{
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  "proofs": [
    {
      "id": "AM:B0",
      "source": "bounded-derivative-operators.md",
      "source_sha256": "3c2ad8c4992c609500a20f2eb5f82457ff28686360972909ecc0ba7d23087d9c",
      "proof_locator": "## B0.",
      "dependencies": [
        "P3:L1",
        "P3:L2",
        "P3:L3",
        "P3:M4",
        "P3:M7",
        "P1:B3",
        "CT:T1",
        "CR:I1",
        "U001:Q3",
        "U001:U001-A4"
      ],
      "scope": "Actual normalized packets, synthesis integral, measure and Fourier prerequisites"
    },
    {
      "id": "AM:B1",
      "source": "bounded-derivative-operators.md",
      "source_sha256": "3c2ad8c4992c609500a20f2eb5f82457ff28686360972909ecc0ba7d23087d9c",
      "proof_locator": "**Proof of (E23).**",
      "dependencies": [
        "AM:B0"
      ],
      "scope": "Packet isometry, synthesis contraction, exact reconstruction and Schwartz convergence"
    },
    {
      "id": "AM:B2",
      "source": "bounded-derivative-operators.md",
      "source_sha256": "3c2ad8c4992c609500a20f2eb5f82457ff28686360972909ecc0ba7d23087d9c",
      "proof_locator": "Consider the matrix of",
      "dependencies": [
        "AM:B1",
        "U001:F0-DIFF",
        "U001:Q3"
      ],
      "scope": "Exact packet matrix, integrations by parts and uniform 4N derivative estimate"
    },
    {
      "id": "AM:B3",
      "source": "bounded-derivative-operators.md",
      "source_sha256": "3c2ad8c4992c609500a20f2eb5f82457ff28686360972909ecc0ba7d23087d9c",
      "proof_locator": "Since \\(2N>n\\)",
      "dependencies": [
        "AM:B2",
        "P1:B3",
        "P3:L1"
      ],
      "scope": "Integrable phase-space marginals, actual operator identification, bounded extension and fixed-size matrices"
    },
    {
      "id": "AM:A0",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "## 1.",
      "dependencies": [
        "U001:Q3",
        "P3:L2"
      ],
      "scope": "Tempered inverse Fourier definition from a bounded multiplier"
    },
    {
      "id": "AM:A1",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "**Lemma 1.1",
      "dependencies": [
        "AM:A0",
        "U001:U001-2.1",
        "U001:F0-DIFF",
        "U001:P6.5",
        "P3:M3"
      ],
      "scope": "Every cutoff derivative, noncompact polynomial tail, uniform compact smooth convergence and uniqueness"
    },
    {
      "id": "AM:A2",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "**Lemma 1.2",
      "dependencies": [
        "AM:A1",
        "U001:Q1",
        "U001:P15.1",
        "U001:P16.1",
        "U001:P13.2-series",
        "P3:M3",
        "P3:M4"
      ],
      "scope": "Entire contour realization by an explicit dominated complex power series, height independence and real distributional identification"
    },
    {
      "id": "AM:A3",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "Differentiating twice in",
      "dependencies": [
        "AM:A2",
        "U001:F0-DIFF"
      ],
      "scope": "Exact Airy differential equation and its sign"
    },
    {
      "id": "AM:A4",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "**Proof.** For the positive ray",
      "dependencies": [
        "AM:A2",
        "U001:Q1",
        "U001:P14.2-powers"
      ],
      "scope": "Every positive-ray derivative with sharp Gaussian polynomial factor and exponential decay"
    },
    {
      "id": "AM:A5",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "For the negative ray use",
      "dependencies": [
        "AM:A1",
        "U001:U001-5.1",
        "U001:U001-2.1",
        "U001:P16.1"
      ],
      "scope": "Both fixed stationary points, Hessian signs, all differentiated noncompact tails and exact asymptotic remainder"
    },
    {
      "id": "AM:A6",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "The bracket in (1.13)",
      "dependencies": [
        "AM:A4",
        "AM:A5",
        "U001:P14.2-powers",
        "U001:P16.1"
      ],
      "scope": "Uniform all-real derivative bounds and explicit sequence proving each negative-ray exponent optimal"
    },
    {
      "id": "AM:M0",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "## 2.",
      "dependencies": [
        "FA:C0",
        "FA:A1",
        "FA:S2"
      ],
      "scope": "Entire canonical model, actual homogeneous phase and exact two coefficient orders"
    },
    {
      "id": "AM:M1",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "Theorem 1.3 and (2.4)",
      "dependencies": [
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        "AM:A6",
        "U001:Q3",
        "P3:M3"
      ],
      "scope": "Defined Schwartz operator, polynomial growth, smooth outputs and distributional kernel"
    },
    {
      "id": "AM:T0",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "**Lemma 3.1",
      "dependencies": [
        "AM:M1",
        "AM:A1",
        "U001:P14.2-powers"
      ],
      "scope": "Both exact cubic moments, orientation, positive i derivative moment and negative i coefficient"
    },
    {
      "id": "AM:T1",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "**Lemma 3.2",
      "dependencies": [
        "AM:T0",
        "U001:F0-DIFF"
      ],
      "scope": "Separated noncompact-s tail and all inverse phase-derivative mixed bounds"
    },
    {
      "id": "AM:T2",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "Put \\(\\mu=m+1/2\\).",
      "dependencies": [
        "AM:T1",
        "U001:F0-DIFF"
      ],
      "scope": "All transpose mixed frequency orders and s weights with receding smooth cutoffs"
    },
    {
      "id": "AM:T3",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "To justify the integrations,",
      "dependencies": [
        "AM:T2",
        "AM:A1",
        "P3:M3",
        "U001:P6.5"
      ],
      "scope": "Every differentiated smoothing tail, common integrable majorants and exact choices of N"
    },
    {
      "id": "AM:R1",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "**Proof of membership and symbol.**",
      "dependencies": [
        "AM:T0",
        "AM:T3",
        "FA:A1",
        "FA:S2",
        "OSC:O1",
        "OSC:O2"
      ],
      "scope": "Distributional cubic realization, exact order, symbol, odd sign, density and common frame"
    },
    {
      "id": "AM:R2",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "**Proof of surjectivity.**",
      "dependencies": [
        "AM:R1",
        "FD:S1",
        "FD:S2",
        "GS:Z12",
        "U001:P14.2-powers"
      ],
      "scope": "Full even/odd descent, both frequency sides, whole signed interval and exact coefficient orders"
    },
    {
      "id": "AM:R3",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "Here the support induction needs care.",
      "dependencies": [
        "AM:R2",
        "FA:V0",
        "OSC:O2"
      ],
      "scope": "One fixed symmetric support under residual induction; reflected-sheet lower-order terms permitted"
    },
    {
      "id": "AM:R4",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "Apply the imported E6 separately",
      "dependencies": [
        "AM:R3",
        "K:K1",
        "IR:R4"
      ],
      "scope": "Both actual full-frequency symbol sums, every residual order and complete smooth remainder"
    },
    {
      "id": "AM:L1",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "**Lemma 5.1",
      "dependencies": [
        "AM:A6",
        "AM:M0",
        "U001:F0-DIFF"
      ],
      "scope": "Every mixed derivative of Airy amplitude with order m+1/6 and no ordinary frequency gain"
    },
    {
      "id": "AM:L2",
      "source": "airy-functions-and-fold-model-operators.md",
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      "proof_locator": "**Theorem 5.2",
      "dependencies": [
        "AM:L1",
        "AM:B3",
        "AM:R4",
        "P3:M7"
      ],
      "scope": "Full compact-base model L2 theorem at m<=-1/6, explicit finite derivative count and sqrt(2pi) normalization"
    },
    {
      "id": "AM:L3",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "For a localized general model kernel,",
      "dependencies": [
        "AM:L2",
        "CT:T2"
      ],
      "scope": "Compact input/output localization and actual Hilbert-Schmidt remainder"
    },
    {
      "id": "AM:L4",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "**Example 5.3",
      "dependencies": [
        "AM:A5",
        "AM:L1",
        "U001:P16.1"
      ],
      "scope": "Explicit bounded amplitude outside ordinary S0 with exact high-frequency sequence"
    },
    {
      "id": "AM:X1",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "**Exercise 6.1 ",
      "dependencies": [
        "AM:A2",
        "AM:A4"
      ],
      "scope": "Exercise 6.1 and solution"
    },
    {
      "id": "AM:X2",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "**Exercise 6.2 ",
      "dependencies": [
        "AM:A5"
      ],
      "scope": "Exercise 6.2 and solution"
    },
    {
      "id": "AM:X3",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "**Exercise 6.3 ",
      "dependencies": [
        "AM:A5"
      ],
      "scope": "Exercise 6.3 and solution"
    },
    {
      "id": "AM:X4",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "**Exercise 6.4 ",
      "dependencies": [
        "AM:T0"
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      "scope": "Exercise 6.4 and solution"
    },
    {
      "id": "AM:X5",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "**Exercise 6.5 ",
      "dependencies": [
        "AM:T3"
      ],
      "scope": "Exercise 6.5 and solution"
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    {
      "id": "AM:X6",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "**Exercise 6.6 ",
      "dependencies": [
        "AM:M0",
        "AM:R1",
        "AM:L2"
      ],
      "scope": "Exercise 6.6 and solution"
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      "id": "AM:X7",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "**Exercise 6.7 ",
      "dependencies": [
        "AM:R2"
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    {
      "id": "AM:X8",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "**Exercise 6.8 ",
      "dependencies": [
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        "AM:L2"
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      "scope": "Exercise 6.8 and solution"
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    {
      "id": "AM:X9",
      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "**Exercise 6.9 ",
      "dependencies": [
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      "scope": "Exercise 6.9 and solution"
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      "source": "airy-functions-and-fold-model-operators.md",
      "source_sha256": "6550a6fb09f0179ff8ac282adfe683935ff8e685a08564ba38e4f5b65226ef96",
      "proof_locator": "**Exercise 6.10 ",
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      "scope": "Exercise 6.10 and solution"
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