Blame for THEOREM_INDEX.json

{
  "schema_version": "1.0.0",
  "title": "AELYRAITH III: The Nonlinear Continuation Theorem",
  "index_status": "Navigation derived from the supplied release; not an independent proof or novelty certification.",
  "authoritative_source": "manuscript/aelyraith3.tex",
  "pdf": "manuscript/AELYRAITH_III_Nonlinear_Continuation_Theorem.pdf",
  "implementation": "src/continuation.py",
  "global_hypotheses": [
    "States range over the full probability simplex in R^n; n >= 2.",
    "A finite known library contains strictly positive real n by n branch matrices.",
    "Branches act by positive normalization T*x/(e*T*x).",
    "Protected readouts are beta*x/(alpha*x), with every alpha entry strictly positive.",
    "The preserved contract is conditional readout equality for every finite supplied branch word, including the empty word.",
    "Stochastic normalization, when required, is an extra condition per action."
  ],
  "word_convention": {
    "branch_indices": "zero-based",
    "words": "chronological",
    "state_product": "T_aL * ... * T_a1",
    "row_evaluation": "multiply row by matrices in reverse chronological order",
    "explicit_family_labels": {
      "0": "A",
      "1": "B"
    }
  },
  "results": [
    {
      "id": "thm:exterior",
      "title": "Finite nonlinear continuation theorem",
      "kind": "theorem",
      "section": "2",
      "number": "2.1",
      "tex_label": "thm:exterior",
      "equation_labels": [
        "eq:wedge",
        "eq:compound",
        "eq:closure",
        "eq:signature"
      ],
      "conclusion": "Finite exterior orbit closure exactly characterizes complete conditional behavior. The selected ratio signature defines a coarsest sufficient information partition and admits representative-independent branch updates.",
      "additional_hypotheses": [],
      "implementation_entrypoints": [
        "compile_exterior",
        "compile_ratios",
        "ExteriorCertificate.verify",
        "ExteriorCertificate.signature_exact"
      ],
      "related_tests": [
        "AlgebraTests",
        "CertificateIntegrityTests",
        "FamilyTests.test_all_words_six"
      ],
      "status": "written argument in supplied manuscript; see implementation fields for executable support",
      "boundaries": [
        "The signature coordinate count is not its intrinsic dimension or a bit count.",
        "Certificate completeness requires seed coverage, genuine provenance, and transport, not transport alone.",
        "No equality of label distributions is implied."
      ]
    },
    {
      "id": "prop:fiber",
      "title": "Exact fiber geometry",
      "kind": "proposition",
      "section": "3",
      "number": "3.1",
      "tex_label": "prop:fiber",
      "equation_labels": [
        "eq:L",
        "eq:fiber"
      ],
      "conclusion": "Every feasible signature fiber equals the simplex intersected with the affine equations L(z)*x=0 and is compact and convex; any representative gives the same exact signature update.",
      "additional_hypotheses": [],
      "implementation_entrypoints": [
        "ExteriorCertificate.fiber_matrix",
        "ExteriorCertificate.reconstruct_fiber_float",
        "ExteriorCertificate.update_float"
      ],
      "related_tests": [
        "FamilyTests.test_lp_representative_update"
      ],
      "status": "written argument in supplied manuscript; see implementation fields for executable support",
      "boundaries": [
        "The delivered representative solver is floating-point linear programming, not an exact feasibility certificate.",
        "A representative need not be the true hidden state.",
        "Temporary workspace can remain n-dimensional."
      ]
    },
    {
      "id": "thm:dimension",
      "title": "Exact local continuous dimension",
      "kind": "theorem",
      "section": "3",
      "number": "3.2",
      "tex_label": "thm:dimension",
      "equation_labels": [
        "eq:rank"
      ],
      "conclusion": "At each interior regular point, generic signature rank r equals rank L(sigma(x)); r smooth coordinates suffice locally and every continuous exact encoder needs at least r local coordinates.",
      "additional_hypotheses": [
        "Interior regular point of the rational signature; r is its maximum interior rank."
      ],
      "implementation_entrypoints": [
        "jacobian_rank",
        "ExteriorCertificate.fiber_matrix"
      ],
      "related_tests": [
        "FamilyTests.test_local_rank_two",
        "RewriteTests.test_new_local_rank_full"
      ],
      "status": "written argument in supplied manuscript; see implementation fields for executable support",
      "boundaries": [
        "The generic theorem is local; a globally minimal chart is not asserted.",
        "jacobian_rank evaluates one state and does not by itself prove regularity or generic maximal rank.",
        "Discontinuous real encodings are not bounded by this continuous dimension statement."
      ]
    },
    {
      "id": "lem:ratio",
      "title": "Affine information required by a nonconstant ratio",
      "kind": "lemma",
      "section": "4",
      "number": "4.1",
      "tex_label": "lem:ratio",
      "equation_labels": [],
      "conclusion": "Any fixed affine row space containing e and permitting arbitrary decoding of a nonconstant positive-denominator ratio must contain both its numerator and denominator.",
      "additional_hypotheses": [
        "Nonconstant ratio on the full simplex; affine encoding row span contains e."
      ],
      "implementation_entrypoints": [],
      "related_tests": [],
      "status": "written argument in supplied manuscript; see implementation fields for executable support",
      "boundaries": [
        "An initialization statement about fixed affine information; it does not impose recurrence."
      ]
    },
    {
      "id": "thm:two",
      "title": "Sharp affine-snapshot law",
      "kind": "theorem",
      "section": "4",
      "number": "4.2",
      "tex_label": "thm:two",
      "equation_labels": [
        "eq:S",
        "eq:factor2"
      ],
      "conclusion": "The minimum affine initial-snapshot dimension is s = dim span(e,a_i,b_i)-1. With generic nonlinear dimension r, r <= s <= 2*r, and both endpoints are attained.",
      "additional_hypotheses": [],
      "implementation_entrypoints": [
        "snapshot_space",
        "factor_two_fixture"
      ],
      "related_tests": [
        "SnapshotTests.test_factor_two_sharp",
        "SnapshotTests.test_family_snapshot_three",
        "SnapshotTests.test_snapshot_initializes_compact"
      ],
      "status": "written argument in supplied manuscript; see implementation fields for executable support",
      "boundaries": [
        "Snapshot coordinates need not update from themselves.",
        "The theorem rules out an unbounded nonlinear scalar advantage for affine initialization under this contract."
      ]
    },
    {
      "id": "thm:family",
      "title": "Unbounded recurrent affine gap",
      "kind": "theorem",
      "section": "5",
      "number": "5.1",
      "tex_label": "thm:family",
      "equation_labels": [
        "eq:params",
        "eq:LA",
        "eq:LB",
        "eq:physical",
        "eq:compact",
        "eq:updates"
      ],
      "conclusion": "An explicit strictly positive instrument has branch probabilities in [3/8,5/8], minimum continuous nonlinear memory 2, minimum affine snapshot 3, and minimum fixed affine recurrent memory n-1.",
      "additional_hypotheses": [
        "n is a power of two and n >= 4.",
        "Explicit Hadamard construction; c=1/2, tau=1/4, gamma=kappa=1/8, lambda=1/(8*n); delta=0."
      ],
      "implementation_entrypoints": [
        "hadamard_exact",
        "separation_family",
        "Family.signature",
        "Family.compact_step",
        "affine_closure"
      ],
      "related_tests": [
        "FamilyTests",
        "CertificateIntegrityTests.test_mode_aware_same_evidence_control"
      ],
      "status": "written argument in supplied manuscript; see implementation fields for executable support",
      "boundaries": [
        "The affine recurrence lower bound applies to a fixed affine encoder on the entire state base.",
        "A mode-aware conventional controller matches outputs with three initial affine values and subsequently at most one scalar plus a mode flag.",
        "No superiority over all conventional controllers is established."
      ]
    },
    {
      "id": "thm:rewrite",
      "title": "Nonlinear memory dimension transition",
      "kind": "theorem",
      "section": "6",
      "number": "6.1",
      "tex_label": "thm:rewrite",
      "equation_labels": [
        "eq:perturb",
        "eq:inverse",
        "eq:recover"
      ],
      "conclusion": "An arbitrarily small positivity-preserving rank-one rewrite keeps branch probability functions and individual/aggregate characteristic polynomials unchanged, but makes complete conditional behavior injective and raises minimum continuous exact memory to n-1.",
      "additional_hypotheses": [
        "Explicit Hadamard family with 0 < delta < 1/8."
      ],
      "implementation_entrypoints": [
        "separation_family",
        "reconstruct_hidden_from_B_readouts"
      ],
      "related_tests": [
        "RewriteTests.test_rank_one_and_equal_spectra",
        "RewriteTests.test_uniform_stationary",
        "RewriteTests.test_reconstruction_inversion",
        "RewriteTests.test_rewrite_distinguishes_old_memory",
        "RewriteTests.test_new_local_rank_full"
      ],
      "status": "written argument in supplied manuscript; see implementation fields for executable support",
      "boundaries": [
        "Exact inverse recovery can be ill-conditioned for small delta and deep coordinates.",
        "The result does not imply a large error at a fixed practical tolerance.",
        "The isospectral statement concerns the explicit family."
      ]
    },
    {
      "id": "thm:approxjump",
      "title": "Uniform approximation across the dimension jump",
      "kind": "theorem",
      "section": "7",
      "number": "7.1",
      "tex_label": "thm:approxjump",
      "equation_labels": [
        "eq:uniformdelta"
      ],
      "conclusion": "The old compact updates, initialized with the true old signature, retain output error at most 8*delta/3 uniformly over states, supplied label words, time, and dimension.",
      "additional_hypotheses": [
        "Explicit perturbed Hadamard family; true old signature at initialization."
      ],
      "implementation_entrypoints": [
        "Family.signature",
        "Family.compact_step",
        "separation_family"
      ],
      "related_tests": [
        "RewriteTests.test_perturbation_output_bound"
      ],
      "status": "written argument in supplied manuscript; see implementation fields for executable support",
      "boundaries": [
        "Approximate protected-output guarantee, not exact compatibility or full-state recovery.",
        "This family-specific bound is not a generic robustness theorem."
      ]
    },
    {
      "id": "finite-bit-implementation",
      "title": "Finite-bit recurrent implementation",
      "kind": "derived bound",
      "section": "7",
      "number": null,
      "tex_label": null,
      "equation_labels": [
        "eq:quant"
      ],
      "conclusion": "For the explicit contractive compact maps, rounding on spacing Delta yields uniform coordinate error at most Delta/(2*(1-kappa)); deterministic worst-state state-bit cost has leading term 2*log2(1/epsilon).",
      "additional_hypotheses": [
        "Unperturbed explicit family; kappa=1/8; specified rounding and initialization error.",
        "Matching packing lower bound uses current output and next-A output accuracy."
      ],
      "implementation_entrypoints": [
        "Family.compact_step"
      ],
      "related_tests": [],
      "status": "written argument in supplied manuscript; see implementation fields for executable support",
      "boundaries": [
        "The argument is in an unnumbered subsection, not a separately numbered theorem.",
        "Code, model, acquisition, and metadata costs are excluded.",
        "No claim for a different average-error or randomized encoding contract."
      ]
    },
    {
      "id": "thm:admit",
      "title": "Exact nonlinear rewrite admission",
      "kind": "theorem",
      "section": "8",
      "number": "8.1",
      "tex_label": "thm:admit",
      "equation_labels": [
        "eq:QE",
        "eq:suff"
      ],
      "conclusion": "A proposed positive branch admits updates from the old signature exactly when every transported exterior row annihilates all feasible equivalent simplex pairs. An analogous pair test governs new readouts.",
      "additional_hypotheses": [
        "Proposed branch is strictly positive."
      ],
      "implementation_entrypoints": [],
      "related_tests": [
        "RewriteTests.test_span_admission_not_necessary"
      ],
      "status": "written argument in supplied manuscript; see implementation fields for executable support",
      "boundaries": [
        "General real quantifier elimination is a theoretical decision route and is not implemented.",
        "Linear exterior transport invariance is sufficient but not necessary.",
        "A failed span test alone is not a valid impossibility certificate."
      ]
    },
    {
      "id": "prop:all",
      "title": "No exact compression under arbitrary positive rewrites",
      "kind": "proposition",
      "section": "8",
      "number": "8.2",
      "tex_label": "prop:all",
      "equation_labels": [],
      "conclusion": "Preserving a nonconstant affine readout under every strictly positive stochastic rewrite forces injectivity on the simplex; continuous encoders need at least n-1 real coordinates.",
      "additional_hypotheses": [
        "Every strictly positive stochastic rewrite is permitted.",
        "At least one protected affine readout is nonconstant."
      ],
      "implementation_entrypoints": [],
      "related_tests": [],
      "status": "written argument in supplied manuscript; see implementation fields for executable support",
      "boundaries": [
        "Restricted rewrite libraries and controlled approximate memory are not ruled out."
      ]
    },
    {
      "id": "thm:robust",
      "title": "Finite continuation-error certificate",
      "kind": "theorem",
      "section": "9",
      "number": "9.1",
      "tex_label": "thm:robust",
      "equation_labels": [
        "eq:residual",
        "eq:boundC",
        "eq:robust"
      ],
      "conclusion": "Valid transport residual and denominator bounds give a componentwise one-step signature error bound; a common weighted contraction gives a geometric all-horizon bound with implementation error charges.",
      "additional_hypotheses": [
        "Declared feasible compact state set and valid positive denominator lower bounds.",
        "For the lifetime claim, common positive vector v, q<1, and C_a*v <= q*v.",
        "Represented-state updates have genuine weighted signature error charges."
      ],
      "implementation_entrypoints": [],
      "related_tests": [],
      "status": "written argument in supplied manuscript; see implementation fields for executable support",
      "boundaries": [
        "No generic robust certificate solver is delivered.",
        "Ambient box quantization is not automatically feasible for a general signature image.",
        "Small residual alone does not establish all-horizon stability."
      ]
    },
    {
      "id": "successor-gain-gate",
      "title": "Successor decisions and a sound recursive deployment rule",
      "kind": "protocol and counterexample",
      "section": "10",
      "number": null,
      "tex_label": null,
      "equation_labels": [
        "eq:gate"
      ],
      "conclusion": "A positive full-set gain lower bound with valid uniform model errors, common anchor, charged costs, compatible memory, and separate operational requirements supports the stated bounded promotion rule. Vertex-only evaluation can be unsound.",
      "additional_hypotheses": [
        "Valid common uncertainty event and transition model.",
        "Actual infimum bound over the full feasible uncertainty set.",
        "Uniform error guarantees and actual information acquisition when needed."
      ],
      "implementation_entrypoints": [],
      "related_tests": [
        "RewriteTests.test_vertex_promotion_failure"
      ],
      "status": "written argument in supplied manuscript; see implementation fields for executable support",
      "boundaries": [
        "No general nonlinear gain minimizer is implemented.",
        "No guarantee of useful proposals, recursive acceleration, or inclusion of every human requirement.",
        "The protocol is not a separately numbered theorem."
      ]
    }
  ],
  "unestablished_claims": [
    "Unknown-model or arbitrary neural-agent faithful realization.",
    "Measured frontier-agent performance or research-productivity advantage.",
    "Universal superiority over known methods.",
    "Exhaustively established publication priority.",
    "Proof-assistant formalization or external expert certification.",
    "A justified T9 significance classification."
  ]
}