{
  "schema": "AN04-restored-singular-ray-proof-map/v1",
  "proofs": [
    {
      "id": "SR:P0",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "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": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "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": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "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": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "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": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "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": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "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": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "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": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "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": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "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": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "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": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "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": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "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": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "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": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "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": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "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": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "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": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "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": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "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": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "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": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "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": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "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": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "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": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "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": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "proof_locator": "### Exercise 4.1.",
      "dependencies": [
        "SR:P1",
        "SR:P2",
        "SR:P7",
        "P3:M4",
        "U001:Q2",
        "U001:L5.4.1"
      ],
      "scope": "Unchanged full critical-exponent solution with both all-real weighted bounds and nonattained endpoint"
    },
    {
      "id": "SR:X2",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "proof_locator": "### Exercise 4.2.",
      "dependencies": [
        "SR:P7",
        "RP:W0",
        "RP:W1",
        "SP:C2",
        "RC:C1",
        "CH:W1",
        "U001:Q3"
      ],
      "scope": "Unchanged full sharp-indicator obstruction, both temporal signs, proper averaging and closed base limit"
    },
    {
      "id": "SR:X3",
      "source": "prescribed-finite-singular-rays.md",
      "source_sha256": "de71e178b7a2273326303b38f170eed7d48a53ca2a31a918b18a3d7aaa683773",
      "proof_locator": "### Exercise 4.3.",
      "dependencies": [
        "SR:T1",
        "SR:T2",
        "GE:D7",
        "SP:S4"
      ],
      "scope": "Unchanged 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"
  ],
  "component_proofs_complete": true,
  "U037_restored": true,
  "full_course_complete": false,
  "public_release_authorized": false
}