{
  "schema": "AN04-restored-global-signed-proof-map/v1",
  "proofs": [
    {
      "id": "GP:N0",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "Let \\(X\\) be a Hausdorff",
      "dependencies": [
        "GF:N0",
        "GF:N1",
        "GF:N2",
        "GF:N3",
        "GF:Q5",
        "RC:C0"
      ],
      "scope": "Full global real-principal-type, compact-return and half-density hypotheses; actual closed relation and nonzero endpoints"
    },
    {
      "id": "GP:P0",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "Choose a locally finite cover",
      "dependencies": [
        "GP:N0",
        "PS:PS5",
        "U001:F0-COMP",
        "LP:C0",
        "LP:C1"
      ],
      "scope": "Base-locally-finite coordinate and conic cover with finite compact angular subdivisions and every ordinary derivative bound"
    },
    {
      "id": "GP:P1",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "Quantize each",
      "dependencies": [
        "GP:P0",
        "P2:OP4",
        "P2:OP5",
        "CH:W4",
        "K:K1a"
      ],
      "scope": "Proper full-microsupport quantization, actual locally finite sum and both proper projections"
    },
    {
      "id": "GP:P2",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "The programme elliptic inverse theorem",
      "dependencies": [
        "GP:P1",
        "GE:G0",
        "K:K1",
        "K:K2",
        "K:K3"
      ],
      "scope": "Complete proper two-sided order-zero inverse and retention of the full smooth partition error"
    },
    {
      "id": "GP:P3",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "A simultaneous choice can be described",
      "dependencies": [
        "GP:P0",
        "GP:P2",
        "PS:PS5",
        "P2:OP4",
        "P2:OP6",
        "U001:F0-COMP"
      ],
      "scope": "Locally finite closed forbidden union, one simultaneous proper diagonal cutoff and preservation of both inverse identities"
    },
    {
      "id": "GP:P4",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "Set \\(T_j=Q_jB\\).",
      "dependencies": [
        "GP:P1",
        "GP:P2",
        "GP:P3",
        "CH:W4",
        "K:K4",
        "RC:C9"
      ],
      "scope": "Actual compactly supported corrected partition, full conic separation and principal weight with exact smooth residual"
    },
    {
      "id": "GP:L0",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "On an elliptic patch",
      "dependencies": [
        "GP:P4",
        "K:K3",
        "P2:OP4",
        "CH:W4",
        "CM:W3",
        "CM:W4"
      ],
      "scope": "Full conic elliptic inverse and all support-cutoff commutators smoothing"
    },
    {
      "id": "GP:L1",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "On a characteristic patch",
      "dependencies": [
        "GP:P4",
        "LP:C3",
        "LP:C4",
        "LP:C5",
        "LP:F0",
        "LP:F1",
        "LP:F2",
        "LP:F3",
        "LP:F4",
        "LP:F5"
      ],
      "scope": "Actual compact local signed pieces, fixed travel gap, all ordinary residual terms and every-real mapping"
    },
    {
      "id": "GP:L2",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "The first sum is finite",
      "dependencies": [
        "GP:L0",
        "GP:L1",
        "GP:P4",
        "P2:OP5",
        "RC:C1"
      ],
      "scope": "Actual proper base-locally-finite kernel sum and locally finite full residual after applying proper P"
    },
    {
      "id": "GP:L3",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "The order statement is local",
      "dependencies": [
        "GP:L2",
        "LP:F3",
        "OSC:O2",
        "IR:R1"
      ],
      "scope": "Strict global residual excludes diagonal accumulation by a finite common normalized neighborhood"
    },
    {
      "id": "GP:L4",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "The elliptic pieces cancel exactly.",
      "dependencies": [
        "GP:L0",
        "GP:L1",
        "GP:L2",
        "GE:D7",
        "LP:M2",
        "IF:M2"
      ],
      "scope": "Ordinary difference and actual finite compact-local bounds for every positive, negative and noninteger Sobolev order"
    },
    {
      "id": "GP:C0",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "Let \\(S_\\pm=",
      "dependencies": [
        "GP:L3",
        "KC:C0",
        "DT:H0",
        "DT:H1",
        "DT:H2",
        "DT:H3"
      ],
      "scope": "Closed strict residual, full controlled correction and closed reflexive directional hull including its forcing boundary"
    },
    {
      "id": "GP:C1",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "We verify the geometric hypotheses",
      "dependencies": [
        "GP:N0",
        "G2:G0",
        "CR:L1",
        "CR:R1",
        "GF:N2",
        "U001:P2"
      ],
      "scope": "Exact corank two of the pulled-back symplectic form and exclusion of both individual lifted radial vectors"
    },
    {
      "id": "GP:C2",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "The sufficient corank theorem now gives",
      "dependencies": [
        "GP:C0",
        "GP:C1",
        "CR:A1",
        "CR:A2",
        "CR:A3",
        "GF:N3"
      ],
      "scope": "All-real correction bound through finite closed normalized phase cover; no false properness requirement"
    },
    {
      "id": "GP:C3",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "Set \\(E_\\pm=F_\\pm-G_\\pm\\).",
      "dependencies": [
        "GP:L4",
        "GP:C0",
        "GP:C2",
        "RC:C1"
      ],
      "scope": "Actual signed right kernels, exact residual cancellation, ordinary difference and compact-input domain"
    },
    {
      "id": "GP:A0",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "The construction above applies to",
      "dependencies": [
        "GP:C3",
        "P2:OP3",
        "RC:C2",
        "CM:A0",
        "CM:A1"
      ],
      "scope": "Actual opposite-sign constructions for P adjoint supply matching left inverses and both identities"
    },
    {
      "id": "GP:A1",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "If another right or left parametrix",
      "dependencies": [
        "GP:A0",
        "CM:I1",
        "CM:I2"
      ],
      "scope": "Unconditional same-sign uniqueness through the constructed opposite-sided parametrix"
    },
    {
      "id": "GP:D0",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "Here is its construction and the compatibility check.",
      "dependencies": [
        "LP:F4",
        "LP:D0",
        "FN:R1",
        "G2:G1",
        "G2:F2",
        "GS:Z10",
        "GS:Z11",
        "RC:C8",
        "GE:D9"
      ],
      "scope": "Exact common jump section, tau-preserving transition, reduced symplectic volume and cancellation of graph, density and Maslov factors"
    },
    {
      "id": "GP:D1",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "For a local characteristic piece",
      "dependencies": [
        "GP:D0",
        "GP:P4",
        "GP:L4",
        "LP:F4",
        "RC:C8",
        "GS:Z12"
      ],
      "scope": "Every partition weight multiplies the same symbol class; local finite sum has no diagonal cancellation"
    },
    {
      "id": "GP:D2",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "In a compact symbol-line trivialization",
      "dependencies": [
        "GP:D1",
        "K:K2",
        "U001:F0-CALC",
        "U001:F0-COMP"
      ],
      "scope": "Uniform high-frequency lower bound and every reciprocal derivative estimate at the diagonal"
    },
    {
      "id": "GP:D3",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "The correction kernels \\(G_\\pm\\)",
      "dependencies": [
        "GP:C0",
        "GP:C3",
        "GP:D2",
        "DT:B0",
        "DT:B1",
        "DT:B2",
        "DT:B3",
        "DT:B4",
        "ST:T8",
        "ST:T9",
        "ST:T10",
        "KC:P2"
      ],
      "scope": "Smoothing of corrections near the diagonal, actual homogeneous degree-zero frame and ordinary complex transport equation"
    },
    {
      "id": "GP:D4",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "Fix a finite characteristic interval",
      "dependencies": [
        "GP:D3",
        "GF:Q5",
        "DT:M0",
        "DT:T0",
        "DT:T1",
        "DT:T2",
        "DT:T3",
        "KC:T1",
        "KC:T2"
      ],
      "scope": "Variation of constants in the actual interval domain, full moving-endpoint estimates, nonzero compact lower bound and all inverse symbol derivatives"
    },
    {
      "id": "GP:D5",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "The implication from a noncharacteristic symbol",
      "dependencies": [
        "GP:D4",
        "IR:R2",
        "IR:R3",
        "IR:R4",
        "GS:Z12",
        "RP:F2",
        "CH:W4"
      ],
      "scope": "Global noncharacteristic difference and exact signed wavefront equality including every full-diagonal covector and empty characteristic set"
    },
    {
      "id": "GP:O0",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "Let now \\(P\\in\\Psi^m_",
      "dependencies": [
        "GP:A0",
        "P2:OP3",
        "P2:OP5",
        "P2:OP6",
        "K:K3",
        "PS:PS5",
        "GF:T0"
      ],
      "scope": "Positive all-real-order normalization, full ordinary order-zero lower terms and preservation of maximal oriented strips"
    },
    {
      "id": "GP:O1",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "This is an actual input-side composition:",
      "dependencies": [
        "GP:O0",
        "CM:W3",
        "CM:W4",
        "RC:C1",
        "RC:C9"
      ],
      "scope": "Correct input-side E1 Q, actual proper products and both inverse identities with every smooth error retained"
    },
    {
      "id": "GP:O2",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "For every real \\(s\\), the precise scale",
      "dependencies": [
        "GP:O1",
        "GP:C2",
        "GE:D7",
        "RC:C8",
        "RC:C9",
        "GP:D5"
      ],
      "scope": "Exact gain m-1, noncharacteristic difference order 1/2-m and full wavefront for every real operator order"
    },
    {
      "id": "GP:O3",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "Same-sign uniqueness for the order-",
      "dependencies": [
        "GP:O1",
        "GP:A1",
        "CM:I1",
        "CM:W4"
      ],
      "scope": "All-real uniqueness and alternate right-inverse reduction with full proper smooth error control"
    },
    {
      "id": "GP:M0",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "Suppose \\(N=N_+",
      "dependencies": [
        "GP:N0",
        "G2:NF7",
        "PS:PS5"
      ],
      "scope": "Automatic conicity and whole-strip invariance of both relatively clopen characteristic parts"
    },
    {
      "id": "GP:M1",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "To spell out that full-symbol choice,",
      "dependencies": [
        "GP:M0",
        "GP:P0",
        "PS:PS5",
        "CH:T0",
        "CH:T3",
        "CH:W4",
        "P2:OP4",
        "P2:OP5",
        "P2:OP6",
        "CM:W3"
      ],
      "scope": "Actual full-symbol identity/zero separation and exact residual split with all orders and proper support controlled"
    },
    {
      "id": "GP:M2",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "Apply the forward correction to",
      "dependencies": [
        "GP:M1",
        "GP:C0",
        "GP:C2",
        "GP:A0",
        "CM:R4",
        "CM:I1"
      ],
      "scope": "Both directional closed hulls, both right and left constructions and mixed-sign compact-return comparison"
    },
    {
      "id": "GP:M3",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "At the characteristic diagonal the first",
      "dependencies": [
        "GP:M0",
        "GP:M2",
        "GP:D1",
        "GP:D4",
        "GP:D5",
        "GP:O1",
        "GP:O2",
        "GP:O3"
      ],
      "scope": "Signed common jump, no cancellation, noncharacteristic full difference, exact mixed relations, gain and uniqueness"
    },
    {
      "id": "GP:W0",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "On \\(X=\\mathbb R_t",
      "dependencies": [
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:P15.1",
        "P3:L2",
        "RP:K0",
        "RP:K1",
        "RP:F2",
        "RP:F3"
      ],
      "scope": "Exact complex weighted model, compact return and both differentiated inverse identities"
    },
    {
      "id": "GP:W1",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "To verify the real-order bound directly",
      "dependencies": [
        "GP:W0",
        "LP:M2",
        "GE:D7",
        "IF:M2"
      ],
      "scope": "Actual localized finite-measure factorization and all-real norm bounds with compact inverse weights"
    },
    {
      "id": "GP:V0",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "*Exact coordinate section.*",
      "dependencies": [
        "GP:C0",
        "GP:D1",
        "GP:W0",
        "U001:P2",
        "U001:P15.1"
      ],
      "scope": "Exact affine hull/source polygons, reflexive boundary witness, noncancelling partition weights and complex model coefficients"
    },
    {
      "id": "GP:X1",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "**1. Lower terms in a partition",
      "dependencies": [
        "GP:P1",
        "GP:P2",
        "GP:P4",
        "GP:L3",
        "K:K1"
      ],
      "scope": "Complete unchanged solution retaining and removing the nonsmooth lower-order partition defect"
    },
    {
      "id": "GP:X2",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "**2. Strict errors can accumulate",
      "dependencies": [
        "GP:L3",
        "P3:M3",
        "P3:M4",
        "RP:F2",
        "RP:F3"
      ],
      "scope": "Complete unchanged rescaled strict kernels converging distributionally to the full identity diagonal and failure of local finiteness"
    },
    {
      "id": "GP:X3",
      "source": "global-signed-parametrices-from-local-delayed-errors.md",
      "source_sha256": "ef6f1b721ec878124658a09b8b63008ee83894eb231f0ccd36d524f86291ccbe",
      "proof_locator": "**3. Corank and real operator order",
      "dependencies": [
        "GP:C1",
        "GP:C2",
        "GP:O1",
        "GP:O2"
      ],
      "scope": "Complete unchanged corank threshold, correctly ordered parametrix and every-real gain calculation"
    }
  ],
  "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",
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    "../20261005-restored-prescribed-coordinates/proof-map.json",
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