{
  "schema": "AN04-restored-singular-ray-proof-map/v1",
  "proofs": [
    {
      "id": "SR:P0",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "Put \\(d=n-1",
      "dependencies": [
        "U001:Q3",
        "U001:Q4",
        "U001:P14.3",
        "U001:P14.2-powers",
        "P3:M4"
      ],
      "scope": "Positive Gaussian Fourier profile, empty-dimensional convention and actual tempered inverse"
    },
    {
      "id": "SR:P1",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "**Sobolev norms.**",
      "dependencies": [
        "SR:P0",
        "U001:Q2",
        "P3:M4",
        "P3:L1",
        "P3:L2",
        "U001:L5.4.1"
      ],
      "scope": "Exact Gaussian scaling, every negative Sobolev order and logarithmic failure at zero"
    },
    {
      "id": "SR:P2",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "**Smoothness and L2 away",
      "dependencies": [
        "SR:P0",
        "U001:Q2",
        "U001:Q3",
        "U001:Q4",
        "P3:M3",
        "P3:M4"
      ],
      "scope": "Actual Gaussian inversion, exact constants, compact cutoff Fubini and distributional limit"
    },
    {
      "id": "SR:P3",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "When \\(y\\ne0\\)",
      "dependencies": [
        "SR:P2",
        "U001:Q1",
        "U001:Q2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Every mixed derivative and Gaussian bound, including all finite cutoff-band terms"
    },
    {
      "id": "SR:P4",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "For a precise oscillatory justification",
      "dependencies": [
        "SR:P3",
        "U001:Q3",
        "P3:M3"
      ],
      "scope": "Actual cutoff integration by parts with explicit vanishing tail for every derivative"
    },
    {
      "id": "SR:P5",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "For \\(v\\ne0\\), (SR5)",
      "dependencies": [
        "SR:P3",
        "SR:P4",
        "P3:M4",
        "P3:M5",
        "SR:P1"
      ],
      "scope": "Full L2 estimate outside any origin neighborhood including k0 and all unbounded directions"
    },
    {
      "id": "SR:P6",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "**The full wavefront direction",
      "dependencies": [
        "SR:P0",
        "SR:P4",
        "CH:W1",
        "U001:Q2"
      ],
      "scope": "All differentiated rapid decay outside the single positive cone, actual compact localization and base smoothness"
    },
    {
      "id": "SR:P7",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "It is also not microlocally L2",
      "dependencies": [
        "SR:P5",
        "SR:P6",
        "SP:S3",
        "CH:W1",
        "P3:M4",
        "P3:M5",
        "P3:L1",
        "U001:F0-COMP"
      ],
      "scope": "Weighted convolution estimate, finite sphere cover, exact microlocal failure and positive conicity"
    },
    {
      "id": "SR:R0",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "**An explicit partition",
      "dependencies": [
        "U001:P14.3",
        "U001:P3",
        "U001:F0-COMP",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Explicit rational time coordinate, positive locally finite integer partition, shrinking transverse supports and exact neighborhood sum"
    },
    {
      "id": "SR:R1",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "Put \\(u_j=",
      "dependencies": [
        "SR:P1",
        "SR:R0",
        "RP:W1",
        "RP:W2",
        "CE:G1",
        "P3:M4",
        "CH:W1"
      ],
      "scope": "Actual compact tensor Sobolev norms and exact allowed wavefront direction"
    },
    {
      "id": "SR:R2",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "Let \\(q_j=",
      "dependencies": [
        "SR:R1",
        "U001:P14.3",
        "P3:M3",
        "U001:Q3",
        "RC:K1"
      ],
      "scope": "Compact mollifier residuals, support constraints and dominated-convergence norm control"
    },
    {
      "id": "SR:R3",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "These are finitely many requirements",
      "dependencies": [
        "SR:R2",
        "SR:R1",
        "CH:W1",
        "U001:F0-COMP",
        "U001:Q3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Finite compact base/angular covers, all covector derivatives and simultaneous weighted conic seminorm convergence"
    },
    {
      "id": "SR:R4",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "converges in every \\(H^{-\\epsilon}",
      "dependencies": [
        "SR:R2",
        "SR:R3",
        "P3:M6",
        "P3:L1",
        "P3:L2",
        "P3:L3",
        "P3:M5"
      ],
      "scope": "Summable Sobolev series, completeness and one unique common tempered-distribution limit with compact support"
    },
    {
      "id": "SR:R5",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "On any closed cone avoiding",
      "dependencies": [
        "SR:R3",
        "SR:R4",
        "U001:Q3",
        "CH:W1"
      ],
      "scope": "All differentiated rapid-decay seminorms of the actual Fourier sum, including endpoint covectors"
    },
    {
      "id": "SR:R6",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "Away from the two endpoint base points",
      "dependencies": [
        "SR:R0",
        "SR:R2",
        "SR:R4",
        "SR:R5",
        "RC:K1",
        "SR:P4"
      ],
      "scope": "Actual convolution smoothness, support proof, local finiteness and complete interior smooth difference"
    },
    {
      "id": "SR:R7",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "The inclusion was just proved.",
      "dependencies": [
        "SR:R6",
        "SR:P7",
        "SP:C1",
        "SP:C2",
        "SP:S2",
        "CH:W1"
      ],
      "scope": "Interior equality and exact L2 failure, endpoint persistence by open fixed-order regularity and closed wavefront"
    },
    {
      "id": "SR:R8",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "The derivative \\(D_tU\\)",
      "dependencies": [
        "SR:R7",
        "RP:T4",
        "CH:W4",
        "CH:W1"
      ],
      "scope": "Both actual endpoint forcing rays by pseudolocality and complete model propagation across each endpoint"
    },
    {
      "id": "SR:F0",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "*At the fixed model covector",
      "dependencies": [
        "SR:R5",
        "SR:R7",
        "SR:R8"
      ],
      "scope": "Exact normalized segment/endpoints diagram and scaled frequency-cone section; no boundary or support misidentification"
    },
    {
      "id": "SR:T0",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "**Theorem 3.1",
      "dependencies": [
        "SR:T1",
        "SR:T2",
        "SR:T3",
        "SR:T4",
        "TUBE:Q1"
      ],
      "scope": "Full arbitrary-real-order theorem with exact threshold, both endpoint forcing rays, complex lower terms and injective cosphere hypothesis"
    },
    {
      "id": "SR:T1",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "**Proof.** First take \\(m=1\\)",
      "dependencies": [
        "SR:R7",
        "TUBE:E4",
        "TUBE:M4",
        "TUBE:M5",
        "GE:D7",
        "RC:C1"
      ],
      "scope": "Proper graph order minus s, all lower Sobolev orders and both inclusions for full solution wavefront"
    },
    {
      "id": "SR:T2",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "If \\(u\\) were microlocally",
      "dependencies": [
        "SR:T1",
        "SP:S4",
        "SR:R7"
      ],
      "scope": "Actual failure at prescribed order s by the full inverse of order s"
    },
    {
      "id": "SR:T3",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "The full intertwining defect",
      "dependencies": [
        "SR:T1",
        "SR:R8",
        "TUBE:E4",
        "TUBE:M4",
        "TUBE:M5",
        "RC:C1",
        "CH:W4",
        "P2:OP6"
      ],
      "scope": "Both full intertwining identities, nested complete graph cutoffs and exact endpoint forcing equality"
    },
    {
      "id": "SR:T4",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "For order \\(m\\)",
      "dependencies": [
        "SR:T3",
        "SP:P2",
        "RP:P1",
        "G2:NF1",
        "U001:F0-COMP",
        "U001:P3",
        "CH:W4"
      ],
      "scope": "Positive all-real-order reduction, compact positive time reparametrization and exact elliptic wavefront equivalence"
    },
    {
      "id": "SR:X1",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "### Exercise 4.1.",
      "dependencies": [
        "SR:P1",
        "SR:P2",
        "SR:P7",
        "P3:M4",
        "U001:Q2",
        "U001:L5.4.1"
      ],
      "scope": "full critical-exponent solution with both all-real weighted bounds and nonattained endpoint"
    },
    {
      "id": "SR:X2",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "### Exercise 4.2.",
      "dependencies": [
        "SR:P7",
        "RP:W0",
        "RP:W1",
        "SP:C2",
        "RC:C1",
        "CH:W1",
        "U001:Q3"
      ],
      "scope": "full sharp-indicator obstruction, both temporal signs, proper averaging and closed base limit"
    },
    {
      "id": "SR:X3",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "988afe0ba842a37e63220889aece311c778b41f25a64e7774ba75ced606906e9",
      "proof_locator": "### Exercise 4.3.",
      "dependencies": [
        "SR:T1",
        "SR:T2",
        "GE:D7",
        "SP:S4"
      ],
      "scope": "full graph-order threshold and both signs via actual inverse"
    }
  ],
  "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",
    "../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"
  ],
  "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
  }
}