{
  "schema": "AN04-restored-intrinsic-regularity-proof-map/v1",
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    {
      "id": "IR:R0",
      "source": "intrinsic-lagrangian-regularity.md",
      "source_sha256": "39d72d6bb19325dcd939671603a7b02abd3d120f597ace7ebac51bb34495327e",
      "proof_locator": "## 1. The endpoint",
      "dependencies": [
        "P3:M6",
        "P3:M7",
        "P3:L3",
        "P1:B1",
        "CH:T3"
      ],
      "scope": "L2 extension, exact dyadic endpoint, differentiated restriction ideal, chart/frame invariance and finite-word definition"
    },
    {
      "id": "IR:R1",
      "source": "intrinsic-lagrangian-regularity.md",
      "source_sha256": "39d72d6bb19325dcd939671603a7b02abd3d120f597ace7ebac51bb34495327e",
      "proof_locator": "**Theorem 2.1.",
      "dependencies": [
        "IR:R0",
        "K:K6",
        "K:K7",
        "CH:W5"
      ],
      "scope": "Wavefront, matrix commutators, finite elliptic reconstruction and localized intrinsic class"
    },
    {
      "id": "IR:R2",
      "source": "intrinsic-lagrangian-regularity.md",
      "source_sha256": "39d72d6bb19325dcd939671603a7b02abd3d120f597ace7ebac51bb34495327e",
      "proof_locator": "**Proof of the forward implication.**",
      "dependencies": [
        "IR:R0",
        "IR:R1",
        "GEO:C4",
        "GR:G5",
        "GR:G12-forward",
        "P1:B2"
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      "scope": "Proper real phase regularization, exact balanced-word gauge and every symbol derivative"
    },
    {
      "id": "IR:R3",
      "source": "intrinsic-lagrangian-regularity.md",
      "source_sha256": "39d72d6bb19325dcd939671603a7b02abd3d120f597ace7ebac51bb34495327e",
      "proof_locator": "**Proof of the converse.**",
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        "IR:R1",
        "GR:G12-reverse",
        "OSC:O4"
      ],
      "scope": "General ordinary matrix tests, Taylor decomposition, cutoffs, repeated order preservation and noncompact local converse"
    },
    {
      "id": "IR:R4",
      "source": "intrinsic-lagrangian-regularity.md",
      "source_sha256": "39d72d6bb19325dcd939671603a7b02abd3d120f597ace7ebac51bb34495327e",
      "proof_locator": "Locally, the intersection makes the reduced symbol",
      "dependencies": [
        "IR:R2",
        "IR:R3",
        "P3:L2",
        "CH:W1"
      ],
      "scope": "All-order intersection is smooth, in charts and bundle frames"
    },
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      "source": "intrinsic-lagrangian-regularity.md",
      "source_sha256": "39d72d6bb19325dcd939671603a7b02abd3d120f597ace7ebac51bb34495327e",
      "proof_locator": "## 4. Every local phase",
      "dependencies": [
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        "IR:R3",
        "CH:T3",
        "OSC:O4"
      ],
      "scope": "Nondegenerate and clean phase membership with the preserved excess-dependent normalization"
    },
    {
      "id": "IR:R6",
      "source": "intrinsic-lagrangian-regularity.md",
      "source_sha256": "39d72d6bb19325dcd939671603a7b02abd3d120f597ace7ebac51bb34495327e",
      "proof_locator": "**Lemma 4.1",
      "dependencies": [
        "HT:H2",
        "HT:H5"
      ],
      "scope": "Proper homogeneous pullback, compact normalized support and zero extension"
    },
    {
      "id": "IR:R7",
      "source": "intrinsic-lagrangian-regularity.md",
      "source_sha256": "39d72d6bb19325dcd939671603a7b02abd3d120f597ace7ebac51bb34495327e",
      "proof_locator": "**Proof for a nondegenerate phase.**",
      "dependencies": [
        "IR:R5",
        "IR:R6",
        "PH:F4",
        "PH:F5",
        "PH:F6"
      ],
      "scope": "Critical restriction lift, exact determinant degree, support-preserving asymptotic construction and smooth error"
    },
    {
      "id": "IR:R8",
      "source": "intrinsic-lagrangian-regularity.md",
      "source_sha256": "39d72d6bb19325dcd939671603a7b02abd3d120f597ace7ebac51bb34495327e",
      "proof_locator": "**Proof for a clean phase.**",
      "dependencies": [
        "IR:R7",
        "PH:F3",
        "PH:F4",
        "PH:F5",
        "PH:F6",
        "OSC:O4"
      ],
      "scope": "Normalized clean-fiber right inverse, excess/order constants and full differentiated reconstruction"
    },
    {
      "id": "IR:R9",
      "source": "intrinsic-lagrangian-regularity.md",
      "source_sha256": "39d72d6bb19325dcd939671603a7b02abd3d120f597ace7ebac51bb34495327e",
      "proof_locator": "**Theorem 5.1.",
      "dependencies": [
        "IR:R1",
        "IR:R2",
        "IR:R3",
        "SB:S1",
        "SB:S2",
        "SB:S3"
      ],
      "scope": "Compatible microlocal cutoff, coordinate Sobolev invariance and adjustable Fourier split for every strict improved order"
    },
    {
      "id": "IR:R10",
      "source": "intrinsic-lagrangian-regularity.md",
      "source_sha256": "39d72d6bb19325dcd939671603a7b02abd3d120f597ace7ebac51bb34495327e",
      "proof_locator": "**The point mass.**",
      "dependencies": [
        "IR:R2",
        "IR:R3",
        "P1:B1",
        "SB:S-EX1"
      ],
      "scope": "Exact point-mass intrinsic, Besov and strict Sobolev thresholds"
    },
    {
      "id": "IR:E1",
      "source": "intrinsic-lagrangian-regularity.md",
      "source_sha256": "39d72d6bb19325dcd939671603a7b02abd3d120f597ace7ebac51bb34495327e",
      "proof_locator": "**Exercise 6.1 ",
      "dependencies": [
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        "P1:B1"
      ],
      "scope": "Preserved Exercise 6.1 and complete solution"
    },
    {
      "id": "IR:E2",
      "source": "intrinsic-lagrangian-regularity.md",
      "source_sha256": "39d72d6bb19325dcd939671603a7b02abd3d120f597ace7ebac51bb34495327e",
      "proof_locator": "**Exercise 6.2 ",
      "dependencies": [
        "IR:R1"
      ],
      "scope": "Preserved Exercise 6.2 and complete solution"
    },
    {
      "id": "IR:E3",
      "source": "intrinsic-lagrangian-regularity.md",
      "source_sha256": "39d72d6bb19325dcd939671603a7b02abd3d120f597ace7ebac51bb34495327e",
      "proof_locator": "**Exercise 6.3 ",
      "dependencies": [
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      "scope": "Preserved Exercise 6.3 and complete solution"
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      "id": "IR:E4",
      "source": "intrinsic-lagrangian-regularity.md",
      "source_sha256": "39d72d6bb19325dcd939671603a7b02abd3d120f597ace7ebac51bb34495327e",
      "proof_locator": "**Exercise 6.4 ",
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      "source": "intrinsic-lagrangian-regularity.md",
      "source_sha256": "39d72d6bb19325dcd939671603a7b02abd3d120f597ace7ebac51bb34495327e",
      "proof_locator": "**Exercise 6.5 ",
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        "OSC:O2",
        "K:K7"
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