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "modular arithmetic", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A1DCD328739C", "name": "Hensel's lemma", "statement_latex": "f(x) = 0 (mod p^n)", "assumptions": "All displayed arithmetic expressions are integers; the modulus is a positive integer."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(0 (mod p^n)) = (f(x))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic expressions are integers; the modulus is a positive integer.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(x) = 0 (mod p^n).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f(x) = 0 (mod p^n)", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic expressions are integers; the modulus is a positive integer.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(x) = 0 (mod p^n).\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_8c40a070bc60595b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "Algebra", "subfield": "Lie algebras", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AE35A2B1281A", "name": "Jacobi identity", "statement_latex": "x * (y * z) + y * (z * x) + z * (x * y) = 0", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(x * (y * z) + y * (z * x) + z * (x * y))\\}=\\{(0)\\}", "to_expression": "x * (y * z) + y * (z * x) + z * (x * y) = 0", "justification": "singletons equal iff elements equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as x * (y * z) + y * (z * x) + z * (x * y) = 0.\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
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{"schema_version": "1.0", "item_id": "rcd2_f19c28495a7a9c45", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal combinatorics", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C321BE7114E8", "name": "Ahlswede–Khachatrian theorem", "statement_latex": "|F| <= max_{r: t+2r <= n} |F_{n,k,t,r}|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(max_{r: t+2r ≤ n} |F_{n,k,t,r}|)\\le t\\Rightarrow (|F|)\\le t", "to_expression": "|F| <= max_{r: t+2r <= n} |F_{n,k,t,r}|", "justification": "converse uses t=Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |F| ≤ max_{r: t+2r ≤ n} |F_{n,k,t,r}|.\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_1ba34d11172bbc57", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "vector calculus", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2E42699EEB3D", "name": "Gradient theorem", "statement_latex": "∫_{γ} ∇φ(r) · dr = φ(q) - φ(p)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((∫_{γ} ∇φ(r) · dr),z)=((φ(q) - φ(p)),z)", "to_expression": "∫_{γ} ∇φ(r) · dr = φ(q) - φ(p)", "justification": "pair equality forces first coordinates equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∫_{γ} ∇φ(r) · dr = φ(q) - φ(p).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_469f7d82e4f620fa", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "Ramsey theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_884099ae89e156b1", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "prime numbers", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a11ace3716918907", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "measure theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3D75E63AA146", "name": "Vitale's random Brunn–Minkowski inequality", "statement_latex": "(vol_n(E[X]))^(1/n) >= E[vol_n(X)^(1/n)]", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (E[vol_n(X)^(1/n)]) \\le x \\land x \\le ((vol_n(E[X]))^(1/n))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (vol_n(E[X]))^(1/n) >= E[vol_n(X)^(1/n)].\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (vol_n(E[X]))^(1/n) >= E[vol_n(X)^(1/n)].\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "(vol_n(E[X]))^(1/n) >= E[vol_n(X)^(1/n)]", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (vol_n(E[X]))^(1/n) >= E[vol_n(X)^(1/n)].\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_aff78b3ee4c134d6", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_79d9cce6d53dc0bc", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "harmonic analysis", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_9e4c850b6373d8cc", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "approximation theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_986f5b990e782a17", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "numerical analysis, finite element method", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a28a2ddb9397fdfc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "physics", "subfield": "quantum field theory", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5C335B571C48", "name": "Gell-Mann and Low theorem", "statement_latex": "H = H_0 + gV", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(H)=z\\;\\land\\;z=(H_0 + gV)", "to_expression": "H = H_0 + gV", "justification": "common witness iff equality", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as H = H_0 + gV.\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_690ffce80d5a8a0f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_EAE642263ABE", "name": "Catalan's conjecture", "statement_latex": "x^a - y^b = 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(1) = (x^a - y^b)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover x^a - y^b = 1.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "x^a - y^b = 1", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover x^a - y^b = 1.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_55c48d14e457a143", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "permutation patterns", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8A6780273F9B", "name": "Stanley–Wilf conjecture", "statement_latex": "lim_{n→∞} |S_n(β)|^(1/n)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\forall m\\in\\mathbb{N}_{>0}\\exists N_m\\forall n\\ge N_m:(lim_{n)\\in B((∞} |S_n(β)|^(1/n)),1/m)", "to_expression": "lim_{n→∞} |S_n(β)|^(1/n)", "justification": "balls of radius 1/m form base", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{n→∞} |S_n(β)|^(1/n).\"]"}], "source_equivalence_identity": "balls of radius 1/m form base"}
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{"schema_version": "1.0", "item_id": "rcd2_cf1092f1cb4000ca", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "additive number theory", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_870FF97782B2", "name": "Goldbach's weak conjecture", "statement_latex": "n = p1 + p2 + p3", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(n)=z\\;\\land\\;z=(p1 + p2 + p3)", "to_expression": "n = p1 + p2 + p3", "justification": "common witness iff equality", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as n = p1 + p2 + p3.\"]"}], "source_equivalence_identity": "common witness iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_c750863af7dac028", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "number theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A54D9D349D9E", "name": "Candido's identity", "statement_latex": "[x^2 + y^2 + (x+y)^2]^2 = 2[x^4 + y^4 + (x+y)^4]", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,([x^2 + y^2 + (x+y)^2]^2))+\\delta_D(z,(2[x^4 + y^4 + (x+y)^4])))=0", "to_expression": "[x^2 + y^2 + (x+y)^2]^2 = 2[x^4 + y^4 + (x+y)^4]", "justification": "zero loss iff some z equals both", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as [x^2 + y^2 + (x+y)^2]^2 = 2[x^4 + y^4 + (x+y)^4].\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_a353a9af95c4895f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "metric geometry", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BDB71158A694", "name": "Besicovitch inequality", "statement_latex": "prod_{i} d_{i} <= Vol(M)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(prod_{i} d_{i})}{\\exp(prod_{i} d_{i})+\\exp(Vol(M))}\\le\\frac{1}{2}", "to_expression": "prod_{i} d_{i} <= Vol(M)", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as prod_{i} d_{i} ≤ Vol(M).\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_5ab802b8a09b4284", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_69A821A9D880", "name": "Hockey-stick identity", "statement_latex": "\\sum_{i=r}^{n} \\binom{i}{r} = \\binom{n+1}{r+1}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(\\sum_{i=r}^{n} \\binom{i}{r})\\Longleftrightarrow z=(\\binom{n+1}{r+1}))", "to_expression": "\\sum_{i=r}^{n} \\binom{i}{r} = \\binom{n+1}{r+1}", "justification": "same equality fiber iff equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\sum_{i=r}^{n} \\\\binom{i}{r} = \\\\binom{n+1}{r+1}.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_5ed72a11b9890e90", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "probability theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_27AD75237F2D", "name": "Entropy power inequality", "statement_latex": "N(X+Y) >= N(X) + N(Y)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (N(X) + N(Y)) \\le x \\land x \\le (N(X+Y))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover N(X+Y) >= N(X) + N(Y).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover N(X+Y) >= N(X) + N(Y).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "N(X+Y) >= N(X) + N(Y)", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover N(X+Y) >= N(X) + N(Y).\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_b5de283aabc12eb5", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_DB1D004A0930", "name": "Cramer's rule", "statement_latex": "x_i = \\det(A_i) / \\det(A)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(x_i))+\\delta_D(z,(\\det(A_i) / \\det(A))))=0", "to_expression": "x_i = \\det(A_i) / \\det(A)", "justification": "zero loss iff some z equals both", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as x_i = \\\\det(A_i) / \\\\det(A).\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
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{"schema_version": "1.0", "item_id": "rcd2_2a6e5fc8efa4c8e8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "combinatorics", "subfield": "combinatorial geometry", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_23E48784FFFD", "name": "Littlewood–Offord problem", "statement_latex": "P(S_n in A) <= binom(n,floor(n/2))/2^n", "assumptions": "S_n is the signed sum in the Littlewood-Offord setup; A is an interval or target set as specified by the theorem; n is positive."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(P(S_n in A))+s=(binom(n,floor(n/2))/2^n)", "to_expression": "P(S_n in A) <= binom(n,floor(n/2))/2^n", "justification": "nonnegative slack exists iff X≤Y", "preconditions": "S_n is the signed sum in the Littlewood-Offord setup; A is an interval or target set as specified by the theorem; n is positive.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(S_n in A) ≤ binom(n,floor(n/2))/2^n.\"]"}], "source_equivalence_identity": "nonnegative slack exists iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_61d71babcecb614d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "geometry of numbers", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_638EC6734424", "name": "Minkowski–Hlawka theorem", "statement_latex": "Δ ≥ ζ(n) / 2^(n-1)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (ζ(n) / 2^(n-1)) \\le x \\land x \\le (Δ)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Δ >= ζ(n) / 2^(n-1).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Δ >= ζ(n) / 2^(n-1).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "Δ ≥ ζ(n) / 2^(n-1)", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Δ >= ζ(n) / 2^(n-1).\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
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{"schema_version": "1.0", "item_id": "rcd2_bfc69b1311bb3c2a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "Ramsey theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_05F0F9A888B8", "name": "Erdős–Szekeres theorem", "statement_latex": "n = (r - 1)(s - 1) + 1", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((n),z):z\\in D\\}=\\{(((r - 1)(s - 1) + 1),z):z\\in D\\}", "to_expression": "n = (r - 1)(s - 1) + 1", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as n = (r - 1)(s - 1) + 1.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_338164c54fc253cf", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "representation theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1292421DFB1E", "name": "Schur's lemma", "statement_latex": "ρ_W(g) ∘ f = f ∘ ρ_V(g)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "(\\{(ρ_W(g) ∘ f)\\}\\times\\{(f ∘ ρ_V(g))\\})\\cap\\{(z,z):z\\in D\\}=\\{((ρ_W(g) ∘ f),(f ∘ ρ_V(g)))\\}", "to_expression": "ρ_W(g) ∘ f = f ∘ ρ_V(g)", "justification": "intersection is singleton iff pair diagonal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ρ_W(g) ∘ f = f ∘ ρ_V(g).\"]"}], "source_equivalence_identity": "intersection is singleton iff pair diagonal"}
{"schema_version": "1.0", "item_id": "rcd2_fd43ff4b857f5551", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "large deviations theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B2BA1A96D211", "name": "Varadhan's lemma", "statement_latex": "lim_{ε→0} ε log E[exp(φ(Z_ε)/ε)] = sup_{x∈X} (φ(x) - I(x))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((lim_{ε→0} ε log E[exp(φ(Z_ε)/ε)]),z)=((sup_{x∈X} (φ(x) - I(x))),z)", "to_expression": "lim_{ε→0} ε log E[exp(φ(Z_ε)/ε)] = sup_{x∈X} (φ(x) - I(x))", "justification": "pair equality forces first coordinates equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{ε→0} ε log E[exp(φ(Z_ε)/ε)] = sup_{x∈X} (φ(x) - I(x)).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
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{"schema_version": "1.0", "item_id": "rcd2_0f8624cc27e697cf", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3DB69D9F31E5", "name": "Reynolds transport theorem", "statement_latex": "d/dt integral_{Omega(t)} f dV = integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(d/dt integral_{Omega(t)} f dV))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA))", "to_expression": "d/dt integral_{Omega(t)} f dV = integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA", "justification": "each argmin set is a singleton", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as d/dt integral_{Omega(t)} f dV = integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_399272e1d9f6529c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "prime number theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BFDCE2A9DBB2", "name": "Bertrand's postulate", "statement_latex": "exists prime p such that n < p < 2*n", "assumptions": "n is an integer greater than 1."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\neg\\forall x\\in D,x\\notin \\{prime p\\in D:n < p < 2*n\\}", "to_expression": "exists prime p such that n < p < 2*n", "justification": "not all nonmembers iff member exists", "preconditions": "n is an integer greater than 1.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as exists prime p such that n < p < 2*n.\"]"}], "source_equivalence_identity": "not all nonmembers iff member exists"}
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{"schema_version": "1.0", "item_id": "rcd2_5e2f163808974868", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "graph theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_57B1E9BA3579", "name": "Crossing number inequality", "statement_latex": "cr(G) >= e^3 / (29n^2)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(e^3 / (29n^2))-(cr(G))\\}=0", "to_expression": "cr(G) >= e^3 / (29n^2)", "justification": "positive part vanishes iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as cr(G) ≥ e^3 / (29n^2).\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_8c9824c4c90bb513", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "cryptography", "subfield": "cryptanalysis", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E35608C3DE5E", "name": "Piling-up lemma", "statement_latex": "ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn) = 2^{n-1}∏_{i=1}^{n}ε(Xi)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn) = 2^{n-1}∏_{i=1}^{n}ε(Xi)))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn) = 2^{n-1}∏_{i=1}^{n}ε(Xi)", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
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{"schema_version": "1.0", "item_id": "rcd2_9b42300ea16d377d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "calculus", "subfield": "integral calculus", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F8E7F707512F", "name": "Bioche's rules", "statement_latex": "ω(-t) = ω(t)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((ω(-t)) \\in S \\land (ω(t)) \\notin S) \\lor ((ω(t)) \\in S \\land (ω(-t)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover ω(-t) = ω(t).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "ω(-t) = ω(t)", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover ω(-t) = ω(t).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
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{"schema_version": "1.0", "item_id": "rcd2_064bdc489c193261", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "algebraic combinatorics", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_95918AB9C155", "name": "Kruskal–Katona theorem", "statement_latex": "N^{(i-1)}=\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(N^{(i-1)})\\Longleftrightarrow z=(\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1}))", "to_expression": "N^{(i-1)}=\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1}", "justification": "same equality fiber iff equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as N^{(i-1)}=\\\\binom{n_i}{i-1}+\\\\binom{n_{i-1}}{i-2}+\\\\dots+\\\\binom{n_j}{j-1}.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
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{"schema_version": "1.0", "item_id": "rcd2_570a1a3a1885a1ab", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "graph theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_37489533E622", "name": "Dinitz theorem", "statement_latex": "chi'_l(K_{n,n}) = n", "assumptions": "K_{n,n} is the complete bipartite graph; chi_l prime denotes list chromatic index; n is a positive integer."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(chi'_l(K_{n,n})) \\in {(n)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "K_{n,n} is the complete bipartite graph; chi_l prime denotes list chromatic index; n is a positive integer.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover chi'_l(K_{n,n}) = n.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "chi'_l(K_{n,n}) = n", "justification": "L in {R} iff L = R.", "preconditions": "K_{n,n} is the complete bipartite graph; chi_l prime denotes list chromatic index; n is a positive integer.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover chi'_l(K_{n,n}) = n.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
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{"schema_version": "1.0", "item_id": "rcd2_a786d18fc023cf5a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_773C0118E21F", "name": "Darboux's theorem", "statement_latex": "f'(x) = y", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((f'(x)),z)=((y),z)", "to_expression": "f'(x) = y", "justification": "pair equality forces first coordinates equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f'(x) = y.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_9d4ab1bd41b4534d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9F50AE0DD0D5", "name": "Cauchy–Davenport theorem", "statement_latex": "|A+B| >= min(|A|+|B|-1, p)", "assumptions": "A and B are nonempty finite subsets of Z/pZ; p is prime."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(min(|A|+|B|-1, p))+s=(|A+B|)", "to_expression": "|A+B| >= min(|A|+|B|-1, p)", "justification": "nonnegative slack exists iff X≤Y", "preconditions": "A and B are nonempty finite subsets of Z/pZ; p is prime.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p).\"]"}], "source_equivalence_identity": "nonnegative slack exists iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_db129acb42e793d7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "computer science", "subfield": "theoretical computer science", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_30A4875B9AAE", "name": "Isolation lemma", "statement_latex": "w(S) = sum_{x in S} w(x)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((w(S)))=F((sum_{x in S} w(x)))", "to_expression": "w(S) = sum_{x in S} w(x)", "justification": "indicator of {X} separates unequal objects", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as w(S) = sum_{x in S} w(x).\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_c1a5cce5015a50b4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum mechanics", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9078CE9B3CD4", "name": "Ehrenfest theorem", "statement_latex": "F = -V'(x)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((F),z)=((-V'(x)),z)", "to_expression": "F = -V'(x)", "justification": "pair equality forces first coordinates equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as F = -V'(x).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
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{"schema_version": "1.0", "item_id": "rcd2_59ffa794b1b0fc28", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "analysis", "subfield": "complex analysis", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_065364B1C6FB", "name": "Arakelyan's theorem", "statement_latex": "|g − f| < ε", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (|g − f|) < x \\land x < (ε)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |g − f| < ε.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |g − f| < ε.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "|g − f| < ε", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |g − f| < ε.\"]"}], "source_equivalence_identity": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B."}
{"schema_version": "1.0", "item_id": "rcd2_5e6ebce1d593e46b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "probabilistic number theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_65b0dd81bde89cb0", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "combinatorics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6bad360f9ff91e22", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "harmonic analysis", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d64809a4cf732c0c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_4c78bce5fc70cce2", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum information theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c9120a719f8fb7d8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "graph theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_37489533E622", "name": "Dinitz theorem", "statement_latex": "chi'_l(K_{n,n}) = n", "assumptions": "K_{n,n} is the complete bipartite graph; chi_l prime denotes list chromatic index; n is a positive integer."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((chi'_l(K_{n,n})) \\in S \\land (n) \\notin S) \\lor ((n) \\in S \\land (chi'_l(K_{n,n})) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "K_{n,n} is the complete bipartite graph; chi_l prime denotes list chromatic index; n is a positive integer.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover chi'_l(K_{n,n}) = n.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "chi'_l(K_{n,n}) = n", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "K_{n,n} is the complete bipartite graph; chi_l prime denotes list chromatic index; n is a positive integer.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover chi'_l(K_{n,n}) = n.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_3c62d3ba713a12c9", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "quadratic forms", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_4cb6986228ff3daa", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_763D96A1B9F8", "name": "Rouché–Capelli theorem", "statement_latex": "rank(A) = rank([A|b])", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((rank(A)),z)=((rank([A|b])),z)", "to_expression": "rank(A) = rank([A|b])", "justification": "pair equality forces first coordinates equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as rank(A) = rank([A|b]).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_e4a323b1dcf865cc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "numerical linear algebra", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FBD08849D0F3", "name": "Kreiss matrix theorem", "statement_latex": "K(A) = sup_{|z|>1} (|z|-1) ||(z-A)^-1||", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "-(K(A) = sup_{|z|) < -(1} (|z|-1) ||(z-A)^-1||)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "x -> -x is strictly decreasing, so A<B iff -B<-A.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover K(A) = sup_{|z| > 1} (|z|-1) ||(z-A)^-1||.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "x -> -x is strictly decreasing, so A<B iff -B<-A.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover K(A) = sup_{|z| > 1} (|z|-1) ||(z-A)^-1||.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "K(A) = sup_{|z|>1} (|z|-1) ||(z-A)^-1||", "justification": "x -> -x is strictly decreasing, so A<B iff -B<-A.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover K(A) = sup_{|z| > 1} (|z|-1) ||(z-A)^-1||.\"]"}], "source_equivalence_identity": "x -> -x is strictly decreasing, so A<B iff -B<-A."}
{"schema_version": "1.0", "item_id": "rcd2_78deff716579e260", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "Algebra", "subfield": "Lie algebras", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AE35A2B1281A", "name": "Jacobi identity", "statement_latex": "x * (y * z) + y * (z * x) + z * (x * y) = 0", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(0) = (x * (y * z) + y * (z * x) + z * (x * y))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover x * (y * z) + y * (z * x) + z * (x * y) = 0.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "x * (y * z) + y * (z * x) + z * (x * y) = 0", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover x * (y * z) + y * (z * x) + z * (x * y) = 0.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_a806c3cddb1f1ef5", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "statistics", "subfield": "empirical Bayes methods", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2C7BEBF4ECBC", "name": "Robbins lemma", "statement_latex": "E(Xf(X-1)) = λE(f(X))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(λE(f(X))) = (E(Xf(X-1)))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E(Xf(X-1)) = λE(f(X)).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "E(Xf(X-1)) = λE(f(X))", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E(Xf(X-1)) = λE(f(X)).\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_978741094d647b86", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "geometry", "subfield": "discrete geometry", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e5e0eca5e4e7de72", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum field theory", "subfield": "axiomatic quantum field theory", "primary_mechanism": "quantifier", "transformation_category": "counting_combinatorial", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FEF97AED413B", "name": "Cluster decomposition property", "statement_latex": "S_βα → S_β1α1 * S_β2α2", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall k\\in\\mathbb{N}_{>0}\\exists N\\forall n\\ge N:d((S_βα),(S_β1α1 * S_β2α2))<1/k", "to_expression": "S_βα → S_β1α1 * S_β2α2", "justification": "1/k base is cofinal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as S_βα → S_β1α1 * S_β2α2.\"]"}], "source_equivalence_identity": "1/k base is cofinal"}
{"schema_version": "1.0", "item_id": "rcd2_45307f61cce73a11", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "matching theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_74807724c1680ac1", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "Algebra", "subfield": "Lie algebras", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0315a7307fe31636", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "matroid theory", "subfield": "combinatorics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_09298FD96036", "name": "Ingleton's inequality", "statement_latex": "ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4) ≤ ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4))\\}", "to_expression": "ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4) ≤ ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4)", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ρ(X1)+ρ(X2) + ρ(X1∪X2∪X3) + ρ(X1∪X2∪X4) + ρ(X3∪X4) ≤ ρ(X1∪X2) + ρ(X1∪X3) + ρ(X1∪X4) + ρ(X2∪X3) + ρ(X2∪X4).\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_5b023de788e96cb3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "combinatorics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_843AE41690C1", "name": "q-Vandermonde identity", "statement_latex": "binom(m+n, k)_q = sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(binom(m+n, k)_q) \\in {(sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)})}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover binom(m+n, k)_q = sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "binom(m+n, k)_q = sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)}", "justification": "L in {R} iff L = R.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover binom(m+n, k)_q = sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)}.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_f836b9749f40cb1e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "harmonic analysis", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_ECE01C7A6C37", "name": "Wiener's lemma", "statement_latex": "lim_{N->inf} (1/(2N+1)) sum_{n=-N}^{N} |widehat{mu}(n)|^2 = sum_{j} |c_{j}|^2", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((lim_{N→inf} (1/(2N+1)) sum_{n=-N}^{N} |widehat{mu}(n)|^2),z)=((sum_{j} |c_{j}|^2),z)", "to_expression": "lim_{N->inf} (1/(2N+1)) sum_{n=-N}^{N} |widehat{mu}(n)|^2 = sum_{j} |c_{j}|^2", "justification": "pair equality forces first coordinates equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{N→inf} (1/(2N+1)) sum_{n=-N}^{N} |widehat{mu}(n)|^2 = sum_{j} |c_{j}|^2.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
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{"schema_version": "1.0", "item_id": "rcd2_1dd558d1fc5193d4", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_4d5cd037c5142016", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "measure theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3D75E63AA146", "name": "Vitale's random Brunn–Minkowski inequality", "statement_latex": "(vol_n(E[X]))^(1/n) >= E[vol_n(X)^(1/n)]", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((E[vol_n(X)^(1/n)])-(vol_n(E[X]))^(1/n),(-\\infty,0])=0", "to_expression": "(vol_n(E[X]))^(1/n) >= E[vol_n(X)^(1/n)]", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (vol_n(E[X]))^(1/n) ≥ E[vol_n(X)^(1/n)].\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
{"schema_version": "1.0", "item_id": "rcd2_5f8c10395ed3fac7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0506260C92DB", "name": "Bennett's inequality", "statement_latex": "Pr(S_n > t) <= exp(-(sigma^2 / a^2) * h(at / sigma^2))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (exp(-(sigma^2 / a^2) * h(at / sigma^2)))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (Pr(S_n > t))\\}", "to_expression": "Pr(S_n > t) <= exp(-(sigma^2 / a^2) * h(at / sigma^2))", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr(S_n > t) ≤ exp(-(sigma^2 / a^2) * h(at / sigma^2)).\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_9a6b1a387ebda813", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "inequalities", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ef2c122056fa57e0", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal set theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e4455088088ee3b9", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "quadratic residues", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_451c3a4a04197187", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_3ec6a643f47a1bed", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "functional analysis", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_3c99b65646cd5a84", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "symbolic dynamics", "subfield": "cellular automata", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6b3934d55292fb7e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "Ramsey theory", "subfield": "discrete mathematics", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3F2BB2E376A3", "name": "Rado's theorem", "statement_latex": "Ax = 0", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((Ax),z):z\\in D\\}=\\{((0),z):z\\in D\\}", "to_expression": "Ax = 0", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Ax = 0.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_f76c548144ed8a19", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "field theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_6E9382B37B92", "name": "Primitive element theorem", "statement_latex": "E = F(α)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((E),(F(α)))=0", "to_expression": "E = F(α)", "justification": "zero discrete distance iff equality", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as E = F(α).\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_4b6606b13db175c1", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "computer algebra", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_816a334f7186db49", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "Ramsey theory", "subfield": "discrete mathematics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4B3C8E83990", "name": "Schur's theorem", "statement_latex": "x + y = z", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((x + y),(z))=0", "to_expression": "x + y = z", "justification": "zero discrete distance iff equality", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as x + y = z.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_2fe8bcfaf55d0a59", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "graph theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_37489533E622", "name": "Dinitz theorem", "statement_latex": "chi'_l(K_{n,n}) = n", "assumptions": "K_{n,n} is the complete bipartite graph; chi_l prime denotes list chromatic index; n is a positive integer."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(n) = (chi'_l(K_{n,n}))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "K_{n,n} is the complete bipartite graph; chi_l prime denotes list chromatic index; n is a positive integer.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover chi'_l(K_{n,n}) = n.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "chi'_l(K_{n,n}) = n", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "K_{n,n} is the complete bipartite graph; chi_l prime denotes list chromatic index; n is a positive integer.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover chi'_l(K_{n,n}) = n.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_f841faaa26d2852b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematical analysis", "subfield": "analytic number theory", "primary_mechanism": "distance", "transformation_category": "residual_nonnegativity", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6af427263bc3c0dc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_41098B279612", "name": "Girsanov theorem", "statement_latex": "E(X)_t = exp(X_t - 1/2 [X]_t)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\exp(X_t - 1/2 [X]_t)) = (E(X)_t)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E(X)_t = exp(X_t - 1/2 [X]_t).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "E(X)_t = exp(X_t - 1/2 [X]_t)", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E(X)_t = exp(X_t - 1/2 [X]_t).\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_f7ff13fc4dc34fd4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "quadratic forms", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_434E3E7B6A54", "name": "Sylvester's law of inertia", "statement_latex": "B = SAS^T", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((B),(SAS^T))=0", "to_expression": "B = SAS^T", "justification": "zero discrete distance iff equality", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as B = SAS^T.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_1aa9903bc437b253", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "calculus", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_88C089C7703F", "name": "Leibniz integral rule", "statement_latex": "d/dx integral_{a(x)}^{b(x)} f(x,t) dt = f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((d/dx integral_{a(x)}^{b(x)} f(x,t) dt),(f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt))\\in\\{(z,z):z\\in D\\}", "to_expression": "d/dx integral_{a(x)}^{b(x)} f(x,t) dt = f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt", "justification": "pair in diagonal iff X=Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as d/dx integral_{a(x)}^{b(x)} f(x,t) dt = f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_b10f5d086a7c4aed", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5795F6B3279F", "name": "Chain rule (probability)", "statement_latex": "P(A ∩ B) = P(B|A)*P(A)", "assumptions": "A and B are events with P(A)>0 for the displayed conditional form."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((P(A ∩ B)))=F((P(B|A)*P(A)))", "to_expression": "P(A ∩ B) = P(B|A)*P(A)", "justification": "indicator of {X} separates unequal objects", "preconditions": "A and B are events with P(A)>0 for the displayed conditional form.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(A ∩ B) = P(B|A)*P(A).\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_b8ccc6f4fe5d4ce0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "analysis", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A35964624017", "name": "AM–GM inequality", "statement_latex": "(x + y)/2 >= sqrt(x*y)", "assumptions": "x and y are nonnegative real numbers."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge ((x + y)/2)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (sqrt(x*y))\\}", "to_expression": "(x + y)/2 >= sqrt(x*y)", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "x and y are nonnegative real numbers.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (x + y)/2 ≥ sqrt(x*y).\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_ab2ae4d13aa1be74", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "Ramsey theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_fea84f2021d38582", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_7e3049635da64a19", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "extremal graph theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F0F061CBC3F5", "name": "Blow-up lemma", "statement_latex": "e(X,Y) > delta |X| |Y|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(e(X,Y)) - (delta |X| |Y|) > 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover e(X,Y) > delta |X| |Y|.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover e(X,Y) > delta |X| |Y|.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "e(X,Y) > delta |X| |Y|", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover e(X,Y) > delta |X| |Y|.\"]"}], "source_equivalence_identity": "A<B iff B-A>0 over finite real quantities."}
{"schema_version": "1.0", "item_id": "rcd2_3b12e1a895ef76a0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "calculus of variations", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7F37206494D7", "name": "Isoperimetric inequality", "statement_latex": "4πA ≤ L^2", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(4πA)}{\\exp(4πA)+\\exp(L^2)}\\le\\frac{1}{2}", "to_expression": "4πA ≤ L^2", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as 4πA ≤ L^2.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_5d09f419fdb15695", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "design theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_966aae52c93603bd", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_83A41482F4E9", "name": "Fekete's lemma", "statement_latex": "lim_{n→∞} (a_n / n) = inf_{n∈N} (a_n / n)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((lim_{n→∞} (a_n / n)),z)=((inf_{n∈N} (a_n / n)),z)", "to_expression": "lim_{n→∞} (a_n / n) = inf_{n∈N} (a_n / n)", "justification": "pair equality forces first coordinates equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{n→∞} (a_n / n) = inf_{n∈N} (a_n / n).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_a1bb1c5ca6d2fcf2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_368EF3E8339F", "name": "Summation by parts", "statement_latex": "sum_{k=m}^{n} f_k Delta g_k = (f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} Delta f_k", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (sum_{k=m}^{n} f_k \\Delta g_k = (f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} \\Delta f_k))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "sum_{k=m}^{n} f_k Delta g_k = (f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} Delta f_k", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_e7e6df1a777c6c8e", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_4213f7cc6e2a17c0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "polynomial ring theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BF300C15A52D", "name": "Gauss's lemma (polynomials)", "statement_latex": "F(X) = a_0 + a_1X + ... + a_nX^n", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((F(X)))=F((a_0 + a_1X + ... + a_nX^n))", "to_expression": "F(X) = a_0 + a_1X + ... + a_nX^n", "justification": "indicator of {X} separates unequal objects", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as F(X) = a_0 + a_1X + ... + a_nX^n.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_6a53a4def76f6b55", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3DB69D9F31E5", "name": "Reynolds transport theorem", "statement_latex": "d/dt integral_{Omega(t)} f dV = integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((d/dt integral_{Omega(t)} f dV),z)=((integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA),z)", "to_expression": "d/dt integral_{Omega(t)} f dV = integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA", "justification": "pair equality forces first coordinates equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as d/dt integral_{Omega(t)} f dV = integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_7a3dc061b89e715b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C95D5999DB6B", "name": "Nullity theorem", "statement_latex": "nullity A = nullity H", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(nullity A)\\}=\\{(nullity H)\\}", "to_expression": "nullity A = nullity H", "justification": "singletons equal iff elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as nullity A = nullity H.\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_0526bbd943056984", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "number theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c91439a8b8007ad9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "algebraic graph theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0ABE50B0E12D", "name": "Expander mixing lemma", "statement_latex": "|e(S,T) - (d/n)|S||T|| <= λ√|S||T|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(|e(S,T) - (d/n)|S||T||)-(λ√|S||T|)\\}=0", "to_expression": "|e(S,T) - (d/n)|S||T|| <= λ√|S||T|", "justification": "positive part vanishes iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |e(S,T) - (d/n)|S||T|| ≤ λ√|S||T|.\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_1006fc4babb54512", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "probability theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_27AD75237F2D", "name": "Entropy power inequality", "statement_latex": "N(X+Y) >= N(X) + N(Y)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(N(X+Y)) - (N(X) + N(Y)) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover N(X+Y) >= N(X) + N(Y).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover N(X+Y) >= N(X) + N(Y).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "N(X+Y) >= N(X) + N(Y)", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover N(X+Y) >= N(X) + N(Y).\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
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{"schema_version": "1.0", "item_id": "rcd2_aa83eea0deaff087", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "large deviations theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f4ee3f0ef34141e0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "representation theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1292421DFB1E", "name": "Schur's lemma", "statement_latex": "ρ_W(g) ∘ f = f ∘ ρ_V(g)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((ρ_W(g) ∘ f),z)=((f ∘ ρ_V(g)),z)", "to_expression": "ρ_W(g) ∘ f = f ∘ ρ_V(g)", "justification": "pair equality forces first coordinates equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ρ_W(g) ∘ f = f ∘ ρ_V(g).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_5f1dab8f8e2f733b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "vector calculus", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F8BCAAED7243", "name": "Green's theorem", "statement_latex": "∮_{C}(L dx + M dy) = ∬_{D} (∂M/∂x - ∂L/∂y) dA", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((∮_{C}(L dx + M dy)),z)=((∬_{D} (∂M/∂x - ∂L/∂y) dA),z)", "to_expression": "∮_{C}(L dx + M dy) = ∬_{D} (∂M/∂x - ∂L/∂y) dA", "justification": "pair equality forces first coordinates equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∮_{C}(L dx + M dy) = ∬_{D} (∂M/∂x - ∂L/∂y) dA.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_6f649c3f14a0f12d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_db10902a96b7b3cc", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_373778705e282965", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "calculus of variations", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7F37206494D7", "name": "Isoperimetric inequality", "statement_latex": "4πA ≤ L^2", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(L^2)\\le t\\Rightarrow (4πA)\\le t", "to_expression": "4πA ≤ L^2", "justification": "converse uses t=Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as 4πA ≤ L^2.\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_c13646015b80b48c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_091E7F42D767", "name": "Vitali–Carathéodory theorem", "statement_latex": "∫_X (v - u) dμ < ε", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(∫_X (v - u) dμ)-(ε)\\}=0\\;\\land\\;(∫_X (v - u) dμ)\\ne (ε)", "to_expression": "∫_X (v - u) dμ < ε", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∫_X (v - u) dμ < ε.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_bb88a5ab805380e9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "probabilistic combinatorics", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7C9C39383770", "name": "Ajtai–Komlós–Tusnády theorem", "statement_latex": "P(C_1 n log n < inf_{σ ∈ S_n} sum_{k=1}^n |X_k - Y_{σ(k)}| < C_2 n log n) = 1", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((P(C_1 n log n < inf_{σ ∈ S_n} sum_{k=1}^n |X_k - Y_{σ(k)}| < C_2 n log n)),z)=((1),z)", "to_expression": "P(C_1 n log n < inf_{σ ∈ S_n} sum_{k=1}^n |X_k - Y_{σ(k)}| < C_2 n log n) = 1", "justification": "pair equality forces first coordinates equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(C_1 n log n < inf_{σ ∈ S_n} sum_{k=1}^n |X_k - Y_{σ(k)}| < C_2 n log n) = 1.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_4cbf983e9f1a396b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_319FF857BA97", "name": "Burgess inequality", "statement_latex": "|S_chi(N,H)| <= C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(C_{r,\\varepsilon} H^{1-1/r} q^{(r+1)/(4r^2)+\\varepsilon}) \\in (|S_chi(N,H)|) + [0,\\infty)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |S_chi(N,H)| <= C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |S_chi(N,H)| <= C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon}.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "|S_chi(N,H)| <= C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon}", "justification": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |S_chi(N,H)| <= C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon}.\"]"}], "source_equivalence_identity": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_e9e6ab86d8919fcc", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2848f450efce0587", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F902B76211EA", "name": "Cantelli's inequality", "statement_latex": "Pr(X - E[X] ≥ λ) ≤ σ^2 / (σ^2 + λ^2)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(0 = 0) \\Rightarrow (Pr(X - E[X] ≥ λ) ≤ σ^2 / (σ^2 + λ^2))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Since 0=0 is true, true=>Phi_T has the same truth value as Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "Pr(X - E[X] ≥ λ) ≤ σ^2 / (σ^2 + λ^2)", "justification": "Since 0=0 is true, true=>Phi_T has the same truth value as Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "Since 0=0 is true, true=>Phi_T has the same truth value as Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_e1af51bba914c37f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "topology", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3919D9203C0B", "name": "Cantor's intersection theorem", "statement_latex": "\\bigcap _{k=0}^{\\infty }C_{k}\\neq \\emptyset", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((\\bigcap _{k) \\in S \\land (0}^{\\infty }C_{k}\\neq \\\\emptyset) \\notin S) \\lor ((0}^{\\infty }C_{k}\\neq \\\\emptyset) \\in S \\land (\\bigcap _{k) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\bigcap _{k=0}^{\\\\infty }C_{k}\\\\neq \\\\emptyset.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\bigcap _{k=0}^{\\infty }C_{k}\\neq \\emptyset", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\bigcap _{k=0}^{\\\\infty }C_{k}\\\\neq \\\\emptyset.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_8a6afa5f28bfba70", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "linear algebra", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FF8D78866125", "name": "Binet–Cauchy identity", "statement_latex": "(sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i<j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i)", "assumptions": "All sums are finite over the same index range; entries lie in R."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i)) - ((sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i) > 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All sums are finite over the same index range; entries lie in R.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i < j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All sums are finite over the same index range; entries lie in R.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i < j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "(sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i<j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i)", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All sums are finite over the same index range; entries lie in R.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i < j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i).\"]"}], "source_equivalence_identity": "A<B iff B-A>0 over finite real quantities."}
{"schema_version": "1.0", "item_id": "rcd2_c6576a5e393cae13", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "commutative algebra", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_8908fdc6ffad98bf", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "p-adic analysis", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_32b951eb38ddf2e9", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_00a339cc5d92e99c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "approximation theory", "subfield": "mathematical analysis", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_389D9173C8B7", "name": "Lebesgue's lemma", "statement_latex": "‖v - Pv‖ ≤ (1 + ‖P‖) inf_{u ∈ U} ‖v - u‖", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge ((1 + ‖P‖) inf_{u ∈ U} ‖v - u‖)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (‖v - Pv‖)\\}", "to_expression": "‖v - Pv‖ ≤ (1 + ‖P‖) inf_{u ∈ U} ‖v - u‖", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ‖v - Pv‖ ≤ (1 + ‖P‖) inf_{u ∈ U} ‖v - u‖.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_6adc29e713cc8b7e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "measure theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_448CEF326F89", "name": "Doob–Dynkin lemma", "statement_latex": "f = g ∘ T", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(f) \\in {(g ∘ T)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f = g ∘ T.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f = g ∘ T", "justification": "L in {R} iff L = R.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f = g ∘ T.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_faac4aad7480e56f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "analysis", "subfield": "complex analysis", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_5da63915ee9a539a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "physics", "subfield": "quantum field theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5C335B571C48", "name": "Gell-Mann and Low theorem", "statement_latex": "H = H_0 + gV", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(H))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(H_0 + gV))", "to_expression": "H = H_0 + gV", "justification": "each argmin set is a singleton", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as H = H_0 + gV.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_9ff65c5ae2c9252a", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "geometry", "subfield": "Euclidean geometry", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e2acf071f89adf73", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "computer science", "subfield": "probabilistic combinatorics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ed1b6d93aeaf47c3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "Ramsey theory", "subfield": "discrete mathematics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3F2BB2E376A3", "name": "Rado's theorem", "statement_latex": "Ax = 0", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((Ax) \\in S \\land (0) \\notin S) \\lor ((0) \\in S \\land (Ax) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover Ax = 0.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "Ax = 0", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover Ax = 0.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_04a375dd31cb4e23", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "graph theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_57B1E9BA3579", "name": "Crossing number inequality", "statement_latex": "cr(G) >= e^3 / (29n^2)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (cr(G))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (e^3 / (29n^2))\\}", "to_expression": "cr(G) >= e^3 / (29n^2)", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as cr(G) ≥ e^3 / (29n^2).\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_4f8a8e15a9039133", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "vector calculus", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_24234F1E6547", "name": "Divergence theorem", "statement_latex": "\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S} = \\int_{V} (\\nabla \\cdot \\mathbf{F}) dV", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S}),(\\int_{V} (\\nabla \\cdot \\mathbf{F}) dV))=0", "to_expression": "\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S} = \\int_{V} (\\nabla \\cdot \\mathbf{F}) dV", "justification": "zero discrete distance iff equality", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\oint_{S} \\\\mathbf{F} \\\\cdot d\\\\mathbf{S} = \\\\int_{V} (\\\\nabla \\\\cdot \\\\mathbf{F}) dV.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_fa497770d8bd57a7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "inequalities", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9822BD8B9BE6", "name": "Muirhead's inequality", "statement_latex": "[a] >= [b] for a majorizing b", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;([a])\\le t\\Rightarrow ([b] for a majorizing b)\\le t", "to_expression": "[a] >= [b] for a majorizing b", "justification": "converse uses t=Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as [a] ≥ [b] for a majorizing b.\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_357ecb3ed435909d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "point processes", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C81FC5D06BF2", "name": "Campbell's theorem", "statement_latex": "E[sum_{x in N} f(x)] = integral f(x) dLambda(x)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((E[sum_{x in N} f(x)]))=F((integral f(x) dLambda(x)))", "to_expression": "E[sum_{x in N} f(x)] = integral f(x) dLambda(x)", "justification": "indicator of {X} separates unequal objects", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as E[sum_{x in N} f(x)] = integral f(x) dLambda(x).\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
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{"schema_version": "1.0", "item_id": "rcd2_3990bf766129f0e4", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "topology", "subfield": "algebraic topology", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0daf444f555983ad", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "theoretical computer science", "subfield": "algebraic complexity theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_46E2A92077B9", "name": "Schwartz–Zippel lemma", "statement_latex": "Pr[P(r1, ..., rn) = 0] ≤ d/|S|", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(Pr[P(r1, ..., rn) = 0])-(d/|S|)\\}=0", "to_expression": "Pr[P(r1, ..., rn) = 0] ≤ d/|S|", "justification": "positive part vanishes iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr[P(r1, ..., rn) = 0] ≤ d/|S|.\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_0da020e185cd0f70", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "discrete mathematics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2F917150CBA3", "name": "Pigeonhole principle", "statement_latex": "max_{1<=i<=m} |B_i| >= ceil(n/m)", "assumptions": "n objects are distributed among m nonempty boxes B_1,...,B_m; m>=1."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (max_{1\\lei\\lem} |B_i| \\ge ceil(n/m)))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "n objects are distributed among m nonempty boxes B_1,...,B_m; m>=1.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "max_{1<=i<=m} |B_i| >= ceil(n/m)", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "n objects are distributed among m nonempty boxes B_1,...,B_m; m>=1.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_ba2835aa40a5e152", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "field theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4E2C0DED1A31", "name": "Joubert's theorem", "statement_latex": "p(t) = t^6 + c_4t^4 + c_2t^2 + c_1t + c_0", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((p(t)),z):z\\in D\\}=\\{((t^6 + c_4t^4 + c_2t^2 + c_1t + c_0),z):z\\in D\\}", "to_expression": "p(t) = t^6 + c_4t^4 + c_2t^2 + c_1t + c_0", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p(t) = t^6 + c_4t^4 + c_2t^2 + c_1t + c_0.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_3356f870086adeb7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "measure theory", "subfield": "stochastic analysis", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FB5E9395CC7C", "name": "Malliavin's absolute continuity lemma", "statement_latex": "|∫ Dφ(y)(x) dμ(y)| ≤ C(x) ‖φ‖_{∞}", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(C(x) ‖φ‖_{∞})\\le t\\Rightarrow (|∫ Dφ(y)(x) dμ(y)|)\\le t", "to_expression": "|∫ Dφ(y)(x) dμ(y)| ≤ C(x) ‖φ‖_{∞}", "justification": "converse uses t=Y", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |∫ Dφ(y)(x) dμ(y)| ≤ C(x) ‖φ‖_{∞}.\"]"}], "source_equivalence_identity": "converse uses t=Y"}
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{"schema_version": "1.0", "item_id": "rcd2_e5b5e164dd68fcc2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9F50AE0DD0D5", "name": "Cauchy–Davenport theorem", "statement_latex": "|A+B| >= min(|A|+|B|-1, p)", "assumptions": "A and B are nonempty finite subsets of Z/pZ; p is prime."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(|A+B|)\\le t\\Rightarrow (min(|A|+|B|-1, p))\\le t", "to_expression": "|A+B| >= min(|A|+|B|-1, p)", "justification": "converse uses t=Y", "preconditions": "A and B are nonempty finite subsets of Z/pZ; p is prime.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p).\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_97526f876e8bfb7b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "sieve theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5DC5BE909937", "name": "Fundamental lemma of sieve theory", "statement_latex": "|A_d| = \\frac{w(d)}{d}X + R_d", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(|A_d|)\\Longleftrightarrow z=(\\frac{w(d)}{d}X + R_d))", "to_expression": "|A_d| = \\frac{w(d)}{d}X + R_d", "justification": "same equality fiber iff equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |A_d| = \\\\frac{w(d)}{d}X + R_d.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_7a5d51029339da1e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_319FF857BA97", "name": "Burgess inequality", "statement_latex": "|S_chi(N,H)| <= C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (|S_chi(N,H)|) \\le x \\land x \\le (C_{r,\\varepsilon} H^{1-1/r} q^{(r+1)/(4r^2)+\\varepsilon})", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |S_chi(N,H)| <= C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |S_chi(N,H)| <= C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon}.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "|S_chi(N,H)| <= C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon}", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |S_chi(N,H)| <= C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon}.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_ee011a959990dc45", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ddb4ef63aecacf2b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3DB69D9F31E5", "name": "Reynolds transport theorem", "statement_latex": "d/dt integral_{Omega(t)} f dV = integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(d/dt integral_{Omega(t)} f dV)\\Longleftrightarrow z=(integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA))", "to_expression": "d/dt integral_{Omega(t)} f dV = integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA", "justification": "same equality fiber iff equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as d/dt integral_{Omega(t)} f dV = integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
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{"schema_version": "1.0", "item_id": "rcd2_458610c8ac5dd02b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ae61e5d62dd86eb2", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6270cd93e4912f62", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "combinatorial number theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_bcb83fe244060818", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_95775DC75EC2", "name": "Oppenheim conjecture", "statement_latex": "|Q(x)| < \\epsilon", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (|Q(x)|) < x \\land x < (\\\\varepsilon)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |Q(x)| < \\\\epsilon.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |Q(x)| < \\\\epsilon.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "|Q(x)| < \\epsilon", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |Q(x)| < \\\\epsilon.\"]"}], "source_equivalence_identity": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B."}
{"schema_version": "1.0", "item_id": "rcd2_862a1c6a19ce8a02", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "Combinatorics", "subfield": "Order theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_93a3e0328467e974", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "graph theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d146f4d752d85f52", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "number theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9793AD078B54", "name": "Degen's eight-square identity", "statement_latex": "(∑_{i=1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2) = ∑_{i=1}^{8} c_i^2", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg ((∑_{i=1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2) = ∑_{i=1}^{8} c_i^2))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "(∑_{i=1}^{8} a_i^2)(∑_{i=1}^{8} b_i^2) = ∑_{i=1}^{8} c_i^2", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_02d668c2393f2fd2", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_4a3e31a78e155710", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "calculus", "subfield": "integral calculus", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F8E7F707512F", "name": "Bioche's rules", "statement_latex": "ω(-t) = ω(t)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(ω(-t))\\Longleftrightarrow z=(ω(t)))", "to_expression": "ω(-t) = ω(t)", "justification": "same equality fiber iff equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ω(-t) = ω(t).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_bc76aa03e18c8fe5", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AFD3A111ECE4", "name": "Boole's inequality", "statement_latex": "P(∪_{i=1}^∞ A_i) ≤ ∑_{i=1}^∞ P(A_i)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(P(∪_{i=1}^∞ A_i))+s=(∑_{i=1}^∞ P(A_i))", "to_expression": "P(∪_{i=1}^∞ A_i) ≤ ∑_{i=1}^∞ P(A_i)", "justification": "nonnegative slack exists iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(∪_{i=1}^∞ A_i) ≤ ∑_{i=1}^∞ P(A_i).\"]"}], "source_equivalence_identity": "nonnegative slack exists iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_dff887497f71e7d8", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "linear algebra", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_570b94d9085240fd", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "discrete mathematics", "subfield": "combinatorics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AE455D38D038", "name": "Bregman–Minc inequality", "statement_latex": "per(A) <= prod_{i=1}^n (r_i!)^(1/r_i)", "assumptions": "A is an n by n 0-1 matrix; r_i is the sum of row i; terms with r_i=0 are handled by the standard zero-permanent convention."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (per(A)) \\le x \\land x \\le (prod_{i=1}^n (r_i!)^(1/r_i))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "A is an n by n 0-1 matrix; r_i is the sum of row i; terms with r_i=0 are handled by the standard zero-permanent convention.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover per(A) <= prod_{i=1}^n (r_i!)^(1/r_i).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "A is an n by n 0-1 matrix; r_i is the sum of row i; terms with r_i=0 are handled by the standard zero-permanent convention.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover per(A) <= prod_{i=1}^n (r_i!)^(1/r_i).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "per(A) <= prod_{i=1}^n (r_i!)^(1/r_i)", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "A is an n by n 0-1 matrix; r_i is the sum of row i; terms with r_i=0 are handled by the standard zero-permanent convention.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover per(A) <= prod_{i=1}^n (r_i!)^(1/r_i).\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_d8464cfcdfc0fc5f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "differential calculus", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F1B027EB50E5", "name": "Interior extremum theorem", "statement_latex": "f'(x_0) = 0", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(f'(x_0)) \\in {(0)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f'(x_0) = 0.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f'(x_0) = 0", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f'(x_0) = 0.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
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{"schema_version": "1.0", "item_id": "rcd2_5bf4786eb6e5a051", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "extremal graph theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_204a4b7181bdd7fc", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_89b41888688c0576", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_69A821A9D880", "name": "Hockey-stick identity", "statement_latex": "\\sum_{i=r}^{n} \\binom{i}{r} = \\binom{n+1}{r+1}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((\\sum_{i=r}^{n} \\binom{i}{r}),z):z\\in D\\}=\\{((\\binom{n+1}{r+1}),z):z\\in D\\}", "to_expression": "\\sum_{i=r}^{n} \\binom{i}{r} = \\binom{n+1}{r+1}", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\sum_{i=r}^{n} \\\\binom{i}{r} = \\\\binom{n+1}{r+1}.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_1a42ba03879a1194", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_54f3e0d05ebd8957", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "infinite-dimensional vector spaces", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_17441119733191e6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0FE477D29636", "name": "Hardy–Ramanujan theorem", "statement_latex": "|ω(n)−loglogn|<(loglogn)^(1/2+ε)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "((loglogn)^(1/2+ε)) - (|ω(n)−loglogn|) > 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |ω(n)−loglogn| < (loglogn)^(1/2+ε).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |ω(n)−loglogn| < (loglogn)^(1/2+ε).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "|ω(n)−loglogn|<(loglogn)^(1/2+ε)", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |ω(n)−loglogn| < (loglogn)^(1/2+ε).\"]"}], "source_equivalence_identity": "A<B iff B-A>0 over finite real quantities."}
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{"schema_version": "1.0", "item_id": "rcd2_aa25ee2da4fe2523", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "combinatorics", "subfield": "combinatorial geometry", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_17dd9ea788acb589", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "combinatorics on words", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_8eb12a694aac79ed", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "matrix analysis", "subfield": "linear algebra", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D76BA21A085C", "name": "Crouzeix's conjecture", "statement_latex": "||f(A)|| <= 2 sup_{z in W(A)} |f(z)|", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (2 sup_{z in W(A)} |f(z)|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (||f(A)||)\\}", "to_expression": "||f(A)|| <= 2 sup_{z in W(A)} |f(z)|", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ||f(A)|| ≤ 2 sup_{z in W(A)} |f(z)|.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_d94da739294469ea", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "representation theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CAF4E18AA4C8", "name": "Brauer's height zero conjecture", "statement_latex": "\\nu(\\chi(1)) = \\nu(|G:D|)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\nu(\\chi(1))))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\nu(|G:D|)))", "to_expression": "\\nu(\\chi(1)) = \\nu(|G:D|)", "justification": "each argmin set is a singleton", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\nu(\\\\chi(1)) = \\\\nu(|G:D|).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_d60b1b18c12e59eb", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_EE0D43BAAED5", "name": "Ahlswede–Daykin inequality", "statement_latex": "f_1(X)f_2(Y) ≤ f_3(X ∨ Y)f_4(X ∧ Y)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(f_3(X ∨ Y)f_4(X ∧ Y)) - (f_1(X)f_2(Y)) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover f_1(X)f_2(Y) <= f_3(X ∨ Y)f_4(X ∧ Y).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover f_1(X)f_2(Y) <= f_3(X ∨ Y)f_4(X ∧ Y).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "f_1(X)f_2(Y) ≤ f_3(X ∨ Y)f_4(X ∧ Y)", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover f_1(X)f_2(Y) <= f_3(X ∨ Y)f_4(X ∧ Y).\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_39fe66d38fe9cdfa", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "queueing theory", "subfield": "probability", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b39dfc79c77aa66b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8DC5B9EC1AA5", "name": "Dimension theorem for vector spaces", "statement_latex": "|B_1| = |B_2|", "assumptions": "B_1 and B_2 are bases of the same finite-dimensional vector space."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(|B_1|) \\in {(|B_2|)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "B_1 and B_2 are bases of the same finite-dimensional vector space.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover |B_1| = |B_2|.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "|B_1| = |B_2|", "justification": "L in {R} iff L = R.", "preconditions": "B_1 and B_2 are bases of the same finite-dimensional vector space.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover |B_1| = |B_2|.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_4465aff701049117", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "zeta and L-functions", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CEA3FEEC415F", "name": "Selberg's zeta function conjecture", "statement_latex": "N(T + H) - N(T) ≥ cH log T", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(cH log T)}{\\exp(cH log T)+\\exp(N(T + H) - N(T))}\\le\\frac{1}{2}", "to_expression": "N(T + H) - N(T) ≥ cH log T", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as N(T + H) - N(T) ≥ cH log T.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_ebd58781b6721f8f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_368EF3E8339F", "name": "Summation by parts", "statement_latex": "sum_{k=m}^{n} f_k Delta g_k = (f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} Delta f_k", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((sum_{k=m}^{n} f_k Delta g_k),z):z\\in D\\}=\\{(((f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} Delta f_k),z):z\\in D\\}", "to_expression": "sum_{k=m}^{n} f_k Delta g_k = (f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} Delta f_k", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{k=m}^{n} f_k Delta g_k = (f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} Delta f_k.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_398e11a5aaf0a255", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic functions", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FDD3E2DFBB65", "name": "Hermite's identity", "statement_latex": "sum_{k=0}^{n-1} floor(x+k/n) = floor(n*x)", "assumptions": "n is a positive integer; x is real."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(sum_{k=0}^{n-1} floor(x+k/n))\\Longleftrightarrow z=(floor(n*x)))", "to_expression": "sum_{k=0}^{n-1} floor(x+k/n) = floor(n*x)", "justification": "same equality fiber iff equal", "preconditions": "n is a positive integer; x is real.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{k=0}^{n-1} floor(x+k/n) = floor(n*x).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_307f43cf9ab9d397", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "enumerative combinatorics", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_1355a711c1bec7a7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_68687FF65AAB", "name": "Landsberg–Schaar relation", "statement_latex": "1/sqrt(p) * sum_{n=0}^{p-1} exp(2*pi*i*n^2*q/p) = e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (1/sqrt(p) * sum_{n=0}^{p-1} \\exp(2*\\pi*i*n^2*q/p) = e^(1/4 * \\pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} \\exp(-\\pi*i*n^2*p/(2q))))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "1/sqrt(p) * sum_{n=0}^{p-1} exp(2*pi*i*n^2*q/p) = e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q))", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_83393a6d080330a5", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematical analysis", "subfield": "analytic number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_88a07c0719d7ad42", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "inequalities", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2a44d80584b324cc", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "modular arithmetic", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_4d2731460b2b1ea9", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "physics", "subfield": "general relativity", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2762ab20f86ed416", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "commutative algebra", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0EBFF1A397C6", "name": "Local criterion for flatness", "statement_latex": "Tor_1^A(C, M) = 0", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(0) = (Tor_1^A(C, M))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover Tor_1^A(C, M) = 0.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "Tor_1^A(C, M) = 0", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover Tor_1^A(C, M) = 0.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_eb91aa0070ccc0db", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5A3D942AED87", "name": "Markov's inequality", "statement_latex": "P(X >= a) <= E[X] / a", "assumptions": "X is a nonnegative random variable; a>0; E[X] is finite."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((P(X ≥ a))-(E[X] / a),(-\\infty,0])=0", "to_expression": "P(X >= a) <= E[X] / a", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "X is a nonnegative random variable; a>0; E[X] is finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(X ≥ a) ≤ E[X] / a.\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
{"schema_version": "1.0", "item_id": "rcd2_2065ad2dc59855ac", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_68687FF65AAB", "name": "Landsberg–Schaar relation", "statement_latex": "1/sqrt(p) * sum_{n=0}^{p-1} exp(2*pi*i*n^2*q/p) = e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((1/sqrt(p) * sum_{n=0}^{p-1} exp(2*pi*i*n^2*q/p)))=F((e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q))))", "to_expression": "1/sqrt(p) * sum_{n=0}^{p-1} exp(2*pi*i*n^2*q/p) = e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q))", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as 1/sqrt(p) * sum_{n=0}^{p-1} exp(2*pi*i*n^2*q/p) = e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q)).\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_399e8381b2d5ea24", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "algebra", "subfield": "algebraic combinatorics", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_396E0535E013", "name": "n! conjecture", "statement_latex": "dim(H_μ) = n!", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((dim(H_μ)),z)=((n!),z)", "to_expression": "dim(H_μ) = n!", "justification": "pair equality forces first coordinates equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as dim(H_μ) = n!.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
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{"schema_version": "1.0", "item_id": "rcd2_b3f0df3b43b77504", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5795F6B3279F", "name": "Chain rule (probability)", "statement_latex": "P(A ∩ B) = P(B|A)*P(A)", "assumptions": "A and B are events with P(A)>0 for the displayed conditional form."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(P(A ∩ B)) \\in {(P(B|A)*P(A))}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "A and B are events with P(A)>0 for the displayed conditional form.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover P(A ∩ B) = P(B|A)*P(A).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "P(A ∩ B) = P(B|A)*P(A)", "justification": "L in {R} iff L = R.", "preconditions": "A and B are events with P(A)>0 for the displayed conditional form.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover P(A ∩ B) = P(B|A)*P(A).\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_1a225a016cee4e0c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2B79341A5DAC", "name": "Doob's martingale inequality", "statement_latex": "P[max_{1<=i<=n} X_i >= C] <= E[max(X_n, 0)] / C", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (P[max_{1\\lei\\len} X_i \\ge C] \\le E[\\max(X_n, 0)] / C))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "P[max_{1<=i<=n} X_i >= C] <= E[max(X_n, 0)] / C", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_0f5195926cedcd0c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b21d61526f25ff7e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0C6AE95E0EE0", "name": "Hoeffding's inequality", "statement_latex": "P(S_n - E[S_n] >= t) <= exp(-2*t^2 / sum_i (b_i-a_i)^2)", "assumptions": "The summands are independent and bounded with X_i in [a_i,b_i]; t>=0; denominator is positive."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(P(S_n - E[S_n] ≥ t))+s=(exp(-2*t^2 / sum_i (b_i-a_i)^2))", "to_expression": "P(S_n - E[S_n] >= t) <= exp(-2*t^2 / sum_i (b_i-a_i)^2)", "justification": "nonnegative slack exists iff X≤Y", "preconditions": "The summands are independent and bounded with X_i in [a_i,b_i]; t>=0; denominator is positive.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(S_n - E[S_n] ≥ t) ≤ exp(-2*t^2 / sum_i (b_i-a_i)^2).\"]"}], "source_equivalence_identity": "nonnegative slack exists iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_8e0c8bea8017e8c3", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "combinatorics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c65ca9806806e914", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "algebraic combinatorics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_95918AB9C155", "name": "Kruskal–Katona theorem", "statement_latex": "N^{(i-1)}=\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(N^{(i-1)}) \\in {(\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1})}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover N^{(i-1)}=\\\\binom{n_i}{i-1}+\\\\binom{n_{i-1}}{i-2}+\\\\dots+\\\\binom{n_j}{j-1}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "N^{(i-1)}=\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1}", "justification": "L in {R} iff L = R.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover N^{(i-1)}=\\\\binom{n_i}{i-1}+\\\\binom{n_{i-1}}{i-2}+\\\\dots+\\\\binom{n_j}{j-1}.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_1916ee0ece58636f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "discrete mathematics", "subfield": "combinatorics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e5a7f9def1c83164", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "discrete mathematics", "subfield": "combinatorics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AE455D38D038", "name": "Bregman–Minc inequality", "statement_latex": "per(A) <= prod_{i=1}^n (r_i!)^(1/r_i)", "assumptions": "A is an n by n 0-1 matrix; r_i is the sum of row i; terms with r_i=0 are handled by the standard zero-permanent convention."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(prod_{i=1}^n (r_i!)^(1/r_i)) \\in (per(A)) + [0,\\infty)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B.", "preconditions": "A is an n by n 0-1 matrix; r_i is the sum of row i; terms with r_i=0 are handled by the standard zero-permanent convention.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover per(A) <= prod_{i=1}^n (r_i!)^(1/r_i).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B.", "preconditions": "A is an n by n 0-1 matrix; r_i is the sum of row i; terms with r_i=0 are handled by the standard zero-permanent convention.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover per(A) <= prod_{i=1}^n (r_i!)^(1/r_i).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "per(A) <= prod_{i=1}^n (r_i!)^(1/r_i)", "justification": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B.", "preconditions": "A is an n by n 0-1 matrix; r_i is the sum of row i; terms with r_i=0 are handled by the standard zero-permanent convention.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover per(A) <= prod_{i=1}^n (r_i!)^(1/r_i).\"]"}], "source_equivalence_identity": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_7dbd0e223ce5bd06", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "calculus of variations", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_70ad24cd768215c3", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "polynomial ring theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b58d6c04ec8be979", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "differential calculus", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_9811f4cc1e361b7d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "representation theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c9e45aa870868bce", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "approximation theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_DC96C747092B", "name": "Müntz–Szász theorem", "statement_latex": "sum_{n in S} (1/n) = infinity", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(sum_{n \\in S} (1/n)) \\in {(\\infty)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover sum_{n in S} (1/n) = infinity.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "sum_{n in S} (1/n) = infinity", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover sum_{n in S} (1/n) = infinity.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_78440324aa48b75c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_4f83895a0d0cfa1f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "complex analysis", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_484796D95DA9", "name": "Abel's theorem", "statement_latex": "lim_{x -> 1^-} G(x) = sum_{k=0}^infinity a_k", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (1^-} G(x) = sum_{k=0}^\\infty a_k) < x \\land x < (lim_{x -)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover lim_{x - > 1^-} G(x) = sum_{k=0}^infinity a_k.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover lim_{x - > 1^-} G(x) = sum_{k=0}^infinity a_k.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "lim_{x -> 1^-} G(x) = sum_{k=0}^infinity a_k", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover lim_{x - > 1^-} G(x) = sum_{k=0}^infinity a_k.\"]"}], "source_equivalence_identity": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B."}
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{"schema_version": "1.0", "item_id": "rcd2_ac77d5a441f1e417", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_DB1D004A0930", "name": "Cramer's rule", "statement_latex": "x_i = \\det(A_i) / \\det(A)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(x_i)\\}=\\{(\\det(A_i) / \\det(A))\\}", "to_expression": "x_i = \\det(A_i) / \\det(A)", "justification": "singletons equal iff elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as x_i = \\\\det(A_i) / \\\\det(A).\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_d8e63f16602a3555", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "discrete mathematics", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2F917150CBA3", "name": "Pigeonhole principle", "statement_latex": "max_{1<=i<=m} |B_i| >= ceil(n/m)", "assumptions": "n objects are distributed among m nonempty boxes B_1,...,B_m; m>=1."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(ceil(n/m))}{\\exp(ceil(n/m))+\\exp(max_{1≤i≤m} |B_i|)}\\le\\frac{1}{2}", "to_expression": "max_{1<=i<=m} |B_i| >= ceil(n/m)", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "n objects are distributed among m nonempty boxes B_1,...,B_m; m>=1.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as max_{1≤i≤m} |B_i| ≥ ceil(n/m).\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_3d0047bb3c4b6ab3", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "statistical mechanics", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6273117847e8b39d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "elementary number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0F6BD7E91884", "name": "Bézout's identity", "statement_latex": "ax + by = gcd(a, b)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((ax + by),(gcd(a, b)))\\in\\{(z,z):z\\in D\\}", "to_expression": "ax + by = gcd(a, b)", "justification": "pair in diagonal iff X=Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ax + by = gcd(a, b).\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
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{"schema_version": "1.0", "item_id": "rcd2_ea2b698fd89cd252", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "elementary number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a61166594d73c73d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum information theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3795DDB6E155", "name": "NLTS conjecture", "statement_latex": "H = \\sum_{i=1}^{m} H_{i}", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((H),(\\sum_{i=1}^{m} H_{i}))=0", "to_expression": "H = \\sum_{i=1}^{m} H_{i}", "justification": "zero discrete distance iff equality", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as H = \\\\sum_{i=1}^{m} H_{i}.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_208231715833f2a1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_94B20D8A1D74", "name": "Chebyshev's inequality", "statement_latex": "Pr(|X-mu| >= k*sigma) <= 1/k^2", "assumptions": "X has mean mu and finite nonzero standard deviation sigma; k>0."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(1/k^2)\\le t\\Rightarrow (Pr(|X-mu| ≥ k*sigma))\\le t", "to_expression": "Pr(|X-mu| >= k*sigma) <= 1/k^2", "justification": "converse uses t=Y", "preconditions": "X has mean mu and finite nonzero standard deviation sigma; k>0.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr(|X-mu| ≥ k*sigma) ≤ 1/k^2.\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_ee04996690fd5613", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5795F6B3279F", "name": "Chain rule (probability)", "statement_latex": "P(A ∩ B) = P(B|A)*P(A)", "assumptions": "A and B are events with P(A)>0 for the displayed conditional form."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(P(B|A)*P(A)) = (P(A ∩ B))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "A and B are events with P(A)>0 for the displayed conditional form.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover P(A ∩ B) = P(B|A)*P(A).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "P(A ∩ B) = P(B|A)*P(A)", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "A and B are events with P(A)>0 for the displayed conditional form.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover P(A ∩ B) = P(B|A)*P(A).\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_85b615da766c0f0a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C95D5999DB6B", "name": "Nullity theorem", "statement_latex": "nullity A = nullity H", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "(\\{(nullity A)\\}\\times\\{(nullity H)\\})\\cap\\{(z,z):z\\in D\\}=\\{((nullity A),(nullity H))\\}", "to_expression": "nullity A = nullity H", "justification": "intersection is singleton iff pair diagonal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as nullity A = nullity H.\"]"}], "source_equivalence_identity": "intersection is singleton iff pair diagonal"}
{"schema_version": "1.0", "item_id": "rcd2_9bf9d2830f83d8c8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "calculus", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_819FFAB136D4", "name": "L'Hôpital's rule", "statement_latex": "lim_{x->c} f(x)/g(x) = lim_{x->c} f'(x)/g'(x)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(lim_{x→c} f(x)/g(x))=z\\;\\land\\;z=(lim_{x→c} f'(x)/g'(x))", "to_expression": "lim_{x->c} f(x)/g(x) = lim_{x->c} f'(x)/g'(x)", "justification": "common witness iff equality", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{x→c} f(x)/g(x) = lim_{x→c} f'(x)/g'(x).\"]"}], "source_equivalence_identity": "common witness iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_2461b0407b1aa21b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_41972AC1B752", "name": "Rothe–Hagen identity", "statement_latex": "\\sum _{k=0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k}={\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "(\\{(\\sum _{k=0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k})\\}\\times\\{({\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n})\\})\\cap\\{(z,z):z\\in D\\}=\\{((\\sum _{k=0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k}),({\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n}))\\}", "to_expression": "\\sum _{k=0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k}={\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n}", "justification": "intersection is singleton iff pair diagonal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\sum _{k=0}^{n}{\\\\frac {x}{x+kz}}{x+kz \\\\choose k}{\\\\frac {y}{y+(n-k)z}}{y+(n-k)z \\\\choose n-k}={\\\\frac {x+y}{x+y+nz}}{x+y+nz \\\\choose n}.\"]"}], "source_equivalence_identity": "intersection is singleton iff pair diagonal"}
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{"schema_version": "1.0", "item_id": "rcd2_932a9f593481f1f6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "commutative algebra", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_666997CDC49C", "name": "Noether normalization lemma", "statement_latex": "A = k[y_1, ..., y_m]", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(A))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(k[y_1, ..., y_m]))", "to_expression": "A = k[y_1, ..., y_m]", "justification": "each argmin set is a singleton", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as A = k[y_1, ..., y_m].\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_7a3b62640f51d92a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_319FF857BA97", "name": "Burgess inequality", "statement_latex": "|S_chi(N,H)| <= C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon})\\le t\\Rightarrow (|S_chi(N,H)|)\\le t", "to_expression": "|S_chi(N,H)| <= C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon}", "justification": "converse uses t=Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |S_chi(N,H)| ≤ C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon}.\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_70a9e3842fbc5759", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "probabilistic method", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8F1DAFA4A16E", "name": "Lovász local lemma", "statement_latex": "e*p*(d+1) <= 1", "assumptions": "Symmetric Lovasz local lemma setting; each bad event has probability at most p and dependency degree at most d."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(e*p*(d+1))}{\\exp(e*p*(d+1))+\\exp(1)}\\le\\frac{1}{2}", "to_expression": "e*p*(d+1) <= 1", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "Symmetric Lovasz local lemma setting; each bad event has probability at most p and dependency degree at most d.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as e*p*(d+1) ≤ 1.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_6c4c5e7ee0edc8cc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "field theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4E2C0DED1A31", "name": "Joubert's theorem", "statement_latex": "p(t) = t^6 + c_4t^4 + c_2t^2 + c_1t + c_0", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(p(t))\\Longleftrightarrow z=(t^6 + c_4t^4 + c_2t^2 + c_1t + c_0))", "to_expression": "p(t) = t^6 + c_4t^4 + c_2t^2 + c_1t + c_0", "justification": "same equality fiber iff equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p(t) = t^6 + c_4t^4 + c_2t^2 + c_1t + c_0.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
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{"schema_version": "1.0", "item_id": "rcd2_e790faa26118510e", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "enumerative combinatorics", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_8a5280444d33e475", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "time-reversal symmetry", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BACF4A4770A1", "name": "Kramers' theorem", "statement_latex": "T^2 = -1", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(-1) = (T^2)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover T^2 = -1.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "T^2 = -1", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover T^2 = -1.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_b5599511a186ac35", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "automorphic forms", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_64D8463F3766", "name": "Fundamental lemma", "statement_latex": "SO_{\\gamma _{H}}(1_{K_{H}})=\\Delta (\\gamma _{H},\\gamma _{G})O_{\\gamma _{G}}^{\\kappa }(1_{K_{G}})", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(SO_{\\gamma _{H}}(1_{K_{H}})))+\\delta_D(z,(\\Delta (\\gamma _{H},\\gamma _{G})O_{\\gamma _{G}}^{\\kappa }(1_{K_{G}}))))=0", "to_expression": "SO_{\\gamma _{H}}(1_{K_{H}})=\\Delta (\\gamma _{H},\\gamma _{G})O_{\\gamma _{G}}^{\\kappa }(1_{K_{G}})", "justification": "zero loss iff some z equals both", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as SO_{\\\\gamma _{H}}(1_{K_{H}})=\\\\Delta (\\\\gamma _{H},\\\\gamma _{G})O_{\\\\gamma _{G}}^{\\\\kappa }(1_{K_{G}}).\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_5e24a9ab500e2476", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic calculus", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4D5944961D8", "name": "Itô's lemma", "statement_latex": "dX_t = mu_t dt + sigma_t dB_t", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((dX_t),z)=((mu_t dt + sigma_t dB_t),z)", "to_expression": "dX_t = mu_t dt + sigma_t dB_t", "justification": "pair equality forces first coordinates equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as dX_t = mu_t dt + sigma_t dB_t.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_61188a7f5a5f913a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "harmonic analysis", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_ECE01C7A6C37", "name": "Wiener's lemma", "statement_latex": "lim_{N->inf} (1/(2N+1)) sum_{n=-N}^{N} |widehat{mu}(n)|^2 = sum_{j} |c_{j}|^2", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((lim_{N→inf} (1/(2N+1)) sum_{n=-N}^{N} |widehat{mu}(n)|^2),z):z\\in D\\}=\\{((sum_{j} |c_{j}|^2),z):z\\in D\\}", "to_expression": "lim_{N->inf} (1/(2N+1)) sum_{n=-N}^{N} |widehat{mu}(n)|^2 = sum_{j} |c_{j}|^2", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{N→inf} (1/(2N+1)) sum_{n=-N}^{N} |widehat{mu}(n)|^2 = sum_{j} |c_{j}|^2.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_9330e4e163980fed", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "extremal graph theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F0F061CBC3F5", "name": "Blow-up lemma", "statement_latex": "e(X,Y) > delta |X| |Y|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(delta |X| |Y|)-(e(X,Y))\\}=0\\;\\land\\;(delta |X| |Y|)\\ne (e(X,Y))", "to_expression": "e(X,Y) > delta |X| |Y|", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as e(X,Y) > delta |X| |Y|.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
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{"schema_version": "1.0", "item_id": "rcd2_b759ca8bf50925a8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_773C0118E21F", "name": "Darboux's theorem", "statement_latex": "f'(x) = y", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((f'(x)),z):z\\in D\\}=\\{((y),z):z\\in D\\}", "to_expression": "f'(x) = y", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f'(x) = y.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_2465778485baff60", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematical analysis", "subfield": "number theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_49b723f3d88c0ed6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "calculus", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_88C089C7703F", "name": "Leibniz integral rule", "statement_latex": "d/dx integral_{a(x)}^{b(x)} f(x,t) dt = f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((d/dx integral_{a(x)}^{b(x)} f(x,t) dt),(f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt))=0", "to_expression": "d/dx integral_{a(x)}^{b(x)} f(x,t) dt = f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt", "justification": "zero discrete distance iff equality", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as d/dx integral_{a(x)}^{b(x)} f(x,t) dt = f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_377616037ca74545", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0FE477D29636", "name": "Hardy–Ramanujan theorem", "statement_latex": "|ω(n)−loglogn|<(loglogn)^(1/2+ε)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(|ω(n)−loglogn|)}{\\exp(|ω(n)−loglogn|)+\\exp(loglogn)^(1/2+ε)}<\\frac{1}{2}", "to_expression": "|ω(n)−loglogn|<(loglogn)^(1/2+ε)", "justification": "equiv exp(X)<exp(Y)", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |ω(n)−loglogn|<(loglogn)^(1/2+ε).\"]"}], "source_equivalence_identity": "equiv exp(X)<exp(Y)"}
{"schema_version": "1.0", "item_id": "rcd2_a4fb3cdd1336abd0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "combinatorics", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_EE0D43BAAED5", "name": "Ahlswede–Daykin inequality", "statement_latex": "f_1(X)f_2(Y) ≤ f_3(X ∨ Y)f_4(X ∧ Y)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(f_1(X)f_2(Y))}{\\exp(f_1(X)f_2(Y))+\\exp(f_3(X ∨ Y)f_4(X ∧ Y))}\\le\\frac{1}{2}", "to_expression": "f_1(X)f_2(Y) ≤ f_3(X ∨ Y)f_4(X ∧ Y)", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f_1(X)f_2(Y) ≤ f_3(X ∨ Y)f_4(X ∧ Y).\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_fa3d510252da466c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "vector calculus", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2E42699EEB3D", "name": "Gradient theorem", "statement_latex": "∫_{γ} ∇φ(r) · dr = φ(q) - φ(p)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((∫_{γ} ∇φ(r) · dr))=F((φ(q) - φ(p)))", "to_expression": "∫_{γ} ∇φ(r) · dr = φ(q) - φ(p)", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∫_{γ} ∇φ(r) · dr = φ(q) - φ(p).\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_cc49b13c94c52052", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "algebraic combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_95918AB9C155", "name": "Kruskal–Katona theorem", "statement_latex": "N^{(i-1)}=\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((N^{(i-1)}),(\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1}))=0", "to_expression": "N^{(i-1)}=\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1}", "justification": "zero discrete distance iff equality", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as N^{(i-1)}=\\\\binom{n_i}{i-1}+\\\\binom{n_{i-1}}{i-2}+\\\\dots+\\\\binom{n_j}{j-1}.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_bb440288026b4961", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "large deviations theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_53869962DDDE", "name": "Laplace's principle", "statement_latex": "\\lim_{\\theta \\to \\infty}\\frac{1}{\\theta}\\log\\int_{A}e^{-\\theta\\varphi(x)}\\,dx=-\\mathop{\\mathrm{ess\\,inf}}_{x\\in A}\\varphi(x)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\lim_{\\theta → \\infty}\\frac{1}{\\theta}\\log\\int_{A}e^{-\\theta\\varphi(x)}\\,dx))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(-\\mathop{\\mathrm{ess\\,inf}}_{x\\in A}\\varphi(x)))", "to_expression": "\\lim_{\\theta \\to \\infty}\\frac{1}{\\theta}\\log\\int_{A}e^{-\\theta\\varphi(x)}\\,dx=-\\mathop{\\mathrm{ess\\,inf}}_{x\\in A}\\varphi(x)", "justification": "each argmin set is a singleton", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\lim_{\\\\theta → \\\\infty}\\\\frac{1}{\\\\theta}\\\\log\\\\int_{A}e^{-\\\\theta\\\\varphi(x)}\\\\,dx=-\\\\mathop{\\\\mathrm{ess\\\\,inf}}_{x\\\\in A}\\\\varphi(x).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_2195a70ec794a6a9", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum information theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0813262767f07d23", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "design theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f7ab720779f8269b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "measure theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FC57BDFBA0EA", "name": "Fernique's theorem", "statement_latex": "integral_{X} exp(alpha * ||x||^2) dmu(x) < +infinity", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(integral_{X} exp(alpha * ||x||^2) dmu(x))-(+infinity)\\}=0\\;\\land\\;(integral_{X} exp(alpha * ||x||^2) dmu(x))\\ne (+infinity)", "to_expression": "integral_{X} exp(alpha * ||x||^2) dmu(x) < +infinity", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as integral_{X} exp(alpha * ||x||^2) dmu(x) < +infinity.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_482c80dec40d7566", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_617ACB7CFB1C", "name": "Specht's theorem", "statement_latex": "tr W(A, A*) = tr W(B, B*)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(tr W(B, B*)) = (tr W(A, A*))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover tr W(A, A*) = tr W(B, B*).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "tr W(A, A*) = tr W(B, B*)", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover tr W(A, A*) = tr W(B, B*).\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_66cab5f6d936d09f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e2dba4a0470e3814", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "Combinatorics", "subfield": "Order theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B5F5A5581E5D", "name": "Kruskal's tree theorem", "statement_latex": "T1 <= T2", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((T1)-(T2),(-\\infty,0])=0", "to_expression": "T1 <= T2", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as T1 ≤ T2.\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
{"schema_version": "1.0", "item_id": "rcd2_030ac09b7511d767", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_9965e29bf7704ec5", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "representation theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1292421DFB1E", "name": "Schur's lemma", "statement_latex": "ρ_W(g) ∘ f = f ∘ ρ_V(g)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(ρ_W(g) ∘ f))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(f ∘ ρ_V(g)))", "to_expression": "ρ_W(g) ∘ f = f ∘ ρ_V(g)", "justification": "each argmin set is a singleton", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ρ_W(g) ∘ f = f ∘ ρ_V(g).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_86b0c56b8ac175d1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "inclusion-exclusion principle", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_58BBE6141124", "name": "Schuette–Nesbitt formula", "statement_latex": "sum_{k=0}^m c_k 1_{N=k} = sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((sum_{k=0}^m c_k 1_{N=k}),(sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k))=0", "to_expression": "sum_{k=0}^m c_k 1_{N=k} = sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k", "justification": "zero discrete distance iff equality", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{k=0}^m c_k 1_{N=k} = sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_65644afbd1e3e4ce", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic geometry", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_49F57EB1100D", "name": "Faltings' theorem", "statement_latex": "g > 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (1) < x \\land x < (g)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover g > 1.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover g > 1.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "g > 1", "justification": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover g > 1.\"]"}], "source_equivalence_identity": "(A,B) nonempty iff exists x with A<x<B, which is equivalent to A<B."}
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{"schema_version": "1.0", "item_id": "rcd2_09c367dcd9f8db8a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "calculus", "subfield": "vector calculus / electrostatics", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D5081E95CCA1", "name": "Uniqueness theorem for Poisson's equation", "statement_latex": "∇^2φ = -ρ_f/ε_0", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(∇^2φ)\\}=\\{(-ρ_f/ε_0)\\}", "to_expression": "∇^2φ = -ρ_f/ε_0", "justification": "singletons equal iff elements equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∇^2φ = -ρ_f/ε_0.\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
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{"schema_version": "1.0", "item_id": "rcd2_513b1598b9baa859", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_051344808DC5", "name": "Gauss congruence", "statement_latex": "sum_{d|n} mu(d)*a_{n/d} ≡ 0 (mod n)", "assumptions": "n is a positive integer; the displayed sum is an integer; mu is the Möbius function."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(n) \\mid (sum_{d|n} mu(d)*a_{n/d}) - (0)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "alpha ≡ beta mod m iff m divides alpha-beta.", "preconditions": "n is a positive integer; the displayed sum is an integer; mu is the Möbius function.", "inverse_mapping_note": "[\"Step 1: Substitute alpha, beta, and m from the variant formula.\",\"Step 2: Convert the witness, ideal-membership, quotient, residue, or distance condition to m | (alpha-beta).\",\"Step 3: Use the definition of congruence modulo m to recover sum_{d|n} mu(d)*a_{n/d} ≡ 0 (mod n).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "sum_{d|n} mu(d)*a_{n/d} ≡ 0 (mod n)", "justification": "alpha ≡ beta mod m iff m divides alpha-beta.", "preconditions": "n is a positive integer; the displayed sum is an integer; mu is the Möbius function.", "inverse_mapping_note": "[\"Step 1: Substitute alpha, beta, and m from the variant formula.\",\"Step 2: Convert the witness, ideal-membership, quotient, residue, or distance condition to m | (alpha-beta).\",\"Step 3: Use the definition of congruence modulo m to recover sum_{d|n} mu(d)*a_{n/d} ≡ 0 (mod n).\"]"}], "source_equivalence_identity": "alpha ≡ beta mod m iff m divides alpha-beta."}
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{"schema_version": "1.0", "item_id": "rcd2_62b414b9d18b3dd4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1C2EEE35D0F6", "name": "Fischer's inequality", "statement_latex": "det(M) <= det(A)*det(C)", "assumptions": "M is positive semidefinite and block-partitioned with principal blocks A and C."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\operatorname{softplus}((det(M))-(det(A)*det(C)))\\le\\log 2", "to_expression": "det(M) <= det(A)*det(C)", "justification": "softplus is increasing and softplus(0)=log2", "preconditions": "M is positive semidefinite and block-partitioned with principal blocks A and C.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as det(M) ≤ det(A)*det(C).\"]"}], "source_equivalence_identity": "softplus is increasing and softplus(0)=log2"}
{"schema_version": "1.0", "item_id": "rcd2_20ccc4fed064828a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "extremal graph theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4D5DD853BD7", "name": "Szemerédi regularity lemma", "statement_latex": "|d(X,Y) - d(A,B)| <= ε", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(|d(X,Y) - d(A,B)|)-(ε)\\}=0", "to_expression": "|d(X,Y) - d(A,B)| <= ε", "justification": "positive part vanishes iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |d(X,Y) - d(A,B)| ≤ ε.\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_f308238a1ba07a9c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_cead893eee6bb0ad", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_45A3C68C1B71", "name": "Linnik's theorem", "statement_latex": "p(a,d) < c * d^L", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(p(a,d))-(c * d^L)\\}=0\\;\\land\\;(p(a,d))\\ne (c * d^L)", "to_expression": "p(a,d) < c * d^L", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p(a,d) < c * d^L.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_1d07f441412c4ea9", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "calculus", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b81291ea2f3f8929", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_aab6f364324e2818", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "percolation theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6f0762f2ce2af172", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematical analysis", "subfield": "number theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_8899bbee82c3142f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_7654f8274afb3628", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "representation theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_fc98b2e75d3803cc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_42828E58305E", "name": "Pascal's rule", "statement_latex": "\\binom{n-1}{k} + \\binom{n-1}{k-1} = \\binom{n}{k}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((\\binom{n-1}{k} + \\binom{n-1}{k-1}) \\in S \\land (\\binom{n}{k}) \\notin S) \\lor ((\\binom{n}{k}) \\in S \\land (\\binom{n-1}{k} + \\binom{n-1}{k-1}) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\binom{n-1}{k} + \\\\binom{n-1}{k-1} = \\\\binom{n}{k}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\binom{n-1}{k} + \\binom{n-1}{k-1} = \\binom{n}{k}", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\binom{n-1}{k} + \\\\binom{n-1}{k-1} = \\\\binom{n}{k}.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_019d6531a75cb753", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum information theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_95ae01b78d09c3f1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum mechanics", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9078CE9B3CD4", "name": "Ehrenfest theorem", "statement_latex": "F = -V'(x)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(F)=z\\;\\land\\;z=(-V'(x))", "to_expression": "F = -V'(x)", "justification": "common witness iff equality", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as F = -V'(x).\"]"}], "source_equivalence_identity": "common witness iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_79f748a9aa349da8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "statistical mechanics", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4D516E39AEB4", "name": "H-theorem", "statement_latex": "H(t) = ∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(H(t)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE))", "to_expression": "H(t) = ∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE", "justification": "each argmin set is a singleton", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as H(t) = ∫ f(E,t) * ln(f(E,t) / sqrt(E) - 1) dE.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
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{"schema_version": "1.0", "item_id": "rcd2_0383c29f5d9ba824", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "approximation theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_DC96C747092B", "name": "Müntz–Szász theorem", "statement_latex": "sum_{n in S} (1/n) = infinity", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((sum_{n in S} (1/n)),(infinity))\\in\\{(z,z):z\\in D\\}", "to_expression": "sum_{n in S} (1/n) = infinity", "justification": "pair in diagonal iff X=Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{n in S} (1/n) = infinity.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_af3f970c00253080", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3DB69D9F31E5", "name": "Reynolds transport theorem", "statement_latex": "d/dt integral_{Omega(t)} f dV = integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(d/dt integral_{Omega(t)} f dV))+\\delta_D(z,(integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA)))=0", "to_expression": "d/dt integral_{Omega(t)} f dV = integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA", "justification": "zero loss iff some z equals both", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as d/dt integral_{Omega(t)} f dV = integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA.\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
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{"schema_version": "1.0", "item_id": "rcd2_45f638c90a3357c0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "geometry", "subfield": "Euclidean geometry", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F38D8FABD394", "name": "Ptolemy's inequality", "statement_latex": "AB * CD + BC * DA >= AC * BD", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\operatorname{softplus}((AC * BD)-(AB * CD + BC * DA))\\le\\log 2", "to_expression": "AB * CD + BC * DA >= AC * BD", "justification": "softplus is increasing and softplus(0)=log2", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as AB * CD + BC * DA ≥ AC * BD.\"]"}], "source_equivalence_identity": "softplus is increasing and softplus(0)=log2"}
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{"schema_version": "1.0", "item_id": "rcd2_276e38004b1fb789", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "commutative algebra", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AB6EC963565B", "name": "Artin–Rees lemma", "statement_latex": "I^nM ∩ N = I^{n-k}(I^kM ∩ N)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((I^nM ∩ N),z):z\\in D\\}=\\{((I^{n-k}(I^kM ∩ N)),z):z\\in D\\}", "to_expression": "I^nM ∩ N = I^{n-k}(I^kM ∩ N)", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as I^nM ∩ N = I^{n-k}(I^kM ∩ N).\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_c1bf5dd809ef3ff4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3BD55A296126", "name": "Rank–nullity theorem", "statement_latex": "rank(T) + nullity(T) = dim(V)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(rank(T) + nullity(T)))+\\delta_D(z,(dim(V))))=0", "to_expression": "rank(T) + nullity(T) = dim(V)", "justification": "zero loss iff some z equals both", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as rank(T) + nullity(T) = dim(V).\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
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{"schema_version": "1.0", "item_id": "rcd2_020dfc2462e1c4b7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "representation theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_13CEC210D728", "name": "Wigner–Eckart theorem", "statement_latex": "⟨j m | T_q^(k) | j' m'⟩ = ⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((⟨j m | T_q^(k) | j' m'⟩),(⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩))=0", "to_expression": "⟨j m | T_q^(k) | j' m'⟩ = ⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩", "justification": "zero discrete distance iff equality", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ⟨j m | T_q^(k) | j' m'⟩ = ⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_801361205d7e2558", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "inclusion-exclusion principle", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_58BBE6141124", "name": "Schuette–Nesbitt formula", "statement_latex": "sum_{k=0}^m c_k 1_{N=k} = sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (sum_{k=0}^m c_k 1_{N=k} = sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "sum_{k=0}^m c_k 1_{N=k} = sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_bab5ea97193b3010", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "representation theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D8B93C65D10E", "name": "McKay conjecture", "statement_latex": "|Irr_{p'}(G)|=|Irr_{p'}(N_{G}(P))|", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((|Irr_{p'}(G)|))=F((|Irr_{p'}(N_{G}(P))|))", "to_expression": "|Irr_{p'}(G)|=|Irr_{p'}(N_{G}(P))|", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |Irr_{p'}(G)|=|Irr_{p'}(N_{G}(P))|.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
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{"schema_version": "1.0", "item_id": "rcd2_6d2b5b4eb6cf4cd9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "representation theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CAF4E18AA4C8", "name": "Brauer's height zero conjecture", "statement_latex": "\\nu(\\chi(1)) = \\nu(|G:D|)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((\\nu(\\chi(1))),z):z\\in D\\}=\\{((\\nu(|G:D|)),z):z\\in D\\}", "to_expression": "\\nu(\\chi(1)) = \\nu(|G:D|)", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\nu(\\\\chi(1)) = \\\\nu(|G:D|).\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_968a638644dc17e3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7BB84422BC5F", "name": "Hadamard's inequality", "statement_latex": "|det(N)| <= prod_{i=1}^n ||v_i||", "assumptions": "N is a square matrix with row or column vectors v_i in an inner-product space; norms are Euclidean."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\operatorname{softplus}((|det(N)|)-(prod_{i=1}^n ||v_i||))\\le\\log 2", "to_expression": "|det(N)| <= prod_{i=1}^n ||v_i||", "justification": "softplus is increasing and softplus(0)=log2", "preconditions": "N is a square matrix with row or column vectors v_i in an inner-product space; norms are Euclidean.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |det(N)| ≤ prod_{i=1}^n ||v_i||.\"]"}], "source_equivalence_identity": "softplus is increasing and softplus(0)=log2"}
{"schema_version": "1.0", "item_id": "rcd2_21b903155a2fe99b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e0ea5be342965139", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "combinatorics on words", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3E0E814BA7D9", "name": "Levi's lemma", "statement_latex": "uv = xy", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((uv) \\in S \\land (xy) \\notin S) \\lor ((xy) \\in S \\land (uv) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover uv = xy.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "uv = xy", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover uv = xy.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_01f884b566f4801a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "enumerative combinatorics", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7EDAF21632BF", "name": "Labelled enumeration theorem", "statement_latex": "F(z,t)=e^{g(z)t}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(F(z,t)))+\\delta_D(z,(e^{g(z)t})))=0", "to_expression": "F(z,t)=e^{g(z)t}", "justification": "zero loss iff some z equals both", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as F(z,t)=e^{g(z)t}.\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_564fb63145ce20ed", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "Sobolev spaces", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_79b7647bad8bcba8", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "inclusion-exclusion principle", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_3d2097ea21807fcf", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "elementary number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0F6BD7E91884", "name": "Bézout's identity", "statement_latex": "ax + by = gcd(a, b)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((ax + by),(gcd(a, b)))=0", "to_expression": "ax + by = gcd(a, b)", "justification": "zero discrete distance iff equality", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ax + by = gcd(a, b).\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_fe40129fd8b57a98", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2DC4FABA16C6", "name": "Kantorovich inequality", "statement_latex": "(Σ p_i x_i) * (Σ p_i / x_i) <= (a+b)^2 / (4ab)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "((a+b)^2 / (4ab)) - ((Σ p_i x_i) * (Σ p_i / x_i)) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (Σ p_i x_i) * (Σ p_i / x_i) <= (a+b)^2 / (4ab).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (Σ p_i x_i) * (Σ p_i / x_i) <= (a+b)^2 / (4ab).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "(Σ p_i x_i) * (Σ p_i / x_i) <= (a+b)^2 / (4ab)", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover (Σ p_i x_i) * (Σ p_i / x_i) <= (a+b)^2 / (4ab).\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_3fab9e4574a2e4ce", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "control theory", "subfield": "geometric control theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2ADEC5E217BE", "name": "Krener's theorem", "statement_latex": "\\dot{q}=f(q,u)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "(\\{(\\dot{q})\\}\\times\\{(f(q,u))\\})\\cap\\{(z,z):z\\in D\\}=\\{((\\dot{q}),(f(q,u)))\\}", "to_expression": "\\dot{q}=f(q,u)", "justification": "intersection is singleton iff pair diagonal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\dot{q}=f(q,u).\"]"}], "source_equivalence_identity": "intersection is singleton iff pair diagonal"}
{"schema_version": "1.0", "item_id": "rcd2_421055a40df188f8", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "coding theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_3cc560f548e45440", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "coding theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0213f7359b431293", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f4579c81146869af", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematical analysis", "subfield": "number theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0AAEED4B3B51", "name": "Goldbach–Euler theorem", "statement_latex": "sum_{p=2}^{\\infty} 1/(p-1) = 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (sum_{p=2}^{\\infty} 1/(p-1) = 1))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "sum_{p=2}^{\\infty} 1/(p-1) = 1", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_cf7e538ff1d9d0f2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AFD3A111ECE4", "name": "Boole's inequality", "statement_latex": "P(∪_{i=1}^∞ A_i) ≤ ∑_{i=1}^∞ P(A_i)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(∑_{i=1}^∞ P(A_i)) - (P(∪_{i=1}^∞ A_i)) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover P(∪_{i=1}^∞ A_i) <= ∑_{i=1}^∞ P(A_i).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover P(∪_{i=1}^∞ A_i) <= ∑_{i=1}^∞ P(A_i).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "P(∪_{i=1}^∞ A_i) ≤ ∑_{i=1}^∞ P(A_i)", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover P(∪_{i=1}^∞ A_i) <= ∑_{i=1}^∞ P(A_i).\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_18e670434eae62a6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "algebraic K-theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_582614A6491A", "name": "Whitehead's lemma", "statement_latex": "[[u, 0], [0, u^-1]] = e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)) = ([[u, 0], [0, u^-1]])", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover [[u, 0], [0, u^-1]] = e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "[[u, 0], [0, u^-1]] = e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover [[u, 0], [0, u^-1]] = e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1).\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_e41c36c446bdf551", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "Mathematical analysis", "subfield": "Sequences and series", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_921a76f6edb470bb", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CCCCB5088787", "name": "Mean value theorem", "statement_latex": "f'(c) = (f(b)-f(a))/(b-a)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(f'(c)) \\in {((f(b)-f(a))/(b-a))}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f'(c) = (f(b)-f(a))/(b-a).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f'(c) = (f(b)-f(a))/(b-a)", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f'(c) = (f(b)-f(a))/(b-a).\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
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{"schema_version": "1.0", "item_id": "rcd2_e4f1f7b37dce275c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "measure theory", "subfield": "stochastic analysis", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_fdc4befaa8e4a544", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "graph theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_07CDE34AC57A", "name": "Counting lemma", "statement_latex": "γ <= ε^h / 4h", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((γ)-(ε^h / 4h),(-\\infty,0])=0", "to_expression": "γ <= ε^h / 4h", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as γ ≤ ε^h / 4h.\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
{"schema_version": "1.0", "item_id": "rcd2_34ec5865ac0efbad", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "analysis", "subfield": "approximation theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e1e4ebb749df1d3e", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_16598d9b5c092193", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2b56ec612096c89a", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c5ae805707bb3d85", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0506260C92DB", "name": "Bennett's inequality", "statement_latex": "Pr(S_n > t) <= exp(-(sigma^2 / a^2) * h(at / sigma^2))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(Pr(S_n > t)) - (\\exp(-(sigma^2 / a^2) * h(at / sigma^2))) \\le 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A-B <= 0 iff A <= B by adding B to both sides.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Pr(S_n > t) <= exp(-(sigma^2 / a^2) * h(at / sigma^2)).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A-B <= 0 iff A <= B by adding B to both sides.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Pr(S_n > t) <= exp(-(sigma^2 / a^2) * h(at / sigma^2)).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "Pr(S_n > t) <= exp(-(sigma^2 / a^2) * h(at / sigma^2))", "justification": "A-B <= 0 iff A <= B by adding B to both sides.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Pr(S_n > t) <= exp(-(sigma^2 / a^2) * h(at / sigma^2)).\"]"}], "source_equivalence_identity": "A-B <= 0 iff A <= B by adding B to both sides."}
{"schema_version": "1.0", "item_id": "rcd2_04992f36dc4c8cda", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CCCCB5088787", "name": "Mean value theorem", "statement_latex": "f'(c) = (f(b)-f(a))/(b-a)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((f'(c)) \\in S \\land ((f(b)-f(a))/(b-a)) \\notin S) \\lor (((f(b)-f(a))/(b-a)) \\in S \\land (f'(c)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f'(c) = (f(b)-f(a))/(b-a).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f'(c) = (f(b)-f(a))/(b-a)", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f'(c) = (f(b)-f(a))/(b-a).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_bba38e27fb61a0b2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_617ACB7CFB1C", "name": "Specht's theorem", "statement_latex": "tr W(A, A*) = tr W(B, B*)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(tr W(A, A*)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(tr W(B, B*)))", "to_expression": "tr W(A, A*) = tr W(B, B*)", "justification": "each argmin set is a singleton", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as tr W(A, A*) = tr W(B, B*).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
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{"schema_version": "1.0", "item_id": "rcd2_3fa7fe27c8bb3e44", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "graph theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2a362a899d5e2dc9", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "probabilistic combinatorics", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0f5adc18d213ab75", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "Ramsey theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A40EE0DC8F58", "name": "Van der Waerden's theorem", "statement_latex": "W(r,k) ≤ 2^{2^{r^{2^{2^{k+9}}}}}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(2^{2^{r^{2^{2^{k+9}}}}}) - (W(r,k)) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover W(r,k) <= 2^{2^{r^{2^{2^{k+9}}}}}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover W(r,k) <= 2^{2^{r^{2^{2^{k+9}}}}}.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "W(r,k) ≤ 2^{2^{r^{2^{2^{k+9}}}}}", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover W(r,k) <= 2^{2^{r^{2^{2^{k+9}}}}}.\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_9587acb5bcacbe28", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "topology", "subfield": "combinatorial topology", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CB3E317B00C5", "name": "Tucker's lemma", "statement_latex": "L(-v) = -L(v)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((L(-v)) \\in S \\land (-L(v)) \\notin S) \\lor ((-L(v)) \\in S \\land (L(-v)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover L(-v) = -L(v).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "L(-v) = -L(v)", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover L(-v) = -L(v).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_125aa2859642e72a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "quadratic residues", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A18C6C225F06", "name": "Zolotarev's lemma", "statement_latex": "(a/p) = epsilon(pi_a)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: ((((a/p)) \\in S \\land (\\varepsilon(pi_a)) \\notin S) \\lor ((\\varepsilon(pi_a)) \\in S \\land ((a/p)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover (a/p) = epsilon(pi_a).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "(a/p) = epsilon(pi_a)", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover (a/p) = epsilon(pi_a).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_c71ad74a0911080b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "sieve theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5DC5BE909937", "name": "Fundamental lemma of sieve theory", "statement_latex": "|A_d| = \\frac{w(d)}{d}X + R_d", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((|A_d|))=F((\\frac{w(d)}{d}X + R_d))", "to_expression": "|A_d| = \\frac{w(d)}{d}X + R_d", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |A_d| = \\\\frac{w(d)}{d}X + R_d.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_9ba8f667c7f84eef", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "intersection", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7C6C20BC69B4", "name": "Slepian's lemma", "statement_latex": "Pr[Intersection(X_i <= u_i)] <= Pr[Intersection(Y_i <= u_i)]", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\frac{\\exp(Pr[Intersection(X_i ≤ u_i)])}{\\exp(Pr[Intersection(X_i ≤ u_i)])+\\exp(Pr[Intersection(Y_i ≤ u_i)])}\\le\\frac{1}{2}", "to_expression": "Pr[Intersection(X_i <= u_i)] <= Pr[Intersection(Y_i <= u_i)]", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr[Intersection(X_i ≤ u_i)] ≤ Pr[Intersection(Y_i ≤ u_i)].\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_2157838478ca50db", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AFD3A111ECE4", "name": "Boole's inequality", "statement_latex": "P(∪_{i=1}^∞ A_i) ≤ ∑_{i=1}^∞ P(A_i)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((P(∪_{i=1}^∞ A_i))-(∑_{i=1}^∞ P(A_i)),(-\\infty,0])=0", "to_expression": "P(∪_{i=1}^∞ A_i) ≤ ∑_{i=1}^∞ P(A_i)", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(∪_{i=1}^∞ A_i) ≤ ∑_{i=1}^∞ P(A_i).\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
{"schema_version": "1.0", "item_id": "rcd2_e341ce7bfbc967e4", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "combinatorics", "subfield": "combinatorial geometry", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_5d3ec810f8f3d5e7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "combinatorial number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3FBB7DDC427F", "name": "Erdős–Graham problem", "statement_latex": "sum_{n in S} (1/n) = 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(sum_{n \\in S} (1/n)) \\in {(1)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover sum_{n in S} (1/n) = 1.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "sum_{n in S} (1/n) = 1", "justification": "L in {R} iff L = R.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover sum_{n in S} (1/n) = 1.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_933d1f7c4a570618", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "calculus of variations", "subfield": "Morse theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5B2379DEDFD8", "name": "Morse–Palais lemma", "statement_latex": "f(x) = ⟨Aφ(x), φ(x)⟩", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((f(x)))=F((⟨Aφ(x), φ(x)⟩))", "to_expression": "f(x) = ⟨Aφ(x), φ(x)⟩", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f(x) = ⟨Aφ(x), φ(x)⟩.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_911dbed887181438", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "Number theory", "subfield": "Additive number theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ade69f2fe4b188c6", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_914ba9e86b167a6d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "additive number theory", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_041f731cd54aaa3f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "approximation theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9336CBED6851", "name": "Bernstein's theorem (polynomials)", "statement_latex": "max_{|z|=1}|P'(z)| <= n max_{|z|=1}|P(z)|", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (n max_{|z|=1}|P(z)|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (max_{|z|=1}|P'(z)|)\\}", "to_expression": "max_{|z|=1}|P'(z)| <= n max_{|z|=1}|P(z)|", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as max_{|z|=1}|P'(z)| ≤ n max_{|z|=1}|P(z)|.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_3f1f578d0fa8eb45", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "extremal graph theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0f0179732403d141", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0506260C92DB", "name": "Bennett's inequality", "statement_latex": "Pr(S_n > t) <= exp(-(sigma^2 / a^2) * h(at / sigma^2))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\exp(-(sigma^2 / a^2) * h(at / sigma^2))) - (Pr(S_n > t)) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Pr(S_n > t) <= exp(-(sigma^2 / a^2) * h(at / sigma^2)).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Pr(S_n > t) <= exp(-(sigma^2 / a^2) * h(at / sigma^2)).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "Pr(S_n > t) <= exp(-(sigma^2 / a^2) * h(at / sigma^2))", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover Pr(S_n > t) <= exp(-(sigma^2 / a^2) * h(at / sigma^2)).\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_c2ffadc42182df9b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "algebra", "subfield": "combinatorics", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_275e02b7ca817fd0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematical economics", "subfield": "linear algebra", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1A4ADB6F0297", "name": "Hawkins–Simon condition", "statement_latex": "[I - A]x = d", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{(([I - A]x),z):z\\in D\\}=\\{((d),z):z\\in D\\}", "to_expression": "[I - A]x = d", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as [I - A]x = d.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_19a65c1daf3586a1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "matching theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_69DD0D246726", "name": "Tutte–Berge formula", "statement_latex": "max_matching_size = 1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "(\\{(max_matching_size)\\}\\times\\{(1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|))\\})\\cap\\{(z,z):z\\in D\\}=\\{((max_matching_size),(1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)))\\}", "to_expression": "max_matching_size = 1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)", "justification": "intersection is singleton iff pair diagonal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as max_matching_size = 1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|).\"]"}], "source_equivalence_identity": "intersection is singleton iff pair diagonal"}
{"schema_version": "1.0", "item_id": "rcd2_34a380b8d5e4a737", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A54D9D349D9E", "name": "Candido's identity", "statement_latex": "[x^2 + y^2 + (x+y)^2]^2 = 2[x^4 + y^4 + (x+y)^4]", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D(([x^2 + y^2 + (x+y)^2]^2),(2[x^4 + y^4 + (x+y)^4]))=0", "to_expression": "[x^2 + y^2 + (x+y)^2]^2 = 2[x^4 + y^4 + (x+y)^4]", "justification": "zero discrete distance iff equality", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as [x^2 + y^2 + (x+y)^2]^2 = 2[x^4 + y^4 + (x+y)^4].\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_576a83b0949109be", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2bc9a87801d3aafc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7922A15B98B4", "name": "Kolmogorov's inequality", "statement_latex": "Pr(max_{1<=k<=n}|S_{k}|>=λ) <= (1/λ^2) * Var[S_{n}]", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(Pr(max_{1≤k≤n}|S_{k}|≥λ))-((1/λ^2) * Var[S_{n}])\\}=0", "to_expression": "Pr(max_{1<=k<=n}|S_{k}|>=λ) <= (1/λ^2) * Var[S_{n}]", "justification": "positive part vanishes iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr(max_{1≤k≤n}|S_{k}|≥λ) ≤ (1/λ^2) * Var[S_{n}].\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_2993bb8d22801e8a", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_893dd8586f6d9ac5", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "elementary algebra", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a52ae08957c305a0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "algebraic K-theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_582614A6491A", "name": "Whitehead's lemma", "statement_latex": "[[u, 0], [0, u^-1]] = e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: ((([[u, 0], [0, u^-1]]) \\in S \\land (e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)) \\notin S) \\lor ((e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)) \\in S \\land ([[u, 0], [0, u^-1]]) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover [[u, 0], [0, u^-1]] = e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "[[u, 0], [0, u^-1]] = e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover [[u, 0], [0, u^-1]] = e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_e2db6828dafcb10c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "geometry", "subfield": "Euclidean geometry", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F38D8FABD394", "name": "Ptolemy's inequality", "statement_latex": "AB * CD + BC * DA >= AC * BD", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (AB * CD + BC * DA)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (AC * BD)\\}", "to_expression": "AB * CD + BC * DA >= AC * BD", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as AB * CD + BC * DA ≥ AC * BD.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_55d8c68928a5f177", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "hypergeometric series", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_7ce3f6051346c22b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "automorphic forms", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_577277997be6db38", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "Sobolev spaces", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_bb52f4c29cad4ac4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_94B20D8A1D74", "name": "Chebyshev's inequality", "statement_latex": "Pr(|X-mu| >= k*sigma) <= 1/k^2", "assumptions": "X has mean mu and finite nonzero standard deviation sigma; k>0."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(Pr(|X-mu| ≥ k*sigma))}{\\exp(Pr(|X-mu| ≥ k*sigma))+\\exp(1/k^2)}\\le\\frac{1}{2}", "to_expression": "Pr(|X-mu| >= k*sigma) <= 1/k^2", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "X has mean mu and finite nonzero standard deviation sigma; k>0.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr(|X-mu| ≥ k*sigma) ≤ 1/k^2.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_c9ad6db8a7f22b02", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "geometry, probability theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_60cf00adafbbc993", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_19b594f55f72dcd5", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "queueing theory", "subfield": "probability", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2552d3569686b8b8", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "probability theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_52f5c8ef79bb0d67", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "special functions", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CA3873AC4ACD", "name": "Ramanujan's master theorem", "statement_latex": "∫[0,∞] x^{s-1} f(x) dx = Γ(s) φ(-s)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((∫[0,∞] x^{s-1} f(x) dx),z)=((Γ(s) φ(-s)),z)", "to_expression": "∫[0,∞] x^{s-1} f(x) dx = Γ(s) φ(-s)", "justification": "pair equality forces first coordinates equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∫[0,∞] x^{s-1} f(x) dx = Γ(s) φ(-s).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_dead898a1bad17a9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "vector calculus", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_24234F1E6547", "name": "Divergence theorem", "statement_latex": "\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S} = \\int_{V} (\\nabla \\cdot \\mathbf{F}) dV", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S}),z)=((\\int_{V} (\\nabla \\cdot \\mathbf{F}) dV),z)", "to_expression": "\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S} = \\int_{V} (\\nabla \\cdot \\mathbf{F}) dV", "justification": "pair equality forces first coordinates equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\oint_{S} \\\\mathbf{F} \\\\cdot d\\\\mathbf{S} = \\\\int_{V} (\\\\nabla \\\\cdot \\\\mathbf{F}) dV.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_1125bb34b24596e3", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_86355f4b84e0ed6a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic geometry", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BB5E3AF7EFD7", "name": "Modularity theorem", "statement_latex": "L(E, s) = sum_{n=1}^∞ a_n / n^s", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(L(E, s) = sum_{n=1}^∞ a_n / n^s) \\land (0 = 0)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "L(E, s) = sum_{n=1}^∞ a_n / n^s", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "0=0 is true, so Phi_T and true iff Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_79286948df3bcab7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "algebraic combinatorics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_95918AB9C155", "name": "Kruskal–Katona theorem", "statement_latex": "N^{(i-1)}=\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((N^{(i-1)}),(\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1}))\\in\\{(z,z):z\\in D\\}", "to_expression": "N^{(i-1)}=\\binom{n_i}{i-1}+\\binom{n_{i-1}}{i-2}+\\dots+\\binom{n_j}{j-1}", "justification": "pair in diagonal iff X=Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as N^{(i-1)}=\\\\binom{n_i}{i-1}+\\\\binom{n_{i-1}}{i-2}+\\\\dots+\\\\binom{n_j}{j-1}.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_3c6e0e6d859f0877", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "representation theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_13CEC210D728", "name": "Wigner–Eckart theorem", "statement_latex": "⟨j m | T_q^(k) | j' m'⟩ = ⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(⟨j m | T_q^(k) | j' m'⟩) \\in {(⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover ⟨j m | T_q^(k) | j' m'⟩ = ⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "⟨j m | T_q^(k) | j' m'⟩ = ⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover ⟨j m | T_q^(k) | j' m'⟩ = ⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
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{"schema_version": "1.0", "item_id": "rcd2_0b17490087fc2380", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_893C1D2D0065", "name": "Fermat's Last Theorem", "statement_latex": "a^n + b^n = c^n", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((a^n + b^n),(c^n))\\in\\{(z,z):z\\in D\\}", "to_expression": "a^n + b^n = c^n", "justification": "pair in diagonal iff X=Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as a^n + b^n = c^n.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_43ebfc9c5f2bf842", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "infinite-dimensional vector spaces", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B23BCA4AB254", "name": "Erdős–Kaplansky theorem", "statement_latex": "\\dim(E^*) = \\text{card}(\\mathbb{K}^I)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\text{card}(\\mathbb{K}^I)) = (\\dim(E^*))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\dim(E^*) = \\\\text{card}(\\\\mathbb{K}^I).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\dim(E^*) = \\text{card}(\\mathbb{K}^I)", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\dim(E^*) = \\\\text{card}(\\\\mathbb{K}^I).\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_c7a342fd7d865d7d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "combinatorics on words", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_692f5ed38c16157a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E6F93BEA2A51", "name": "Cassini and Catalan identities", "statement_latex": "F_{n-1}*F_{n+1} - F_n^2 = (-1)^n", "assumptions": "F_n denotes the Fibonacci sequence with the indexing convention used by the theorem; n is an integer in the stated range."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "((-1)^n) = (F_{n-1}*F_{n+1} - F_n^2)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "F_n denotes the Fibonacci sequence with the indexing convention used by the theorem; n is an integer in the stated range.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover F_{n-1}*F_{n+1} - F_n^2 = (-1)^n.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "F_{n-1}*F_{n+1} - F_n^2 = (-1)^n", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "F_n denotes the Fibonacci sequence with the indexing convention used by the theorem; n is an integer in the stated range.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover F_{n-1}*F_{n+1} - F_n^2 = (-1)^n.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_e4f7da0f4be53d58", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f8a294fba4a8dd4b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "algebra", "subfield": "combinatorics", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E24E6480323C", "name": "Multinomial theorem", "statement_latex": "(x_{1}+x_{2}+\\cdots +x_{m})^{n}=\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;(((x_{1}+x_{2}+\\cdots +x_{m})^{n}),z)=((\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}}),z)", "to_expression": "(x_{1}+x_{2}+\\cdots +x_{m})^{n}=\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}}", "justification": "pair equality forces first coordinates equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (x_{1}+x_{2}+\\\\cdots +x_{m})^{n}=\\\\sum _{k_{1}+k_{2}+\\\\cdots +k_{m}=n} \\\\binom{n}{k_1, k_2, \\\\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\\\cdots x_{m}^{k_{m}}.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
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{"schema_version": "1.0", "item_id": "rcd2_30b6e21721c27e48", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "physics", "subfield": "quantum field theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_373f29e202e68b1d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "infinite-dimensional vector spaces", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_8bdfbdf4986bdfc7", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "probabilistic combinatorics", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f11f844daf54447d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_1594bd9a579dfe2a", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "commutative algebra", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_40048c68d319c79f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "inequalities", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_91b1fe1dc43a7827", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "field theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_6E9382B37B92", "name": "Primitive element theorem", "statement_latex": "E = F(α)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(E) \\in {(F(α))}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E = F(α).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "E = F(α)", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E = F(α).\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_75bbf59e9f885039", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_3ead0fb236adbcf7", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic functions", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6151f6b035c13da9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "matching theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_69DD0D246726", "name": "Tutte–Berge formula", "statement_latex": "max_matching_size = 1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((max_matching_size),z)=((1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)),z)", "to_expression": "max_matching_size = 1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)", "justification": "pair equality forces first coordinates equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as max_matching_size = 1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_61fa5ba1681bf179", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_203ad24190faf7f7", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d688439786af3bd7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "representation theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D8B93C65D10E", "name": "McKay conjecture", "statement_latex": "|Irr_{p'}(G)|=|Irr_{p'}(N_{G}(P))|", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(|Irr_{p'}(N_{G}(P))|) = (|Irr_{p'}(G)|)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover |Irr_{p'}(G)|=|Irr_{p'}(N_{G}(P))|.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "|Irr_{p'}(G)|=|Irr_{p'}(N_{G}(P))|", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover |Irr_{p'}(G)|=|Irr_{p'}(N_{G}(P))|.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_6631bf8f668d1b46", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "Ramsey theory", "subfield": "discrete mathematics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4B3C8E83990", "name": "Schur's theorem", "statement_latex": "x + y = z", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(z) = (x + y)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover x + y = z.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "x + y = z", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover x + y = z.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_a4804a253e45aa9d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_79DC6C43DD64", "name": "Perron–Frobenius theorem", "statement_latex": "Av = rv", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((Av),(rv))=0", "to_expression": "Av = rv", "justification": "zero discrete distance iff equality", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Av = rv.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_86d3ce6ddcb8f4de", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum information theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B8493CAFC989", "name": "No-deleting theorem", "statement_latex": "U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C = |\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C)\\Longleftrightarrow z=(|\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C))", "to_expression": "U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C = |\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C", "justification": "same equality fiber iff equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as U|\\\\psi \\\\rangle_A |\\\\psi \\\\rangle_B |A \\\\rangle_C = |\\\\psi \\\\rangle_A |0 \\\\rangle_B |A' \\\\rangle_C.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
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{"schema_version": "1.0", "item_id": "rcd2_0bb6ed38a9a784a7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "algebra", "subfield": "modular arithmetic", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A1DCD328739C", "name": "Hensel's lemma", "statement_latex": "f(x) = 0 (mod p^n)", "assumptions": "All displayed arithmetic expressions are integers; the modulus is a positive integer."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((f(x)) \\in S \\land (0 (mod p^n)) \\notin S) \\lor ((0 (mod p^n)) \\in S \\land (f(x)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic expressions are integers; the modulus is a positive integer.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(x) = 0 (mod p^n).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f(x) = 0 (mod p^n)", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed arithmetic expressions are integers; the modulus is a positive integer.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(x) = 0 (mod p^n).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_2584a59ed2dda73e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "algebra", "subfield": "polynomials", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_235D1B5AD510", "name": "Routh–Hurwitz theorem", "statement_latex": "p - q = (1/π) Δ arg f(iy)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(p - q) \\in {((1/π) Δ arg f(iy))}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover p - q = (1/π) Δ arg f(iy).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "p - q = (1/π) Δ arg f(iy)", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover p - q = (1/π) Δ arg f(iy).\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_fd594362b7e46094", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "algebraic K-theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_582614A6491A", "name": "Whitehead's lemma", "statement_latex": "[[u, 0], [0, u^-1]] = e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F(([[u, 0], [0, u^-1]]))=F((e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)))", "to_expression": "[[u, 0], [0, u^-1]] = e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)", "justification": "indicator of {X} separates unequal objects", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as [[u, 0], [0, u^-1]] = e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1).\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_9c44462f20de09b1", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "graph theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b61cdf2bc3734510", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_5292e4401ab86b3e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "large deviations theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B2BA1A96D211", "name": "Varadhan's lemma", "statement_latex": "lim_{ε→0} ε log E[exp(φ(Z_ε)/ε)] = sup_{x∈X} (φ(x) - I(x))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((lim_{ε→0} ε log E[exp(φ(Z_ε)/ε)]),(sup_{x∈X} (φ(x) - I(x))))=0", "to_expression": "lim_{ε→0} ε log E[exp(φ(Z_ε)/ε)] = sup_{x∈X} (φ(x) - I(x))", "justification": "zero discrete distance iff equality", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{ε→0} ε log E[exp(φ(Z_ε)/ε)] = sup_{x∈X} (φ(x) - I(x)).\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_53a039950c75f367", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "combinatorics", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_847fe3890b71a87c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_303E62405496", "name": "Lorden's inequality", "statement_latex": "E(R_b) <= E(X^2) / E(X)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(E(R_b))+s=(E(X^2) / E(X))", "to_expression": "E(R_b) <= E(X^2) / E(X)", "justification": "nonnegative slack exists iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as E(R_b) ≤ E(X^2) / E(X).\"]"}], "source_equivalence_identity": "nonnegative slack exists iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_720e1f7be2e10e65", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "computer science", "subfield": "theoretical computer science", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_30A4875B9AAE", "name": "Isolation lemma", "statement_latex": "w(S) = sum_{x in S} w(x)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(w(S))\\Longleftrightarrow z=(sum_{x in S} w(x)))", "to_expression": "w(S) = sum_{x in S} w(x)", "justification": "same equality fiber iff equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as w(S) = sum_{x in S} w(x).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
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{"schema_version": "1.0", "item_id": "rcd2_64b60b306ea5cd50", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "inequalities", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9822BD8B9BE6", "name": "Muirhead's inequality", "statement_latex": "[a] >= [b] for a majorizing b", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "([a]) - ([b] for a majorizing b) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover [a] >= [b] for a majorizing b.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover [a] >= [b] for a majorizing b.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "[a] >= [b] for a majorizing b", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover [a] >= [b] for a majorizing b.\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
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{"schema_version": "1.0", "item_id": "rcd2_2440b0622d7ef1e8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "differential calculus", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F1B027EB50E5", "name": "Interior extremum theorem", "statement_latex": "f'(x_0) = 0", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((f'(x_0)),(0))=0", "to_expression": "f'(x_0) = 0", "justification": "zero discrete distance iff equality", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f'(x_0) = 0.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_e3b3df24cf201b86", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "Mathematical analysis", "subfield": "Sequences and series", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F3C0470E95D9", "name": "Abel's inequality", "statement_latex": "|∑ a_k b_k| ≤ max|B_k|(|a_n| + |a_n - a_1|)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(max|B_k|(|a_n| + |a_n - a_1|))\\le t\\Rightarrow (|∑ a_k b_k|)\\le t", "to_expression": "|∑ a_k b_k| ≤ max|B_k|(|a_n| + |a_n - a_1|)", "justification": "converse uses t=Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |∑ a_k b_k| ≤ max|B_k|(|a_n| + |a_n - a_1|).\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_e7b343249d9140b4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_68687FF65AAB", "name": "Landsberg–Schaar relation", "statement_latex": "1/sqrt(p) * sum_{n=0}^{p-1} exp(2*pi*i*n^2*q/p) = e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((1/sqrt(p) * sum_{n=0}^{p-1} exp(2*pi*i*n^2*q/p)),z)=((e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q))),z)", "to_expression": "1/sqrt(p) * sum_{n=0}^{p-1} exp(2*pi*i*n^2*q/p) = e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q))", "justification": "pair equality forces first coordinates equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as 1/sqrt(p) * sum_{n=0}^{p-1} exp(2*pi*i*n^2*q/p) = e^(1/4 * pi * i)/sqrt(2q) * sum_{n=0}^{2q-1} exp(-pi*i*n^2*p/(2q)).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_b0ffed41821d312b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "discrepancy theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E6FD55ACB598", "name": "Beck–Fiala theorem", "statement_latex": "|x(S_j)| <= 2t - 1", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(|x(S_j)|)}{\\exp(|x(S_j)|)+\\exp(2t - 1)}\\le\\frac{1}{2}", "to_expression": "|x(S_j)| <= 2t - 1", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |x(S_j)| ≤ 2t - 1.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_e29c151c223de9c7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_37DDB6F9F636", "name": "Duffin–Schaeffer theorem", "statement_latex": "\\sum_{q=1}^{\\infty}\\varphi(q)\\frac{f(q)}{q}=\\infty", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((\\sum_{q=1}^{\\infty}\\varphi(q)\\frac{f(q)}{q}),z):z\\in D\\}=\\{((\\infty),z):z\\in D\\}", "to_expression": "\\sum_{q=1}^{\\infty}\\varphi(q)\\frac{f(q)}{q}=\\infty", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\sum_{q=1}^{\\\\infty}\\\\varphi(q)\\\\frac{f(q)}{q}=\\\\infty.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_8c52647fa2a4dd41", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "metric geometry", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BDB71158A694", "name": "Besicovitch inequality", "statement_latex": "prod_{i} d_{i} <= Vol(M)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (Vol(M))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (prod_{i} d_{i})\\}", "to_expression": "prod_{i} d_{i} <= Vol(M)", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as prod_{i} d_{i} ≤ Vol(M).\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_925770f9c0d7615f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "complex analysis", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_484796D95DA9", "name": "Abel's theorem", "statement_latex": "lim_{x -> 1^-} G(x) = sum_{k=0}^infinity a_k", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(lim_{x -) - (1^-} G(x) = sum_{k=0}^\\infty a_k) > 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover lim_{x - > 1^-} G(x) = sum_{k=0}^infinity a_k.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover lim_{x - > 1^-} G(x) = sum_{k=0}^infinity a_k.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "lim_{x -> 1^-} G(x) = sum_{k=0}^infinity a_k", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover lim_{x - > 1^-} G(x) = sum_{k=0}^infinity a_k.\"]"}], "source_equivalence_identity": "A<B iff B-A>0 over finite real quantities."}
{"schema_version": "1.0", "item_id": "rcd2_d702dd1cb8bf9934", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "vector calculus", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_24234F1E6547", "name": "Divergence theorem", "statement_latex": "\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S} = \\int_{V} (\\nabla \\cdot \\mathbf{F}) dV", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S}),z):z\\in D\\}=\\{((\\int_{V} (\\nabla \\cdot \\mathbf{F}) dV),z):z\\in D\\}", "to_expression": "\\oint_{S} \\mathbf{F} \\cdot d\\mathbf{S} = \\int_{V} (\\nabla \\cdot \\mathbf{F}) dV", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\oint_{S} \\\\mathbf{F} \\\\cdot d\\\\mathbf{S} = \\\\int_{V} (\\\\nabla \\\\cdot \\\\mathbf{F}) dV.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_2956e002fed20ab3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "discrepancy theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E6FD55ACB598", "name": "Beck–Fiala theorem", "statement_latex": "|x(S_j)| <= 2t - 1", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (|x(S_j)|) \\le x \\land x \\le (2t - 1)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |x(S_j)| <= 2t - 1.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |x(S_j)| <= 2t - 1.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "|x(S_j)| <= 2t - 1", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |x(S_j)| <= 2t - 1.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
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{"schema_version": "1.0", "item_id": "rcd2_3ed80810ff18d883", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A00EEAE41A77", "name": "Intermediate value theorem", "statement_latex": "f(x) = s", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((f(x)),z)=((s),z)", "to_expression": "f(x) = s", "justification": "pair equality forces first coordinates equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f(x) = s.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_a254932073f897c8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum field theory", "subfield": "lattice field theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7FD1BE25E27D", "name": "Nielsen–Ninomiya theorem", "statement_latex": "H = ∑_{x,y} ψ^†(x) F(x, y) ψ(y)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(H)\\}=\\{(∑_{x,y} ψ^†(x) F(x, y) ψ(y))\\}", "to_expression": "H = ∑_{x,y} ψ^†(x) F(x, y) ψ(y)", "justification": "singletons equal iff elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as H = ∑_{x,y} ψ^†(x) F(x, y) ψ(y).\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_a109039f86ec3e52", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "algebra", "subfield": "algebraic combinatorics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_396E0535E013", "name": "n! conjecture", "statement_latex": "dim(H_μ) = n!", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((dim(H_μ)) \\in S \\land (n!) \\notin S) \\lor ((n!) \\in S \\land (dim(H_μ)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover dim(H_μ) = n!.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "dim(H_μ) = n!", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover dim(H_μ) = n!.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_db888f2b9c2f6445", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_da425914fdcfab0c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "commutative algebra", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AB6EC963565B", "name": "Artin–Rees lemma", "statement_latex": "I^nM ∩ N = I^{n-k}(I^kM ∩ N)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "(\\{(I^nM ∩ N)\\}\\times\\{(I^{n-k}(I^kM ∩ N))\\})\\cap\\{(z,z):z\\in D\\}=\\{((I^nM ∩ N),(I^{n-k}(I^kM ∩ N)))\\}", "to_expression": "I^nM ∩ N = I^{n-k}(I^kM ∩ N)", "justification": "intersection is singleton iff pair diagonal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as I^nM ∩ N = I^{n-k}(I^kM ∩ N).\"]"}], "source_equivalence_identity": "intersection is singleton iff pair diagonal"}
{"schema_version": "1.0", "item_id": "rcd2_95c6eb39434b8c12", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "geometry of numbers", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_9563b3e84bb19824", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "probabilistic number theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9D5ECA10E501", "name": "Turán–Kubilius inequality", "statement_latex": "(1/x) * sum_{n<=x} |f(n) - A(x)|^2 <= (2 + epsilon(x)) * B(x)^2", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp((1/x) * sum_{n≤x} |f(n) - A(x)|^2)}{\\exp((1/x) * sum_{n≤x} |f(n) - A(x)|^2)+\\exp((2 + epsilon(x)) * B(x)^2)}\\le\\frac{1}{2}", "to_expression": "(1/x) * sum_{n<=x} |f(n) - A(x)|^2 <= (2 + epsilon(x)) * B(x)^2", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (1/x) * sum_{n≤x} |f(n) - A(x)|^2 ≤ (2 + epsilon(x)) * B(x)^2.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_9753fa09ed0b8e9a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D13DCB7F188A", "name": "Kuznetsov trace formula", "statement_latex": "∑_{c≡0 mod N}c^{-r}K(m,n,c)g(4π√mn/c)=Integral transform+Spectral terms", "assumptions": "All displayed arithmetic expressions are integers; the modulus is a positive integer."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(∑_{c≡0 mod N}c^{-r}K(m,n,c)g(4π√mn/c)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(Integral transform+Spectral terms))", "to_expression": "∑_{c≡0 mod N}c^{-r}K(m,n,c)g(4π√mn/c)=Integral transform+Spectral terms", "justification": "each argmin set is a singleton", "preconditions": "All displayed arithmetic expressions are integers; the modulus is a positive integer.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∑_{c≡0 mod N}c^{-r}K(m,n,c)g(4π√mn/c)=Integral transform+Spectral terms.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_f2fcb4f38217cc6a", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "algebraic K-theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_fdde63d7e0c1d462", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_5346D791A6AE", "name": "Siegel's lemma", "statement_latex": "max|X_j| <= (D^{-1}*sqrt(det(AA^T)))^(1/(N-M))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(max|X_j|)}{\\exp(max|X_j|)+\\exp(D^{-1}*sqrt(det(AA^T)))^(1/(N-M))}\\le\\frac{1}{2}", "to_expression": "max|X_j| <= (D^{-1}*sqrt(det(AA^T)))^(1/(N-M))", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as max|X_j| ≤ (D^{-1}*sqrt(det(AA^T)))^(1/(N-M)).\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_ef7ab71ec22ef43e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "measure theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_448CEF326F89", "name": "Doob–Dynkin lemma", "statement_latex": "f = g ∘ T", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((f) \\in S \\land (g ∘ T) \\notin S) \\lor ((g ∘ T) \\in S \\land (f) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f = g ∘ T.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f = g ∘ T", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f = g ∘ T.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_07e383b362aa8a51", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "queueing theory", "subfield": "probability", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_91967E3D2C5F", "name": "Little's law", "statement_latex": "L = lambda*W", "assumptions": "The queueing system is stable and in steady state; L, lambda, and W are finite real quantities."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(L)=z\\;\\land\\;z=(lambda*W)", "to_expression": "L = lambda*W", "justification": "common witness iff equality", "preconditions": "The queueing system is stable and in steady state; L, lambda, and W are finite real quantities.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as L = lambda*W.\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_fac3875e17459c54", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "algebra", "subfield": "combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E24E6480323C", "name": "Multinomial theorem", "statement_latex": "(x_{1}+x_{2}+\\cdots +x_{m})^{n}=\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D(((x_{1}+x_{2}+\\cdots +x_{m})^{n}),(\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}}))=0", "to_expression": "(x_{1}+x_{2}+\\cdots +x_{m})^{n}=\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}}", "justification": "zero discrete distance iff equality", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (x_{1}+x_{2}+\\\\cdots +x_{m})^{n}=\\\\sum _{k_{1}+k_{2}+\\\\cdots +k_{m}=n} \\\\binom{n}{k_1, k_2, \\\\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\\\cdots x_{m}^{k_{m}}.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_e0f36614bc9c9d98", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "complex analysis", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_70d2fbb16e7c30fd", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2cb28ca6f59a2372", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "hypergraph theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0E38B7F67353", "name": "Baranyai's theorem", "statement_latex": "\\binom{k}{r} \\frac{r}{k} = \\binom{k-1}{r-1}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((\\binom{k}{r} \\frac{r}{k}) \\in S \\land (\\binom{k-1}{r-1}) \\notin S) \\lor ((\\binom{k-1}{r-1}) \\in S \\land (\\binom{k}{r} \\frac{r}{k}) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\binom{k}{r} \\\\frac{r}{k} = \\\\binom{k-1}{r-1}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\binom{k}{r} \\frac{r}{k} = \\binom{k-1}{r-1}", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\binom{k}{r} \\\\frac{r}{k} = \\\\binom{k-1}{r-1}.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_0078c099eead9d38", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum information theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B8493CAFC989", "name": "No-deleting theorem", "statement_latex": "U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C = |\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C))=F((|\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C))", "to_expression": "U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C = |\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C", "justification": "indicator of {X} separates unequal objects", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as U|\\\\psi \\\\rangle_A |\\\\psi \\\\rangle_B |A \\\\rangle_C = |\\\\psi \\\\rangle_A |0 \\\\rangle_B |A' \\\\rangle_C.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_2ca7dfac045ea5b3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F881715F08CD", "name": "Sylvester's determinant identity", "statement_latex": "det(A) = det(A_v^u) * det(A_tilde_v^u)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(det(A)) \\in {(det(A_v^u) * det(A_tilde_v^u))}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover det(A) = det(A_v^u) * det(A_tilde_v^u).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "det(A) = det(A_v^u) * det(A_tilde_v^u)", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover det(A) = det(A_v^u) * det(A_tilde_v^u).\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_bf94654b4f9a2707", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_379368c4ab265f7d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "special functions", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e28d890b7d0071d2", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "prime numbers", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_85b60f83c70c950f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "geometry", "subfield": "discrete geometry", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_DCDC0E01EC8B", "name": "Covering problem of Rado", "statement_latex": "F(X) = inf_S sup_I (|I|/|S|)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "(\\{(F(X))\\}\\times\\{(inf_S sup_I (|I|/|S|))\\})\\cap\\{(z,z):z\\in D\\}=\\{((F(X)),(inf_S sup_I (|I|/|S|)))\\}", "to_expression": "F(X) = inf_S sup_I (|I|/|S|)", "justification": "intersection is singleton iff pair diagonal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as F(X) = inf_S sup_I (|I|/|S|).\"]"}], "source_equivalence_identity": "intersection is singleton iff pair diagonal"}
{"schema_version": "1.0", "item_id": "rcd2_43745c941b0f0976", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "control theory", "subfield": "geometric control theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_3c74fee1471bead6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "analysis", "subfield": "approximation theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_538DF6DD562D", "name": "Favard's theorem", "statement_latex": "y_{n+1}=(x-c_{n})y_{n}-d_{n}y_{n-1}", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(y_{n+1}) \\in {((x-c_{n})y_{n}-d_{n}y_{n-1})}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover y_{n+1}=(x-c_{n})y_{n}-d_{n}y_{n-1}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "y_{n+1}=(x-c_{n})y_{n}-d_{n}y_{n-1}", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover y_{n+1}=(x-c_{n})y_{n}-d_{n}y_{n-1}.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_040e0c69fc864cb1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum information theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B8493CAFC989", "name": "No-deleting theorem", "statement_latex": "U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C = |\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C) \\in S \\land (|\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C) \\notin S) \\lor ((|\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C) \\in S \\land (U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover U|\\\\psi \\\\rangle_A |\\\\psi \\\\rangle_B |A \\\\rangle_C = |\\\\psi \\\\rangle_A |0 \\\\rangle_B |A' \\\\rangle_C.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "U|\\psi \\rangle_A |\\psi \\rangle_B |A \\rangle_C = |\\psi \\rangle_A |0 \\rangle_B |A' \\rangle_C", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover U|\\\\psi \\\\rangle_A |\\\\psi \\\\rangle_B |A \\\\rangle_C = |\\\\psi \\\\rangle_A |0 \\\\rangle_B |A' \\\\rangle_C.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_55b78034bad68fbd", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "recreational mathematics", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_96F33F66821C", "name": "Tupper's self-referential formula", "statement_latex": "1/2 < floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2))", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,\\infty)}\\lambda((floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2)))-(1/2))=+\\infty", "to_expression": "1/2 < floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2))", "justification": "unbounded iff Y-X>0", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as 1/2 < floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2)).\"]"}], "source_equivalence_identity": "unbounded iff Y-X>0"}
{"schema_version": "1.0", "item_id": "rcd2_7585478edbd6d0c2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "prime number theory", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BFDCE2A9DBB2", "name": "Bertrand's postulate", "statement_latex": "exists prime p such that n < p < 2*n", "assumptions": "n is an integer greater than 1."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists x\\in D:x\\in \\{prime p\\in D:n < p < 2*n\\}", "to_expression": "exists prime p such that n < p < 2*n", "justification": "nonempty iff element exists", "preconditions": "n is an integer greater than 1.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as exists prime p such that n < p < 2*n.\"]"}], "source_equivalence_identity": "nonempty iff element exists"}
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{"schema_version": "1.0", "item_id": "rcd2_9bdd936dd6f7807f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "calculus", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_819FFAB136D4", "name": "L'Hôpital's rule", "statement_latex": "lim_{x->c} f(x)/g(x) = lim_{x->c} f'(x)/g'(x)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(lim_{x→c} f(x)/g(x))\\Longleftrightarrow z=(lim_{x→c} f'(x)/g'(x)))", "to_expression": "lim_{x->c} f(x)/g(x) = lim_{x->c} f'(x)/g'(x)", "justification": "same equality fiber iff equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{x→c} f(x)/g(x) = lim_{x→c} f'(x)/g'(x).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_a4257263dce9415a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal set theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4D5A3A69C1D", "name": "Sauer–Shelah lemma", "statement_latex": "|F| <= sum_{i=0}^{d} binom(n,i)", "assumptions": "F is a family of subsets of an n-element set with VC dimension d."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(sum_{i=0}^{d} binom(n,i))\\le t\\Rightarrow (|F|)\\le t", "to_expression": "|F| <= sum_{i=0}^{d} binom(n,i)", "justification": "converse uses t=Y", "preconditions": "F is a family of subsets of an n-element set with VC dimension d.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |F| ≤ sum_{i=0}^{d} binom(n,i).\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_f4256470099e4754", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "symbolic dynamics", "subfield": "cellular automata", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_03C6BB7FE5A2", "name": "Curtis–Hedlund–Lyndon theorem", "statement_latex": "f(s(x)) = s(f(x))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(s(f(x))) = (f(s(x)))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(s(x)) = s(f(x)).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f(s(x)) = s(f(x))", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(s(x)) = s(f(x)).\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_003b90410ebd7a92", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_8e85f992e49f523e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "probabilistic number theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9D5ECA10E501", "name": "Turán–Kubilius inequality", "statement_latex": "(1/x) * sum_{n<=x} |f(n) - A(x)|^2 <= (2 + epsilon(x)) * B(x)^2", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg ((1/x) * sum_{n\\lex} |f(n) - A(x)|^2 \\le (2 + \\varepsilon(x)) * B(x)^2))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "(1/x) * sum_{n<=x} |f(n) - A(x)|^2 <= (2 + epsilon(x)) * B(x)^2", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_8008892ac7412c22", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "measure theory", "subfield": "stochastic analysis", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FB5E9395CC7C", "name": "Malliavin's absolute continuity lemma", "statement_latex": "|∫ Dφ(y)(x) dμ(y)| ≤ C(x) ‖φ‖_{∞}", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(|∫ Dφ(y)(x) dμ(y)|)-(C(x) ‖φ‖_{∞})\\}=0", "to_expression": "|∫ Dφ(y)(x) dμ(y)| ≤ C(x) ‖φ‖_{∞}", "justification": "positive part vanishes iff X≤Y", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |∫ Dφ(y)(x) dμ(y)| ≤ C(x) ‖φ‖_{∞}.\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_34cc1f8a3a253cf8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "geometry of numbers", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_638EC6734424", "name": "Minkowski–Hlawka theorem", "statement_latex": "Δ ≥ ζ(n) / 2^(n-1)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((ζ(n) / 2^(n-1))-(Δ),(-\\infty,0])=0", "to_expression": "Δ ≥ ζ(n) / 2^(n-1)", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Δ ≥ ζ(n) / 2^(n-1).\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
{"schema_version": "1.0", "item_id": "rcd2_8792021373e89269", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "p-adic analysis", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2bceec7978731fc6", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "martingale theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c7ac6cc761d45e5f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "combinatorics", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_9179421b66927cd1", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "statistical mechanics", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_70bb68cb1e918a99", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "geometry", "subfield": "Euclidean geometry", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F38D8FABD394", "name": "Ptolemy's inequality", "statement_latex": "AB * CD + BC * DA >= AC * BD", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\le (AC * BD)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\le (AB * CD + BC * DA)\\}", "to_expression": "AB * CD + BC * DA >= AC * BD", "justification": "lower-cut inclusion iff X≤Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as AB * CD + BC * DA ≥ AC * BD.\"]"}], "source_equivalence_identity": "lower-cut inclusion iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_5e7780cb999e6894", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "recreational mathematics", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b2d7a71a952e771c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A6D639595426", "name": "Inverse function theorem", "statement_latex": "(f^{-1})'(b) = 1 / f'(a)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(f^{-1})'(b)=z\\;\\land\\;z=(1 / f'(a))", "to_expression": "(f^{-1})'(b) = 1 / f'(a)", "justification": "common witness iff equality", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (f^{-1})'(b) = 1 / f'(a).\"]"}], "source_equivalence_identity": "common witness iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_eae07cc59e08625f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "combinatorial number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_8d2f66ad3b4dbba8", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic geometry", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ffec2bc0daca0392", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "additive combinatorics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_081265713252", "name": "Erdős–Szemerédi theorem", "statement_latex": "max(|A + A|, |A * A|) >= c|A|^{1 + ε}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (max(|A + A|, |A * A|))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (c|A|^{1 + ε})\\}", "to_expression": "max(|A + A|, |A * A|) >= c|A|^{1 + ε}", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as max(|A + A|, |A * A|) ≥ c|A|^{1 + ε}.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_4cd8d9cae9ab5d29", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "analysis", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
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{"schema_version": "1.0", "item_id": "rcd2_ccbf35656f90e8fb", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "complex analysis", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2C71326A0896", "name": "Area theorem", "statement_latex": "\\sum_{n=0}^{\\infty} n|a_n|^2 \\leq 1", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(\\sum_{n=0}^{\\infty} n|a_n|^2)}{\\exp(\\sum_{n=0}^{\\infty} n|a_n|^2)+\\exp(1)}\\le\\frac{1}{2}", "to_expression": "\\sum_{n=0}^{\\infty} n|a_n|^2 \\leq 1", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\sum_{n=0}^{\\\\infty} n|a_n|^2 ≤ 1.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_bdfeb27a0cfd2b26", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "point processes", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C81FC5D06BF2", "name": "Campbell's theorem", "statement_latex": "E[sum_{x in N} f(x)] = integral f(x) dLambda(x)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((E[sum_{x in N} f(x)]),z):z\\in D\\}=\\{((integral f(x) dLambda(x)),z):z\\in D\\}", "to_expression": "E[sum_{x in N} f(x)] = integral f(x) dLambda(x)", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as E[sum_{x in N} f(x)] = integral f(x) dLambda(x).\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_c7e9f232585dc265", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "computer science", "subfield": "probabilistic combinatorics", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f60a443a376b7450", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_1debddf48a890c05", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "representation theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b46056b00fdcea6d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_ABB949903DF7", "name": "Vandermonde's identity", "statement_latex": "{m+n \\choose r}=\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,({m+n \\choose r}))+\\delta_D(z,(\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k})))=0", "to_expression": "{m+n \\choose r}=\\sum _{k=0}^{r}{m \\choose k}{n \\choose r-k}", "justification": "zero loss iff some z equals both", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as {m+n \\\\choose r}=\\\\sum _{k=0}^{r}{m \\\\choose k}{n \\\\choose r-k}.\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_23ee629edc8aa69e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "topology", "subfield": "general topology", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_04953D782E69", "name": "Tube lemma", "statement_latex": "X × Y → X", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall r\\in\\mathbb{Q}_{>0}\\exists N\\forall n\\ge N:d((X × Y),(X))<r", "to_expression": "X × Y → X", "justification": "positive rationals cofinal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as X × Y → X.\"]"}], "source_equivalence_identity": "positive rationals cofinal"}
{"schema_version": "1.0", "item_id": "rcd2_55c96b773598218d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "analysis", "subfield": "moment problem", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A82D2EB23B1F", "name": "Carleman's condition", "statement_latex": "\\sum _{n=1}^{\\infty }m_{2n}^{-{1/2n}}=+\\infty", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(0 = 0) \\Rightarrow (\\sum _{n=1}^{\\infty }m_{2n}^{-{1/2n}}=+\\infty)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Since 0=0 is true, true=>Phi_T has the same truth value as Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\sum _{n=1}^{\\infty }m_{2n}^{-{1/2n}}=+\\infty", "justification": "Since 0=0 is true, true=>Phi_T has the same truth value as Phi_T.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "Since 0=0 is true, true=>Phi_T has the same truth value as Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_77f3b7e8e090a90b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "decision theory", "subfield": "information theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c3c28791d9330cd1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "algebra", "subfield": "algebraic combinatorics", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_396E0535E013", "name": "n! conjecture", "statement_latex": "dim(H_μ) = n!", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(dim(H_μ))\\}=\\{(n!)\\}", "to_expression": "dim(H_μ) = n!", "justification": "singletons equal iff elements equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as dim(H_μ) = n!.\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_be38ac27e88f1710", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "numerical analysis", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_DC75D2014574", "name": "Céa's lemma", "statement_latex": "||u - u_h|| <= (gamma / alpha) * ||u - v||", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(||u - u_h||)+s=((gamma / alpha) * ||u - v||)", "to_expression": "||u - u_h|| <= (gamma / alpha) * ||u - v||", "justification": "nonnegative slack exists iff X≤Y", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ||u - u_h|| ≤ (gamma / alpha) * ||u - v||.\"]"}], "source_equivalence_identity": "nonnegative slack exists iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_8289d9b44400e9d4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "metric geometry", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BDB71158A694", "name": "Besicovitch inequality", "statement_latex": "prod_{i} d_{i} <= Vol(M)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(Vol(M)) - (prod_{i} d_{i}) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover prod_{i} d_{i} <= Vol(M).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover prod_{i} d_{i} <= Vol(M).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "prod_{i} d_{i} <= Vol(M)", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover prod_{i} d_{i} <= Vol(M).\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_55b0385ad7c1fd53", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_98220A4F883B", "name": "Siegel identity", "statement_latex": "\\frac{x_3-x_1}{x_2-x_1} + \\frac{x_2-x_3}{x_2-x_1} = 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((\\frac{x_3-x_1}{x_2-x_1} + \\frac{x_2-x_3}{x_2-x_1}),(1))\\in\\{(z,z):z\\in D\\}", "to_expression": "\\frac{x_3-x_1}{x_2-x_1} + \\frac{x_2-x_3}{x_2-x_1} = 1", "justification": "pair in diagonal iff X=Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\frac{x_3-x_1}{x_2-x_1} + \\\\frac{x_2-x_3}{x_2-x_1} = 1.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_1a2451b5c0918434", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "commutative algebra", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0EBFF1A397C6", "name": "Local criterion for flatness", "statement_latex": "Tor_1^A(C, M) = 0", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(Tor_1^A(C, M)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(0))", "to_expression": "Tor_1^A(C, M) = 0", "justification": "each argmin set is a singleton", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Tor_1^A(C, M) = 0.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
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{"schema_version": "1.0", "item_id": "rcd2_4ffd9988125d6814", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7BB84422BC5F", "name": "Hadamard's inequality", "statement_latex": "|det(N)| <= prod_{i=1}^n ||v_i||", "assumptions": "N is a square matrix with row or column vectors v_i in an inner-product space; norms are Euclidean."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(prod_{i=1}^n ||v_i||)\\le t\\Rightarrow (|det(N)|)\\le t", "to_expression": "|det(N)| <= prod_{i=1}^n ||v_i||", "justification": "converse uses t=Y", "preconditions": "N is a square matrix with row or column vectors v_i in an inner-product space; norms are Euclidean.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |det(N)| ≤ prod_{i=1}^n ||v_i||.\"]"}], "source_equivalence_identity": "converse uses t=Y"}
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{"schema_version": "1.0", "item_id": "rcd2_6319fee0953e466b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0a3faa27e9ce3baa", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_EAE642263ABE", "name": "Catalan's conjecture", "statement_latex": "x^a - y^b = 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((x^a - y^b))=F((1))", "to_expression": "x^a - y^b = 1", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as x^a - y^b = 1.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_737e7a1095d0606b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d773d2d3ac0f85fa", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "enumerative combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_93d43fde52adb3f7", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "topology", "subfield": "combinatorial topology", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_3006f70578de9908", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0b461641dcf6cbe9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "complex analysis", "subfield": "complex geometry", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A0642B3B222A", "name": "Schwarz lemma", "statement_latex": "|f(z)| <= |z|", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((|f(z)|)-(|z|),(-\\infty,0])=0", "to_expression": "|f(z)| <= |z|", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |f(z)| ≤ |z|.\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
{"schema_version": "1.0", "item_id": "rcd2_896a94659a66a436", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "automorphic forms", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_64D8463F3766", "name": "Fundamental lemma", "statement_latex": "SO_{\\gamma _{H}}(1_{K_{H}})=\\Delta (\\gamma _{H},\\gamma _{G})O_{\\gamma _{G}}^{\\kappa }(1_{K_{G}})", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(SO_{\\gamma _{H}}(1_{K_{H}}))\\}=\\{(\\Delta (\\gamma _{H},\\gamma _{G})O_{\\gamma _{G}}^{\\kappa }(1_{K_{G}}))\\}", "to_expression": "SO_{\\gamma _{H}}(1_{K_{H}})=\\Delta (\\gamma _{H},\\gamma _{G})O_{\\gamma _{G}}^{\\kappa }(1_{K_{G}})", "justification": "singletons equal iff elements equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as SO_{\\\\gamma _{H}}(1_{K_{H}})=\\\\Delta (\\\\gamma _{H},\\\\gamma _{G})O_{\\\\gamma _{G}}^{\\\\kappa }(1_{K_{G}}).\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_b5588c2f68ca06a3", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "Algebra", "subfield": "Lie algebras", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_4a1d9c12f5d58cf4", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "additive number theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b5855a2c124d4168", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "combinatorics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_843AE41690C1", "name": "q-Vandermonde identity", "statement_latex": "binom(m+n, k)_q = sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((binom(m+n, k)_q) \\in S \\land (sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)}) \\notin S) \\lor ((sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)}) \\in S \\land (binom(m+n, k)_q) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover binom(m+n, k)_q = sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "binom(m+n, k)_q = sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)}", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover binom(m+n, k)_q = sum_j binom(m, k-j)_q * binom(n, j)_q * q^{j(m-k+j)}.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_5e7294e4885399f8", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "combinatorics on words", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_cd5096d20438a806", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "combinatorics", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E6F93BEA2A51", "name": "Cassini and Catalan identities", "statement_latex": "F_{n-1}*F_{n+1} - F_n^2 = (-1)^n", "assumptions": "F_n denotes the Fibonacci sequence with the indexing convention used by the theorem; n is an integer in the stated range."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((F_{n-1}*F_{n+1} - F_n^2),z):z\\in D\\}=\\{(((-1)^n),z):z\\in D\\}", "to_expression": "F_{n-1}*F_{n+1} - F_n^2 = (-1)^n", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "F_n denotes the Fibonacci sequence with the indexing convention used by the theorem; n is an integer in the stated range.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as F_{n-1}*F_{n+1} - F_n^2 = (-1)^n.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_39f2d598e043ba48", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal set theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3B6CADC238DF", "name": "Lubell–Yamamoto–Meshalkin inequality", "statement_latex": "sum_{k=0}^n a_k/binom(n,k) <= 1", "assumptions": "The family is Sperner; a_k counts chosen k-subsets of an n-element ground set."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((sum_{k=0}^n a_k/binom(n,k))-(1),(-\\infty,0])=0", "to_expression": "sum_{k=0}^n a_k/binom(n,k) <= 1", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "The family is Sperner; a_k counts chosen k-subsets of an n-element ground set.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{k=0}^n a_k/binom(n,k) ≤ 1.\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
{"schema_version": "1.0", "item_id": "rcd2_c5c3600d230fb93a", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "coding theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e73c668286c8e7e7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F881715F08CD", "name": "Sylvester's determinant identity", "statement_latex": "det(A) = det(A_v^u) * det(A_tilde_v^u)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((det(A)))=F((det(A_v^u) * det(A_tilde_v^u)))", "to_expression": "det(A) = det(A_v^u) * det(A_tilde_v^u)", "justification": "indicator of {X} separates unequal objects", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as det(A) = det(A_v^u) * det(A_tilde_v^u).\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_992ae5dfbe51271d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_933f4f9ba8f494a1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_736807BC5C8F", "name": "Brun–Titchmarsh theorem", "statement_latex": "pi(x;q,a) <= 2x / (phi(q) log(x/q))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(pi(x;q,a))}{\\exp(pi(x;q,a))+\\exp(2x / (phi(q) log(x/q)))}\\le\\frac{1}{2}", "to_expression": "pi(x;q,a) <= 2x / (phi(q) log(x/q))", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as pi(x;q,a) ≤ 2x / (phi(q) log(x/q)).\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_8286230baebd6c45", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "vector calculus", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_128437815487b55b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3DB69D9F31E5", "name": "Reynolds transport theorem", "statement_latex": "d/dt integral_{Omega(t)} f dV = integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(d/dt integral_{Omega(t)} f dV)=z\\;\\land\\;z=(integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA)", "to_expression": "d/dt integral_{Omega(t)} f dV = integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA", "justification": "common witness iff equality", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as d/dt integral_{Omega(t)} f dV = integral_{Omega(t)} (partial f / partial t) dV + integral_{partial Omega(t)} (v_b dot n) f dA.\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_28d1c509e38b5959", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "elementary algebra", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D16BD70DAEA0", "name": "Difference of two squares", "statement_latex": "a^2 - b^2 = (a + b)(a - b)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((a^2 - b^2) \\in S \\land ((a + b)(a - b)) \\notin S) \\lor (((a + b)(a - b)) \\in S \\land (a^2 - b^2) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover a^2 - b^2 = (a + b)(a - b).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "a^2 - b^2 = (a + b)(a - b)", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover a^2 - b^2 = (a + b)(a - b).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_942e12e712bbf374", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "polynomial factorization", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_299359E3C599", "name": "Sophie Germain's identity", "statement_latex": "x^{4}+4y^{4}=(x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2})", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(x^{4}+4y^{4}))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2}))", "to_expression": "x^{4}+4y^{4}=(x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2})", "justification": "each argmin set is a singleton", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as x^{4}+4y^{4}=(x^{2}+2xy+2y^{2})(x^{2}-2xy+2y^{2}).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_63abd39b034fdec8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_69A821A9D880", "name": "Hockey-stick identity", "statement_latex": "\\sum_{i=r}^{n} \\binom{i}{r} = \\binom{n+1}{r+1}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((\\sum_{i=r}^{n} \\binom{i}{r}),z)=((\\binom{n+1}{r+1}),z)", "to_expression": "\\sum_{i=r}^{n} \\binom{i}{r} = \\binom{n+1}{r+1}", "justification": "pair equality forces first coordinates equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\sum_{i=r}^{n} \\\\binom{i}{r} = \\\\binom{n+1}{r+1}.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_59e258f17bbbb660", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "measure theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_43493e93f21a6269", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic geometry", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BB5E3AF7EFD7", "name": "Modularity theorem", "statement_latex": "L(E, s) = sum_{n=1}^∞ a_n / n^s", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(L(E, s)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(sum_{n=1}^∞ a_n / n^s))", "to_expression": "L(E, s) = sum_{n=1}^∞ a_n / n^s", "justification": "each argmin set is a singleton", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as L(E, s) = sum_{n=1}^∞ a_n / n^s.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_7c09cdce6a70d80a", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "convex geometry", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_58a79061f633fe57", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "theoretical computer science", "subfield": "algebraic complexity theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ed64dda77e8c0e3c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "p-adic analysis", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_47325fe0d2beb9c9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "representation theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1292421DFB1E", "name": "Schur's lemma", "statement_latex": "ρ_W(g) ∘ f = f ∘ ρ_V(g)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((ρ_W(g) ∘ f),(f ∘ ρ_V(g)))\\in\\{(z,z):z\\in D\\}", "to_expression": "ρ_W(g) ∘ f = f ∘ ρ_V(g)", "justification": "pair in diagonal iff X=Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ρ_W(g) ∘ f = f ∘ ρ_V(g).\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_9bf391a03bd81a63", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "analytic number theory", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C97AADA78BC1", "name": "Vaughan's identity", "statement_latex": "-zeta'(s)/zeta(s) = F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(-zeta'(s)/zeta(s))=z\\;\\land\\;z=(F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s)))", "to_expression": "-zeta'(s)/zeta(s) = F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s))", "justification": "common witness iff equality", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as -zeta'(s)/zeta(s) = F(s) - zeta(s)F(s)G(s) - zeta'(s)G(s) + (-zeta'(s)/zeta(s) - F(s))(1 - zeta(s)G(s)).\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_9317f094283fe416", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_979a02ed5e86a5e6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "convex geometry", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4F00365B64E5", "name": "Minkowski's first inequality for convex bodies", "statement_latex": "V_1(K, L) ≥ V(K)^((n-1)/n) V(L)^(1/n)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(V_1(K, L)) - (V(K)^((n-1)/n) V(L)^(1/n)) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover V_1(K, L) >= V(K)^((n-1)/n) V(L)^(1/n).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover V_1(K, L) >= V(K)^((n-1)/n) V(L)^(1/n).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "V_1(K, L) ≥ V(K)^((n-1)/n) V(L)^(1/n)", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover V_1(K, L) >= V(K)^((n-1)/n) V(L)^(1/n).\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_4bf77880ecf45445", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic calculus", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4D5944961D8", "name": "Itô's lemma", "statement_latex": "dX_t = mu_t dt + sigma_t dB_t", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(dX_t) \\in {(mu_t dt + sigma_t dB_t)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover dX_t = mu_t dt + sigma_t dB_t.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "dX_t = mu_t dt + sigma_t dB_t", "justification": "L in {R} iff L = R.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover dX_t = mu_t dt + sigma_t dB_t.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_dc89fc01824c8f34", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "linear algebra", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FF8D78866125", "name": "Binet–Cauchy identity", "statement_latex": "(sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i<j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i)", "assumptions": "All sums are finite over the same index range; entries lie in R."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(sum_i a_i*c_i)*(sum_j b_j*d_j)=z\\;\\land\\;z=(sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i<j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i)", "to_expression": "(sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i<j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i)", "justification": "common witness iff equality", "preconditions": "All sums are finite over the same index range; entries lie in R.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (sum_i a_i*c_i)*(sum_j b_j*d_j) = (sum_i a_i*d_i)*(sum_j b_j*c_j) + sum_{i<j}(a_i*b_j-a_j*b_i)*(c_i*d_j-c_j*d_i).\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_a7d1b5c45e0ab4d0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum optics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BFC6929833DF", "name": "Optical equivalence theorem", "statement_latex": "\\operatorname{tr}(\\hat{\\rho} \\cdot g_{\\Omega}(\\hat{a}, \\hat{a}^{\\dagger})) = \\int f_{\\bar{\\Omega}}(\\alpha, \\alpha^*) g_{\\Omega}(\\alpha, \\alpha^*) d^2\\alpha", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((\\operatorname{tr}(\\hat{\\rho} \\cdot g_{\\Omega}(\\hat{a}, \\hat{a}^{\\dagger}))),(\\int f_{\\bar{\\Omega}}(\\alpha, \\alpha^*) g_{\\Omega}(\\alpha, \\alpha^*) d^2\\alpha))\\in\\{(z,z):z\\in D\\}", "to_expression": "\\operatorname{tr}(\\hat{\\rho} \\cdot g_{\\Omega}(\\hat{a}, \\hat{a}^{\\dagger})) = \\int f_{\\bar{\\Omega}}(\\alpha, \\alpha^*) g_{\\Omega}(\\alpha, \\alpha^*) d^2\\alpha", "justification": "pair in diagonal iff X=Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\operatorname{tr}(\\\\hat{\\\\rho} \\\\cdot g_{\\\\Omega}(\\\\hat{a}, \\\\hat{a}^{\\\\dagger})) = \\\\int f_{\\\\bar{\\\\Omega}}(\\\\alpha, \\\\alpha^*) g_{\\\\Omega}(\\\\alpha, \\\\alpha^*) d^2\\\\alpha.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
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{"schema_version": "1.0", "item_id": "rcd2_9e94669458c1bfc0", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "Number theory", "subfield": "Elliptic functions", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_210c31d97cbd01f5", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2ae77902a37e5439", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "measure theory", "subfield": "stochastic analysis", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f96a60888ce1de15", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_06cc2cc1acce42b7", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "queueing theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f633899b8152ae8f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "inner product spaces", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_45074ae4634aa92a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_99B889CC4005", "name": "Hoeffding's lemma", "statement_latex": "E[exp(lambda*(X-E[X]))] <= exp(lambda^2*(b-a)^2/8)", "assumptions": "X is almost surely in [a,b]; lambda is real; expectations are finite."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (exp(lambda^2*(b-a)^2/8))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (E[exp(lambda*(X-E[X]))])\\}", "to_expression": "E[exp(lambda*(X-E[X]))] <= exp(lambda^2*(b-a)^2/8)", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "X is almost surely in [a,b]; lambda is real; expectations are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as E[exp(lambda*(X-E[X]))] ≤ exp(lambda^2*(b-a)^2/8).\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_093f9a4917a8a7b7", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c736bfd81fc10a4e", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "computer science", "subfield": "formal language theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c37c92e14f11df36", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "inequalities", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_50b6391773166e37", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "geometry", "subfield": "discrete geometry", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_DCDC0E01EC8B", "name": "Covering problem of Rado", "statement_latex": "F(X) = inf_S sup_I (|I|/|S|)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((F(X)),z)=((inf_S sup_I (|I|/|S|)),z)", "to_expression": "F(X) = inf_S sup_I (|I|/|S|)", "justification": "pair equality forces first coordinates equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as F(X) = inf_S sup_I (|I|/|S|).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_618952c56acc3476", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_3f953fae1bf03687", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "graph theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ba5331b2ca7379d8", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "Combinatorics", "subfield": "Order theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_8bab135a939b0de4", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_261a97b5ffce201c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "combinatorics on words", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3E0E814BA7D9", "name": "Levi's lemma", "statement_latex": "uv = xy", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\forall h:D\\to{0,1}: h((uv)) = h((xy))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "All Boolean functions agree on L and R exactly when L=R.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover uv = xy.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "uv = xy", "justification": "All Boolean functions agree on L and R exactly when L=R.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover uv = xy.\"]"}], "source_equivalence_identity": "All Boolean functions agree on L and R exactly when L=R."}
{"schema_version": "1.0", "item_id": "rcd2_ca9b23c8a4167963", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_319FF857BA97", "name": "Burgess inequality", "statement_latex": "|S_chi(N,H)| <= C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon})\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|S_chi(N,H)|)\\}", "to_expression": "|S_chi(N,H)| <= C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon}", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |S_chi(N,H)| ≤ C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon}.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_90a4d4469bc29872", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic geometry", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BB5E3AF7EFD7", "name": "Modularity theorem", "statement_latex": "L(E, s) = sum_{n=1}^∞ a_n / n^s", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(L(E, s))\\Longleftrightarrow z=(sum_{n=1}^∞ a_n / n^s))", "to_expression": "L(E, s) = sum_{n=1}^∞ a_n / n^s", "justification": "same equality fiber iff equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as L(E, s) = sum_{n=1}^∞ a_n / n^s.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_605424c227742759", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "elementary algebra", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_40DB1B003904", "name": "Binomial theorem", "statement_latex": "(x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "((x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k) \\lor (0 = 1)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "0=1 is false, so Phi_T or false iff Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "(x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k", "justification": "0=1 is false, so Phi_T or false iff Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "0=1 is false, so Phi_T or false iff Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_01f296a0215e5989", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum field theory", "subfield": "lattice field theory", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7FD1BE25E27D", "name": "Nielsen–Ninomiya theorem", "statement_latex": "H = ∑_{x,y} ψ^†(x) F(x, y) ψ(y)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(H)=z\\;\\land\\;z=(∑_{x,y} ψ^†(x) F(x, y) ψ(y))", "to_expression": "H = ∑_{x,y} ψ^†(x) F(x, y) ψ(y)", "justification": "common witness iff equality", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as H = ∑_{x,y} ψ^†(x) F(x, y) ψ(y).\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_3633d058ec15608c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "recreational mathematics", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_96F33F66821C", "name": "Tupper's self-referential formula", "statement_latex": "1/2 < floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2))", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{>0}\\;\\text{such that}\\;(1/2)+s=(floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2)))", "to_expression": "1/2 < floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2))", "justification": "positive slack iff X<Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as 1/2 < floor(mod(floor(y/17) * 2^(-17*floor(x) - mod(floor(y), 17)), 2)).\"]"}], "source_equivalence_identity": "positive slack iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_006a8f8829a36255", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "measure theory", "subfield": "stochastic analysis", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FB5E9395CC7C", "name": "Malliavin's absolute continuity lemma", "statement_latex": "|∫ Dφ(y)(x) dμ(y)| ≤ C(x) ‖φ‖_{∞}", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (C(x) ‖φ‖_{∞})\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|∫ Dφ(y)(x) dμ(y)|)\\}", "to_expression": "|∫ Dφ(y)(x) dμ(y)| ≤ C(x) ‖φ‖_{∞}", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |∫ Dφ(y)(x) dμ(y)| ≤ C(x) ‖φ‖_{∞}.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_5fc73b6db1b9142b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum optics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BFC6929833DF", "name": "Optical equivalence theorem", "statement_latex": "\\operatorname{tr}(\\hat{\\rho} \\cdot g_{\\Omega}(\\hat{a}, \\hat{a}^{\\dagger})) = \\int f_{\\bar{\\Omega}}(\\alpha, \\alpha^*) g_{\\Omega}(\\alpha, \\alpha^*) d^2\\alpha", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\operatorname{tr}(\\hat{\\rho} \\cdot g_{\\Omega}(\\hat{a}, \\hat{a}^{\\dagger}))) \\in {(\\int f_{\\bar{\\Omega}}(\\alpha, \\alpha^*) g_{\\Omega}(\\alpha, \\alpha^*) d^2\\alpha)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\operatorname{tr}(\\\\hat{\\\\rho} \\\\cdot g_{\\\\Omega}(\\\\hat{a}, \\\\hat{a}^{\\\\dagger})) = \\\\int f_{\\\\bar{\\\\Omega}}(\\\\alpha, \\\\alpha^*) g_{\\\\Omega}(\\\\alpha, \\\\alpha^*) d^2\\\\alpha.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\operatorname{tr}(\\hat{\\rho} \\cdot g_{\\Omega}(\\hat{a}, \\hat{a}^{\\dagger})) = \\int f_{\\bar{\\Omega}}(\\alpha, \\alpha^*) g_{\\Omega}(\\alpha, \\alpha^*) d^2\\alpha", "justification": "L in {R} iff L = R.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\operatorname{tr}(\\\\hat{\\\\rho} \\\\cdot g_{\\\\Omega}(\\\\hat{a}, \\\\hat{a}^{\\\\dagger})) = \\\\int f_{\\\\bar{\\\\Omega}}(\\\\alpha, \\\\alpha^*) g_{\\\\Omega}(\\\\alpha, \\\\alpha^*) d^2\\\\alpha.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_6384211fb4bf176f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_86d3f5f0ac797f18", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "computational number theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A541E6B5FCB3", "name": "Lamé's theorem", "statement_latex": "n < 5 * log_10(min(u, v))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(n)-(5 * log_10(min(u, v)))\\}=0\\;\\land\\;(n)\\ne (5 * log_10(min(u, v)))", "to_expression": "n < 5 * log_10(min(u, v))", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as n < 5 * log_10(min(u, v)).\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_d0fd5002e0805541", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "quadratic forms", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2d5c7f9fec7f21fd", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_23309e8582c56057", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum information theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3795DDB6E155", "name": "NLTS conjecture", "statement_latex": "H = \\sum_{i=1}^{m} H_{i}", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(H))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(\\sum_{i=1}^{m} H_{i}))", "to_expression": "H = \\sum_{i=1}^{m} H_{i}", "justification": "each argmin set is a singleton", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as H = \\\\sum_{i=1}^{m} H_{i}.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_601e3b7aae2f0eb2", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_085b25c77e5a116b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "combinatorics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_8798bfa9bc8afc7c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal set theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b5775b0cdd83a44c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b3e9d3088067423d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal set theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_673474bd6517e3b2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AFD3A111ECE4", "name": "Boole's inequality", "statement_latex": "P(∪_{i=1}^∞ A_i) ≤ ∑_{i=1}^∞ P(A_i)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(∑_{i=1}^∞ P(A_i))\\le t\\Rightarrow (P(∪_{i=1}^∞ A_i))\\le t", "to_expression": "P(∪_{i=1}^∞ A_i) ≤ ∑_{i=1}^∞ P(A_i)", "justification": "converse uses t=Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(∪_{i=1}^∞ A_i) ≤ ∑_{i=1}^∞ P(A_i).\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_6528db9dff1e4be1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic functions", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FDD3E2DFBB65", "name": "Hermite's identity", "statement_latex": "sum_{k=0}^{n-1} floor(x+k/n) = floor(n*x)", "assumptions": "n is a positive integer; x is real."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(sum_{k=0}^{n-1} floor(x+k/n)))+\\delta_D(z,(floor(n*x))))=0", "to_expression": "sum_{k=0}^{n-1} floor(x+k/n) = floor(n*x)", "justification": "zero loss iff some z equals both", "preconditions": "n is a positive integer; x is real.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{k=0}^{n-1} floor(x+k/n) = floor(n*x).\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_8763026c14110c72", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a79bc5ce30756026", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "discrepancy theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E6FD55ACB598", "name": "Beck–Fiala theorem", "statement_latex": "|x(S_j)| <= 2t - 1", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(|x(S_j)|)-(2t - 1)\\}=0", "to_expression": "|x(S_j)| <= 2t - 1", "justification": "positive part vanishes iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |x(S_j)| ≤ 2t - 1.\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_3828007b9ae465e1", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_3c8a440dcfa3c37d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "inner product spaces", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2EE2B752D03D", "name": "Polarization identity", "statement_latex": "\\operatorname{Re} \\langle x, y \\rangle = \\frac{1}{2} (\\|x + y\\|^2 - \\|x\\|^2 - \\|y\\|^2)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\frac{1}{2} (\\|x + y\\|^2 - \\|x\\|^2 - \\|y\\|^2)) = (\\operatorname{Re} \\langle x, y \\rangle)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\operatorname{Re} \\\\langle x, y \\\\rangle = \\\\frac{1}{2} (\\\\|x + y\\\\|^2 - \\\\|x\\\\|^2 - \\\\|y\\\\|^2).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\operatorname{Re} \\langle x, y \\rangle = \\frac{1}{2} (\\|x + y\\|^2 - \\|x\\|^2 - \\|y\\|^2)", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\operatorname{Re} \\\\langle x, y \\\\rangle = \\\\frac{1}{2} (\\\\|x + y\\\\|^2 - \\\\|x\\\\|^2 - \\\\|y\\\\|^2).\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_955c53e9f8a00bdb", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "linear algebra", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_8d3190f9ad183499", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "queueing theory", "subfield": "probability", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_bb76bf5af6ab77d7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1C2EEE35D0F6", "name": "Fischer's inequality", "statement_latex": "det(M) <= det(A)*det(C)", "assumptions": "M is positive semidefinite and block-partitioned with principal blocks A and C."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(det(A)*det(C)) \\in (det(M)) + [0,\\infty)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B.", "preconditions": "M is positive semidefinite and block-partitioned with principal blocks A and C.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover det(M) <= det(A)*det(C).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B.", "preconditions": "M is positive semidefinite and block-partitioned with principal blocks A and C.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover det(M) <= det(A)*det(C).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "det(M) <= det(A)*det(C)", "justification": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B.", "preconditions": "M is positive semidefinite and block-partitioned with principal blocks A and C.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover det(M) <= det(A)*det(C).\"]"}], "source_equivalence_identity": "B in A+[0,infinity) iff B=A+s for some s>=0 iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_fd6f7bcda99e8066", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "geometry", "subfield": "Euclidean geometry", "primary_mechanism": "monotone", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_96b57d8e36fa706f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "queueing theory", "subfield": "probability", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_666939b00ebe2b08", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "representation theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CAF4E18AA4C8", "name": "Brauer's height zero conjecture", "statement_latex": "\\nu(\\chi(1)) = \\nu(|G:D|)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(\\nu(\\chi(1))))+\\delta_D(z,(\\nu(|G:D|))))=0", "to_expression": "\\nu(\\chi(1)) = \\nu(|G:D|)", "justification": "zero loss iff some z equals both", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\nu(\\\\chi(1)) = \\\\nu(|G:D|).\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_e97d1a100a56b298", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "probability theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8F3F3785BDA2", "name": "Gibbs' inequality", "statement_latex": "-Σ p_i log p_i <= -Σ p_i log q_i", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(-Σ p_i log q_i) - (-Σ p_i log p_i) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover -Σ p_i log p_i <= -Σ p_i log q_i.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover -Σ p_i log p_i <= -Σ p_i log q_i.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "-Σ p_i log p_i <= -Σ p_i log q_i", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover -Σ p_i log p_i <= -Σ p_i log q_i.\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_d62c97fba729c3f0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2B79341A5DAC", "name": "Doob's martingale inequality", "statement_latex": "P[max_{1<=i<=n} X_i >= C] <= E[max(X_n, 0)] / C", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(P[max_{1≤i≤n} X_i ≥ C])+s=(E[max(X_n, 0)] / C)", "to_expression": "P[max_{1<=i<=n} X_i >= C] <= E[max(X_n, 0)] / C", "justification": "nonnegative slack exists iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P[max_{1≤i≤n} X_i ≥ C] ≤ E[max(X_n, 0)] / C.\"]"}], "source_equivalence_identity": "nonnegative slack exists iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_89c6dc0e4977804c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "Fourier analysis", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B7E6169619F4", "name": "Dini test", "statement_latex": "\\int_{0}^{\\pi} \\frac{1}{\\delta} \\omega_{f}(\\delta; t) \\, d\\delta < \\infty", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(\\int_{0}^{\\pi} \\frac{1}{\\delta} \\omega_{f}(\\delta; t) \\, d\\delta)-(\\infty)\\}=0\\;\\land\\;(\\int_{0}^{\\pi} \\frac{1}{\\delta} \\omega_{f}(\\delta; t) \\, d\\delta)\\ne (\\infty)", "to_expression": "\\int_{0}^{\\pi} \\frac{1}{\\delta} \\omega_{f}(\\delta; t) \\, d\\delta < \\infty", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\int_{0}^{\\\\pi} \\\\frac{1}{\\\\delta} \\\\omega_{f}(\\\\delta; t) \\\\, d\\\\delta < \\\\infty.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_a7a22381d77d3495", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "elementary algebra", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_40DB1B003904", "name": "Binomial theorem", "statement_latex": "(x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "((x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k) \\land (0 = 0)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "(x+y)^n = \\sum_{k=0}^n \\binom{n}{k} x^{n-k} y^k", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "0=0 is true, so Phi_T and true iff Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_2fbe3a783dba2f54", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "finite fields", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_15C9D05A3E27", "name": "Carlitz–Wan conjecture", "statement_latex": "gcd(d, q − 1) > 1", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(1)-(gcd(d, q − 1))\\}=0\\;\\land\\;(1)\\ne (gcd(d, q − 1))", "to_expression": "gcd(d, q − 1) > 1", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as gcd(d, q − 1) > 1.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_8296d29446935029", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "Ramsey theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6414897882d1e8e2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "calculus", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_88C089C7703F", "name": "Leibniz integral rule", "statement_latex": "d/dx integral_{a(x)}^{b(x)} f(x,t) dt = f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt) = (d/dx integral_{a(x)}^{b(x)} f(x,t) dt)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover d/dx integral_{a(x)}^{b(x)} f(x,t) dt = f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "d/dx integral_{a(x)}^{b(x)} f(x,t) dt = f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover d/dx integral_{a(x)}^{b(x)} f(x,t) dt = f(x, b(x))b'(x) - f(x, a(x))a'(x) + integral_{a(x)}^{b(x)} partial/partial x f(x,t) dt.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_dfb99f09db23a771", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_5b6096cfdc0f6e14", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7922A15B98B4", "name": "Kolmogorov's inequality", "statement_latex": "Pr(max_{1<=k<=n}|S_{k}|>=λ) <= (1/λ^2) * Var[S_{n}]", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(Pr(max_{1\\lek\\len}|S_{k}|\\geλ) \\le (1/λ^2) * Var[S_{n}]) \\land (0 = 0)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "Pr(max_{1<=k<=n}|S_{k}|>=λ) <= (1/λ^2) * Var[S_{n}]", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "0=0 is true, so Phi_T and true iff Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_eaef315dd90ea15c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "discrepancy theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E6FD55ACB598", "name": "Beck–Fiala theorem", "statement_latex": "|x(S_j)| <= 2t - 1", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((|x(S_j)|)-(2t - 1))=0", "to_expression": "|x(S_j)| <= 2t - 1", "justification": "sup equals max(0,X-Y)", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |x(S_j)| ≤ 2t - 1.\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
{"schema_version": "1.0", "item_id": "rcd2_7187cb212b639fd0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "physics", "subfield": "general relativity", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FAE91EE049E6", "name": "Lovelock's theorem", "statement_latex": "A^{\\mu \\nu} = aG^{\\mu \\nu} + bg^{\\mu \\nu}", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(A^{\\mu \\nu}) \\in {(aG^{\\mu \\nu} + bg^{\\mu \\nu})}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover A^{\\\\mu \\\\nu} = aG^{\\\\mu \\\\nu} + bg^{\\\\mu \\\\nu}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "A^{\\mu \\nu} = aG^{\\mu \\nu} + bg^{\\mu \\nu}", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover A^{\\\\mu \\\\nu} = aG^{\\\\mu \\\\nu} + bg^{\\\\mu \\\\nu}.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_9c7a49dd77921ecd", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic calculus", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6c12d52aa2fae8fb", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "calculus", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_19594575fa5d2a05", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_f0409e23e3a57028", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "topology", "subfield": "algebraic topology", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_48CD393CFABB", "name": "Borsuk–Ulam theorem", "statement_latex": "f(-x) = f(x)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(f(-x)) \\in {(f(x))}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(-x) = f(x).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "f(-x) = f(x)", "justification": "L in {R} iff L = R.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover f(-x) = f(x).\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_df218b86b5bf0587", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "vector calculus", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_9fe489080289dc6b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "combinatorics", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_30AD154E7704", "name": "Cyclotomic identity", "statement_latex": "1/(1-alpha*z) = product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\{(1/(1-alpha*z))\\}=\\{(product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)})\\}", "to_expression": "1/(1-alpha*z) = product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}", "justification": "singletons equal iff elements equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as 1/(1-alpha*z) = product_{j=1}^{infinity} (1/(1-z^j))^{M(alpha, j)}.\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_ca9f78da5f076245", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "statistics", "subfield": "statistical hypothesis testing", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_69B4EB18D461", "name": "Neyman–Pearson lemma", "statement_latex": "α = Pr_θ0(X ∈ R)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (α = Pr_θ0(X ∈ R)))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "α = Pr_θ0(X ∈ R)", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_5d42b4706c37da8d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "percolation theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4AB97BAC2783", "name": "Van den Berg–Kesten inequality", "statement_latex": "P(A square B) <= P(A)P(B)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(P(A)P(B))\\le t\\Rightarrow (P(A square B))\\le t", "to_expression": "P(A square B) <= P(A)P(B)", "justification": "converse uses t=Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(A square B) ≤ P(A)P(B).\"]"}], "source_equivalence_identity": "converse uses t=Y"}
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{"schema_version": "1.0", "item_id": "rcd2_3687e5847afc2b4f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "Ramsey theory", "subfield": "discrete mathematics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4B3C8E83990", "name": "Schur's theorem", "statement_latex": "x + y = z", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((x + y))=F((z))", "to_expression": "x + y = z", "justification": "indicator of {X} separates unequal objects", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as x + y = z.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
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{"schema_version": "1.0", "item_id": "rcd2_75a7042a694c60c7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "measure theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4450EEEA850", "name": "Borel–Cantelli lemma", "statement_latex": "limsup_{n->infinity} E_n = intersection_{n=1 to infinity} union_{k=n to infinity} E_k", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(limsup_{n-) - (\\infty} E_n = intersection_{n=1 to \\infty} union_{k=n to \\infty} E_k) > 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover limsup_{n- > infinity} E_n = intersection_{n=1 to infinity} union_{k=n to infinity} E_k.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover limsup_{n- > infinity} E_n = intersection_{n=1 to infinity} union_{k=n to infinity} E_k.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "limsup_{n->infinity} E_n = intersection_{n=1 to infinity} union_{k=n to infinity} E_k", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover limsup_{n- > infinity} E_n = intersection_{n=1 to infinity} union_{k=n to infinity} E_k.\"]"}], "source_equivalence_identity": "A<B iff B-A>0 over finite real quantities."}
{"schema_version": "1.0", "item_id": "rcd2_ebb7ef80087ca749", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "Ramsey theory", "subfield": "discrete mathematics", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3F2BB2E376A3", "name": "Rado's theorem", "statement_latex": "Ax = 0", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(Ax)\\Longleftrightarrow z=(0))", "to_expression": "Ax = 0", "justification": "same equality fiber iff equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Ax = 0.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
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{"schema_version": "1.0", "item_id": "rcd2_c51fe3e74cef7f1f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B63439ADF46F", "name": "Rising sun lemma", "statement_latex": "E = \\bigcup_{k}(a_{k},b_{k})", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(E)\\Longleftrightarrow z=(\\bigcup_{k}(a_{k},b_{k})))", "to_expression": "E = \\bigcup_{k}(a_{k},b_{k})", "justification": "same equality fiber iff equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as E = \\\\bigcup_{k}(a_{k},b_{k}).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
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{"schema_version": "1.0", "item_id": "rcd2_4eac0c67855426c5", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "approximation theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F6B5A5657F99", "name": "Fejér's theorem", "statement_latex": "lim_{n→∞} σ_n(f,x) = f(x)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((lim_{n→∞} σ_n(f,x)),(f(x)))=0", "to_expression": "lim_{n→∞} σ_n(f,x) = f(x)", "justification": "zero discrete distance iff equality", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{n→∞} σ_n(f,x) = f(x).\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_9265d8795be426d0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D13DCB7F188A", "name": "Kuznetsov trace formula", "statement_latex": "∑_{c≡0 mod N}c^{-r}K(m,n,c)g(4π√mn/c)=Integral transform+Spectral terms", "assumptions": "All displayed arithmetic expressions are integers; the modulus is a positive integer."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(∑_{c≡0 mod N}c^{-r}K(m,n,c)g(4π√mn/c)))+\\delta_D(z,(Integral transform+Spectral terms)))=0", "to_expression": "∑_{c≡0 mod N}c^{-r}K(m,n,c)g(4π√mn/c)=Integral transform+Spectral terms", "justification": "zero loss iff some z equals both", "preconditions": "All displayed arithmetic expressions are integers; the modulus is a positive integer.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∑_{c≡0 mod N}c^{-r}K(m,n,c)g(4π√mn/c)=Integral transform+Spectral terms.\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
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{"schema_version": "1.0", "item_id": "rcd2_58b221892db2f04a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "geometry, probability theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3B9F2B85DC25", "name": "Gaussian isoperimetric inequality", "statement_latex": "liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)} >= φ(Φ^{-1}(γ^n(A)))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((φ(Φ^{-1}(γ^n(A))))-(liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)}))=0", "to_expression": "liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)} >= φ(Φ^{-1}(γ^n(A)))", "justification": "sup equals max(0,X-Y)", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as liminf_{ε→+0} ε^{-1} {γ^n(A_ε) - γ^n(A)} ≥ φ(Φ^{-1}(γ^n(A))).\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
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{"schema_version": "1.0", "item_id": "rcd2_1774a14a0cf3b2e0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "complex analysis", "subfield": "complex geometry", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_1CFF4C1B7932", "name": "Andreotti–Grauert theorem", "statement_latex": "dim_C H^i(X, F) < infinity", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\infty) - (dim_C H^i(X, F)) > 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover dim_C H^i(X, F) < infinity.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover dim_C H^i(X, F) < infinity.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "dim_C H^i(X, F) < infinity", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover dim_C H^i(X, F) < infinity.\"]"}], "source_equivalence_identity": "A<B iff B-A>0 over finite real quantities."}
{"schema_version": "1.0", "item_id": "rcd2_ec77aae0654d9ded", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "computer algebra", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F94B0806E6BB", "name": "Landau–Mignotte bound", "statement_latex": "||h||_1 <= 2^n * ||f||_1", "assumptions": "h is a factor of the integer polynomial f of degree n under the standard Landau-Mignotte hypotheses; l1 coefficient norms are finite."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(||h||_1)}{\\exp(||h||_1)+\\exp(2^n * ||f||_1)}\\le\\frac{1}{2}", "to_expression": "||h||_1 <= 2^n * ||f||_1", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "h is a factor of the integer polynomial f of degree n under the standard Landau-Mignotte hypotheses; l1 coefficient norms are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ||h||_1 ≤ 2^n * ||f||_1.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_ca3b309d2341539f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "enumerative combinatorics", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c508ae452f376b16", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "differential equations", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c4e14250c1a931a1", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0e88700fe08373b3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "geometric probability", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F57E82952899", "name": "Wendel's theorem", "statement_latex": "p_{n,N}=2^{-N+1}\\sum _{k=0}^{n-1}{\\binom {N-1}{k}}", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((p_{n,N}),(2^{-N+1}\\sum _{k=0}^{n-1}{\\binom {N-1}{k}}))=0", "to_expression": "p_{n,N}=2^{-N+1}\\sum _{k=0}^{n-1}{\\binom {N-1}{k}}", "justification": "zero discrete distance iff equality", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p_{n,N}=2^{-N+1}\\\\sum _{k=0}^{n-1}{\\\\binom {N-1}{k}}.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_00a3e39014ef960c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "differential calculus", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_d03a5a5b1732d780", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "enumerative combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_30BF4DFA8E4C", "name": "MacMahon's master theorem", "statement_latex": "G(k1, ..., km) = [x1^k1 ... xm^km] Prod_{i=1 to m} (Sum_{j=1 to m} a_{ij} x_j)^k_i", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (G(k1, ..., km) = [x1^k1 ... xm^km] Prod_{i=1 to m} (Sum_{j=1 to m} a_{ij} x_j)^k_i))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "G(k1, ..., km) = [x1^k1 ... xm^km] Prod_{i=1 to m} (Sum_{j=1 to m} a_{ij} x_j)^k_i", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_bff1dd29617a4445", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "Algebra", "subfield": "Lie algebras", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_416ddc3b3cb7fd08", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "representation theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_13CEC210D728", "name": "Wigner–Eckart theorem", "statement_latex": "⟨j m | T_q^(k) | j' m'⟩ = ⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(⟨j m | T_q^(k) | j' m'⟩))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩))", "to_expression": "⟨j m | T_q^(k) | j' m'⟩ = ⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩", "justification": "each argmin set is a singleton", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ⟨j m | T_q^(k) | j' m'⟩ = ⟨j' m' k q | j m⟩ ⟨j || T^(k) || j'⟩.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_c31394e3f36e4529", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "calculus", "subfield": "vector calculus / electrostatics", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e9cc05ca252f26b0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D236ADD607AB", "name": "Chernoff bound", "statement_latex": "P(X ≥ a) ≤ inf_{t>0} M(t)e^{-ta}", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(inf_{t>0} M(t)e^{-ta})\\le t\\Rightarrow (P(X ≥ a))\\le t", "to_expression": "P(X ≥ a) ≤ inf_{t>0} M(t)e^{-ta}", "justification": "converse uses t=Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(X ≥ a) ≤ inf_{t>0} M(t)e^{-ta}.\"]"}], "source_equivalence_identity": "converse uses t=Y"}
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{"schema_version": "1.0", "item_id": "rcd2_46d19bbbdc767b74", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "metric geometry", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0df2a2bca5533c02", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "commutative algebra", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_666997CDC49C", "name": "Noether normalization lemma", "statement_latex": "A = k[y_1, ..., y_m]", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((A),(k[y_1, ..., y_m]))=0", "to_expression": "A = k[y_1, ..., y_m]", "justification": "zero discrete distance iff equality", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as A = k[y_1, ..., y_m].\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_ee69d5d3535ee66a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "statistical mechanics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_63998BA24BBA", "name": "Fluctuation theorem", "statement_latex": "Pr(Σ_t=A) / Pr(Σ_t=-A) = e^(At)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((Pr(Σ_t=A) / Pr(Σ_t=-A)))=F((e^(At)))", "to_expression": "Pr(Σ_t=A) / Pr(Σ_t=-A) = e^(At)", "justification": "indicator of {X} separates unequal objects", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr(Σ_t=A) / Pr(Σ_t=-A) = e^(At).\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_1104b6afd1528fc9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "Sobolev spaces", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_35353A67F17E", "name": "Pólya–Szegő inequality", "statement_latex": "∫ |∇u*|^p dx <= ∫ |∇u|^p dx", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(∫ |∇u*|^p dx)}{\\exp(∫ |∇u*|^p dx)+\\exp(∫ |∇u|^p dx)}\\le\\frac{1}{2}", "to_expression": "∫ |∇u*|^p dx <= ∫ |∇u|^p dx", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∫ |∇u*|^p dx ≤ ∫ |∇u|^p dx.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_7f17732e394111df", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_41098B279612", "name": "Girsanov theorem", "statement_latex": "E(X)_t = exp(X_t - 1/2 [X]_t)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((E(X)_t))=F((exp(X_t - 1/2 [X]_t)))", "to_expression": "E(X)_t = exp(X_t - 1/2 [X]_t)", "justification": "indicator of {X} separates unequal objects", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as E(X)_t = exp(X_t - 1/2 [X]_t).\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
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{"schema_version": "1.0", "item_id": "rcd2_90cdfc2108eaa7a0", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "statistical mechanics", "subfield": "non-equilibrium thermodynamics", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c9fd8a24ebc79989", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_684A9634E4F6", "name": "Hypergeometric identity", "statement_latex": "sum_{i=0}^n binom(n,i) = 2^n", "assumptions": "n is a nonnegative integer."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((sum_{i=0}^n binom(n,i)),(2^n))=0", "to_expression": "sum_{i=0}^n binom(n,i) = 2^n", "justification": "zero discrete distance iff equality", "preconditions": "n is a nonnegative integer.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{i=0}^n binom(n,i) = 2^n.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_d0b64b7d8b301dd0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "permutation patterns", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8A6780273F9B", "name": "Stanley–Wilf conjecture", "statement_latex": "lim_{n→∞} |S_n(β)|^(1/n)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (lim_{n→∞} |S_n(β)|^(1/n)))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "lim_{n→∞} |S_n(β)|^(1/n)", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_d92451ac8aeab523", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "Combinatorics", "subfield": "Order theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B5F5A5581E5D", "name": "Kruskal's tree theorem", "statement_latex": "T1 <= T2", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (T1) \\le x \\land x \\le (T2)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover T1 <= T2.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover T1 <= T2.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "T1 <= T2", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover T1 <= T2.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_92c3733caa9498dc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9F50AE0DD0D5", "name": "Cauchy–Davenport theorem", "statement_latex": "|A+B| >= min(|A|+|B|-1, p)", "assumptions": "A and B are nonempty finite subsets of Z/pZ; p is prime."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(min(|A|+|B|-1, p))}{\\exp(min(|A|+|B|-1, p))+\\exp(|A+B|)}\\le\\frac{1}{2}", "to_expression": "|A+B| >= min(|A|+|B|-1, p)", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "A and B are nonempty finite subsets of Z/pZ; p is prime.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p).\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_79f715e10ba2e910", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_41098B279612", "name": "Girsanov theorem", "statement_latex": "E(X)_t = exp(X_t - 1/2 [X]_t)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(E(X)_t) \\in {(\\exp(X_t - 1/2 [X]_t))}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E(X)_t = exp(X_t - 1/2 [X]_t).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "E(X)_t = exp(X_t - 1/2 [X]_t)", "justification": "L in {R} iff L = R.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E(X)_t = exp(X_t - 1/2 [X]_t).\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_a1785274e4932901", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "approximation theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_9d641c718f6f8f8a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_79DC6C43DD64", "name": "Perron–Frobenius theorem", "statement_latex": "Av = rv", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((Av))=F((rv))", "to_expression": "Av = rv", "justification": "indicator of {X} separates unequal objects", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Av = rv.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_fc860fc35c90c78f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7922A15B98B4", "name": "Kolmogorov's inequality", "statement_latex": "Pr(max_{1<=k<=n}|S_{k}|>=λ) <= (1/λ^2) * Var[S_{n}]", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((Pr(max_{1≤k≤n}|S_{k}|≥λ))-((1/λ^2) * Var[S_{n}]),(-\\infty,0])=0", "to_expression": "Pr(max_{1<=k<=n}|S_{k}|>=λ) <= (1/λ^2) * Var[S_{n}]", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr(max_{1≤k≤n}|S_{k}|≥λ) ≤ (1/λ^2) * Var[S_{n}].\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
{"schema_version": "1.0", "item_id": "rcd2_4a3926af433ce763", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "harmonic analysis", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_ECE01C7A6C37", "name": "Wiener's lemma", "statement_latex": "lim_{N->inf} (1/(2N+1)) sum_{n=-N}^{N} |widehat{mu}(n)|^2 = sum_{j} |c_{j}|^2", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((lim_{N→inf} (1/(2N+1)) sum_{n=-N}^{N} |widehat{mu}(n)|^2))=F((sum_{j} |c_{j}|^2))", "to_expression": "lim_{N->inf} (1/(2N+1)) sum_{n=-N}^{N} |widehat{mu}(n)|^2 = sum_{j} |c_{j}|^2", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{N→inf} (1/(2N+1)) sum_{n=-N}^{N} |widehat{mu}(n)|^2 = sum_{j} |c_{j}|^2.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_e3905b75cb9f9a54", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "matching theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_5b820a81edca1626", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "discrete mathematics", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_75050c5debae4399", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "probability theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_8d46366b211cf04c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "calculus", "subfield": "vector calculus / electrostatics", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D5081E95CCA1", "name": "Uniqueness theorem for Poisson's equation", "statement_latex": "∇^2φ = -ρ_f/ε_0", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((∇^2φ),z)=((-ρ_f/ε_0),z)", "to_expression": "∇^2φ = -ρ_f/ε_0", "justification": "pair equality forces first coordinates equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∇^2φ = -ρ_f/ε_0.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_ab10577063a4e0a8", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "calculus", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b42ccc9944097d26", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "topology", "subfield": "combinatorial topology", "primary_mechanism": "distance", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CB3E317B00C5", "name": "Tucker's lemma", "statement_latex": "L(-v) = -L(v)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists c \\in D: (L(-v)) = c \\land (-L(v)) = c", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A common c exists iff L and R are equal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover L(-v) = -L(v).\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "L(-v) = -L(v)", "justification": "A common c exists iff L and R are equal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover L(-v) = -L(v).\"]"}], "source_equivalence_identity": "A common c exists iff L and R are equal."}
{"schema_version": "1.0", "item_id": "rcd2_3793ec4e449bcfb1", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "combinatorics", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_35ab090831c7bed9", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "field theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_7b6c1761c9fb3d31", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic geometry", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b9c3caf7eeb09be1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "incidence geometry", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B3E12B4FAA4C", "name": "Loomis–Whitney inequality", "statement_latex": "|| \\prod_{j=1}^{d} g_j \\circ \\pi_j ||_{L^1} \\le \\prod_{j=1}^{d} ||g_j||_{L^{d-1}}", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (\\prod_{j=1}^{d} ||g_j||_{L^{d-1}})\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|| \\prod_{j=1}^{d} g_j \\circ \\pi_j ||_{L^1})\\}", "to_expression": "|| \\prod_{j=1}^{d} g_j \\circ \\pi_j ||_{L^1} \\le \\prod_{j=1}^{d} ||g_j||_{L^{d-1}}", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as || \\\\prod_{j=1}^{d} g_j \\\\circ \\\\pi_j ||_{L^1} ≤ \\\\prod_{j=1}^{d} ||g_j||_{L^{d-1}}.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_091cf4cfdc133751", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal set theory", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4D5A3A69C1D", "name": "Sauer–Shelah lemma", "statement_latex": "|F| <= sum_{i=0}^{d} binom(n,i)", "assumptions": "F is a family of subsets of an n-element set with VC dimension d."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(|F|)+s=(sum_{i=0}^{d} binom(n,i))", "to_expression": "|F| <= sum_{i=0}^{d} binom(n,i)", "justification": "nonnegative slack exists iff X≤Y", "preconditions": "F is a family of subsets of an n-element set with VC dimension d.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |F| ≤ sum_{i=0}^{d} binom(n,i).\"]"}], "source_equivalence_identity": "nonnegative slack exists iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_a19e551d25cf6be7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "functional analysis", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_462346AD2251", "name": "Ehrling's lemma", "statement_latex": "||x||_Y <= ε||x||_X + C(ε)||x||_Z", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (ε||x||_X + C(ε)||x||_Z)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (||x||_Y)\\}", "to_expression": "||x||_Y <= ε||x||_X + C(ε)||x||_Z", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ||x||_Y ≤ ε||x||_X + C(ε)||x||_Z.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_48b1f26e61628302", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "analytic number theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_625e0abe41881798", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_00ff77ca0ea9d005", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "analysis", "subfield": "complex analysis", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_065364B1C6FB", "name": "Arakelyan's theorem", "statement_latex": "|g − f| < ε", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(|g − f|)}{\\exp(|g − f|)+\\exp(ε)}<\\frac{1}{2}", "to_expression": "|g − f| < ε", "justification": "equiv exp(X)<exp(Y)", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |g − f| < ε.\"]"}], "source_equivalence_identity": "equiv exp(X)<exp(Y)"}
{"schema_version": "1.0", "item_id": "rcd2_d78913a50028459f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "probabilistic method", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8F1DAFA4A16E", "name": "Lovász local lemma", "statement_latex": "e*p*(d+1) <= 1", "assumptions": "Symmetric Lovasz local lemma setting; each bad event has probability at most p and dependency degree at most d."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(1) - (e*p*(d+1)) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "Symmetric Lovasz local lemma setting; each bad event has probability at most p and dependency degree at most d.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover e*p*(d+1) <= 1.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "Symmetric Lovasz local lemma setting; each bad event has probability at most p and dependency degree at most d.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover e*p*(d+1) <= 1.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "e*p*(d+1) <= 1", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "Symmetric Lovasz local lemma setting; each bad event has probability at most p and dependency degree at most d.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover e*p*(d+1) <= 1.\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_8f27d2bd238554b6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_42ED352AD252", "name": "Bernstein inequalities", "statement_latex": "P(|(1/n) * sum_{i=1}^n X_i| > epsilon) <= 2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3)))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((P(|(1/n) * sum_{i=1}^n X_i| > epsilon))-(2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3)))),(-\\infty,0])=0", "to_expression": "P(|(1/n) * sum_{i=1}^n X_i| > epsilon) <= 2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3)))", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(|(1/n) * sum_{i=1}^n X_i| > epsilon) ≤ 2 * exp(-n * epsilon^2 / (2 * (1 + epsilon/3))).\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
{"schema_version": "1.0", "item_id": "rcd2_ea68a688ef82946c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "design theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_41032C4E68AC", "name": "Bruck–Ryser–Chowla theorem", "statement_latex": "x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2} = 0", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2}) \\in S \\land (0) \\notin S) \\lor ((0) \\in S \\land (x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2}) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover x^{2} - (k - \\\\lambda)y^{2} - (-1)^{(v-1)/2} \\\\lambda z^{2} = 0.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "x^{2} - (k - \\lambda)y^{2} - (-1)^{(v-1)/2} \\lambda z^{2} = 0", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover x^{2} - (k - \\\\lambda)y^{2} - (-1)^{(v-1)/2} \\\\lambda z^{2} = 0.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_c748ef21ea98e68a", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "computer science", "subfield": "theoretical computer science", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b5be8ba8b14f81c2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "enumerative combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E1416BDA0AD6", "name": "Pólya enumeration theorem", "statement_latex": "|Y^X/G| = (1/|G|) * Σ_{g∈G} m^{c(g)}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(|Y^X/G| = (1/|G|) * Σ_{g∈G} m^{c(g)}) \\lor (0 = 1)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "0=1 is false, so Phi_T or false iff Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "|Y^X/G| = (1/|G|) * Σ_{g∈G} m^{c(g)}", "justification": "0=1 is false, so Phi_T or false iff Phi_T.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "0=1 is false, so Phi_T or false iff Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_d6790e714d3c59fc", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "geometric probability", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_8c89efa758b8894a", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "approximation theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ad050d2f4a4c223c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "Mathematical analysis", "subfield": "Sequences and series", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F3C0470E95D9", "name": "Abel's inequality", "statement_latex": "|∑ a_k b_k| ≤ max|B_k|(|a_n| + |a_n - a_1|)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;t\\le (|∑ a_k b_k|)\\Rightarrow t\\le (max|B_k|(|a_n| + |a_n - a_1|))", "to_expression": "|∑ a_k b_k| ≤ max|B_k|(|a_n| + |a_n - a_1|)", "justification": "converse uses t=X", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |∑ a_k b_k| ≤ max|B_k|(|a_n| + |a_n - a_1|).\"]"}], "source_equivalence_identity": "converse uses t=X"}
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{"schema_version": "1.0", "item_id": "rcd2_807e5797968cd010", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "arithmetic geometry", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_49F57EB1100D", "name": "Faltings' theorem", "statement_latex": "g > 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(1)-(g)\\}=0\\;\\land\\;(1)\\ne (g)", "to_expression": "g > 1", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as g > 1.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_d247b4611bc7bf35", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F902B76211EA", "name": "Cantelli's inequality", "statement_latex": "Pr(X - E[X] ≥ λ) ≤ σ^2 / (σ^2 + λ^2)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(Pr(X - E[X] ≥ λ) ≤ σ^2 / (σ^2 + λ^2)) \\land (0 = 0)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "Pr(X - E[X] ≥ λ) ≤ σ^2 / (σ^2 + λ^2)", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "0=0 is true, so Phi_T and true iff Phi_T."}
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{"schema_version": "1.0", "item_id": "rcd2_381a4e1c8e8c37f9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "linear algebra", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_08D175804776", "name": "Cayley–Hamilton theorem", "statement_latex": "p_A(A) = 0", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((p_A(A)),(0))\\in\\{(z,z):z\\in D\\}", "to_expression": "p_A(A) = 0", "justification": "pair in diagonal iff X=Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p_A(A) = 0.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
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{"schema_version": "1.0", "item_id": "rcd2_24f4c0e266b95061", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "quadratic forms", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_434E3E7B6A54", "name": "Sylvester's law of inertia", "statement_latex": "B = SAS^T", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\{(B)\\}=\\{(SAS^T)\\}", "to_expression": "B = SAS^T", "justification": "singletons equal iff elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as B = SAS^T.\"]"}], "source_equivalence_identity": "singletons equal iff elements equal"}
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{"schema_version": "1.0", "item_id": "rcd2_544de3a8d329d8f7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9DBE3057770F", "name": "Kneser's theorem", "statement_latex": "|A+B|=|A+H|+|B+H|-|H|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "(\\{(|A+B|)\\}\\times\\{(|A+H|+|B+H|-|H|)\\})\\cap\\{(z,z):z\\in D\\}=\\{((|A+B|),(|A+H|+|B+H|-|H|))\\}", "to_expression": "|A+B|=|A+H|+|B+H|-|H|", "justification": "intersection is singleton iff pair diagonal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |A+B|=|A+H|+|B+H|-|H|.\"]"}], "source_equivalence_identity": "intersection is singleton iff pair diagonal"}
{"schema_version": "1.0", "item_id": "rcd2_c58911d821657cf3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "geometry of numbers", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0EF9844EAE12", "name": "Minkowski's theorem", "statement_latex": "Vol(S) > 2^n d(L)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(2^n d(L))-(Vol(S))\\}=0\\;\\land\\;(2^n d(L))\\ne (Vol(S))", "to_expression": "Vol(S) > 2^n d(L)", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Vol(S) > 2^n d(L).\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_9796c5eb59708d18", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "scattering theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0C5D6466FC1D", "name": "Levinson's theorem", "statement_latex": "φ(+∞) − φ(0) = −π(1/2*n0 + nb)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "(\\{(φ(+∞) − φ(0))\\}\\times\\{(−π(1/2*n0 + nb))\\})\\cap\\{(z,z):z\\in D\\}=\\{((φ(+∞) − φ(0)),(−π(1/2*n0 + nb)))\\}", "to_expression": "φ(+∞) − φ(0) = −π(1/2*n0 + nb)", "justification": "intersection is singleton iff pair diagonal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as φ(+∞) − φ(0) = −π(1/2*n0 + nb).\"]"}], "source_equivalence_identity": "intersection is singleton iff pair diagonal"}
{"schema_version": "1.0", "item_id": "rcd2_032f8ed374400db2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_936D29F3BF52", "name": "Hsu–Robbins–Erdős theorem", "statement_latex": "∑_{n>=1}P(|S_{n}|>εn)<∞", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(∑_{n≥1}P(|S_{n}|>εn))}{\\exp(∑_{n≥1}P(|S_{n}|>εn))+\\exp(∞)}<\\frac{1}{2}", "to_expression": "∑_{n>=1}P(|S_{n}|>εn)<∞", "justification": "equiv exp(X)<exp(Y)", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∑_{n≥1}P(|S_{n}|>εn)<∞.\"]"}], "source_equivalence_identity": "equiv exp(X)<exp(Y)"}
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{"schema_version": "1.0", "item_id": "rcd2_09e5c6c8945da19a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_89D2662A27BE", "name": "Freiman's theorem", "statement_latex": "|A+A| <= K|A|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(|A+A|)-(K|A|)\\}=0", "to_expression": "|A+A| <= K|A|", "justification": "positive part vanishes iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |A+A| ≤ K|A|.\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_6902297255aee68d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "enumerative combinatorics", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7EDAF21632BF", "name": "Labelled enumeration theorem", "statement_latex": "F(z,t)=e^{g(z)t}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "(\\{(F(z,t))\\}\\times\\{(e^{g(z)t})\\})\\cap\\{(z,z):z\\in D\\}=\\{((F(z,t)),(e^{g(z)t}))\\}", "to_expression": "F(z,t)=e^{g(z)t}", "justification": "intersection is singleton iff pair diagonal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as F(z,t)=e^{g(z)t}.\"]"}], "source_equivalence_identity": "intersection is singleton iff pair diagonal"}
{"schema_version": "1.0", "item_id": "rcd2_e95355111f622553", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "measure theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_448CEF326F89", "name": "Doob–Dynkin lemma", "statement_latex": "f = g ∘ T", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((f))=F((g ∘ T))", "to_expression": "f = g ∘ T", "justification": "indicator of {X} separates unequal objects", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f = g ∘ T.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_2d644345eebc9aa7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "calculus of variations", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7F37206494D7", "name": "Isoperimetric inequality", "statement_latex": "4πA ≤ L^2", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((4πA)-(L^2))=0", "to_expression": "4πA ≤ L^2", "justification": "sup equals max(0,X-Y)", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as 4πA ≤ L^2.\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
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{"schema_version": "1.0", "item_id": "rcd2_72d4aaccbf09b00b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "field theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4E2C0DED1A31", "name": "Joubert's theorem", "statement_latex": "p(t) = t^6 + c_4t^4 + c_2t^2 + c_1t + c_0", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((p(t)),z)=((t^6 + c_4t^4 + c_2t^2 + c_1t + c_0),z)", "to_expression": "p(t) = t^6 + c_4t^4 + c_2t^2 + c_1t + c_0", "justification": "pair equality forces first coordinates equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p(t) = t^6 + c_4t^4 + c_2t^2 + c_1t + c_0.\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
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{"schema_version": "1.0", "item_id": "rcd2_f64ef8fa13ec9a2f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "algebra", "subfield": "combinatorics", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E24E6480323C", "name": "Multinomial theorem", "statement_latex": "(x_{1}+x_{2}+\\cdots +x_{m})^{n}=\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=((x_{1}+x_{2}+\\cdots +x_{m})^{n})\\Longleftrightarrow z=(\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}}))", "to_expression": "(x_{1}+x_{2}+\\cdots +x_{m})^{n}=\\sum _{k_{1}+k_{2}+\\cdots +k_{m}=n} \\binom{n}{k_1, k_2, \\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\cdots x_{m}^{k_{m}}", "justification": "same equality fiber iff equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (x_{1}+x_{2}+\\\\cdots +x_{m})^{n}=\\\\sum _{k_{1}+k_{2}+\\\\cdots +k_{m}=n} \\\\binom{n}{k_1, k_2, \\\\dots, k_m} x_{1}^{k_{1}} x_{2}^{k_{2}} \\\\cdots x_{m}^{k_{m}}.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
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{"schema_version": "1.0", "item_id": "rcd2_bd7f329048f4aef8", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "statistics", "subfield": "empirical Bayes methods", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2C7BEBF4ECBC", "name": "Robbins lemma", "statement_latex": "E(Xf(X-1)) = λE(f(X))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((E(Xf(X-1))) \\in S \\land (λE(f(X))) \\notin S) \\lor ((λE(f(X))) \\in S \\land (E(Xf(X-1))) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E(Xf(X-1)) = λE(f(X)).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "E(Xf(X-1)) = λE(f(X))", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E(Xf(X-1)) = λE(f(X)).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_8899f2615a98dede", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "calculus", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_819FFAB136D4", "name": "L'Hôpital's rule", "statement_latex": "lim_{x->c} f(x)/g(x) = lim_{x->c} f'(x)/g'(x)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(lim_{x→c} f(x)/g(x)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(lim_{x→c} f'(x)/g'(x)))", "to_expression": "lim_{x->c} f(x)/g(x) = lim_{x->c} f'(x)/g'(x)", "justification": "each argmin set is a singleton", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{x→c} f(x)/g(x) = lim_{x→c} f'(x)/g'(x).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_1e138c54ad42079c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "graph theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2f01dbcd182ca94c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "complex analysis", "subfield": "complex geometry", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e8b86e80b7aa551f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "queueing theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D0BC02092015", "name": "Bartlett's theorem", "statement_latex": "mu(t) = integral_{-infinity}^{t} A(s)p(s,t) ds", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(integral_{-\\infty}^{t} A(s)p(s,t) ds) = (mu(t))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover mu(t) = integral_{-infinity}^{t} A(s)p(s,t) ds.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "mu(t) = integral_{-infinity}^{t} A(s)p(s,t) ds", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover mu(t) = integral_{-infinity}^{t} A(s)p(s,t) ds.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_c5d499bcf17a35fd", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "probabilistic combinatorics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_32a935350dfd460e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "statistics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4273AD2C24D7", "name": "Bayes' theorem", "statement_latex": "P(A|B) = P(B|A)*P(A)/P(B)", "assumptions": "A and B are events with P(B)>0; conditional probabilities are defined."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((P(A|B)) \\in S \\land (P(B|A)*P(A)/P(B)) \\notin S) \\lor ((P(B|A)*P(A)/P(B)) \\in S \\land (P(A|B)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "A and B are events with P(B)>0; conditional probabilities are defined.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover P(A|B) = P(B|A)*P(A)/P(B).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "P(A|B) = P(B|A)*P(A)/P(B)", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "A and B are events with P(B)>0; conditional probabilities are defined.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover P(A|B) = P(B|A)*P(A)/P(B).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_cce50ad5d4c9be17", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_24b021eff602e94c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "differential topology", "subfield": "geometry", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CC0CC8596114", "name": "Morse theory", "statement_latex": "M^a = {x in M | f(x) <= a}", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(a}) - (M^a = {x \\in M | f(x)) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover M^a = {x in M | f(x) <= a}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover M^a = {x in M | f(x) <= a}.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "M^a = {x in M | f(x) <= a}", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover M^a = {x in M | f(x) <= a}.\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_8e0fdd5fbb78076e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "quadratic forms", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_434E3E7B6A54", "name": "Sylvester's law of inertia", "statement_latex": "B = SAS^T", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((B),(SAS^T))\\in\\{(z,z):z\\in D\\}", "to_expression": "B = SAS^T", "justification": "pair in diagonal iff X=Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as B = SAS^T.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_b027f30b3eff894a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_319FF857BA97", "name": "Burgess inequality", "statement_latex": "|S_chi(N,H)| <= C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(|S_chi(N,H)|)}{\\exp(|S_chi(N,H)|)+\\exp(C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon})}\\le\\frac{1}{2}", "to_expression": "|S_chi(N,H)| <= C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon}", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |S_chi(N,H)| ≤ C_{r,epsilon} H^{1-1/r} q^{(r+1)/(4r^2)+epsilon}.\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_a963b2e62fedf1f3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic calculus", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_58173198312A", "name": "Boué–Dupuis formula", "statement_latex": "-log E[e^{-f(B)}] = inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)]", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(-log E[e^{-f(B)}]))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)]))", "to_expression": "-log E[e^{-f(B)}] = inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)]", "justification": "each argmin set is a singleton", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as -log E[e^{-f(B)}] = inf_V E[1/2 * integral_0^1 ||V_t||^2 dt + f(B + integral_0^. V_t dt)].\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_ba83a20cbd35cc26", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "combinatorics", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_684A9634E4F6", "name": "Hypergeometric identity", "statement_latex": "sum_{i=0}^n binom(n,i) = 2^n", "assumptions": "n is a nonnegative integer."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(sum_{i=0}^n binom(n,i))\\Longleftrightarrow z=(2^n))", "to_expression": "sum_{i=0}^n binom(n,i) = 2^n", "justification": "same equality fiber iff equal", "preconditions": "n is a nonnegative integer.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{i=0}^n binom(n,i) = 2^n.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_760d41071ebdd66b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "complex analysis", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_484796D95DA9", "name": "Abel's theorem", "statement_latex": "lim_{x -> 1^-} G(x) = sum_{k=0}^infinity a_k", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(1^-} G(x) = sum_{k=0}^\\infty a_k) - (lim_{x -) < 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A-B<0 iff A<B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover lim_{x - > 1^-} G(x) = sum_{k=0}^infinity a_k.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A-B<0 iff A<B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover lim_{x - > 1^-} G(x) = sum_{k=0}^infinity a_k.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "lim_{x -> 1^-} G(x) = sum_{k=0}^infinity a_k", "justification": "A-B<0 iff A<B.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover lim_{x - > 1^-} G(x) = sum_{k=0}^infinity a_k.\"]"}], "source_equivalence_identity": "A-B<0 iff A<B."}
{"schema_version": "1.0", "item_id": "rcd2_2e05c976dc9dfcea", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "real analysis", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A00EEAE41A77", "name": "Intermediate value theorem", "statement_latex": "f(x) = s", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(f(x))\\Longleftrightarrow z=(s))", "to_expression": "f(x) = s", "justification": "same equality fiber iff equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f(x) = s.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_e52ccb51868dbea3", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "commutative algebra", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ee23d2e2cadfb10a", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "percolation theory", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_7c6918fae05833ef", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "physics", "subfield": "general relativity", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FAE91EE049E6", "name": "Lovelock's theorem", "statement_latex": "A^{\\mu \\nu} = aG^{\\mu \\nu} + bg^{\\mu \\nu}", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(A^{\\mu \\nu})\\Longleftrightarrow z=(aG^{\\mu \\nu} + bg^{\\mu \\nu}))", "to_expression": "A^{\\mu \\nu} = aG^{\\mu \\nu} + bg^{\\mu \\nu}", "justification": "same equality fiber iff equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as A^{\\\\mu \\\\nu} = aG^{\\\\mu \\\\nu} + bg^{\\\\mu \\\\nu}.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_45df59e4184f4f62", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "representation theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_78311cbffa371883", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "statistical mechanics", "subfield": "non-equilibrium thermodynamics", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_12a48816da906fa3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "zeta and L-functions", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CEA3FEEC415F", "name": "Selberg's zeta function conjecture", "statement_latex": "N(T + H) - N(T) ≥ cH log T", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (N(T + H) - N(T))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (cH log T)\\}", "to_expression": "N(T + H) - N(T) ≥ cH log T", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as N(T + H) - N(T) ≥ cH log T.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_21516ccfc875d21f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "linear algebra", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_827a4ed5d9dfc34d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "polyhedral combinatorics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_687D9EB61DA6", "name": "Eberhard's theorem", "statement_latex": "sum (6-i)p_i = 12", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(sum (6-i)p_i) \\in {(12)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover sum (6-i)p_i = 12.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "sum (6-i)p_i = 12", "justification": "L in {R} iff L = R.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover sum (6-i)p_i = 12.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_57e6976665f4b54c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "topology", "subfield": "combinatorial topology", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_504590f16026ee8e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_DB1D004A0930", "name": "Cramer's rule", "statement_latex": "x_i = \\det(A_i) / \\det(A)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(x_i)=z\\;\\land\\;z=(\\det(A_i) / \\det(A))", "to_expression": "x_i = \\det(A_i) / \\det(A)", "justification": "common witness iff equality", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as x_i = \\\\det(A_i) / \\\\det(A).\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_1fed824937488500", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9DBE3057770F", "name": "Kneser's theorem", "statement_latex": "|A+B|=|A+H|+|B+H|-|H|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((|A+B|))=F((|A+H|+|B+H|-|H|))", "to_expression": "|A+B|=|A+H|+|B+H|-|H|", "justification": "indicator of {X} separates unequal objects", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |A+B|=|A+H|+|B+H|-|H|.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_e37a685c1f8d26fc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_95775DC75EC2", "name": "Oppenheim conjecture", "statement_latex": "|Q(x)| < \\epsilon", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(|Q(x)|)-(\\epsilon)\\}=0\\;\\land\\;(|Q(x)|)\\ne (\\epsilon)", "to_expression": "|Q(x)| < \\epsilon", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |Q(x)| < \\\\epsilon.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_5b74536588dc91de", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum field theory", "subfield": "axiomatic quantum field theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_FEF97AED413B", "name": "Cluster decomposition property", "statement_latex": "S_βα → S_β1α1 * S_β2α2", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(S_βα → S_β1α1 * S_β2α2) \\land (0 = 0)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "S_βα → S_β1α1 * S_β2α2", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "0=0 is true, so Phi_T and true iff Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_d8ff045cfcb20686", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "binomial coefficients", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_41972AC1B752", "name": "Rothe–Hagen identity", "statement_latex": "\\sum _{k=0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k}={\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n}", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((\\sum _{k=0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k}),z):z\\in D\\}=\\{(({\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n}),z):z\\in D\\}", "to_expression": "\\sum _{k=0}^{n}{\\frac {x}{x+kz}}{x+kz \\choose k}{\\frac {y}{y+(n-k)z}}{y+(n-k)z \\choose n-k}={\\frac {x+y}{x+y+nz}}{x+y+nz \\choose n}", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\sum _{k=0}^{n}{\\\\frac {x}{x+kz}}{x+kz \\\\choose k}{\\\\frac {y}{y+(n-k)z}}{y+(n-k)z \\\\choose n-k}={\\\\frac {x+y}{x+y+nz}}{x+y+nz \\\\choose n}.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_8911219443e82873", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_303E62405496", "name": "Lorden's inequality", "statement_latex": "E(R_b) <= E(X^2) / E(X)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(E(R_b))}{\\exp(E(R_b))+\\exp(E(X^2) / E(X))}\\le\\frac{1}{2}", "to_expression": "E(R_b) <= E(X^2) / E(X)", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as E(R_b) ≤ E(X^2) / E(X).\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_8fa39731f1229953", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "combinatorics on words", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a33f7a23541f91a2", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "extremal graph theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4D5DD853BD7", "name": "Szemerédi regularity lemma", "statement_latex": "|d(X,Y) - d(A,B)| <= ε", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (|d(X,Y) - d(A,B)|) \\le x \\land x \\le (ε)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |d(X,Y) - d(A,B)| <= ε.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |d(X,Y) - d(A,B)| <= ε.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "|d(X,Y) - d(A,B)| <= ε", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |d(X,Y) - d(A,B)| <= ε.\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
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{"schema_version": "1.0", "item_id": "rcd2_cb917cbac3c8a01d", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "linear algebra", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_08D175804776", "name": "Cayley–Hamilton theorem", "statement_latex": "p_A(A) = 0", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((p_A(A)),z):z\\in D\\}=\\{((0),z):z\\in D\\}", "to_expression": "p_A(A) = 0", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p_A(A) = 0.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_a2f359f7b29417ef", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "analysis", "subfield": "approximation theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_538DF6DD562D", "name": "Favard's theorem", "statement_latex": "y_{n+1}=(x-c_{n})y_{n}-d_{n}y_{n-1}", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((y_{n+1}) \\in S \\land ((x-c_{n})y_{n}-d_{n}y_{n-1}) \\notin S) \\lor (((x-c_{n})y_{n}-d_{n}y_{n-1}) \\in S \\land (y_{n+1}) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover y_{n+1}=(x-c_{n})y_{n}-d_{n}y_{n-1}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "y_{n+1}=(x-c_{n})y_{n}-d_{n}y_{n-1}", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover y_{n+1}=(x-c_{n})y_{n}-d_{n}y_{n-1}.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_0260013d7872f8dd", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "Mathematical analysis", "subfield": "Sequences and series", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F3C0470E95D9", "name": "Abel's inequality", "statement_latex": "|∑ a_k b_k| ≤ max|B_k|(|a_n| + |a_n - a_1|)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(|∑ a_k b_k|)}{\\exp(|∑ a_k b_k|)+\\exp(max|B_k|(|a_n| + |a_n - a_1|))}\\le\\frac{1}{2}", "to_expression": "|∑ a_k b_k| ≤ max|B_k|(|a_n| + |a_n - a_1|)", "justification": "logistic comparison is ≤1/2 iff X≤Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |∑ a_k b_k| ≤ max|B_k|(|a_n| + |a_n - a_1|).\"]"}], "source_equivalence_identity": "logistic comparison is ≤1/2 iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_e2dfeb6973ab8df7", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6f04ad8a4fa1b45d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "algebraic combinatorics", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c9e09f800d22c5b7", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum information theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_39d52fd2e3c1f09f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "symplectic geometry", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C62ED9627235", "name": "Williamson theorem", "statement_latex": "SMS^T = D ⊕ D", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(D ⊕ D) = (SMS^T)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover SMS^T = D ⊕ D.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "SMS^T = D ⊕ D", "justification": "R = L iff L = R by symmetry of equality.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover SMS^T = D ⊕ D.\"]"}], "source_equivalence_identity": "R = L iff L = R by symmetry of equality."}
{"schema_version": "1.0", "item_id": "rcd2_7a425253f5cee270", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum field theory", "subfield": "axiomatic quantum field theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e8e51414a5398874", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "Mathematical analysis", "subfield": "Sequences and series", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_af11668e8febf88a", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "discrepancy theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0b97786a44c8dc7a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "topology", "subfield": "differential topology", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2151776292B6", "name": "Thom's second isotopy lemma", "statement_latex": "ker(d(f|_Y)_y) subset tau", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "(ker(d(f|_Y)_y))\\cap(D\\setminus (tau))=\\emptyset", "to_expression": "ker(d(f|_Y)_y) subset tau", "justification": "no intersection with complement iff subset", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ker(d(f|_Y)_y) ⊆ tau.\"]"}], "source_equivalence_identity": "no intersection with complement iff subset"}
{"schema_version": "1.0", "item_id": "rcd2_5ed14c24a712c862", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "numerical analysis, finite element method", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7C7FF749E01A", "name": "Bramble–Hilbert lemma", "statement_latex": "inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)} <= C |u|_{W^{m,p}(\\Omega)}", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(C |u|_{W^{m,p}(\\Omega)}) - (\\inf_{v \\in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)}) \\ge 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\\\Omega)} <= C |u|_{W^{m,p}(\\\\Omega)}.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\\\Omega)} <= C |u|_{W^{m,p}(\\\\Omega)}.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)} <= C |u|_{W^{m,p}(\\Omega)}", "justification": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals.", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\\\Omega)} <= C |u|_{W^{m,p}(\\\\Omega)}.\"]"}], "source_equivalence_identity": "Substitute A and B; verify A <= B iff B-A >= 0 over finite reals."}
{"schema_version": "1.0", "item_id": "rcd2_f5874bb9cf0f0400", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "algebraic combinatorics", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B8C05F31E817", "name": "Dyson conjecture", "statement_latex": "constant_term(prod_{1<=i!=j<=n} (1-t_i/t_j)^{a_i}) = (sum a_i)! / prod(a_i!)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "(\\{(constant_term(prod_{1≤i≠j≤n} (1-t_i/t_j)^{a_i}))\\}\\times\\{(sum a_i)! / prod(a_i!)\\})\\cap\\{(z,z):z\\in D\\}=\\{((constant_term(prod_{1≤i≠j≤n} (1-t_i/t_j)^{a_i})),(sum a_i)! / prod(a_i!))\\}", "to_expression": "constant_term(prod_{1<=i!=j<=n} (1-t_i/t_j)^{a_i}) = (sum a_i)! / prod(a_i!)", "justification": "intersection is singleton iff pair diagonal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as constant_term(prod_{1≤i≠j≤n} (1-t_i/t_j)^{a_i}) = (sum a_i)! / prod(a_i!).\"]"}], "source_equivalence_identity": "intersection is singleton iff pair diagonal"}
{"schema_version": "1.0", "item_id": "rcd2_f37576059ed3476f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "elementary number theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_677EBF92D0D7", "name": "Lifting-the-exponent lemma", "statement_latex": "nu_p(x^n - y^n) = nu_p(x - y) + nu_p(n)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\inf_{z\\in D}(\\delta_D(z,(nu_p(x^n - y^n)))+\\delta_D(z,(nu_p(x - y) + nu_p(n))))=0", "to_expression": "nu_p(x^n - y^n) = nu_p(x - y) + nu_p(n)", "justification": "zero loss iff some z equals both", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as nu_p(x^n - y^n) = nu_p(x - y) + nu_p(n).\"]"}], "source_equivalence_identity": "zero loss iff some z equals both"}
{"schema_version": "1.0", "item_id": "rcd2_030f2109906efc1e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "elementary number theory", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0F6BD7E91884", "name": "Bézout's identity", "statement_latex": "ax + by = gcd(a, b)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(ax + by))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(gcd(a, b)))", "to_expression": "ax + by = gcd(a, b)", "justification": "each argmin set is a singleton", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ax + by = gcd(a, b).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_f0ab35299e26b591", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "topology", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3919D9203C0B", "name": "Cantor's intersection theorem", "statement_latex": "\\bigcap _{k=0}^{\\infty }C_{k}\\neq \\emptyset", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\operatorname{Id}_D[(\\bigcap _{k=0}^{\\infty }C_{k})]\\ne\\emptyset", "to_expression": "\\bigcap _{k=0}^{\\infty }C_{k}\\neq \\emptyset", "justification": "identity image equals S", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\bigcap _{k=0}^{\\\\infty }C_{k}≠ \\\\emptyset.\"]"}], "source_equivalence_identity": "identity image equals S"}
{"schema_version": "1.0", "item_id": "rcd2_4a3d719d57f786c7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "differential equations", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_891C2603FAE3", "name": "Grönwall's inequality", "statement_latex": "u(t) <= u(a) * exp(integral from a to t of beta(s) ds)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(u(t))+s=(u(a) * exp(integral from a to t of beta(s) ds))", "to_expression": "u(t) <= u(a) * exp(integral from a to t of beta(s) ds)", "justification": "nonnegative slack exists iff X≤Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as u(t) ≤ u(a) * exp(integral from a to t of beta(s) ds).\"]"}], "source_equivalence_identity": "nonnegative slack exists iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_c0d532e4ec1803f3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "information theory", "subfield": "probability theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8F3F3785BDA2", "name": "Gibbs' inequality", "statement_latex": "-Σ p_i log p_i <= -Σ p_i log q_i", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(-Σ p_i log p_i)-(-Σ p_i log q_i)\\}=0", "to_expression": "-Σ p_i log p_i <= -Σ p_i log q_i", "justification": "positive part vanishes iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as -Σ p_i log p_i ≤ -Σ p_i log q_i.\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_fc2dfca8940654f4", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "topology", "primary_mechanism": "distance", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3919D9203C0B", "name": "Cantor's intersection theorem", "statement_latex": "\\bigcap _{k=0}^{\\infty }C_{k}\\neq \\emptyset", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists c \\in D: (\\bigcap _{k) = c \\land (0}^{\\infty }C_{k}\\neq \\\\emptyset) = c", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A common c exists iff L and R are equal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\bigcap _{k=0}^{\\\\infty }C_{k}\\\\neq \\\\emptyset.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\bigcap _{k=0}^{\\infty }C_{k}\\neq \\emptyset", "justification": "A common c exists iff L and R are equal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\bigcap _{k=0}^{\\\\infty }C_{k}\\\\neq \\\\emptyset.\"]"}], "source_equivalence_identity": "A common c exists iff L and R are equal."}
{"schema_version": "1.0", "item_id": "rcd2_844445219902925b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "design theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_45A9C577BD36", "name": "Fisher's inequality", "statement_latex": "b >= v", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(v)-(b)\\}=0", "to_expression": "b >= v", "justification": "positive part vanishes iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as b ≥ v.\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_e5312cb8830b8634", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "queueing theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e2a138b91cf755e9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "statistics", "subfield": "empirical Bayes methods", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2C7BEBF4ECBC", "name": "Robbins lemma", "statement_latex": "E(Xf(X-1)) = λE(f(X))", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(E(Xf(X-1))))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(λE(f(X))))", "to_expression": "E(Xf(X-1)) = λE(f(X))", "justification": "each argmin set is a singleton", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as E(Xf(X-1)) = λE(f(X)).\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_d29e7e462a8ebffb", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_08a30bba0dad5ba0", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "Fourier analysis", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_6096afefdcfde387", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "computer science", "subfield": "formal language theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_505976359610bca9", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "combinatorics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2511f84106342be0", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "vector calculus", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2E42699EEB3D", "name": "Gradient theorem", "statement_latex": "∫_{γ} ∇φ(r) · dr = φ(q) - φ(p)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((∫_{γ} ∇φ(r) · dr),z):z\\in D\\}=\\{((φ(q) - φ(p)),z):z\\in D\\}", "to_expression": "∫_{γ} ∇φ(r) · dr = φ(q) - φ(p)", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ∫_{γ} ∇φ(r) · dr = φ(q) - φ(p).\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_79c5ad2f0be082cb", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F881715F08CD", "name": "Sylvester's determinant identity", "statement_latex": "det(A) = det(A_v^u) * det(A_tilde_v^u)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((det(A)) \\in S \\land (det(A_v^u) * det(A_tilde_v^u)) \\notin S) \\lor ((det(A_v^u) * det(A_tilde_v^u)) \\in S \\land (det(A)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover det(A) = det(A_v^u) * det(A_tilde_v^u).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "det(A) = det(A_v^u) * det(A_tilde_v^u)", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover det(A) = det(A_v^u) * det(A_tilde_v^u).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
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{"schema_version": "1.0", "item_id": "rcd2_389bf01ea0c097b6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "concentration inequalities", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_99B889CC4005", "name": "Hoeffding's lemma", "statement_latex": "E[exp(lambda*(X-E[X]))] <= exp(lambda^2*(b-a)^2/8)", "assumptions": "X is almost surely in [a,b]; lambda is real; expectations are finite."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "\\operatorname{dist}((E[exp(lambda*(X-E[X]))])-(exp(lambda^2*(b-a)^2/8)),(-\\infty,0])=0", "to_expression": "E[exp(lambda*(X-E[X]))] <= exp(lambda^2*(b-a)^2/8)", "justification": "zero distance to closed halfline iff X-Y≤0", "preconditions": "X is almost surely in [a,b]; lambda is real; expectations are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as E[exp(lambda*(X-E[X]))] ≤ exp(lambda^2*(b-a)^2/8).\"]"}], "source_equivalence_identity": "zero distance to closed halfline iff X-Y≤0"}
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{"schema_version": "1.0", "item_id": "rcd2_dbef72863679bbf3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "physics", "subfield": "general relativity", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_737BDAA0F832", "name": "Stewart–Walker lemma", "statement_latex": "ΔδT = 0", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(ΔδT)\\Longleftrightarrow z=(0))", "to_expression": "ΔδT = 0", "justification": "same equality fiber iff equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ΔδT = 0.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_61533a435ccb26cc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "control theory", "subfield": "geometric control theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2ADEC5E217BE", "name": "Krener's theorem", "statement_latex": "\\dot{q}=f(q,u)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\dot{q}) \\in {(f(q,u))}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\dot{q}=f(q,u).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\dot{q}=f(q,u)", "justification": "L in {R} iff L = R.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\dot{q}=f(q,u).\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_57870989e5abbfb1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "inclusion-exclusion principle", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_58BBE6141124", "name": "Schuette–Nesbitt formula", "statement_latex": "sum_{k=0}^m c_k 1_{N=k} = sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(sum_{k=0}^m c_k 1_{N=k})\\Longleftrightarrow z=(sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k))", "to_expression": "sum_{k=0}^m c_k 1_{N=k} = sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k", "justification": "same equality fiber iff equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{k=0}^m c_k 1_{N=k} = sum_{k=0}^m sum_{l=0}^k (-1)^{k-l} binomial(k,l) c_l N_k.\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
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{"schema_version": "1.0", "item_id": "rcd2_47028f0d3e750a6c", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "field theory", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4E2C0DED1A31", "name": "Joubert's theorem", "statement_latex": "p(t) = t^6 + c_4t^4 + c_2t^2 + c_1t + c_0", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "(\\{(p(t))\\}\\times\\{(t^6 + c_4t^4 + c_2t^2 + c_1t + c_0)\\})\\cap\\{(z,z):z\\in D\\}=\\{((p(t)),(t^6 + c_4t^4 + c_2t^2 + c_1t + c_0))\\}", "to_expression": "p(t) = t^6 + c_4t^4 + c_2t^2 + c_1t + c_0", "justification": "intersection is singleton iff pair diagonal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p(t) = t^6 + c_4t^4 + c_2t^2 + c_1t + c_0.\"]"}], "source_equivalence_identity": "intersection is singleton iff pair diagonal"}
{"schema_version": "1.0", "item_id": "rcd2_39b812435cf06b92", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "approximation theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F6B5A5657F99", "name": "Fejér's theorem", "statement_latex": "lim_{n→∞} σ_n(f,x) = f(x)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((lim_{n→∞} σ_n(f,x)),z)=((f(x)),z)", "to_expression": "lim_{n→∞} σ_n(f,x) = f(x)", "justification": "pair equality forces first coordinates equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{n→∞} σ_n(f,x) = f(x).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
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{"schema_version": "1.0", "item_id": "rcd2_67e7057310b53514", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "approximation theory", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F6B5A5657F99", "name": "Fejér's theorem", "statement_latex": "lim_{n→∞} σ_n(f,x) = f(x)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((lim_{n→∞} σ_n(f,x)),z):z\\in D\\}=\\{((f(x)),z):z\\in D\\}", "to_expression": "lim_{n→∞} σ_n(f,x) = f(x)", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as lim_{n→∞} σ_n(f,x) = f(x).\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_39283332258c7b87", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "representation theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_0b343c1bdc0eef30", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic calculus", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_D4D5944961D8", "name": "Itô's lemma", "statement_latex": "dX_t = mu_t dt + sigma_t dB_t", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((dX_t),(mu_t dt + sigma_t dB_t))=0", "to_expression": "dX_t = mu_t dt + sigma_t dB_t", "justification": "zero discrete distance iff equality", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as dX_t = mu_t dt + sigma_t dB_t.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_b869bbd5ffbeba84", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic calculus", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_9fee4834e9550c10", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "additive combinatorics", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9F50AE0DD0D5", "name": "Cauchy–Davenport theorem", "statement_latex": "|A+B| >= min(|A|+|B|-1, p)", "assumptions": "A and B are nonempty finite subsets of Z/pZ; p is prime."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda((min(|A|+|B|-1, p))-(|A+B|))=0", "to_expression": "|A+B| >= min(|A|+|B|-1, p)", "justification": "sup equals max(0,X-Y)", "preconditions": "A and B are nonempty finite subsets of Z/pZ; p is prime.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |A+B| ≥ min(|A|+|B|-1, p).\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
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{"schema_version": "1.0", "item_id": "rcd2_97607a763199c977", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_41098B279612", "name": "Girsanov theorem", "statement_latex": "E(X)_t = exp(X_t - 1/2 [X]_t)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((E(X)_t) \\in S \\land (\\exp(X_t - 1/2 [X]_t)) \\notin S) \\lor ((\\exp(X_t - 1/2 [X]_t)) \\in S \\land (E(X)_t) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E(X)_t = exp(X_t - 1/2 [X]_t).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "E(X)_t = exp(X_t - 1/2 [X]_t)", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E(X)_t = exp(X_t - 1/2 [X]_t).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
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{"schema_version": "1.0", "item_id": "rcd2_430e527ddd56e16e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "finite fields", "primary_mechanism": "monotone", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_24697C70978F", "name": "Chevalley–Warning theorem", "statement_latex": "n > sum_{j=1}^r d_j", "assumptions": "n is the number of variables; d_j are degrees of the polynomial equations over a finite field. This row encodes the standard degree-sum hypothesis."}, "reference_operations": ["REMOVE_MONOTONE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_MONOTONE_ENCODING", "from_expression": "\\frac{\\exp(sum_{j=1}^r d_j)}{\\exp(sum_{j=1}^r d_j)+\\exp(n)}<\\frac{1}{2}", "to_expression": "n > sum_{j=1}^r d_j", "justification": "equiv exp(X)<exp(Y)", "preconditions": "n is the number of variables; d_j are degrees of the polynomial equations over a finite field. This row encodes the standard degree-sum hypothesis.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as n > sum_{j=1}^r d_j.\"]"}], "source_equivalence_identity": "equiv exp(X)<exp(Y)"}
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{"schema_version": "1.0", "item_id": "rcd2_9dc666c313eea508", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "probabilistic number theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9D5ECA10E501", "name": "Turán–Kubilius inequality", "statement_latex": "(1/x) * sum_{n<=x} |f(n) - A(x)|^2 <= (2 + epsilon(x)) * B(x)^2", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_OPTIMIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_OPTIMIZATION_ENCODING", "from_expression": "\\sup_{\\lambda\\in[0,1]}\\lambda(((1/x) * sum_{n≤x} |f(n) - A(x)|^2)-((2 + epsilon(x)) * B(x)^2))=0", "to_expression": "(1/x) * sum_{n<=x} |f(n) - A(x)|^2 <= (2 + epsilon(x)) * B(x)^2", "justification": "sup equals max(0,X-Y)", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as (1/x) * sum_{n≤x} |f(n) - A(x)|^2 ≤ (2 + epsilon(x)) * B(x)^2.\"]"}], "source_equivalence_identity": "sup equals max(0,X-Y)"}
{"schema_version": "1.0", "item_id": "rcd2_1b3816d113c8d51a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "vector calculus", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_2E42699EEB3D", "name": "Gradient theorem", "statement_latex": "∫_{γ} ∇φ(r) · dr = φ(q) - φ(p)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((∫_{γ} ∇φ(r) · dr) \\in S \\land (φ(q) - φ(p)) \\notin S) \\lor ((φ(q) - φ(p)) \\in S \\land (∫_{γ} ∇φ(r) · dr) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover ∫_{γ} ∇φ(r) · dr = φ(q) - φ(p).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "∫_{γ} ∇φ(r) · dr = φ(q) - φ(p)", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover ∫_{γ} ∇φ(r) · dr = φ(q) - φ(p).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
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{"schema_version": "1.0", "item_id": "rcd2_75a760fa59a65b38", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "mathematics", "subfield": "probability theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F902B76211EA", "name": "Cantelli's inequality", "statement_latex": "Pr(X - E[X] ≥ λ) ≤ σ^2 / (σ^2 + λ^2)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (Pr(X - E[X] ≥ λ) ≤ σ^2 / (σ^2 + λ^2)))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "Pr(X - E[X] ≥ λ) ≤ σ^2 / (σ^2 + λ^2)", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_058f00032d0afb16", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C2E603A410E9", "name": "Uniform limit theorem", "statement_latex": "d_Y(f(x), f(y)) < ε", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\max\\{0,(d_Y(f(x), f(y)))-(ε)\\}=0\\;\\land\\;(d_Y(f(x), f(y)))\\ne (ε)", "to_expression": "d_Y(f(x), f(y)) < ε", "justification": "X≤Y and X≠Y iff X<Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as d_Y(f(x), f(y)) < ε.\"]"}], "source_equivalence_identity": "X≤Y and X≠Y iff X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_4bdd0d1dad6d5863", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "algebra", "subfield": "algebraic combinatorics", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_396E0535E013", "name": "n! conjecture", "statement_latex": "dim(H_μ) = n!", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(dim(H_μ))=z\\;\\land\\;z=(n!)", "to_expression": "dim(H_μ) = n!", "justification": "common witness iff equality", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as dim(H_μ) = n!.\"]"}], "source_equivalence_identity": "common witness iff equality"}
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{"schema_version": "1.0", "item_id": "rcd2_5c340196abb55fa9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "polynomials", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CC8118A54DAF", "name": "Cohn's theorem", "statement_latex": "p(z) = \\omega p^*(z)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((p(z)),(\\omega p^*(z)))\\in\\{(z,z):z\\in D\\}", "to_expression": "p(z) = \\omega p^*(z)", "justification": "pair in diagonal iff X=Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as p(z) = \\\\omega p^*(z).\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_04cf6fc8e142fa2e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "topology", "subfield": "algebraic topology", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_48CD393CFABB", "name": "Borsuk–Ulam theorem", "statement_latex": "f(-x) = f(x)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((f(-x)),(f(x)))\\in\\{(z,z):z\\in D\\}", "to_expression": "f(-x) = f(x)", "justification": "pair in diagonal iff X=Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f(-x) = f(x).\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
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{"schema_version": "1.0", "item_id": "rcd2_56e6cdff355af28a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal combinatorics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_C321BE7114E8", "name": "Ahlswede–Khachatrian theorem", "statement_latex": "|F| <= max_{r: t+2r <= n} |F_{n,k,t,r}|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (max_{r: t+2r ≤ n} |F_{n,k,t,r}|)\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (|F|)\\}", "to_expression": "|F| <= max_{r: t+2r <= n} |F_{n,k,t,r}|", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |F| ≤ max_{r: t+2r ≤ n} |F_{n,k,t,r}|.\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_6f4649aa2503e351", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_9EF5E48E9DAF", "name": "Rolle's theorem", "statement_latex": "f'(c) = 0", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((f'(c)))=F((0))", "to_expression": "f'(c) = 0", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as f'(c) = 0.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
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{"schema_version": "1.0", "item_id": "rcd2_41c7950cd2c472d8", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "computer science", "subfield": "formal language theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ba1e7976f0f3073b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "algebraic K-theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_582614A6491A", "name": "Whitehead's lemma", "statement_latex": "[[u, 0], [0, u^-1]] = e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "([[u, 0], [0, u^-1]]) \\in {(e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1))}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover [[u, 0], [0, u^-1]] = e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "[[u, 0], [0, u^-1]] = e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1)", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover [[u, 0], [0, u^-1]] = e_21(u^-1)e_12(1-u)e_21(-1)e_12(1-u^-1).\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_b355a74166fe4950", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematical analysis", "subfield": "number theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_0AAEED4B3B51", "name": "Goldbach–Euler theorem", "statement_latex": "sum_{p=2}^{\\infty} 1/(p-1) = 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((sum_{p=2}^{\\infty} 1/(p-1)))=F((1))", "to_expression": "sum_{p=2}^{\\infty} 1/(p-1) = 1", "justification": "indicator of {X} separates unequal objects", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{p=2}^{\\\\infty} 1/(p-1) = 1.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
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{"schema_version": "1.0", "item_id": "rcd2_ab93f78ae0f37c73", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "computer science", "subfield": "formal language theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_37D96EA4CFB7", "name": "Pumping lemma for regular languages", "statement_latex": "w = xyz", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((w) \\in S \\land (xyz) \\notin S) \\lor ((xyz) \\in S \\land (w) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover w = xyz.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "w = xyz", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover w = xyz.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_39510de5f32a1d11", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "geometry", "subfield": "incidence geometry", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B3E12B4FAA4C", "name": "Loomis–Whitney inequality", "statement_latex": "|| \\prod_{j=1}^{d} g_j \\circ \\pi_j ||_{L^1} \\le \\prod_{j=1}^{d} ||g_j||_{L^{d-1}}", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(|| \\prod_{j=1}^{d} g_j \\circ \\pi_j ||_{L^1})+s=(\\prod_{j=1}^{d} ||g_j||_{L^{d-1}})", "to_expression": "|| \\prod_{j=1}^{d} g_j \\circ \\pi_j ||_{L^1} \\le \\prod_{j=1}^{d} ||g_j||_{L^{d-1}}", "justification": "nonnegative slack exists iff X≤Y", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as || \\\\prod_{j=1}^{d} g_j \\\\circ \\\\pi_j ||_{L^1} ≤ \\\\prod_{j=1}^{d} ||g_j||_{L^{d-1}}.\"]"}], "source_equivalence_identity": "nonnegative slack exists iff X≤Y"}
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{"schema_version": "1.0", "item_id": "rcd2_6c40bb4c1a9f1184", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "mathematics", "subfield": "representation theory", "primary_mechanism": "pair", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CAF4E18AA4C8", "name": "Brauer's height zero conjecture", "statement_latex": "\\nu(\\chi(1)) = \\nu(|G:D|)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_PAIR_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_PAIR_ENCODING", "from_expression": "\\forall z\\in D,\\;((\\nu(\\chi(1))),z)=((\\nu(|G:D|)),z)", "to_expression": "\\nu(\\chi(1)) = \\nu(|G:D|)", "justification": "pair equality forces first coordinates equal", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\nu(\\\\chi(1)) = \\\\nu(|G:D|).\"]"}], "source_equivalence_identity": "pair equality forces first coordinates equal"}
{"schema_version": "1.0", "item_id": "rcd2_8bbaf18b24b401fb", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_A75F5656C745", "name": "Grosswald–Schnitzer theorem", "statement_latex": "\\zeta(s) = \\prod_{n=1}^{\\infty} \\frac{1}{1-p_n^{-s}}", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\neg(\\neg (\\z\\eta(s) = \\prod_{n=1}^{\\infty} \\frac{1}{1-p_n^{-s}}))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\zeta(s) = \\prod_{n=1}^{\\infty} \\frac{1}{1-p_n^{-s}}", "justification": "The truth table of not(not Phi_T) is identical to Phi_T.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "The truth table of not(not Phi_T) is identical to Phi_T."}
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{"schema_version": "1.0", "item_id": "rcd2_1a24d95e57d454c3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "topology", "subfield": "combinatorial topology", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_CB3E317B00C5", "name": "Tucker's lemma", "statement_latex": "L(-v) = -L(v)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(L(-v)) \\in {(-L(v))}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover L(-v) = -L(v).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "L(-v) = -L(v)", "justification": "L in {R} iff L = R.", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover L(-v) = -L(v).\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
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{"schema_version": "1.0", "item_id": "rcd2_5638e15c37bc243e", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "topology", "subfield": "combinatorial topology", "primary_mechanism": "distance", "transformation_category": "residual_nonnegativity", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_5c0070a906997f7f", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "extremal set theory", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_03FC7B5F26C8", "name": "Erdős–Ko–Rado theorem", "statement_latex": "|A| <= binom(n-1,r-1)", "assumptions": "A is an intersecting family of r-subsets of an n-element set; n>=2r."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(|A|)-(binom(n-1,r-1))\\}=0", "to_expression": "|A| <= binom(n-1,r-1)", "justification": "positive part vanishes iff X≤Y", "preconditions": "A is an intersecting family of r-subsets of an n-element set; n>=2r.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as |A| ≤ binom(n-1,r-1).\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_46cf7c2061e979cd", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "cryptography", "subfield": "cryptanalysis", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E35608C3DE5E", "name": "Piling-up lemma", "statement_latex": "ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn) = 2^{n-1}∏_{i=1}^{n}ε(Xi)", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn)),z):z\\in D\\}=\\{((2^{n-1}∏_{i=1}^{n}ε(Xi)),z):z\\in D\\}", "to_expression": "ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn) = 2^{n-1}∏_{i=1}^{n}ε(Xi)", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as ε(X1 ⊕ X2 ⊕ ⋯ ⊕ Xn) = 2^{n-1}∏_{i=1}^{n}ε(Xi).\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
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{"schema_version": "1.0", "item_id": "rcd2_976bf589ef5ed4ec", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "analysis", "subfield": "moment problem", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_b11b054e51414f55", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "numerical analysis, finite element method", "primary_mechanism": "quantifier", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7C7FF749E01A", "name": "Bramble–Hilbert lemma", "statement_latex": "inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)} <= C |u|_{W^{m,p}(\\Omega)}", "assumptions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite."}, "reference_operations": ["REMOVE_QUANTIFIER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_QUANTIFIER_ENCODING", "from_expression": "\\forall t\\in\\mathbb{R},\\;(C |u|_{W^{m,p}(\\Omega)})\\le t\\Rightarrow (inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)})\\le t", "to_expression": "inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\Omega)} <= C |u|_{W^{m,p}(\\Omega)}", "justification": "converse uses t=Y", "preconditions": "Objects lie in the stated normed or inner-product space; all norms and quantities are finite.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as inf_{v in P_{m-1}} ||u - v||_{W^{k,p}(\\\\Omega)} ≤ C |u|_{W^{m,p}(\\\\Omega)}.\"]"}], "source_equivalence_identity": "converse uses t=Y"}
{"schema_version": "1.0", "item_id": "rcd2_6168ecbd91f0f1fc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "finite fields", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_24697C70978F", "name": "Chevalley–Warning theorem", "statement_latex": "n > sum_{j=1}^r d_j", "assumptions": "n is the number of variables; d_j are degrees of the polynomial equations over a finite field. This row encodes the standard degree-sum hypothesis."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_DISTANCE_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(n) - (sum_{j=1}^r d_j) > 0", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "n is the number of variables; d_j are degrees of the polynomial equations over a finite field. This row encodes the standard degree-sum hypothesis.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover n > sum_{j=1}^r d_j.\"]"}, {"step_no": 2, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "n is the number of variables; d_j are degrees of the polynomial equations over a finite field. This row encodes the standard degree-sum hypothesis.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover n > sum_{j=1}^r d_j.\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_DISTANCE_ENCODING", "to_expression": "n > sum_{j=1}^r d_j", "justification": "A<B iff B-A>0 over finite real quantities.", "preconditions": "n is the number of variables; d_j are degrees of the polynomial equations over a finite field. This row encodes the standard degree-sum hypothesis.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A < B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover n > sum_{j=1}^r d_j.\"]"}], "source_equivalence_identity": "A<B iff B-A>0 over finite real quantities."}
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{"schema_version": "1.0", "item_id": "rcd2_b248c88c5cb80471", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "statistics", "subfield": "statistical hypothesis testing", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_df7c1f727954f4f9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic processes", "primary_mechanism": "intersection", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7C6C20BC69B4", "name": "Slepian's lemma", "statement_latex": "Pr[Intersection(X_i <= u_i)] <= Pr[Intersection(Y_i <= u_i)]", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_INTERSECTION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_INTERSECTION_ENCODING", "from_expression": "\\tanh(Pr[Intersection(X_i ≤ u_i)])\\le\\tanh(Pr[Intersection(Y_i ≤ u_i)])", "to_expression": "Pr[Intersection(X_i <= u_i)] <= Pr[Intersection(Y_i <= u_i)]", "justification": "tanh preserves and reflects order", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as Pr[Intersection(X_i ≤ u_i)] ≤ Pr[Intersection(Y_i ≤ u_i)].\"]"}], "source_equivalence_identity": "tanh preserves and reflects order"}
{"schema_version": "1.0", "item_id": "rcd2_eed330b9e7e150e8", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "symmetric functions", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_25b6f27627b59b8b", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "permutation patterns", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_2aa899f6f232d1f6", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum mechanics", "subfield": "", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_AD101C6FC364", "name": "Adiabatic theorem", "statement_latex": "E = nhv", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "\\forall F:D\\to\\{0,1\\},\\;F((E))=F((nhv))", "to_expression": "E = nhv", "justification": "indicator of {X} separates unequal objects", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as E = nhv.\"]"}], "source_equivalence_identity": "indicator of {X} separates unequal objects"}
{"schema_version": "1.0", "item_id": "rcd2_d510140f6050271d", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "polynomial ring theory", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_4a3defaaa91c5d3f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "polynomial factorization", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_a28f16dde5ea01ac", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "matching theory", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_69DD0D246726", "name": "Tutte–Berge formula", "statement_latex": "max_matching_size = 1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(max_matching_size)\\Longleftrightarrow z=(1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)))", "to_expression": "max_matching_size = 1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|)", "justification": "same equality fiber iff equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as max_matching_size = 1/2 * min_{U⊆V} (|U| - odd(G-U) + |V|).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_02cb61a7b2d6be95", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "mathematics", "subfield": "graph theory", "primary_mechanism": "optimization", "transformation_category": "optimization_projection", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_3fc0fa5bcec0d44a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "combinatorics", "subfield": "algebraic combinatorics", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_B8C05F31E817", "name": "Dyson conjecture", "statement_latex": "constant_term(prod_{1<=i!=j<=n} (1-t_i/t_j)^{a_i}) = (sum a_i)! / prod(a_i!)", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(constant_term(prod_{1≤i≠j≤n} (1-t_i/t_j)^{a_i}))\\Longleftrightarrow z=(sum a_i)! / prod(a_i!))", "to_expression": "constant_term(prod_{1<=i!=j<=n} (1-t_i/t_j)^{a_i}) = (sum a_i)! / prod(a_i!)", "justification": "same equality fiber iff equal", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as constant_term(prod_{1≤i≠j≤n} (1-t_i/t_j)^{a_i}) = (sum a_i)! / prod(a_i!).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_e3e8d953eb986866", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "real analysis", "primary_mechanism": "witness", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_368EF3E8339F", "name": "Summation by parts", "statement_latex": "sum_{k=m}^{n} f_k Delta g_k = (f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} Delta f_k", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists z\\in D\\;\\text{such that}\\;(sum_{k=m}^{n} f_k Delta g_k)=z\\;\\land\\;z=((f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} Delta f_k)", "to_expression": "sum_{k=m}^{n} f_k Delta g_k = (f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} Delta f_k", "justification": "common witness iff equality", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as sum_{k=m}^{n} f_k Delta g_k = (f_{n+1}g_{n+1} - f_m g_m) - sum_{k=m}^{n} g_{k+1} Delta f_k.\"]"}], "source_equivalence_identity": "common witness iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_d11873bf061b0050", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "functional analysis", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_ef041c4ee5b66a6a", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Combinatorics / Graph Theory", "field": "graph theory", "subfield": "extremal graph theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_F0F061CBC3F5", "name": "Blow-up lemma", "statement_latex": "e(X,Y) > delta |X| |Y|", "assumptions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((delta |X| |Y|),(e(X,Y)))\\in\\{(u,v)\\in\\mathbb{R}^2:u<v\\}", "to_expression": "e(X,Y) > delta |X| |Y|", "justification": "membership is exactly X<Y", "preconditions": "All sets, graphs, or matrices are finite and satisfy the hypotheses named in the theorem statement.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as e(X,Y) > delta |X| |Y|.\"]"}], "source_equivalence_identity": "membership is exactly X<Y"}
{"schema_version": "1.0", "item_id": "rcd2_2f5248f6a6c67f47", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "calculus", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_7ec763f912a6588b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "percolation theory", "primary_mechanism": "witness", "transformation_category": "residual_nonnegativity", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4AB97BAC2783", "name": "Van den Berg–Kesten inequality", "statement_latex": "P(A square B) <= P(A)P(B)", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_WITNESS_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_WITNESS_ENCODING", "from_expression": "\\exists s\\in\\mathbb{R}_{\\ge0}\\;\\text{such that}\\;(P(A square B))+s=(P(A)P(B))", "to_expression": "P(A square B) <= P(A)P(B)", "justification": "nonnegative slack exists iff X≤Y", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(A square B) ≤ P(A)P(B).\"]"}], "source_equivalence_identity": "nonnegative slack exists iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_e18e9d5f5232faea", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematics", "subfield": "vector calculus", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_78a9be015a7def2b", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "physics", "subfield": "quantum optics", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_BFC6929833DF", "name": "Optical equivalence theorem", "statement_latex": "\\operatorname{tr}(\\hat{\\rho} \\cdot g_{\\Omega}(\\hat{a}, \\hat{a}^{\\dagger})) = \\int f_{\\bar{\\Omega}}(\\alpha, \\alpha^*) g_{\\Omega}(\\alpha, \\alpha^*) d^2\\alpha", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_DISTANCE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_DISTANCE_ENCODING", "from_expression": "\\delta_D((\\operatorname{tr}(\\hat{\\rho} \\cdot g_{\\Omega}(\\hat{a}, \\hat{a}^{\\dagger}))),(\\int f_{\\bar{\\Omega}}(\\alpha, \\alpha^*) g_{\\Omega}(\\alpha, \\alpha^*) d^2\\alpha))=0", "to_expression": "\\operatorname{tr}(\\hat{\\rho} \\cdot g_{\\Omega}(\\hat{a}, \\hat{a}^{\\dagger})) = \\int f_{\\bar{\\Omega}}(\\alpha, \\alpha^*) g_{\\Omega}(\\alpha, \\alpha^*) d^2\\alpha", "justification": "zero discrete distance iff equality", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as \\\\operatorname{tr}(\\\\hat{\\\\rho} \\\\cdot g_{\\\\Omega}(\\\\hat{a}, \\\\hat{a}^{\\\\dagger})) = \\\\int f_{\\\\bar{\\\\Omega}}(\\\\alpha, \\\\alpha^*) g_{\\\\Omega}(\\\\alpha, \\\\alpha^*) d^2\\\\alpha.\"]"}], "source_equivalence_identity": "zero discrete distance iff equality"}
{"schema_version": "1.0", "item_id": "rcd2_fc82e0aa2f1eb398", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "combinatorics", "primary_mechanism": "cylinder", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3596847807A0", "name": "Coupon collector's problem", "statement_latex": "E(T) = n H_n", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_CYLINDER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CYLINDER_ENCODING", "from_expression": "\\{((E(T)),z):z\\in D\\}=\\{((n H_n),z):z\\in D\\}", "to_expression": "E(T) = n H_n", "justification": "fixed-coordinate cylinders equal iff fixed elements equal", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as E(T) = n H_n.\"]"}], "source_equivalence_identity": "fixed-coordinate cylinders equal iff fixed elements equal"}
{"schema_version": "1.0", "item_id": "rcd2_4571d7b8668a7bcc", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "symmetric functions", "primary_mechanism": "truth_table", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_4E5DD778CBB5", "name": "Newton's identities", "statement_latex": "k e_k = \\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_TRUTH_TABLE_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(k e_k = \\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i) \\land (0 = 0)", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}, {"step_no": 2, "operation_id": "REMOVE_TRUTH_TABLE_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "k e_k = \\sum_{i=1}^k (-1)^{i-1} e_{k-i} p_i", "justification": "0=0 is true, so Phi_T and true iff Phi_T.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Read Phi_T as the canonical theorem condition displayed in original_formula.\",\"Step 2: Apply the stated truth-table, singleton, indicator, or optimization equivalence to recover Phi_T.\",\"Step 3: Substitute the definition of Phi_T to recover the original formula exactly.\"]"}], "source_equivalence_identity": "0=0 is true, so Phi_T and true iff Phi_T."}
{"schema_version": "1.0", "item_id": "rcd2_c116ba2177702021", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_763D96A1B9F8", "name": "Rouché–Capelli theorem", "statement_latex": "rank(A) = rank([A|b])", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((rank(A)) \\in S \\land (rank([A|b])) \\notin S) \\lor ((rank([A|b])) \\in S \\land (rank(A)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover rank(A) = rank([A|b]).\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "rank(A) = rank([A|b])", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover rank(A) = rank([A|b]).\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_5beaf2b2455ca557", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "analytic number theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_736807BC5C8F", "name": "Brun–Titchmarsh theorem", "statement_latex": "pi(x;q,a) <= 2x / (phi(q) log(x/q))", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "\\{t\\in\\mathbb{R}:t\\ge (2x / (phi(q) log(x/q)))\\}\\subseteq\\{t\\in\\mathbb{R}:t\\ge (pi(x;q,a))\\}", "to_expression": "pi(x;q,a) <= 2x / (phi(q) log(x/q))", "justification": "upper-ray inclusion iff X≤Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as pi(x;q,a) ≤ 2x / (phi(q) log(x/q)).\"]"}], "source_equivalence_identity": "upper-ray inclusion iff X≤Y"}
{"schema_version": "1.0", "item_id": "rcd2_499576d936d45b2e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "discrete geometry", "subfield": "geometry", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_E02FAD9C277D", "name": "Erdős–Anning theorem", "statement_latex": "|d(A,X) - d(B,X)| <= d(A,B)", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING", "DENORMALIZE_RELATION"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\exists x \\\\in \\mathbb{R}: (|d(A,X) - d(B,X)|) \\le x \\land x \\le (d(A,B))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |d(A,X) - d(B,X)| <= d(A,B).\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |d(A,X) - d(B,X)| <= d(A,B).\"]"}, {"step_no": 3, "operation_id": "DENORMALIZE_RELATION", "from_expression": "intermediate_after_2_REMOVE_SET_MEMBERSHIP_ENCODING", "to_expression": "|d(A,X) - d(B,X)| <= d(A,B)", "justification": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B.", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: Read the introduced symbols A and B and substitute their displayed definitions.\",\"Step 2: Convert the variant condition back to the normalized order A <= B using the stated equivalence.\",\"Step 3: Undo the normalization of A and B to recover |d(A,X) - d(B,X)| <= d(A,B).\"]"}], "source_equivalence_identity": "[A,B] is nonempty iff there exists x with A<=x<=B, which holds iff A<=B."}
{"schema_version": "1.0", "item_id": "rcd2_60ad6aa5a8354011", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "stochastic calculus", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "TYPE_MATCHED_RHS_SWAP", "corruption_location": "right_hand_side", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_47a59571487def7c", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e48c0330068556b1", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine equations", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_EAE642263ABE", "name": "Catalan's conjecture", "statement_latex": "x^a - y^b = 1", "assumptions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((x^a - y^b),(1))\\in\\{(z,z):z\\in D\\}", "to_expression": "x^a - y^b = 1", "justification": "pair in diagonal iff X=Y", "preconditions": "All displayed arithmetic quantities satisfy the integer and positivity hypotheses of the theorem.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as x^a - y^b = 1.\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_4f03587b8b266105", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Number Theory", "field": "number theory", "subfield": "Diophantine approximation", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "ROUTE_RELATION_DIRECTION_CORRUPTION", "corruption_location": "relation_operator", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_af28ee9857cf2819", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "probabilistic combinatorics", "primary_mechanism": "argmin", "transformation_category": "optimization_projection", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_7C9C39383770", "name": "Ajtai–Komlós–Tusnády theorem", "statement_latex": "P(C_1 n log n < inf_{σ ∈ S_n} sum_{k=1}^n |X_k - Y_{σ(k)}| < C_2 n log n) = 1", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["REMOVE_ARGMIN_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_ARGMIN_ENCODING", "from_expression": "\\operatorname{argmin}_{z\\in D}\\delta_D(z,(P(C_1 n log n < inf_{σ ∈ S_n} sum_{k=1}^n |X_k - Y_{σ(k)}| < C_2 n log n)))=\\operatorname{argmin}_{z\\in D}\\delta_D(z,(1))", "to_expression": "P(C_1 n log n < inf_{σ ∈ S_n} sum_{k=1}^n |X_k - Y_{σ(k)}| < C_2 n log n) = 1", "justification": "each argmin set is a singleton", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as P(C_1 n log n < inf_{σ ∈ S_n} sum_{k=1}^n |X_k - Y_{σ(k)}| < C_2 n log n) = 1.\"]"}], "source_equivalence_identity": "each argmin set is a singleton"}
{"schema_version": "1.0", "item_id": "rcd2_74bd43ca5b47a3b9", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "field theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_6E9382B37B92", "name": "Primitive element theorem", "statement_latex": "E = F(α)", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "((E),(F(α)))\\in\\{(z,z):z\\in D\\}", "to_expression": "E = F(α)", "justification": "pair in diagonal iff X=Y", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as E = F(α).\"]"}], "source_equivalence_identity": "pair in diagonal iff X=Y"}
{"schema_version": "1.0", "item_id": "rcd2_8ffbea40a32a672e", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "combinatorics", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3596847807A0", "name": "Coupon collector's problem", "statement_latex": "E(T) = n H_n", "assumptions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_CONTAINMENT_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "\\nexists S \\subseteq D: (((E(T)) \\in S \\land (n H_n) \\notin S) \\lor ((n H_n) \\in S \\land (E(T)) \\notin S))", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E(T) = n H_n.\"]"}, {"step_no": 2, "operation_id": "REMOVE_CONTAINMENT_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "E(T) = n H_n", "justification": "A separating subset exists iff L and R are unequal.", "preconditions": "All random variables/events/distributions satisfy the theorem hypotheses; probabilities and expectations are finite where displayed.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover E(T) = n H_n.\"]"}], "source_equivalence_identity": "A separating subset exists iff L and R are unequal."}
{"schema_version": "1.0", "item_id": "rcd2_71da5cf19875ae02", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "quantum field theory", "subfield": "lattice field theory", "primary_mechanism": "containment", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_c4d43ed973b4e1d7", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "computer science", "subfield": "formal language theory", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "LOGICAL_NEGATION_SAME_ROUTE", "corruption_location": "surface_formula", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_07666dd646689509", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "linear algebra", "subfield": "matrix theory", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_79DC6C43DD64", "name": "Perron–Frobenius theorem", "statement_latex": "Av = rv", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(Av) \\in {(rv)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover Av = rv.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "Av = rv", "justification": "L in {R} iff L = R.", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover Av = rv.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_7ef1fb1028768aa7", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Analysis", "field": "mathematical analysis", "subfield": "topology", "primary_mechanism": "set_membership", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_3919D9203C0B", "name": "Cantor's intersection theorem", "statement_latex": "\\bigcap _{k=0}^{\\infty }C_{k}\\neq \\emptyset", "assumptions": "All displayed quantities are finite real-valued under the theorem hypotheses."}, "reference_operations": ["EXPAND_ABBREVIATIONS", "REMOVE_SET_MEMBERSHIP_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "EXPAND_ABBREVIATIONS", "from_expression": "(\\bigcap _{k) \\in {(0}^{\\infty }C_{k}\\neq \\\\emptyset)}", "to_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\bigcap _{k=0}^{\\\\infty }C_{k}\\\\neq \\\\emptyset.\"]"}, {"step_no": 2, "operation_id": "REMOVE_SET_MEMBERSHIP_ENCODING", "from_expression": "intermediate_after_1_EXPAND_ABBREVIATIONS", "to_expression": "\\bigcap _{k=0}^{\\infty }C_{k}\\neq \\emptyset", "justification": "L in {R} iff L = R.", "preconditions": "All displayed quantities are finite real-valued under the theorem hypotheses.", "inverse_mapping_note": "[\"Step 1: Substitute the displayed definitions of L and R.\",\"Step 2: Use the stated equality, singleton, diagonal, separator, or discrete-metric characterization to derive L = R.\",\"Step 3: Replace L and R by their definitions to recover \\\\bigcap _{k=0}^{\\\\infty }C_{k}\\\\neq \\\\emptyset.\"]"}], "source_equivalence_identity": "L in {R} iff L = R."}
{"schema_version": "1.0", "item_id": "rcd2_ec003c30ecfe7d6f", "item_role": "constructed_contrast", "original_experiment_label": "negative", "annotation_basis": "construction_recipe", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "algebra", "subfield": "commutative algebra", "primary_mechanism": "distance", "transformation_category": "proof_intermediate", "contrast_recipe": "SYMBOLIC_ENDPOINT_PERTURBATION", "corruption_location": "endpoint_expression", "reference_theorem": null, "reference_operations": [], "reference_route": [], "source_equivalence_identity": null}
{"schema_version": "1.0", "item_id": "rcd2_e81b7cf12baffec3", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Algebra / Linear Algebra", "field": "mathematics", "subfield": "linear algebra", "primary_mechanism": "fiber", "transformation_category": "proof_intermediate", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_763D96A1B9F8", "name": "Rouché–Capelli theorem", "statement_latex": "rank(A) = rank([A|b])", "assumptions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered."}, "reference_operations": ["REMOVE_FIBER_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_FIBER_ENCODING", "from_expression": "\\forall z\\in D,\\;(z=(rank(A))\\Longleftrightarrow z=(rank([A|b])))", "to_expression": "rank(A) = rank([A|b])", "justification": "same equality fiber iff equal", "preconditions": "All matrices, vector spaces, fields, or algebraic objects satisfy the hypotheses named in the theorem statement; displayed scalar quantities are finite real values when ordered.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as rank(A) = rank([A|b]).\"]"}], "source_equivalence_identity": "same equality fiber iff equal"}
{"schema_version": "1.0", "item_id": "rcd2_f04a2c4233fe0178", "item_role": "source_reformulation", "original_experiment_label": "positive", "annotation_basis": "source_metadata", "semantic_certification": "not_independently_established", "family": "Probability / Information Theory", "field": "probability theory", "subfield": "probabilistic method", "primary_mechanism": "characterization", "transformation_category": "characterization_iff", "contrast_recipe": null, "corruption_location": null, "reference_theorem": {"theorem_id": "THM_8F1DAFA4A16E", "name": "Lovász local lemma", "statement_latex": "e*p*(d+1) <= 1", "assumptions": "Symmetric Lovasz local lemma setting; each bad event has probability at most p and dependency degree at most d."}, "reference_operations": ["REMOVE_CHARACTERIZATION_ENCODING"], "reference_route": [{"step_no": 1, "operation_id": "REMOVE_CHARACTERIZATION_ENCODING", "from_expression": "\\max\\{0,(e*p*(d+1))-(1)\\}=0", "to_expression": "e*p*(d+1) <= 1", "justification": "positive part vanishes iff X≤Y", "preconditions": "Symmetric Lovasz local lemma setting; each bad event has probability at most p and dependency degree at most d.", "inverse_mapping_note": "[\"Step 1: identify the encoding family in the variant.\", \"Step 2: apply the stated validation identity to recover the direct source condition.\", \"Step 3: rewrite it as e*p*(d+1) ≤ 1.\"]"}], "source_equivalence_identity": "positive part vanishes iff X≤Y"}
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