| { |
| "release": "2.0.0", |
| "direction": "premise to conclusion", |
| "acyclic_verified": true, |
| "nodes": [ |
| { |
| "id": "foundation:maxwell", |
| "kind": "foundation", |
| "description": "Stated nondispersive canonical Maxwell model and positive material parameters." |
| }, |
| { |
| "id": "foundation:projective", |
| "kind": "foundation", |
| "description": "Direct polar identities, common positive angular velocity, and projective winding." |
| }, |
| { |
| "id": "foundation:ode", |
| "kind": "foundation", |
| "description": "Bounded measurable linear ODEs, determinant one, continuity of transfer matrices." |
| }, |
| { |
| "id": "foundation:fractional", |
| "kind": "foundation", |
| "description": "Exact fractional-linear pointwise subtraction and derivative identities." |
| }, |
| { |
| "id": "foundation:segments", |
| "kind": "foundation", |
| "description": "Explicit constant-generator transfer matrices and their products." |
| }, |
| { |
| "id": "foundation:convex", |
| "kind": "foundation", |
| "description": "Compact convex moment fibers, finite-dimensional support, and weak-* compactness." |
| }, |
| { |
| "id": "foundation:linear_algebra", |
| "kind": "foundation", |
| "description": "Cayley–Hamilton, Vandermonde, Hankel, and singular-value identities." |
| }, |
| { |
| "id": "foundation:inverse_function", |
| "kind": "foundation", |
| "description": "Inverse/constant-rank theorem and local derivative estimates." |
| }, |
| { |
| "id": "foundation:analytic_jets", |
| "kind": "foundation", |
| "description": "Analytic transfer family and finite polynomial/exponential expansions." |
| }, |
| { |
| "id": "external:real_QE", |
| "kind": "external_algorithm", |
| "description": "Effective real-closed-field quantifier elimination; Collins CAD is an external algorithmic input." |
| }, |
| { |
| "id": "lem:linearize", |
| "kind": "lemma", |
| "number": "3.1", |
| "title": "An exact, unrelaxed linearization" |
| }, |
| { |
| "id": "lem:boundary", |
| "kind": "lemma", |
| "number": "3.2", |
| "title": "Boundary moments cannot carry positive gain" |
| }, |
| { |
| "id": "lem:normal", |
| "kind": "lemma", |
| "number": "3.3", |
| "title": "Finite normal duals at every interior moment" |
| }, |
| { |
| "id": "thm:resource", |
| "kind": "theorem", |
| "number": "4.1", |
| "title": "Global resource-constrained extremality and equality" |
| }, |
| { |
| "id": "thm:resourceDefect", |
| "kind": "theorem", |
| "number": "4.2", |
| "title": "Resource defect and unique dual parameters" |
| }, |
| { |
| "id": "thm:concavity", |
| "kind": "theorem", |
| "number": "5.1", |
| "title": "Strict concavity and at most two adjacent winning windings" |
| }, |
| { |
| "id": "thm:resourceFree", |
| "kind": "theorem", |
| "number": "5.2", |
| "title": "Free-period optimum with a fixed long-run resource" |
| }, |
| { |
| "id": "lem:margin", |
| "kind": "lemma", |
| "number": "6.1", |
| "title": "A two-multiplier switching margin" |
| }, |
| { |
| "id": "thm:phasestability", |
| "kind": "theorem", |
| "number": "6.2", |
| "title": "Phase and clock stability from the exact defect" |
| }, |
| { |
| "id": "thm:globalstability", |
| "kind": "theorem", |
| "number": "6.3", |
| "title": "Global resource-constrained square-root rigidity" |
| }, |
| { |
| "id": "prop:resourceSharp", |
| "kind": "proposition", |
| "number": "6.4", |
| "title": "Sharpness while preserving the resource exactly" |
| }, |
| { |
| "id": "thm:main", |
| "kind": "theorem", |
| "number": "7.1", |
| "title": "Preserved exact bounds-only optimum" |
| }, |
| { |
| "id": "thm:defect", |
| "kind": "theorem", |
| "number": "7.2", |
| "title": "Preserved bounds-only exact defect" |
| }, |
| { |
| "id": "cor:boundstability", |
| "kind": "corollary", |
| "number": "7.3", |
| "title": "Preserved bounds-only global stability" |
| }, |
| { |
| "id": "prop:sharp", |
| "kind": "proposition", |
| "number": "7.4", |
| "title": "The exponent $1/2$ cannot generally be improved" |
| }, |
| { |
| "id": "thm:free", |
| "kind": "theorem", |
| "number": "8.1", |
| "title": "Sharp growth per unit time" |
| }, |
| { |
| "id": "thm:quantum", |
| "kind": "theorem", |
| "number": "9.1", |
| "title": "Exact finite-cycle and asymptotic quantum laws" |
| }, |
| { |
| "id": "cor:quantumargmax", |
| "kind": "corollary", |
| "number": "9.2", |
| "title": "Resource-constrained classical--quantum extremality" |
| }, |
| { |
| "id": "prop:quantumfinite", |
| "kind": "proposition", |
| "number": "9.3", |
| "title": "Finite-cycle photon optimization is a different problem" |
| }, |
| { |
| "id": "thm:quantumstability", |
| "kind": "theorem", |
| "number": "9.4", |
| "title": "Finite-observation quantum rigidity" |
| }, |
| { |
| "id": "lem:boxselector", |
| "kind": "lemma", |
| "number": "10.1", |
| "title": "The exact corner selector" |
| }, |
| { |
| "id": "lem:boxclock", |
| "kind": "lemma", |
| "number": "10.2", |
| "title": "A strictly monotone clock" |
| }, |
| { |
| "id": "thm:boxfixed", |
| "kind": "theorem", |
| "number": "10.3", |
| "title": "Complete fixed-period Maxwell corner-purity theorem" |
| }, |
| { |
| "id": "cor:boxfree", |
| "kind": "corollary", |
| "number": "10.4", |
| "title": "Free-period rectangle rate and quantum optimum" |
| }, |
| { |
| "id": "prop:boxstability", |
| "kind": "proposition", |
| "number": "10.5", |
| "title": "Rectangle structural stability" |
| }, |
| { |
| "id": "lem:polylinear", |
| "kind": "lemma", |
| "number": "11.1", |
| "title": "Fractional barycentric linearization" |
| }, |
| { |
| "id": "thm:polyresource", |
| "kind": "theorem", |
| "number": "11.2", |
| "title": "Affine-resource calibration, purity, and finite switching" |
| }, |
| { |
| "id": "thm:abstract", |
| "kind": "theorem", |
| "number": "11.3", |
| "title": "Preserved calibrated rotating-polytope extremality" |
| }, |
| { |
| "id": "thm:boxresource", |
| "kind": "theorem", |
| "number": "12.1", |
| "title": "Explicit two-resource Maxwell extremality" |
| }, |
| { |
| "id": "lem:boxnormal", |
| "kind": "lemma", |
| "number": "12.2", |
| "title": "Boundary moments and normal duals for two Maxwell resources" |
| }, |
| { |
| "id": "lem:boxpalindrome", |
| "kind": "lemma", |
| "number": "12.3", |
| "title": "Time reversal locks the two diagonal visits" |
| }, |
| { |
| "id": "prop:boxresourceconcavity", |
| "kind": "proposition", |
| "number": "12.4", |
| "title": "Resource-ray concavity and rate calibration" |
| }, |
| { |
| "id": "cor:boxresourceStability", |
| "kind": "corollary", |
| "number": "12.5", |
| "title": "Two-resource Maxwell stability and quantum extremality" |
| }, |
| { |
| "id": "prop:inversefailure", |
| "kind": "proposition", |
| "number": "13.1", |
| "title": "Exact two-layer isodiscriminant counterexample" |
| }, |
| { |
| "id": "thm:inverse", |
| "kind": "theorem", |
| "number": "14.1", |
| "title": "Finite derivatives determine every reduced layer" |
| }, |
| { |
| "id": "lem:prony", |
| "kind": "lemma", |
| "number": "14.2", |
| "title": "Finite exponential determination" |
| }, |
| { |
| "id": "lem:peel", |
| "kind": "lemma", |
| "number": "14.3", |
| "title": "Recovering the leading temporal layer" |
| }, |
| { |
| "id": "prop:datacount", |
| "kind": "proposition", |
| "number": "15.1", |
| "title": "Preserved regular-data lower bound" |
| }, |
| { |
| "id": "lem:projection", |
| "kind": "lemma", |
| "number": "15.2", |
| "title": "A direct generic projection bound" |
| }, |
| { |
| "id": "thm:jetCompression", |
| "kind": "theorem", |
| "number": "15.3", |
| "title": "Linear-count compression of exact transfer data" |
| }, |
| { |
| "id": "prop:CAD", |
| "kind": "proposition", |
| "number": "15.4", |
| "title": "Certified projection search for algebraic inputs" |
| }, |
| { |
| "id": "prop:localInverse", |
| "kind": "proposition", |
| "number": "15.5", |
| "title": "Local inverse conditioning at regular points" |
| }, |
| { |
| "id": "thm:rigidityCompression", |
| "kind": "theorem", |
| "number": "15.6", |
| "title": "Saturation is an inverse certificate" |
| }, |
| { |
| "id": "prop:nearNoGo", |
| "kind": "proposition", |
| "number": "15.7", |
| "title": "No finite exact inverse for the entire near-extremal class" |
| } |
| ], |
| "edges": [ |
| { |
| "from": "foundation:projective", |
| "to": "lem:linearize" |
| }, |
| { |
| "from": "foundation:fractional", |
| "to": "lem:linearize" |
| }, |
| { |
| "from": "lem:linearize", |
| "to": "lem:boundary" |
| }, |
| { |
| "from": "foundation:convex", |
| "to": "lem:boundary" |
| }, |
| { |
| "from": "lem:linearize", |
| "to": "lem:normal" |
| }, |
| { |
| "from": "lem:boundary", |
| "to": "lem:normal" |
| }, |
| { |
| "from": "foundation:convex", |
| "to": "lem:normal" |
| }, |
| { |
| "from": "lem:normal", |
| "to": "thm:resource" |
| }, |
| { |
| "from": "foundation:fractional", |
| "to": "thm:resource" |
| }, |
| { |
| "from": "foundation:segments", |
| "to": "thm:resource" |
| }, |
| { |
| "from": "thm:resource", |
| "to": "thm:resourceDefect" |
| }, |
| { |
| "from": "lem:normal", |
| "to": "thm:resourceDefect" |
| }, |
| { |
| "from": "foundation:fractional", |
| "to": "thm:resourceDefect" |
| }, |
| { |
| "from": "lem:normal", |
| "to": "thm:concavity" |
| }, |
| { |
| "from": "thm:resource", |
| "to": "thm:concavity" |
| }, |
| { |
| "from": "foundation:convex", |
| "to": "thm:concavity" |
| }, |
| { |
| "from": "thm:concavity", |
| "to": "thm:resourceFree" |
| }, |
| { |
| "from": "thm:resourceDefect", |
| "to": "thm:resourceFree" |
| }, |
| { |
| "from": "foundation:fractional", |
| "to": "lem:margin" |
| }, |
| { |
| "from": "lem:margin", |
| "to": "thm:phasestability" |
| }, |
| { |
| "from": "thm:resourceDefect", |
| "to": "thm:phasestability" |
| }, |
| { |
| "from": "foundation:projective", |
| "to": "thm:phasestability" |
| }, |
| { |
| "from": "thm:phasestability", |
| "to": "thm:globalstability" |
| }, |
| { |
| "from": "thm:resource", |
| "to": "thm:globalstability" |
| }, |
| { |
| "from": "foundation:segments", |
| "to": "prop:resourceSharp" |
| }, |
| { |
| "from": "thm:resource", |
| "to": "prop:resourceSharp" |
| }, |
| { |
| "from": "foundation:projective", |
| "to": "thm:main" |
| }, |
| { |
| "from": "foundation:fractional", |
| "to": "thm:main" |
| }, |
| { |
| "from": "foundation:segments", |
| "to": "thm:main" |
| }, |
| { |
| "from": "thm:main", |
| "to": "thm:defect" |
| }, |
| { |
| "from": "foundation:fractional", |
| "to": "thm:defect" |
| }, |
| { |
| "from": "thm:main", |
| "to": "cor:boundstability" |
| }, |
| { |
| "from": "thm:defect", |
| "to": "cor:boundstability" |
| }, |
| { |
| "from": "thm:phasestability", |
| "to": "cor:boundstability" |
| }, |
| { |
| "from": "foundation:segments", |
| "to": "prop:sharp" |
| }, |
| { |
| "from": "thm:main", |
| "to": "prop:sharp" |
| }, |
| { |
| "from": "thm:main", |
| "to": "thm:free" |
| }, |
| { |
| "from": "thm:defect", |
| "to": "thm:free" |
| }, |
| { |
| "from": "foundation:maxwell", |
| "to": "thm:quantum" |
| }, |
| { |
| "from": "foundation:linear_algebra", |
| "to": "thm:quantum" |
| }, |
| { |
| "from": "thm:quantum", |
| "to": "cor:quantumargmax" |
| }, |
| { |
| "from": "thm:resource", |
| "to": "cor:quantumargmax" |
| }, |
| { |
| "from": "thm:globalstability", |
| "to": "cor:quantumargmax" |
| }, |
| { |
| "from": "foundation:segments", |
| "to": "prop:quantumfinite" |
| }, |
| { |
| "from": "thm:quantum", |
| "to": "prop:quantumfinite" |
| }, |
| { |
| "from": "thm:resource", |
| "to": "prop:quantumfinite" |
| }, |
| { |
| "from": "thm:quantum", |
| "to": "thm:quantumstability" |
| }, |
| { |
| "from": "thm:globalstability", |
| "to": "thm:quantumstability" |
| }, |
| { |
| "from": "foundation:ode", |
| "to": "thm:quantumstability" |
| }, |
| { |
| "from": "foundation:maxwell", |
| "to": "lem:boxselector" |
| }, |
| { |
| "from": "foundation:fractional", |
| "to": "lem:boxselector" |
| }, |
| { |
| "from": "lem:boxselector", |
| "to": "lem:boxclock" |
| }, |
| { |
| "from": "lem:boxclock", |
| "to": "thm:boxfixed" |
| }, |
| { |
| "from": "lem:boxselector", |
| "to": "thm:boxfixed" |
| }, |
| { |
| "from": "foundation:projective", |
| "to": "thm:boxfixed" |
| }, |
| { |
| "from": "thm:boxfixed", |
| "to": "cor:boxfree" |
| }, |
| { |
| "from": "thm:quantum", |
| "to": "cor:boxfree" |
| }, |
| { |
| "from": "thm:boxfixed", |
| "to": "prop:boxstability" |
| }, |
| { |
| "from": "thm:phasestability", |
| "to": "prop:boxstability" |
| }, |
| { |
| "from": "thm:quantumstability", |
| "to": "prop:boxstability" |
| }, |
| { |
| "from": "foundation:projective", |
| "to": "lem:polylinear" |
| }, |
| { |
| "from": "foundation:fractional", |
| "to": "lem:polylinear" |
| }, |
| { |
| "from": "lem:polylinear", |
| "to": "thm:polyresource" |
| }, |
| { |
| "from": "foundation:convex", |
| "to": "thm:polyresource" |
| }, |
| { |
| "from": "thm:polyresource", |
| "to": "thm:abstract" |
| }, |
| { |
| "from": "lem:polylinear", |
| "to": "lem:boxnormal" |
| }, |
| { |
| "from": "foundation:convex", |
| "to": "lem:boxnormal" |
| }, |
| { |
| "from": "lem:boxselector", |
| "to": "lem:boxnormal" |
| }, |
| { |
| "from": "lem:boxnormal", |
| "to": "lem:boxpalindrome" |
| }, |
| { |
| "from": "foundation:segments", |
| "to": "lem:boxpalindrome" |
| }, |
| { |
| "from": "foundation:projective", |
| "to": "lem:boxpalindrome" |
| }, |
| { |
| "from": "lem:boxnormal", |
| "to": "prop:boxresourceconcavity" |
| }, |
| { |
| "from": "lem:boxpalindrome", |
| "to": "prop:boxresourceconcavity" |
| }, |
| { |
| "from": "foundation:convex", |
| "to": "prop:boxresourceconcavity" |
| }, |
| { |
| "from": "lem:boxnormal", |
| "to": "thm:boxresource" |
| }, |
| { |
| "from": "lem:boxpalindrome", |
| "to": "thm:boxresource" |
| }, |
| { |
| "from": "prop:boxresourceconcavity", |
| "to": "thm:boxresource" |
| }, |
| { |
| "from": "foundation:projective", |
| "to": "thm:boxresource" |
| }, |
| { |
| "from": "thm:boxresource", |
| "to": "cor:boxresourceStability" |
| }, |
| { |
| "from": "prop:boxstability", |
| "to": "cor:boxresourceStability" |
| }, |
| { |
| "from": "thm:quantum", |
| "to": "cor:boxresourceStability" |
| }, |
| { |
| "from": "thm:quantumstability", |
| "to": "cor:boxresourceStability" |
| }, |
| { |
| "from": "foundation:segments", |
| "to": "prop:inversefailure" |
| }, |
| { |
| "from": "foundation:linear_algebra", |
| "to": "lem:prony" |
| }, |
| { |
| "from": "foundation:analytic_jets", |
| "to": "lem:peel" |
| }, |
| { |
| "from": "foundation:segments", |
| "to": "lem:peel" |
| }, |
| { |
| "from": "lem:prony", |
| "to": "thm:inverse" |
| }, |
| { |
| "from": "lem:peel", |
| "to": "thm:inverse" |
| }, |
| { |
| "from": "foundation:analytic_jets", |
| "to": "thm:inverse" |
| }, |
| { |
| "from": "thm:inverse", |
| "to": "prop:datacount" |
| }, |
| { |
| "from": "foundation:inverse_function", |
| "to": "prop:datacount" |
| }, |
| { |
| "from": "foundation:inverse_function", |
| "to": "lem:projection" |
| }, |
| { |
| "from": "thm:inverse", |
| "to": "thm:jetCompression" |
| }, |
| { |
| "from": "lem:projection", |
| "to": "thm:jetCompression" |
| }, |
| { |
| "from": "prop:datacount", |
| "to": "thm:jetCompression" |
| }, |
| { |
| "from": "thm:jetCompression", |
| "to": "prop:CAD" |
| }, |
| { |
| "from": "foundation:analytic_jets", |
| "to": "prop:CAD" |
| }, |
| { |
| "from": "external:real_QE", |
| "to": "prop:CAD" |
| }, |
| { |
| "from": "prop:datacount", |
| "to": "prop:localInverse" |
| }, |
| { |
| "from": "thm:inverse", |
| "to": "prop:localInverse" |
| }, |
| { |
| "from": "thm:jetCompression", |
| "to": "prop:localInverse" |
| }, |
| { |
| "from": "foundation:inverse_function", |
| "to": "prop:localInverse" |
| }, |
| { |
| "from": "thm:resource", |
| "to": "thm:rigidityCompression" |
| }, |
| { |
| "from": "thm:concavity", |
| "to": "thm:rigidityCompression" |
| }, |
| { |
| "from": "thm:quantum", |
| "to": "thm:rigidityCompression" |
| }, |
| { |
| "from": "thm:boxresource", |
| "to": "thm:rigidityCompression" |
| }, |
| { |
| "from": "thm:resource", |
| "to": "prop:nearNoGo" |
| }, |
| { |
| "from": "prop:datacount", |
| "to": "prop:nearNoGo" |
| }, |
| { |
| "from": "foundation:ode", |
| "to": "prop:nearNoGo" |
| } |
| ], |
| "topological_order": [ |
| "foundation:projective", |
| "foundation:fractional", |
| "lem:linearize", |
| "foundation:convex", |
| "lem:boundary", |
| "lem:normal", |
| "foundation:segments", |
| "thm:resource", |
| "thm:resourceDefect", |
| "thm:concavity", |
| "thm:resourceFree", |
| "lem:margin", |
| "thm:phasestability", |
| "thm:globalstability", |
| "prop:resourceSharp", |
| "thm:main", |
| "thm:defect", |
| "cor:boundstability", |
| "prop:sharp", |
| "thm:free", |
| "foundation:maxwell", |
| "foundation:linear_algebra", |
| "thm:quantum", |
| "cor:quantumargmax", |
| "prop:quantumfinite", |
| "foundation:ode", |
| "thm:quantumstability", |
| "lem:boxselector", |
| "lem:boxclock", |
| "thm:boxfixed", |
| "cor:boxfree", |
| "prop:boxstability", |
| "lem:polylinear", |
| "thm:polyresource", |
| "thm:abstract", |
| "lem:boxnormal", |
| "lem:boxpalindrome", |
| "prop:boxresourceconcavity", |
| "thm:boxresource", |
| "cor:boxresourceStability", |
| "prop:inversefailure", |
| "lem:prony", |
| "foundation:analytic_jets", |
| "lem:peel", |
| "thm:inverse", |
| "foundation:inverse_function", |
| "prop:datacount", |
| "lem:projection", |
| "thm:jetCompression", |
| "external:real_QE", |
| "prop:CAD", |
| "prop:localInverse", |
| "thm:rigidityCompression", |
| "prop:nearNoGo" |
| ], |
| "note": "Dependency graph uses proof inputs, not the order in which composite theorems are announced. Some lemmas proving a previously announced theorem occur later. Quantum conclusions do not supply classical premises." |
| } |
|
|
