{
  "schema": "AN04-positivity-foundations-proof-map/v1",
  "proofs": [
    {
      "id": "FP0:S1",
      "source": "scalar-splitting-and-adaptive-scale.md",
      "source_sha256": "71dcc6b6a5289d3a1bd8ab7882b1a36455ed5c09f7066de263337f45f02df1d6",
      "proof_locator": "## S1.",
      "dependencies": [
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:P14.2-powers"
      ],
      "scope": "Full interior nonnegative gradient bound with exact distance-to-boundary dependence and zero-value case"
    },
    {
      "id": "FP0:S2",
      "source": "scalar-splitting-and-adaptive-scale.md",
      "source_sha256": "71dcc6b6a5289d3a1bd8ab7882b1a36455ed5c09f7066de263337f45f02df1d6",
      "proof_locator": "## S2.",
      "dependencies": [
        "U001:F0-COMP",
        "U001:Q5",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full polarization formula, exact symmetric trilinear diagonal norm by compact extremizer and pair improvement"
    },
    {
      "id": "FP0:S3",
      "source": "scalar-splitting-and-adaptive-scale.md",
      "source_sha256": "71dcc6b6a5289d3a1bd8ab7882b1a36455ed5c09f7066de263337f45f02df1d6",
      "proof_locator": "**Normalized jet lemma.**",
      "dependencies": [
        "FP0:S2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full normalized fourth-derivative jet estimate with fixed radius, numerical constants, strict margins and unnormalized upper-bound variant"
    },
    {
      "id": "FP0:S4",
      "source": "scalar-splitting-and-adaptive-scale.md",
      "source_sha256": "71dcc6b6a5289d3a1bd8ab7882b1a36455ed5c09f7066de263337f45f02df1d6",
      "proof_locator": "## S3.",
      "dependencies": [
        "FP0:S1",
        "FP0:S3",
        "U001:Q5",
        "U001:P3",
        "U001:L3.3.8",
        "U001:P14.2-powers",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full uniform scalar splitting, positive Hessian direction, exact critical graph and integral coefficient, all derivative bounds through k from k+2"
    },
    {
      "id": "FP0:S5",
      "source": "scalar-splitting-and-adaptive-scale.md",
      "source_sha256": "71dcc6b6a5289d3a1bd8ab7882b1a36455ed5c09f7066de263337f45f02df1d6",
      "proof_locator": "## S4.",
      "dependencies": [
        "P2:ML1",
        "P2:ML2",
        "P2:ML3",
        "U001:P14.2-powers",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Complete real squared partition with fixed overlap, all finite column derivative bounds and frozen norm comparison; no operator-bound assertion"
    },
    {
      "id": "FP0:S6",
      "source": "scalar-splitting-and-adaptive-scale.md",
      "source_sha256": "71dcc6b6a5289d3a1bd8ab7882b1a36455ed5c09f7066de263337f45f02df1d6",
      "proof_locator": "## S5.",
      "dependencies": [
        "FP0:S3",
        "U001:Q5",
        "U001:P14.2-powers",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full nonsmooth value/Hessian adaptive scale, lambda-uniform slow variation, symplectic dual and same-base two-sided temperateness"
    },
    {
      "id": "FP0:S7",
      "source": "scalar-splitting-and-adaptive-scale.md",
      "source_sha256": "71dcc6b6a5289d3a1bd8ab7882b1a36455ed5c09f7066de263337f45f02df1d6",
      "proof_locator": "**Frozen localized symbols.**",
      "dependencies": [
        "FP0:S4",
        "FP0:S5",
        "FP0:S6",
        "P2:ML4",
        "U001:U001-A4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "All controlled finite derivatives of compact nonnegative localized symbols in moving and frozen metrics; no differentiating H"
    },
    {
      "id": "FP0:S8",
      "source": "scalar-splitting-and-adaptive-scale.md",
      "source_sha256": "71dcc6b6a5289d3a1bd8ab7882b1a36455ed5c09f7066de263337f45f02df1d6",
      "proof_locator": "## S6.",
      "dependencies": [
        "FP0:S4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exact curved critical graph, residual, square and integral coefficient; plot and normalization scope"
    },
    {
      "id": "FP0:M1",
      "source": "finite-matrix-sharp-lower-bound.md",
      "source_sha256": "b9035fda4e36cb672c43a2b70246d61786f6be6630c1d00cdfa4287c31c3519a",
      "proof_locator": "## M1.",
      "dependencies": [
        "CE:G1",
        "P2:OP3",
        "P3:L1",
        "P3:M5",
        "P3:M6",
        "P3:M7"
      ],
      "scope": "Finite matrix Sobolev bounds and density with explicit N factor; every ordered composition and adjoint remainder entry"
    },
    {
      "id": "FP0:M2",
      "source": "finite-matrix-sharp-lower-bound.md",
      "source_sha256": "b9035fda4e36cb672c43a2b70246d61786f6be6630c1d00cdfa4287c31c3519a",
      "proof_locator": "## M2.",
      "dependencies": [
        "FP0:M1",
        "CE:G2",
        "CE:G3",
        "CE:G4",
        "P3:M3",
        "P3:M4"
      ],
      "scope": "Exact nonnegative matrix quadratic form from scalar probe, finite-entry kernel identity, full parameter convergence and positive-order scope"
    },
    {
      "id": "FP0:M3",
      "source": "finite-matrix-sharp-lower-bound.md",
      "source_sha256": "b9035fda4e36cb672c43a2b70246d61786f6be6630c1d00cdfa4287c31c3519a",
      "proof_locator": "## M3.",
      "dependencies": [
        "FP0:M1",
        "FP0:M2",
        "CE:G5",
        "CE:G6",
        "CE:G7"
      ],
      "scope": "Complete finite matrix one-order cancellation with every differentiated remainder, Hermitian decomposition and sharp bound for all real orders"
    },
    {
      "id": "FP0:M4",
      "source": "finite-matrix-sharp-lower-bound.md",
      "source_sha256": "b9035fda4e36cb672c43a2b70246d61786f6be6630c1d00cdfa4287c31c3519a",
      "proof_locator": "## M4.",
      "dependencies": [
        "FP0:M1",
        "FP0:M3",
        "CE:G1",
        "CE:G10",
        "P2:OP3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full order2/H1/2 and order0/H-1/2 energy errors, all real conjugations, finite-family time continuity and explicit nondiagonal example"
    }
  ],
  "earlier_maps": [
    "../20261004-free-stationary-phase/proof-map.json",
    "../20261004-free-intrinsic-graph/proof-map.json",
    "../20261004-free-tangent-zoom/proof-map.json",
    "../20261004-free-canonical-composition/proof-map.json",
    "../20261005-restored-oscillatory/proof-map.json",
    "../20261005-restored-phase-space/proof-map.json",
    "../20261005-restored-phase-equivalence/proof-map.json",
    "../20261005-restored-intrinsic-regularity/proof-map.json",
    "../20261005-restored-tangent-zoom/proof-map.json",
    "../20261005-restored-gaussian-symbols/proof-map.json",
    "../20261005-restored-linear-reduction/proof-map.json",
    "../20261005-restored-analytic-composition/proof-map.json",
    "../20261005-restored-subprincipal-transport/proof-map.json",
    "../20261005-restored-graph-egorov/proof-map.json",
    "../20261005-restored-corank-continuity/proof-map.json",
    "../20261005-restored-submanifolds/proof-map.json",
    "../20261005-restored-cubic-scaling/proof-map.json",
    "../20261005-restored-smooth-descent/proof-map.json",
    "../20261005-restored-simultaneous-reflections/proof-map.json",
    "../20261005-restored-folded-forms/proof-map.json",
    "../20261005-restored-symplectic-reflections/proof-map.json",
    "../20261005-restored-canonical-folds/proof-map.json",
    "../20261005-restored-fold-densities/proof-map.json",
    "../20261005-restored-airy-models/proof-map.json",
    "../20261005-restored-invariant-folds/proof-map.json",
    "../20261005-restored-prescribed-coordinates/proof-map.json",
    "../20261005-restored-clean-pairs/proof-map.json",
    "../20261005-restored-function-normal-forms/proof-map.json",
    "../20261005-restored-quadratic-forms/proof-map.json",
    "../20261005-restored-positive-ideals/proof-map.json",
    "../20261005-restored-complex-fios/proof-map.json",
    "../20261005-restored-maslov-topology/proof-map.json",
    "../20261005-restored-real-principal/proof-map.json",
    "../20261005-cauchy-foundations/proof-map.json",
    "../20261005-restored-first-order-cauchy/proof-map.json",
    "../20261005-mixed-halfspace-foundations/proof-map.json",
    "../20261005-conormal-test-foundations/proof-map.json",
    "../20261005-normal-extension/proof-map.json",
    "../20261005-local-boundary-calculus/proof-map.json",
    "../20261005-positive-boundary-orders/proof-map.json",
    "../20261005-conormal-symbols-and-corners/proof-map.json",
    "../20261005-global-boundary-operators/proof-map.json",
    "../20261005-boundary-wavefront-and-tangential/proof-map.json",
    "../20261005-restored-higher-order-cauchy/proof-map.json",
    "../20261005-hyperbolicity-necessity-foundations/proof-map.json",
    "../20261005-restored-hyperbolicity-necessity/proof-map.json"
  ],
  "component_proofs_complete": true,
  "U033_restored": false,
  "full_course_complete": false,
  "public_release_authorized": false
}