{
  "schema": "AN04-cauchy-foundations-proof-map/v1",
  "proofs": [
    {
      "id": "CE:G1",
      "source": "sharp-lower-bound.md",
      "source_sha256": "8c156e282fbfe365f1607a40350844474213b3a571062e12f8efd0255edcfb47",
      "proof_locator": "## G1.",
      "dependencies": [
        "AM:B3",
        "P2:OP3",
        "P3:M6",
        "P3:M7",
        "P3:L1",
        "P3:L3",
        "P1:B1",
        "U001:P14.2-powers"
      ],
      "scope": "Full global all-real Sobolev bound, actual isometries, simultaneous density, distributional compatibility and dual pairings"
    },
    {
      "id": "CE:G2",
      "source": "sharp-lower-bound.md",
      "source_sha256": "8c156e282fbfe365f1607a40350844474213b3a571062e12f8efd0255edcfb47",
      "proof_locator": "Choose an even",
      "dependencies": [
        "P3:L1",
        "P3:L3",
        "P3:M4",
        "U001:U001-A4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "P2:OP0"
      ],
      "scope": "Actual Schwartz positive probe, parity, full kernel identity and exact integral-one normalization"
    },
    {
      "id": "CE:G3",
      "source": "sharp-lower-bound.md",
      "source_sha256": "8c156e282fbfe365f1607a40350844474213b3a571062e12f8efd0255edcfb47",
      "proof_locator": "For \\(q>0\\)",
      "dependencies": [
        "CE:G2",
        "P3:M4",
        "P3:L1"
      ],
      "scope": "Exact unitary moving-window conjugation, all normalization factors and positive quadratic form"
    },
    {
      "id": "CE:G4",
      "source": "sharp-lower-bound.md",
      "source_sha256": "8c156e282fbfe365f1607a40350844474213b3a571062e12f8efd0255edcfb47",
      "proof_locator": "For a scalar symbol",
      "dependencies": [
        "CE:G3",
        "P3:M3",
        "P3:M4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:P14.2-powers"
      ],
      "scope": "Whole unbounded probe integral, every differentiated local envelope, actual equality of forms and all input tail estimates"
    },
    {
      "id": "CE:G5",
      "source": "sharp-lower-bound.md",
      "source_sha256": "8c156e282fbfe365f1607a40350844474213b3a571062e12f8efd0255edcfb47",
      "proof_locator": "Write \\(\\lambda=",
      "dependencies": [
        "CE:G4",
        "P3:M4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:P14.2-powers"
      ],
      "scope": "Both far and near frequency regions, exact phase-space Jacobian and variable-scale Taylor remainder"
    },
    {
      "id": "CE:G6",
      "source": "sharp-lower-bound.md",
      "source_sha256": "8c156e282fbfe365f1607a40350844474213b3a571062e12f8efd0255edcfb47",
      "proof_locator": "The zeroth moment",
      "dependencies": [
        "CE:G5",
        "P3:M4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Parity and scale-mass cancellations; every degree-zero, one and two moment with full remainder"
    },
    {
      "id": "CE:G7",
      "source": "sharp-lower-bound.md",
      "source_sha256": "8c156e282fbfe365f1607a40350844474213b3a571062e12f8efd0255edcfb47",
      "proof_locator": "To obtain all derivatives,",
      "dependencies": [
        "CE:G6",
        "P3:M4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exact differentiated kernel identities and finite recursive seminorm control for every frequency/base derivative"
    },
    {
      "id": "CE:G8",
      "source": "sharp-lower-bound.md",
      "source_sha256": "8c156e282fbfe365f1607a40350844474213b3a571062e12f8efd0255edcfb47",
      "proof_locator": "There is a useful more precise classical estimate.",
      "dependencies": [
        "CE:G7"
      ],
      "scope": "Full pointwise derivative refinement with original higher-seminorm remainder"
    },
    {
      "id": "CE:G9",
      "source": "sharp-lower-bound.md",
      "source_sha256": "8c156e282fbfe365f1607a40350844474213b3a571062e12f8efd0255edcfb47",
      "proof_locator": "## G6.",
      "dependencies": [
        "CE:G1",
        "CE:G4",
        "CE:G7",
        "P2:OP3"
      ],
      "scope": "Sharp scalar lower bound at every real m, exact imaginary-adjoint sign, H1 extension and all-real energy conjugation"
    },
    {
      "id": "CE:G10",
      "source": "sharp-lower-bound.md",
      "source_sha256": "8c156e282fbfe365f1607a40350844474213b3a571062e12f8efd0255edcfb47",
      "proof_locator": "## G7.",
      "dependencies": [
        "CE:G1",
        "P2:OP1",
        "P3:M0",
        "P3:M3",
        "U001:F0-COMP",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Complete local jet compactness, distributional-to-smooth continuity and actual strong global Sobolev action"
    },
    {
      "id": "CE:H1",
      "source": "integration-and-duality.md",
      "source_sha256": "c869f6983799baa96d4d39f439d0bb41692dd76e10e3353b6608bde48f5aa971",
      "proof_locator": "## 5. Hahn",
      "dependencies": [
        "P3:M0",
        "U001:F0-COMP"
      ],
      "scope": "Full real one-vector extension, Zorn maximal extension, complex conversion and exact scalar norm test"
    },
    {
      "id": "CE:H2",
      "source": "integration-and-duality.md",
      "source_sha256": "c869f6983799baa96d4d39f439d0bb41692dd76e10e3353b6608bde48f5aa971",
      "proof_locator": "### 15.2.",
      "dependencies": [
        "P3:M3",
        "P3:M4",
        "U001:P14.2-powers",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "All-exponent Hölder and Minkowski, every Young endpoint and zero-norm cases"
    },
    {
      "id": "CE:H3",
      "source": "integration-and-duality.md",
      "source_sha256": "c869f6983799baa96d4d39f439d0bb41692dd76e10e3353b6608bde48f5aa971",
      "proof_locator": "### 17.1.",
      "dependencies": [
        "CE:H2",
        "P3:M0",
        "P3:M3"
      ],
      "scope": "Complete strong measurability, explicit finite minimum cells, norm domination and finite-p simple density"
    },
    {
      "id": "CE:H4",
      "source": "integration-and-duality.md",
      "source_sha256": "c869f6983799baa96d4d39f439d0bb41692dd76e10e3353b6608bde48f5aa971",
      "proof_locator": "### 17.2.",
      "dependencies": [
        "CE:H3",
        "P3:M3",
        "U001:F0-COMP"
      ],
      "scope": "Actual vector integral by complete norm limits, independence, norm inequality, bounded maps and dominated convergence"
    },
    {
      "id": "CE:H5",
      "source": "integration-and-duality.md",
      "source_sha256": "c869f6983799baa96d4d39f439d0bb41692dd76e10e3353b6608bde48f5aa971",
      "proof_locator": "### 17.3.",
      "dependencies": [
        "CE:H2",
        "CE:H3",
        "CE:H4",
        "P3:M3",
        "U001:F0-COMP"
      ],
      "scope": "Complete Bochner Lp spaces including infinity, exact Hilbert integral and counterexample to finite-valued Linfinity density"
    },
    {
      "id": "CE:H6",
      "source": "integration-and-duality.md",
      "source_sha256": "c869f6983799baa96d4d39f439d0bb41692dd76e10e3353b6608bde48f5aa971",
      "proof_locator": "## H4.",
      "dependencies": [
        "CE:G1",
        "CE:G10",
        "CE:H3",
        "CE:H4",
        "CE:H5",
        "P3:M7",
        "U001:U001-A4"
      ],
      "scope": "Actual smooth tensor approximation and uniformly bounded strongly continuous operator action on measurable time vectors"
    },
    {
      "id": "CE:H7",
      "source": "integration-and-duality.md",
      "source_sha256": "c869f6983799baa96d4d39f439d0bb41692dd76e10e3353b6608bde48f5aa971",
      "proof_locator": "For \\(g\\in L^1",
      "dependencies": [
        "CE:H4",
        "CE:H2",
        "P3:M3",
        "P3:M4",
        "CE:H1"
      ],
      "scope": "Absolute continuity of the original vector primitive and exact distributional derivative by scalar Fubini and separating duality"
    },
    {
      "id": "CE:H8",
      "source": "integration-and-duality.md",
      "source_sha256": "c869f6983799baa96d4d39f439d0bb41692dd76e10e3353b6608bde48f5aa971",
      "proof_locator": "For completeness the derivative",
      "dependencies": [
        "CE:H6",
        "CE:H7",
        "P3:M2",
        "P3:M3",
        "U001:F0-COMP"
      ],
      "scope": "Complete one-dimensional weak maximal estimate by finite disjoint intervals and compact exhaustion"
    },
    {
      "id": "CE:H9",
      "source": "integration-and-duality.md",
      "source_sha256": "c869f6983799baa96d4d39f439d0bb41692dd76e10e3353b6608bde48f5aa971",
      "proof_locator": "For a continuous approximant",
      "dependencies": [
        "CE:H8",
        "CE:H6",
        "P3:M3"
      ],
      "scope": "Full norm-valued Lebesgue differentiation and actual almost-everywhere primitive derivative"
    },
    {
      "id": "CE:H10",
      "source": "integration-and-duality.md",
      "source_sha256": "c869f6983799baa96d4d39f439d0bb41692dd76e10e3353b6608bde48f5aa971",
      "proof_locator": "If \\(u\\in L^1",
      "dependencies": [
        "CE:H3",
        "CE:H7",
        "CE:H9",
        "U001:U001-A4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "CT:T1"
      ],
      "scope": "Complete vector distributional constant argument, single representative, trace and all-point derivative for continuous forcing"
    },
    {
      "id": "CE:H11",
      "source": "integration-and-duality.md",
      "source_sha256": "c869f6983799baa96d4d39f439d0bb41692dd76e10e3353b6608bde48f5aa971",
      "proof_locator": "## H6.",
      "dependencies": [
        "CE:H1",
        "CE:H2",
        "CE:H3",
        "CE:H5",
        "CT:T1"
      ],
      "scope": "Finite-interval Hilbert L1 duality with full L2 representation and exact measurable norm-attaining test"
    },
    {
      "id": "CE:H12",
      "source": "integration-and-duality.md",
      "source_sha256": "c869f6983799baa96d4d39f439d0bb41692dd76e10e3353b6608bde48f5aa971",
      "proof_locator": "For the Sobolev duality,",
      "dependencies": [
        "CE:H11",
        "CE:G1"
      ],
      "scope": "Exact Sobolev multiplier isometry, original pairing and E2s normalization in the Cauchy receiver"
    }
  ],
  "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"
  ],
  "component_proofs_complete": true,
  "U030_restored": false,
  "full_course_complete": false,
  "public_release_authorized": false
}