{
  "schema": "AN04-restored-global-solvability-proof-map/v1",
  "proofs": [
    {
      "id": "GL:N0",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "**Norm convention.**",
      "dependencies": [
        "PS:PS5",
        "IF:M2",
        "U001:P14.2-powers",
        "P3:M3"
      ],
      "scope": "Actual positive smooth density and global half-density sup norm, with exact compact chart equivalence"
    },
    {
      "id": "GL:G0",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "Fix a positive degree-one cotangent norm",
      "dependencies": [
        "KS:G0",
        "KS:G1",
        "KS:G2"
      ],
      "scope": "Full normalized Hamilton field, compact projected continuation and homogeneous lift at every real order"
    },
    {
      "id": "GL:G1",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "There is no stationary characteristic point",
      "dependencies": [
        "GL:G0",
        "G2:NF1",
        "G2:NF7"
      ],
      "scope": "Global escape excludes zero and periodic projected orbits; every finite interval is injective"
    },
    {
      "id": "GL:B0",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "**(B) implies (C).**",
      "dependencies": [
        "GL:G1",
        "SR:T1",
        "SR:T2",
        "SR:T3",
        "SR:T4",
        "P2:OP3",
        "CH:W1",
        "RC:K8"
      ],
      "scope": "Exact endpoint forcing for the adjoint and projection to every base point of each finite interval"
    },
    {
      "id": "GL:B1",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "**(C) implies (B).**",
      "dependencies": [
        "GL:G0",
        "GL:G1",
        "RP:P0",
        "RP:P4",
        "RP:P5",
        "G2:NF5",
        "G2:NF7",
        "PS:PS5",
        "U001:F0-COMP",
        "CH:W1"
      ],
      "scope": "Full two-direction propagation and actual complete compact limiting orbit, including characteristic and elliptic cases"
    },
    {
      "id": "GL:B2",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "The same argument with forcing smooth everywhere",
      "dependencies": [
        "GL:B1",
        "RP:P0"
      ],
      "scope": "Every compact adjoint distribution with smooth forcing is smooth, without asserting trivial kernel"
    },
    {
      "id": "GL:A0",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "Suppose (C) fails for one compact",
      "dependencies": [
        "PS:PS5",
        "GL:G1",
        "U001:F0-COMP"
      ],
      "scope": "Increasing enlarged exhaustion, escaping marked points, actual last and first closed compact visits"
    },
    {
      "id": "GL:A1",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "The interiors of different",
      "dependencies": [
        "GL:A0",
        "G2:NF1"
      ],
      "scope": "Exact uniqueness and minimal-excursion disjointness, permitting common compact endpoints"
    },
    {
      "id": "GL:A2",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "Apply the exact realization to the half-interval",
      "dependencies": [
        "GL:A0",
        "GL:G1",
        "SR:T1",
        "SR:T2",
        "SR:T3",
        "SR:T4"
      ],
      "scope": "Full half-ray distribution at unattained threshold minus j and exact two forcing rays"
    },
    {
      "id": "GL:A3",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "Here the notation",
      "dependencies": [
        "GL:A2",
        "GL:A1",
        "PS:PS5",
        "CH:W1",
        "RC:K1",
        "RC:K8"
      ],
      "scope": "Exact cutoff separation of endpoint forcing, locally finite supports and one global distribution"
    },
    {
      "id": "GL:A4",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "Suppose \\(Pu-f\\) were smooth",
      "dependencies": [
        "KS:S1",
        "RC:K1",
        "CE:G1",
        "IF:M2",
        "PS:PS5"
      ],
      "scope": "One actual finite negative Sobolev order near a fixed compact set, every Fourier weight and chart transition"
    },
    {
      "id": "GL:A5",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "Choose \\(j\\) with",
      "dependencies": [
        "GL:A1",
        "GL:A2",
        "GL:A3",
        "GL:A4",
        "SP:P1",
        "SP:P5",
        "SP:S1",
        "SP:S3",
        "CH:W1"
      ],
      "scope": "The two original endpoints give opposing Sobolev membership; full forcing-regular shortened interval yields the contradiction at arbitrary m"
    },
    {
      "id": "GL:J0",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "Choose compact exhaustions interleaved",
      "dependencies": [
        "GL:B2",
        "PS:PS5"
      ],
      "scope": "Actual exhaustion and singular-support bounds with the justified empty initial sets"
    },
    {
      "id": "GL:V0",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "This is a Fréchet space.",
      "dependencies": [
        "GL:N0",
        "GL:J0",
        "TG:B2",
        "TG:B3",
        "CE:G1",
        "P2:OP3",
        "P2:OP5",
        "RC:K8",
        "P3:M3"
      ],
      "scope": "Full fixed-support continuous completeness, smooth forcing completeness and distributional graph-limit compatibility"
    },
    {
      "id": "GL:V1",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "By (GS9), restriction",
      "dependencies": [
        "GL:V0",
        "GL:J0",
        "KS:F3"
      ],
      "scope": "Everywhere defined restriction to the precise outer smooth domain, closed graph and every seminorm estimate"
    },
    {
      "id": "GL:V2",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "One consequence will be used explicitly.",
      "dependencies": [
        "GL:V1",
        "TG:B0",
        "TG:B3",
        "PS:PS5",
        "U001:F0-COMP",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Actual finite-grid bound, all-derivative diagonal compactness, compatible derivative limits and coordinate gluing"
    },
    {
      "id": "GL:D0",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "Also choose a countable family",
      "dependencies": [
        "GL:N0",
        "PS:PS5",
        "U001:P14.3",
        "P3:M3",
        "P3:M4",
        "U001:F0-COMP"
      ],
      "scope": "Countable chart bump averages with positive normalization and complete separation of continuous sections"
    },
    {
      "id": "GL:E0",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "Inductively suppose that",
      "dependencies": [
        "GL:J0"
      ],
      "scope": "Full finite-stage estimate on its exact support and vacuous zero initial stage"
    },
    {
      "id": "GL:E1",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "We claim that for some finite",
      "dependencies": [
        "GL:E0",
        "GL:V2",
        "GL:D0",
        "TG:B2"
      ],
      "scope": "Correct normalized failure sequence, vanishing exterior forcing and detectors, and full smooth convergence to zero"
    },
    {
      "id": "GL:E2",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "Choose \\(\\chi\\in\\mathcal D",
      "dependencies": [
        "GL:E1",
        "PS:PS5",
        "P2:OP3",
        "P2:OP5",
        "CE:G1",
        "RC:K8"
      ],
      "scope": "One cutoff has the required interior support; proper all-order continuity controls every remainder and gives the strict finite-stage contradiction"
    },
    {
      "id": "GL:E3",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "Choose positive \\(\\varepsilon_j\\)",
      "dependencies": [
        "GL:E2",
        "U001:P15.1",
        "U001:L5.4.1",
        "U001:F0-COMP"
      ],
      "scope": "Summable relative constant losses give an actual finite uniform global bound"
    },
    {
      "id": "GL:E4",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "At each step the added seminorm",
      "dependencies": [
        "GL:E2",
        "GL:E3",
        "GL:J0",
        "RC:K1",
        "RC:K8"
      ],
      "scope": "Actual finite-on-tests increasing limit seminorm, fixed-support finite-order continuity and locally finite detector supports"
    },
    {
      "id": "GL:H0",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "Define the linear map",
      "dependencies": [
        "GL:E4"
      ],
      "scope": "Correct complex-linear graph map, conjugate-linear image functional and representative independence"
    },
    {
      "id": "GL:H1",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "Apply the complete programme complex seminorm",
      "dependencies": [
        "GL:H0",
        "KS:F4"
      ],
      "scope": "Full arbitrary-subspace complex Hahn-Banach under the original seminorm, with exact conjugation"
    },
    {
      "id": "GL:H2",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "Its restriction to \\(\\mathcal D(X)\\)",
      "dependencies": [
        "GL:H1",
        "GL:E4",
        "RC:K1",
        "RC:K8",
        "U001:F0-COMP"
      ],
      "scope": "The extension defines one global distribution and a bounded conjugate-linear ell-one functional by the actual finite-sequence limit"
    },
    {
      "id": "GL:H3",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "The last sum is locally finite",
      "dependencies": [
        "GL:H2",
        "GL:E4",
        "P2:OP3",
        "RC:K8",
        "TG:B3"
      ],
      "scope": "The exact residual sign and pairing produce one globally smooth locally finite sum and solve modulo smooth"
    },
    {
      "id": "GL:T0",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "**Theorem 1.1.**",
      "dependencies": [
        "GL:B0",
        "GL:B1",
        "GL:A5",
        "GL:H3"
      ],
      "scope": "All three equivalences with exact universal compact and distribution quantifiers, for every real operator order"
    },
    {
      "id": "GL:X1",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "**1. An empty characteristic set.**",
      "dependencies": [
        "GL:T0",
        "KS:D0",
        "KS:S1"
      ],
      "scope": "Unchanged full identity example and the exact generic versus elliptic regularity comparison"
    },
    {
      "id": "GL:X2",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "**2. A missing boundary point",
      "dependencies": [
        "GL:T0",
        "GL:G0",
        "GL:A5",
        "U001:F0-COMP",
        "U001:P2"
      ],
      "scope": "Unchanged slit-domain example, actual complete strips and compact endpoints; positive-distance proof establishes escape and obstruction"
    },
    {
      "id": "GL:X3",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "**3. Which topology controls the extension?**",
      "dependencies": [
        "GL:E4",
        "GL:H2",
        "GL:H3",
        "U001:P14.3",
        "TG:B3"
      ],
      "scope": "Unchanged full test-space versus smooth-space example and exact local finiteness requirement"
    },
    {
      "id": "GL:V3",
      "source": "global-solvability-modulo-smooth.md",
      "source_sha256": "005f659efd9d86f7c1d4e41c00c5b974bcaafcc2a0d1df2e957ec2181c863e7d",
      "proof_locator": "![Base projections",
      "dependencies": [
        "GL:X2",
        "G2:G0"
      ],
      "scope": "Unchanged exact affine slit-domain figure, positive dy covectors, forward flow and stated numerical samples"
    }
  ],
  "earlier_maps": [
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    "../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",
    "../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",
    "../20261005-restored-compact-tube/proof-map.json",
    "../20261005-restored-singular-ray/proof-map.json",
    "../20261005-restored-compact-solvability/proof-map.json"
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  "component_proofs_complete": true,
  "U039_restored": true,
  "full_course_complete": false,
  "public_release_authorized": false
}