{
  "schema": "AN04-restored-compact-tube-proof-map/v1",
  "proofs": [
    {
      "id": "TUBE:Q0",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "## 1.",
      "dependencies": [
        "G2:G0",
        "G2:G41",
        "U001:P2",
        "U001:P3"
      ],
      "scope": "Exact cotangent signs, actual quotient atlas, radial kernel and nontrivial closed-interval hypotheses"
    },
    {
      "id": "TUBE:Q1",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "**Proof.** The homogeneous Hamilton identity",
      "dependencies": [
        "TUBE:Q0",
        "G2:NF1",
        "G2:NF7",
        "FN:R0",
        "U001:P2"
      ],
      "scope": "Projected autonomous uniqueness, nonradialness including endpoints and n1 exclusion"
    },
    {
      "id": "TUBE:D0",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "**Construct the map and its common domain.**",
      "dependencies": [
        "TUBE:Q1",
        "FN:R1",
        "PC:H1",
        "PC:H2",
        "PC:H3",
        "PC:H4",
        "PC:H5"
      ],
      "scope": "Marked homogeneous initial chart, Hamilton transport and normalized transverse coordinates"
    },
    {
      "id": "TUBE:D1",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "Here is why one transverse ball",
      "dependencies": [
        "TUBE:D0",
        "G2:NF0",
        "G2:NF1",
        "G2:NF5",
        "G2:NF6",
        "G2:NF7",
        "U001:F0-COMP"
      ],
      "scope": "Complete finite continuation in both time directions, one parameter neighborhood and all derivatives"
    },
    {
      "id": "TUBE:D2",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "The Hamilton flow commutes",
      "dependencies": [
        "TUBE:D1",
        "G2:G1",
        "G2:G41",
        "G2:NF7"
      ],
      "scope": "Dilation extension for every positive radius, exact central characteristic, p=tau and initial chart agreement"
    },
    {
      "id": "TUBE:S1",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "**Preservation of the symplectic form.**",
      "dependencies": [
        "TUBE:D2",
        "G2:G1",
        "G2:F2"
      ],
      "scope": "Full time-slice symplectic identity under actual flow"
    },
    {
      "id": "TUBE:S2",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "The mixed time components",
      "dependencies": [
        "TUBE:S1",
        "TUBE:D2",
        "G2:G0",
        "U001:P2",
        "U001:P3"
      ],
      "scope": "Correct mixed-time sign, full two-form identity and invertible differential"
    },
    {
      "id": "TUBE:S3",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "Homogeneity identifies the radial fields",
      "dependencies": [
        "TUBE:S2",
        "G2:G41",
        "G2:F1"
      ],
      "scope": "Exact primitive equality by radial contraction without unknown exact term"
    },
    {
      "id": "TUBE:I1",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "**Global injectivity after one shrinking.**",
      "dependencies": [
        "TUBE:Q0",
        "TUBE:S2",
        "TUBE:D1",
        "U001:P3",
        "U001:F0-COMP"
      ],
      "scope": "Projected local inverse and compact limiting-collision contradiction including both endpoints"
    },
    {
      "id": "TUBE:I2",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "Take \\(V\\) to be the conic domain",
      "dependencies": [
        "TUBE:I1",
        "TUBE:D2",
        "TUBE:S2",
        "U001:P3"
      ],
      "scope": "Radial-scale uniqueness, full global injectivity and smooth open inverse"
    },
    {
      "id": "TUBE:I3",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "Finally \\(V\\) is parametrized",
      "dependencies": [
        "TUBE:I2",
        "U001:F0-COMP",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Explicit based contraction CTA1, time convexity, arbitrary graph-neighborhood shrinking and endpoints"
    },
    {
      "id": "TUBE:M1",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "**Proof of the symbol choice.**",
      "dependencies": [
        "TUBE:I3",
        "GS:Z5",
        "GS:Z6",
        "MT:I0",
        "MT:I2",
        "U001:F0-COMP"
      ],
      "scope": "Intrinsic fourth-root flat line, compact-grid homotopy cancellation, actual nonzero global smooth flat section"
    },
    {
      "id": "TUBE:M2",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "Choose the section on the normalized slice",
      "dependencies": [
        "TUBE:M1",
        "GS:Z10",
        "GE:D0",
        "U001:P14.2-powers"
      ],
      "scope": "Dilation-invariant section, exact symplectic half-density degree, all derivatives and inverse-symbol estimates"
    },
    {
      "id": "TUBE:M3",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "Choose a smooth degree-zero cutoff",
      "dependencies": [
        "TUBE:M2",
        "GS:Z11",
        "GS:Z12",
        "GE:D2",
        "U001:P14.3",
        "U001:F0-COMP"
      ],
      "scope": "Finite local phase realization and homogeneous partition, exact transition gluing and arbitrary wavefront confinement"
    },
    {
      "id": "TUBE:M4",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "The input and output base images",
      "dependencies": [
        "TUBE:M3",
        "TUBE:D2",
        "U001:F0-COMP",
        "U001:P14.3",
        "RC:C1",
        "P2:OP5"
      ],
      "scope": "Uniform radial comparability on compact normalized support, no axes, compact proper kernel cutoffs"
    },
    {
      "id": "TUBE:M5",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "**Both inverses on one smaller tube.**",
      "dependencies": [
        "TUBE:M4",
        "GE:G0",
        "GE:D9",
        "RC:C9",
        "K:K3",
        "CH:T2",
        "P2:OP5",
        "P2:OP6"
      ],
      "scope": "Both graph parametrices on one matched tube, ordered inverse comparison and full smoothing ideal"
    },
    {
      "id": "TUBE:E1",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "**Proof.** Take \\(A_1,B_1\\)",
      "dependencies": [
        "TUBE:M5",
        "GE:D10",
        "TUBE:D2",
        "P2:OP6"
      ],
      "scope": "Full ordered Egorov, arbitrary real graph order and actual Psi0 remainder without homogeneous lower-term assumption"
    },
    {
      "id": "TUBE:E2",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "Choose a time interval containing",
      "dependencies": [
        "TUBE:E1",
        "TUBE:I3",
        "GE:D12",
        "GE:D13",
        "GE:D14",
        "GE:D15",
        "GE:D16",
        "GE:D17",
        "GE:D18"
      ],
      "scope": "One finite product cylinder, exact global symbol extension and complete ordered transport/residual construction"
    },
    {
      "id": "TUBE:E3",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "This application uses the complete transport",
      "dependencies": [
        "TUBE:E2",
        "GE:D13",
        "GE:D14",
        "GE:D16",
        "GE:D17",
        "GE:D18"
      ],
      "scope": "Uniform exponential/inverse bounds, all ordinary derivatives, variation of constants and supported asymptotic sum on same domain"
    },
    {
      "id": "TUBE:E4",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "Define \\(A=A_1A_2\\)",
      "dependencies": [
        "TUBE:E3",
        "TUBE:M5",
        "RC:C9",
        "GE:D19",
        "CH:T2",
        "P2:OP6"
      ],
      "scope": "Both inverse and conjugation identities plus exact CT19/CT20 paired intertwining with every defect"
    },
    {
      "id": "TUBE:E5",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "The theorem treats scalar principal direction",
      "dependencies": [
        "TUBE:E4",
        "GE:D13",
        "GE:D19",
        "RP:P1"
      ],
      "scope": "Precise compatible finite-matrix extension and all-real-order reduction with original order-one flow time"
    },
    {
      "id": "TUBE:X1",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "**Exercise 6.1",
      "dependencies": [
        "TUBE:Q0",
        "TUBE:I1",
        "TUBE:I2",
        "TUBE:S3",
        "G2:G0",
        "U001:P16.1",
        "U001:P16.2"
      ],
      "scope": "complete periodic-ray solution including equality endpoint and exact cotangent lift"
    },
    {
      "id": "TUBE:X2",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "**Exercise 6.2",
      "dependencies": [
        "TUBE:Q1",
        "TUBE:D2",
        "G2:G0",
        "U001:P15.1",
        "G2:NF1"
      ],
      "scope": "complete radial exponential example and independent-vector obstruction"
    },
    {
      "id": "TUBE:X3",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "**Exercise 6.3",
      "dependencies": [
        "TUBE:D2",
        "TUBE:S3",
        "U001:P16.1",
        "U001:P16.2",
        "U001:P16.3",
        "U001:P2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "complete tangent tube, exact flow/primitive, inverse, radial norm bounds and missing endpoint"
    },
    {
      "id": "TUBE:F1",
      "source": "compact-homogeneous-characteristic-tubes.md",
      "source_sha256": "96f353b17b78aed91fe001ec8220cf3a6a1bff5b40ec9724b02f985801161f55",
      "proof_locator": "The figure uses the explicit chart",
      "dependencies": [
        "TUBE:X3",
        "U001:P16.1",
        "U001:P16.2",
        "U001:P16.3"
      ],
      "scope": "Exact T1 base/covector diagram, positive minimum, true normalized probe and transverse omissions"
    }
  ],
  "earlier_maps": [
    "../20261004-free-stationary-phase/proof-map.json",
    "../20261004-free-intrinsic-graph/proof-map.json",
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    "../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",
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    "../20261005-restored-folded-forms/proof-map.json",
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    "../20261005-restored-invariant-folds/proof-map.json",
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    "../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",
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    "../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",
    "../20261005-positivity-foundations/proof-map.json",
    "../20261005-metric-weyl-calculus/proof-map.json",
    "../20261005-metric-positivity/proof-map.json",
    "../20261005-restored-principal-boundary/proof-map.json",
    "../20261005-restored-sobolev-propagation/proof-map.json"
  ],
  "external_dependency_policy": {
    "record": "../20261004-free-stationary-phase/external-proof-dependencies.json",
    "notice": "The elementary proof providers have exact earlier-programme routes in C10/C20, with public PDF destinations and editable TeX passages. The author English edition remains available. Human source text and figures stay external to this CC0 export and retain their own terms.",
    "mathematical_proofs_rewritten": false
  }
}