{
  "schema": "AN04-restored-global-flow-proof-map/v1",
  "proofs": [
    {
      "id": "GF:L0",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "Under (A),",
      "dependencies": [
        "G2:NF1",
        "G2:NF5",
        "G2:NF7",
        "PS:PS5",
        "U001:F0-COMP"
      ],
      "scope": "No zero or periodic orbit and no compact half-curve, with full maximal continuation and compact limiting complete flow"
    },
    {
      "id": "GF:P0",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "The differential of (FG2)",
      "dependencies": [
        "G2:NF1",
        "G2:NF5",
        "U001:P2"
      ],
      "scope": "Exact full-source-rank differential and injectivity under nonperiodicity, separately supplied by A or B"
    },
    {
      "id": "GF:P1",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "We prove properness,",
      "dependencies": [
        "GF:L0",
        "GF:P0",
        "G2:NF5",
        "G2:NF7",
        "PS:PS5",
        "U001:F0-COMP",
        "TG:B0"
      ],
      "scope": "Properness in the actual open maximal domain, compact endpoint subsequences, exclusion of infinite times and finite endpoint continuation"
    },
    {
      "id": "GF:P2",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "A proper injective immersion",
      "dependencies": [
        "GF:P1",
        "GF:P0",
        "U001:P3",
        "U001:F0-COMP"
      ],
      "scope": "Actual local graph coordinates, continuous global inverse, smooth embedding and closed image"
    },
    {
      "id": "GF:D0",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "For completeness, (B) also gives",
      "dependencies": [
        "GF:P0",
        "U001:P2",
        "U001:P3",
        "PS:PS5",
        "TG:B0",
        "TG:B1",
        "U001:F0-COMP"
      ],
      "scope": "Automatic relation dimension by smooth embedded factorization, countable compact graph images and full Baire contradiction"
    },
    {
      "id": "GF:B0",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "The inverse function theorem now",
      "dependencies": [
        "GF:D0",
        "U001:P3",
        "G2:NF7",
        "U001:F0-COMP"
      ],
      "scope": "B gives the actual proper diffeomorphism, excludes trapped complete curves and yields compact return through interval interpolation"
    },
    {
      "id": "GF:Q0",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "Assume the equivalent conditions",
      "dependencies": [
        "GF:P2",
        "GF:B0",
        "G2:NF1",
        "G2:NF5"
      ],
      "scope": "Open saturation and actual countable quotient basis"
    },
    {
      "id": "GF:Q1",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "The quotient is Hausdorff.",
      "dependencies": [
        "GF:P2",
        "GF:B0",
        "GF:Q0"
      ],
      "scope": "Closed relation gives disjoint open saturated neighborhoods of different trajectories"
    },
    {
      "id": "GF:Q2",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "Choose a small smooth transversal",
      "dependencies": [
        "GF:P2",
        "GF:B0",
        "GF:Q0",
        "GF:Q1",
        "G2:NF1",
        "G2:NF5",
        "U001:P2",
        "U001:P3"
      ],
      "scope": "Flow-box inverse, genuinely injective transversal charts and full smooth quotient transitions"
    },
    {
      "id": "GF:Q3",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "For the partition in the next paragraph",
      "dependencies": [
        "GF:Q2",
        "PS:PS5",
        "TG:B0"
      ],
      "scope": "Partition inside the actual local section domains and complete zero-dimensional discrete case"
    },
    {
      "id": "GF:Q4",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "Each trajectory has an intrinsic affine time coordinate",
      "dependencies": [
        "GF:Q2",
        "GF:Q3",
        "GF:P2",
        "GF:B0",
        "U001:F0-CALC"
      ],
      "scope": "Affine weighted averaging stays inside each possibly incomplete interval and gives one smooth section, including zero extension of weighted terms"
    },
    {
      "id": "GF:Q5",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "\\Omega=\\{(q,t):",
      "dependencies": [
        "GF:Q4",
        "G2:NF1",
        "G2:NF5",
        "G2:NF7",
        "U001:P3"
      ],
      "scope": "Exact open interval-fiber flow domain, global bijection and compatible smooth inverses"
    },
    {
      "id": "GF:C0",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "Finally, (C) implies (A).",
      "dependencies": [
        "U001:F0-COMP",
        "G2:NF7"
      ],
      "scope": "Complete proof from the assumed C coordinates: no relatively compact maximal curve and compact return interpolation"
    },
    {
      "id": "GF:T0",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "**Theorem 1.1.**",
      "dependencies": [
        "GF:L0",
        "GF:P1",
        "GF:P2",
        "GF:D0",
        "GF:B0",
        "GF:Q5",
        "GF:C0"
      ],
      "scope": "All three equivalences, proper full-domain flow relation and component dimensions with empty case"
    },
    {
      "id": "GF:R0",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "Let \\(M\\) now be a smooth conic manifold",
      "dependencies": [
        "PS:PS5",
        "GF:Q3",
        "U001:F0-CALC"
      ],
      "scope": "Positive homogeneous radius from actual local products and subordinate smooth partition"
    },
    {
      "id": "GF:R1",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "The map \\(y\\mapsto",
      "dependencies": [
        "GF:R0",
        "GF:T0",
        "U001:P3"
      ],
      "scope": "Actual equivariant radial-product diffeomorphism and scaling-invariant radial coefficient"
    },
    {
      "id": "GF:R2",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "The interval from zero to",
      "dependencies": [
        "GF:R1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:P15.1"
      ],
      "scope": "Full smooth negative-integral correction, positive invariant radius and inverse on every original interval fiber"
    },
    {
      "id": "GF:S0",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "These estimates are invariant",
      "dependencies": [
        "GF:R2",
        "U001:F0-CALC",
        "U001:P14.2-powers",
        "U001:F0-COMP"
      ],
      "scope": "Every base/radial derivative under conic coordinate changes, including the bounded low-radius interval"
    },
    {
      "id": "GF:S1",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "For the complex integrating factor",
      "dependencies": [
        "U001:P15.1",
        "U001:P16.1",
        "U001:P16.2",
        "U001:F0-CALC"
      ],
      "scope": "Actual complex exponential, nonvanishing inverse and every compact parameter derivative"
    },
    {
      "id": "GF:S2",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "For \\(f\\in S^\\mu(M)\\),",
      "dependencies": [
        "GF:R2",
        "GF:S0",
        "GF:S1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:F0-COMP"
      ],
      "scope": "Exact complex integrating-factor solution, zero section, compact interpolation and every symbol derivative with exact order"
    },
    {
      "id": "GF:S3",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "The relation manifold in these coordinates",
      "dependencies": [
        "GF:R2",
        "GF:T0",
        "GF:S2"
      ],
      "scope": "Full relation labels retain R; projected quotient labels omit R; the actual open interval-fiber domain gives the full repeated transport construction"
    },
    {
      "id": "GF:N0",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "Let \\(N=\\{p=0\\}\\subset",
      "dependencies": [
        "G2:G0",
        "G2:G41",
        "U001:P2",
        "U001:P3",
        "PS:PS5",
        "KS:G0",
        "KS:G2"
      ],
      "scope": "Nonzero characteristic differential, independent radius differential and actual smooth characteristic/cosphere manifolds at arbitrary order"
    },
    {
      "id": "GF:N1",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "Set \\(\\widetilde p=",
      "dependencies": [
        "GF:N0",
        "KS:G0",
        "KS:G1",
        "KS:G2",
        "GL:T0",
        "PS:PS5",
        "GF:T0"
      ],
      "scope": "Full all-real normalization, no compact projected orbit and transfer of base compact-return bounds to the actual characteristic cosphere"
    },
    {
      "id": "GF:N2",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "If \\(n=1\\),",
      "dependencies": [
        "GF:N0",
        "GF:N1",
        "GF:R2",
        "GF:S3",
        "GF:Q2"
      ],
      "scope": "Exact dimensions, empty one-dimensional characteristic case and closed embedded conic same-trajectory relation"
    },
    {
      "id": "GF:N3",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "We prove closedness also",
      "dependencies": [
        "GF:N2",
        "GF:R2",
        "PS:PS5",
        "U001:F0-COMP"
      ],
      "scope": "Positive invariant-radius ratio bounds and actual ambient closure excluding one zero endpoint covector"
    },
    {
      "id": "GF:L1",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "Use \\(\\omega=",
      "dependencies": [
        "G2:G0",
        "G2:G1",
        "G2:NF2",
        "G2:NF4",
        "G2:NF7",
        "U001:P2",
        "U001:F0-CALC"
      ],
      "scope": "Correct symplectic matrix and variational identity, with finite-chart composition over the full flow interval"
    },
    {
      "id": "GF:L2",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "Parameterize a relation neighborhood",
      "dependencies": [
        "GF:N2",
        "GF:L1",
        "G2:G0"
      ],
      "scope": "Every time/source tangent variation and mixed term, difference-form vanishing and exact Lagrangian dimension"
    },
    {
      "id": "GF:L3",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "The kernel convention identifies",
      "dependencies": [
        "GF:L2",
        "GF:N3",
        "G2:G0",
        "G2:NF1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Correct input covector sign and full maximal-orbit invariance under any real nonvanishing multiplier"
    },
    {
      "id": "GF:L4",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "The construction also respects",
      "dependencies": [
        "GF:L3",
        "G2:G0",
        "G2:G1",
        "G2:NF1"
      ],
      "scope": "Exact symplectic naturality with correct full-domain versus restricted-interval qualification"
    },
    {
      "id": "GF:V0",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "### An exact model",
      "dependencies": [
        "GF:R2",
        "U001:P15.1",
        "U001:L5.4.1",
        "U001:P14.2-powers"
      ],
      "scope": "Exact finite-interval radial drift, unchanged maximal endpoints, invariant coordinate and accurately identified section"
    },
    {
      "id": "GF:X1",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "**1. A quotient that fails",
      "dependencies": [
        "GF:T0",
        "GF:Q1",
        "GL:X2"
      ],
      "scope": "Full unchanged slit-quotient non-Hausdorff example and precise missing return hypothesis"
    },
    {
      "id": "GF:X2",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "**2. Radial drift and resonant",
      "dependencies": [
        "GF:R2",
        "GF:S2",
        "GF:S1",
        "U001:P14.2-powers",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full unchanged all-real-power radial drift solution, complex resonance and every radial derivative"
    },
    {
      "id": "GF:X3",
      "source": "global-time-and-bicharacteristic-relation.md",
      "source_sha256": "c1f4ea05fbe312047171eb54c516e42621380a69d32e53d370fc723c18aaabfe",
      "proof_locator": "**3. The free transport",
      "dependencies": [
        "GF:N2",
        "GF:N3",
        "GF:L1",
        "GF:L2",
        "U001:P2",
        "U001:F0-COMP"
      ],
      "scope": "Full unchanged canonical relation, exact dimension, symplectic cancellation, ambient closure and compact return"
    }
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