{
  "schema": "AN04-restored-fold-densities-proof-map/v1",
  "proofs": [
    {
      "id": "FA:C0",
      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "## 1.",
      "dependencies": [
        "CF:H6",
        "CF:P3"
      ],
      "scope": "Entire two-fold model, sign convention, genuine homogeneous phase and precise compact/conic supports"
    },
    {
      "id": "FA:A1",
      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "**Proposition 1.1",
      "dependencies": [
        "FA:C0",
        "OSC:O3",
        "IR:R6",
        "FD:P0",
        "U001:F0-DIFF"
      ],
      "scope": "Both directions of full symbol equivalence under tau=rho s, radial Jacobian and exact normalized order"
    },
    {
      "id": "FA:D0",
      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "The delta notation has a direct coordinate meaning.",
      "dependencies": [
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        "U001:U001-A5",
        "U001:P2"
      ],
      "scope": "Critical density from actual normal coordinates; variable-dependent constraint matrix rule"
    },
    {
      "id": "FA:D1",
      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "**Proposition 2.1",
      "dependencies": [
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        "FA:A1",
        "U001:F0-DIFF"
      ],
      "scope": "Both independent radial factors, full normal determinant and exact amplitude/half-density cancellation"
    },
    {
      "id": "FA:S1",
      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "**Proposition 3.1.",
      "dependencies": [
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        "FA:D1",
        "U001:F0-DIFF"
      ],
      "scope": "All mixed symbol estimates under critical substitution and homogeneous positive density degree"
    },
    {
      "id": "FA:S2",
      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "The fixed-phase principal-symbol formula",
      "dependencies": [
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        "GS:Z10",
        "GS:Z11",
        "GS:Z12"
      ],
      "scope": "Exact principal symbol in its phase frame, total order m+n/2 and one-order quotient"
    },
    {
      "id": "FA:R0",
      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "## 4.",
      "dependencies": [
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        "FA:D0",
        "CF:H6",
        "U001:P3"
      ],
      "scope": "Reduced homogeneous phase, full critical independence, nonzero differential, product embedding and density"
    },
    {
      "id": "FA:R1",
      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "**Lemma 4.1",
      "dependencies": [
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        "GS:Z9",
        "GS:Z11",
        "OSC:O5",
        "U001:P2"
      ],
      "scope": "Exact degree-one hyperbolic elimination, ordered determinant and zero-signature Gaussian transition"
    },
    {
      "id": "FA:R2",
      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "For a coefficient",
      "dependencies": [
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        "FA:R1",
        "OSC:O1",
        "U001:Q3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exact distributional first-frequency Fourier elimination with test-pair domination and all constants"
    },
    {
      "id": "FA:T1",
      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "**Lemma 5.1",
      "dependencies": [
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        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Every differentiated estimate for first-frequency tail, uniform denominator and rapid decay on full frequencies"
    },
    {
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      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "In particular the conically truncated amplitude",
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        "U001:U001-A4"
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      "scope": "Full conic symbol comparison, critical cutoff equals one and unchanged invariant symbol"
    },
    {
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      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "To make the rapid-decrease assertion explicit,",
      "dependencies": [
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        "CH:W1",
        "IR:R6",
        "U001:F0-DIFF"
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      "scope": "Full microlocal Fourier representatives rapidly decreasing off wavefront, derivatives, graph frames, finite partition and fixed support"
    },
    {
      "id": "FA:V1",
      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "**Proof.** By (4.1)–(4.4)",
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        "FA:V0",
        "GS:Z12",
        "OSC:O2"
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      "scope": "Every principal symbol realized and full residual induction with wavefront retained in a fixed compactly generated support"
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      "proof_locator": "Apply the imported support-preserving",
      "dependencies": [
        "FA:V1",
        "K:K1",
        "IR:R4",
        "OSC:O3"
      ],
      "scope": "Exact support-preserving asymptotic summation in x,s;xi-prime, arbitrary-order remainder and smooth residual"
    },
    {
      "id": "FA:V3",
      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "Finally Fourier inversion",
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        "FA:R2",
        "FA:T2",
        "OSC:O2",
        "U001:Q3",
        "U001:U001-A4"
      ],
      "scope": "Cubic operator identity on compact smooth inputs, absolute convergence and proper base localization modulo smooth kernels"
    },
    {
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      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "## 7.",
      "dependencies": [
        "FA:S2",
        "FA:V3",
        "U001:P14.2-powers"
      ],
      "scope": "Explicit ordinary symbol, ellipticity through fold and exact example orders without claiming continuity"
    },
    {
      "id": "FA:X1",
      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "**Exercise 8.1 ",
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        "FA:S2",
        "FA:R0"
      ],
      "scope": "Exercise 8.1 and solution"
    },
    {
      "id": "FA:X2",
      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "**Exercise 8.2 ",
      "dependencies": [
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        "FA:D1",
        "FA:A1",
        "IR:R6"
      ],
      "scope": "Exercise 8.2 and solution"
    },
    {
      "id": "FA:X3",
      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "**Exercise 8.3 ",
      "dependencies": [
        "FA:S1"
      ],
      "scope": "Exercise 8.3 and solution"
    },
    {
      "id": "FA:X4",
      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "**Exercise 8.4 ",
      "dependencies": [
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        "FA:S2"
      ],
      "scope": "Exercise 8.4 and solution"
    },
    {
      "id": "FA:X5",
      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "**Exercise 8.5 ",
      "dependencies": [
        "FA:R0"
      ],
      "scope": "Exercise 8.5 and solution"
    },
    {
      "id": "FA:X6",
      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "**Exercise 8.6 ",
      "dependencies": [
        "FA:R1",
        "FA:R2"
      ],
      "scope": "Exercise 8.6 and solution"
    },
    {
      "id": "FA:X7",
      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "**Exercise 8.7 ",
      "dependencies": [
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      ],
      "scope": "Exercise 8.7 and solution"
    },
    {
      "id": "FA:X8",
      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "**Exercise 8.8 ",
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      "source": "fold-amplitudes-and-critical-densities.md",
      "source_sha256": "d8c45ff85e6afd9a37242632303f5c6e5dfa6cb9fdb3ca48b4b790f6541620b3",
      "proof_locator": "**Exercise 8.9 ",
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      "proof_locator": "**Exercise 8.10 ",
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  "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
  }
}