{
  "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."
}