{
  "schema": "AN04-restored-invariant-folds-proof-map/v1",
  "proofs": [
    {
      "id": "IF:T0",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "## 1.",
      "dependencies": [
        "RC:C0",
        "RC:C1",
        "P1:B1",
        "P1:B5"
      ],
      "scope": "Full joint-puncture closure, at-most-fold and nonzero one-form hypotheses, exact spaces and kernel action"
    },
    {
      "id": "IF:G0",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "The common pulled-back form",
      "dependencies": [
        "IF:T0",
        "CF:H0",
        "G2:G1"
      ],
      "scope": "Exact common primitive from simultaneous radial contraction"
    },
    {
      "id": "IF:G1",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "**Proof.** Choose a nonzero",
      "dependencies": [
        "IF:G0",
        "CF:G1",
        "CF:G2",
        "CF:G3"
      ],
      "scope": "Common critical set, simple top-wedge zero, distinct transverse kernels and exact two-form radical"
    },
    {
      "id": "IF:G2",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "Since \\(\\iota_{R_C}\\sigma\\ne0\\)",
      "dependencies": [
        "IF:G1",
        "CF:H6"
      ],
      "scope": "Radial independence, entire homogeneous two-target normal form and dimension-one exclusion"
    },
    {
      "id": "IF:G3",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "At a regular point,",
      "dependencies": [
        "IF:G0",
        "U001:P2",
        "GE:D3"
      ],
      "scope": "Regular branches are actual homogeneous canonical graphs"
    },
    {
      "id": "IF:L0",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "## 3.",
      "dependencies": [
        "IF:G2",
        "IF:G3",
        "PS:PS5",
        "RC:C0"
      ],
      "scope": "Compact normalized relation, both frequency comparabilities and finite shrunken cover"
    },
    {
      "id": "IF:L1",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "**Proof.** Work on the kernel manifold",
      "dependencies": [
        "IF:L0",
        "K:K4",
        "CH:T2",
        "CH:W4",
        "P2:OP5",
        "IR:R1"
      ],
      "scope": "Actual full-symbol finite joint-kernel partition in manifold charts, proper cutoffs and class stability"
    },
    {
      "id": "IF:L2",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "Take a compact base multiplier",
      "dependencies": [
        "IF:L1",
        "CH:T2",
        "CH:W4"
      ],
      "scope": "Compact kernels in the chosen wavefront charts and full smooth residual"
    },
    {
      "id": "IF:L3",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "For the concrete warning in",
      "dependencies": [
        "U001:Q3",
        "U001:F0-DIFF",
        "CH:W1",
        "P2:OP1"
      ],
      "scope": "Explicit Fourier convolution remainder showing a principal inverse need not smooth a point mass"
    },
    {
      "id": "IF:L4",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "Here is an axis-limit example",
      "dependencies": [
        "IF:G0",
        "IF:L0",
        "U001:F0-DIFF"
      ],
      "scope": "Canonical regular graph closed only in separate punctures; explicit normalized countersequence"
    },
    {
      "id": "IF:C0",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "For a fold piece,",
      "dependencies": [
        "IF:G2",
        "GS:Z12",
        "GE:D9",
        "GE:D11"
      ],
      "scope": "Actual elliptic order-zero canonical charts and both proper full inverse identities"
    },
    {
      "id": "IF:C1",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "Graph composition is clean with excess zero.",
      "dependencies": [
        "IF:C0",
        "RC:C9",
        "AM:R4",
        "AM:L2",
        "AM:L3"
      ],
      "scope": "Exact zero-excess Airy conjugation, compact supports, common positive radial cone and full model estimate"
    },
    {
      "id": "IF:C2",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "The exact recovery identity is",
      "dependencies": [
        "IF:C1",
        "RC:C1",
        "CH:W4",
        "K:K3",
        "GE:D9"
      ],
      "scope": "Both ordered recovery errors smooth after composition, with neither zero axis contributing"
    },
    {
      "id": "IF:C3",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "All proper factors in (4.4)",
      "dependencies": [
        "IF:C2",
        "IF:G3",
        "IF:L2",
        "GE:D3",
        "CT:T2",
        "P3:M5",
        "P1:B5"
      ],
      "scope": "Finite graph/fold/smooth L2 assembly and finite-rank half-density frame estimates"
    },
    {
      "id": "IF:S0",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "Choose properly supported elliptic order reducers",
      "dependencies": [
        "P1:B4",
        "P1:B5",
        "P1:B6",
        "WM:W6",
        "GE:G0",
        "P2:OP5",
        "PS:PS5",
        "K:K2",
        "CH:T1"
      ],
      "scope": "Positive all-real manifold symbol patching and full proper two-sided inverses"
    },
    {
      "id": "IF:S1",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "The conjugated operator",
      "dependencies": [
        "IF:S0",
        "IF:C3",
        "RC:C9"
      ],
      "scope": "Same relation, compact controlled support and exact order -1/6"
    },
    {
      "id": "IF:S2",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "Let \\(R_Y=I-C_YB_Y\\).",
      "dependencies": [
        "IF:S1",
        "RC:C1",
        "WM:W1",
        "P2:OP6"
      ],
      "scope": "Exact input recovery and every real-order bound of the localized smooth error"
    },
    {
      "id": "IF:S3",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "Let \\(R_X=I-C_XB_X\\).",
      "dependencies": [
        "IF:S2",
        "P1:B4",
        "P1:B6",
        "RC:C1",
        "WM:W1"
      ],
      "scope": "Exact output recovery, both complete Sobolev error bounds and full estimate"
    },
    {
      "id": "IF:S4",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "For any compactly supported \\(u\\in H^s\\)",
      "dependencies": [
        "IF:S3",
        "P1:B1",
        "P1:B4",
        "P1:B5",
        "WM:W6",
        "RC:K7",
        "RC:C1"
      ],
      "scope": "Common-support smooth approximation, actual strong distributional action and all-real compact-to-local theorem"
    },
    {
      "id": "IF:M0",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "## 6.",
      "dependencies": [
        "IF:G2",
        "G2:G1",
        "U001:F0-DIFF"
      ],
      "scope": "Full nonlinear cotangent lifts and exact cancelling common primitive"
    },
    {
      "id": "IF:M1",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "The base diffeomorphisms and their cotangent lifts",
      "dependencies": [
        "IF:M0",
        "IF:L0"
      ],
      "scope": "Preserved folds, transversality and full joint-puncture closure"
    },
    {
      "id": "IF:M2",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "For a half-density, base pullback",
      "dependencies": [
        "IF:M1",
        "GE:D20",
        "P1:B5"
      ],
      "scope": "Exact unitary half-density coordinate pullback and all-real local invariance"
    },
    {
      "id": "IF:N1",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "**Proof.** For \\(w\\in T_cC\\)",
      "dependencies": [
        "IF:G1",
        "IF:G0",
        "G2:G21",
        "CS:N2",
        "IF:C3"
      ],
      "scope": "Both individual radial exclusions and attained sharp universal L2 threshold, with correct quantifier"
    },
    {
      "id": "IF:X1",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "**Exercise 7.1 ",
      "dependencies": [
        "IF:G0"
      ],
      "scope": "Exercise 7.1 with solution"
    },
    {
      "id": "IF:X2",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "**Exercise 7.2 ",
      "dependencies": [
        "IF:G2",
        "IF:G3",
        "GE:D7"
      ],
      "scope": "Exercise 7.2 with solution"
    },
    {
      "id": "IF:X3",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "**Exercise 7.3 ",
      "dependencies": [
        "IF:L4"
      ],
      "scope": "Exercise 7.3 with solution"
    },
    {
      "id": "IF:X4",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "**Exercise 7.4 ",
      "dependencies": [
        "IF:L3"
      ],
      "scope": "Exercise 7.4 with solution"
    },
    {
      "id": "IF:X5",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "**Exercise 7.5 ",
      "dependencies": [
        "IF:C2"
      ],
      "scope": "Exercise 7.5 with solution"
    },
    {
      "id": "IF:X6",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "**Exercise 7.6 ",
      "dependencies": [
        "IF:S1",
        "U001:P14.2-powers"
      ],
      "scope": "Exercise 7.6 with solution"
    },
    {
      "id": "IF:X7",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "**Exercise 7.7 ",
      "dependencies": [
        "IF:S2",
        "IF:S3"
      ],
      "scope": "Exercise 7.7 with solution"
    },
    {
      "id": "IF:X8",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "**Exercise 7.8 ",
      "dependencies": [
        "IF:M0"
      ],
      "scope": "Exercise 7.8 with solution"
    },
    {
      "id": "IF:X9",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "**Exercise 7.9 ",
      "dependencies": [
        "IF:N1"
      ],
      "scope": "Exercise 7.9 with solution"
    },
    {
      "id": "IF:X10",
      "source": "invariant-fold-continuity-and-sobolev-transfer.md",
      "source_sha256": "3349216bfd27a95fbe7113bb6401e883598b2e0ce5ed85d05a0a792541e8c4e9",
      "proof_locator": "**Exercise 7.10 ",
      "dependencies": [
        "IF:C3",
        "P3:M5"
      ],
      "scope": "Exercise 7.10 with solution"
    }
  ],
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    "../20261005-restored-fold-densities/proof-map.json",
    "../20261005-restored-airy-models/proof-map.json"
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    "P10.2-typo",
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    "P10.5-rules",
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    "P10.6-zero",
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    "P11.2-orders",
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    "P12.6",
    "P12.7",
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    "P14.2-inverse",
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    "P15.1",
    "P15.2",
    "P16.1",
    "P16.2",
    "P16.3",
    "P17.1",
    "P17.2",
    "P17.3",
    "P17.4",
    "P17.4-grid",
    "P17.5-order",
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}