{
  "schema": "AN04-restored-complex-fios-proof-map/v1",
  "proofs": [
    {
      "id": "CX:T0",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "## 1.",
      "dependencies": [
        "PI:B0",
        "G2:G0",
        "GS:Z4",
        "RC:K8",
        "U001:F0-ALG"
      ],
      "scope": "Twist, positive canonical ideal, nonzero endpoints and exact half-density/bundle operator type"
    },
    {
      "id": "CX:T1",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "The sesquilinear adjoint uses",
      "dependencies": [
        "CX:T0",
        "RC:K6"
      ],
      "scope": "All conjugate inverse generators, damping, bundle duality and unchanged order"
    },
    {
      "id": "CX:C0",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "## 2.",
      "dependencies": [
        "G2:G21",
        "QF:A0"
      ],
      "scope": "Complex transverse intersection, coisotropic orthogonal, dimensions, injective composition and positivity"
    },
    {
      "id": "CX:C1",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "We need real base coordinates",
      "dependencies": [
        "CX:C0",
        "PI:A0",
        "PI:N0",
        "PI:N2"
      ],
      "scope": "Simultaneous real base shears from two nonzero complex determinant polynomials and normal phases"
    },
    {
      "id": "CX:C2",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "## 3.",
      "dependencies": [
        "CX:C1",
        "PI:D2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Exact real diagonal restriction and intrinsic elimination, including singular real matching sets"
    },
    {
      "id": "CX:C3",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "In these coordinates (2.2)",
      "dependencies": [
        "CX:C2",
        "QF:A0"
      ],
      "scope": "Full middle critical-rank test including both outer Hessian blocks"
    },
    {
      "id": "CX:C4",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "Set \\(\\Phi=\\phi_1+\\phi_2\\)",
      "dependencies": [
        "CX:C3",
        "CX:C11",
        "PI:P9"
      ],
      "scope": "Whole mixed normal-phase independence, positive homogeneous phase and equality with the intrinsic ideal"
    },
    {
      "id": "CX:C5",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "## 4.",
      "dependencies": [
        "CX:C0",
        "PI:P4",
        "CX:C11"
      ],
      "scope": "Direct complex arbitrary-phase rank proof and injectivity of both critical-map differentials"
    },
    {
      "id": "CX:C6",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "We also need equality of smooth ideals",
      "dependencies": [
        "CX:C2",
        "PI:D2",
        "PI:D6",
        "PI:F0",
        "PI:F1",
        "PI:P0",
        "G2:G41"
      ],
      "scope": "Coefficient graph division and every derivative of the flat intermediate-covector residual"
    },
    {
      "id": "CX:C7",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "For completeness, the division is explicit",
      "dependencies": [
        "CX:C6",
        "PI:D6",
        "PI:F2"
      ],
      "scope": "Exact conjugate-generator division, smooth zero extensions, equality of independent ideals and uniform family qualification"
    },
    {
      "id": "CX:C8",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "Here are the support and convergence details.",
      "dependencies": [
        "CX:T0",
        "CX:T1",
        "PI:O0",
        "PI:O3",
        "CH:W1",
        "RC:C1"
      ],
      "scope": "Properly supported kernel composition, smooth factors, wavefront separation and all endpoint derivatives"
    },
    {
      "id": "CX:C9",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "The noncomparable-frequency assertion",
      "dependencies": [
        "CX:C8",
        "PI:CS0",
        "HT:H8",
        "U001:Q3"
      ],
      "scope": "Full two-scale dyadic separation, dominant-frequency estimate and summation of unequal shells"
    },
    {
      "id": "CX:C10",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "Properness in condition 3",
      "dependencies": [
        "CX:C2",
        "CX:C4",
        "CX:C7",
        "CX:C9",
        "U001:F0-COMP",
        "U001:P4"
      ],
      "scope": "Finite matching cover, injective/proper projection and gluing actual local ideal germs"
    },
    {
      "id": "CX:C11",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "On a matching patch use (4.1).",
      "dependencies": [
        "U001:F0-DIFF",
        "G2:G41"
      ],
      "scope": "Exact mixed-to-homogeneous parameter change, inverse, critical Hessian congruence and R-to-the-nY Jacobian"
    },
    {
      "id": "CX:C12",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "Before this Jacobian,",
      "dependencies": [
        "CX:C11",
        "HT:H0"
      ],
      "scope": "All dimensions, symbol derivative losses, amplitude orders and exact two-pi normalization"
    },
    {
      "id": "CX:C13",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "To justify the kernel identity,",
      "dependencies": [
        "CX:C9",
        "CX:C10",
        "CX:C12",
        "PI:O0",
        "P3:M4",
        "RC:K4",
        "RC:K5"
      ],
      "scope": "Actual cutoff/Fubini identity, distributional limits and full transverse composition theorem"
    },
    {
      "id": "CX:L0",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "## 6.",
      "dependencies": [
        "CX:T0",
        "CT:T1"
      ],
      "scope": "Universal scalar and finite-bundle local L2 criterion and its precise quantifiers"
    },
    {
      "id": "CX:N0",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "## 7.",
      "dependencies": [
        "CX:L0",
        "CX:C1",
        "PI:T8",
        "G2:G41",
        "U001:P4"
      ],
      "scope": "Normal generating function, chosen nonzero order-zero amplitude and marked value/gradient normalization"
    },
    {
      "id": "CX:N1",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "For compactly supported smooth",
      "dependencies": [
        "CX:N0",
        "P3:L1",
        "P3:L3",
        "P3:M4"
      ],
      "scope": "Exact norm-preserving packets, transformed pairing and quadratic Taylor limit"
    },
    {
      "id": "CX:N2",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "Here is a uniform domination",
      "dependencies": [
        "CX:N1",
        "PI:P1",
        "P3:M3"
      ],
      "scope": "Whole-frequency polynomial majorant including the moving low-frequency region"
    },
    {
      "id": "CX:N3",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "is consequently a bounded map",
      "dependencies": [
        "CX:N2",
        "CT:T1",
        "P3:M7",
        "P3:L2",
        "RC:K2",
        "RC:K5"
      ],
      "scope": "Riesz extension, nonzero tempered Gaussian kernel and product-test detection"
    },
    {
      "id": "CX:N4",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "The tangent canonical plane is",
      "dependencies": [
        "CX:N3",
        "G2:G0",
        "P3:L2",
        "RC:K6"
      ],
      "scope": "Real one-sided symplectic vector yields the exact distributional transport equation"
    },
    {
      "id": "CX:N5",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "If \\(b=0\\)",
      "dependencies": [
        "CX:N4",
        "P3:M3",
        "P3:L3"
      ],
      "scope": "Pure multiplication and full real quadratic gauge obstructions, hyperplane support and Plancherel"
    },
    {
      "id": "CX:N6",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "There is thus no real output-only vector.",
      "dependencies": [
        "CX:N5",
        "CX:T1",
        "CT:T1"
      ],
      "scope": "Adjoint exclusion of the input-only vector and full necessity"
    },
    {
      "id": "CX:S0",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "## 8.",
      "dependencies": [
        "PI:O3",
        "PI:T8",
        "PS:PS5",
        "PS:PS6",
        "K:K4",
        "AM:L3"
      ],
      "scope": "Finite microlocal localization with exact positive representations and bounded compact smooth remainders"
    },
    {
      "id": "CX:S1",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "Let \\(L=T_pJ\\).",
      "dependencies": [
        "QF:N0",
        "QF:A7"
      ],
      "scope": "Exact positive Hermitian nullspace, real complexification and self-composition transversality"
    },
    {
      "id": "CX:S2",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "We need a particular graph projection",
      "dependencies": [
        "CX:S1",
        "CX:C0"
      ],
      "scope": "Complete mixed-projection proof using both opposite positivity inequalities and both one-sided exclusions"
    },
    {
      "id": "CX:S3",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "The partial Legendre theorem",
      "dependencies": [
        "CX:S2",
        "CX:C5",
        "CX:C7",
        "PI:E3",
        "CX:C10",
        "CX:C13"
      ],
      "scope": "Actual positive partial Legendre phase, unique intermediate graph, local properness and order-zero amplitude"
    },
    {
      "id": "CX:G0",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "## 9.",
      "dependencies": [
        "CX:S3",
        "CX:C5",
        "PI:P4",
        "PI:P5",
        "PI:P6",
        "PI:P7"
      ],
      "scope": "Augmented self-composition, invertible critical Hessian, degree-zero/one graph and ideal-square critical value"
    },
    {
      "id": "CX:G1",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "Put \\(G=\\operatorname{Im}",
      "dependencies": [
        "CX:G0",
        "PI:P0"
      ],
      "scope": "Evaluation at real graph parts, both nonnegative phase values and every imaginary-gradient estimate"
    },
    {
      "id": "CX:G2",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "The critical generators",
      "dependencies": [
        "CX:G1",
        "CX:G0"
      ],
      "scope": "Full real gradient comparison from the three critical derivative groups"
    },
    {
      "id": "CX:G3",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "Write \\(\\phi=\\phi_1+i\\phi_2\\)",
      "dependencies": [
        "CX:G2",
        "CX:S1",
        "CX:L0",
        "PI:P4"
      ],
      "scope": "Exact real derivative map and fixed-x injectivity from the no-real-one-sided hypothesis"
    },
    {
      "id": "CX:G4",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "Select an injective square minor",
      "dependencies": [
        "CX:G3",
        "U001:P2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Uniform inverse comparison on a convex image box with fixed external x"
    },
    {
      "id": "CX:G5",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "Differentiate (9.3)",
      "dependencies": [
        "CX:G4",
        "CX:G0",
        "CX:G1",
        "CX:G2",
        "G2:G41"
      ],
      "scope": "All ideal-square external derivatives, both phase gradients and exact homogeneous radial weights"
    },
    {
      "id": "CX:D0",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "## 10.",
      "dependencies": [
        "CX:G5",
        "U001:F0-DIFF",
        "U001:P15.1"
      ],
      "scope": "Every exponential derivative, high-order homogeneous factors and damping absorption at the half-half endpoint"
    },
    {
      "id": "CX:D1",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "The converse also holds",
      "dependencies": [
        "CX:D0",
        "G2:G41",
        "U001:F0-COMP"
      ],
      "scope": "Exact converse gradient estimate by all-radius rescaling, including zero damping"
    },
    {
      "id": "CX:E0",
      "source": "endpoint-half-order-bound.md",
      "source_sha256": "e1cc8c0ead16146183259b4631e0d732e8e9e761792d8e9b4d0010dbc43bf0dc",
      "proof_locator": "## E0.",
      "dependencies": [
        "HT:H8",
        "U001:P4",
        "P3:L3",
        "P3:M4"
      ],
      "scope": "Complete endpoint statement, finite weighted seminorms, smooth dyadic partition and finite input overlap"
    },
    {
      "id": "CX:E1",
      "source": "endpoint-half-order-bound.md",
      "source_sha256": "e1cc8c0ead16146183259b4631e0d732e8e9e761792d8e9b4d0010dbc43bf0dc",
      "proof_locator": "## E1.",
      "dependencies": [
        "CX:E0",
        "AM:B3",
        "P3:L1",
        "P3:L3",
        "U001:F0-DIFF"
      ],
      "scope": "Exact unitary balanced dilation and uniform full packet bound on every shell"
    },
    {
      "id": "CX:E2",
      "source": "endpoint-half-order-bound.md",
      "source_sha256": "e1cc8c0ead16146183259b4631e0d732e8e9e761792d8e9b4d0010dbc43bf0dc",
      "proof_locator": "## E2.",
      "dependencies": [
        "CX:E0",
        "CX:E1",
        "AM:B0",
        "AM:B3",
        "P3:M4",
        "P3:L3",
        "U001:Q3"
      ],
      "scope": "Actual Fourier kernel, full integration-by-parts decay, geometric separation, both Schur marginals and norm-summable far blocks"
    },
    {
      "id": "CX:E3",
      "source": "endpoint-half-order-bound.md",
      "source_sha256": "e1cc8c0ead16146183259b4631e0d732e8e9e761792d8e9b4d0010dbc43bf0dc",
      "proof_locator": "## E3.",
      "dependencies": [
        "CX:E1",
        "CX:E2",
        "P3:M7",
        "P3:M3",
        "P3:L3",
        "U001:Q3"
      ],
      "scope": "Complete neighboring-block strong convergence, finite overlap constants and distributional identification of the exact endpoint operator"
    },
    {
      "id": "CX:E4",
      "source": "endpoint-half-order-bound.md",
      "source_sha256": "e1cc8c0ead16146183259b4631e0d732e8e9e761792d8e9b4d0010dbc43bf0dc",
      "proof_locator": "## E4.",
      "dependencies": [
        "CX:E3",
        "CX:D0",
        "RC:K2",
        "RC:K4",
        "U001:F0-ALG",
        "U001:Q3"
      ],
      "scope": "Metric seminorm equivalence, localized Fourier modes, finite derivative constants and norm-convergent actual two-base amplitude"
    },
    {
      "id": "CX:B0",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "## 11.",
      "dependencies": [
        "CX:E3",
        "CX:E4",
        "CX:D0"
      ],
      "scope": "Complete actual endpoint provider and precise role of the broader metric comparison"
    },
    {
      "id": "CX:B1",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "Put \\(R=\\langle\\eta\\rangle\\)",
      "dependencies": [
        "QF:A0",
        "U001:F0-DIFF"
      ],
      "scope": "Full metric symplectic dual, Planck function one and equality of normalized derivative seminorms"
    },
    {
      "id": "CX:B2",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "To check slow variation",
      "dependencies": [
        "CX:B1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "All-center slow variation and constant-four exponent-one symplectic temperateness"
    },
    {
      "id": "CX:B3",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "We still justify the two-base amplitude",
      "dependencies": [
        "CX:E4",
        "CX:B0",
        "CX:B2",
        "CX:D0",
        "RC:K2",
        "RC:K4"
      ],
      "scope": "Every amplitude Fourier coefficient, unchanged compact support and summable endpoint operator series"
    },
    {
      "id": "CX:B4",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "Finally for compact smooth",
      "dependencies": [
        "CX:B3",
        "CX:C13",
        "CX:T1",
        "CX:S0",
        "CT:T1",
        "P3:M7"
      ],
      "scope": "Legitimate A-star-A identity on compact tests, density and full compact-to-local sufficiency"
    },
    {
      "id": "CX:M0",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "### 12.1.",
      "dependencies": [
        "CX:C3",
        "CX:C4",
        "CX:C12",
        "U001:P3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Entire conic positive composition rank, unique positive root, homogeneous minimum and exact Schur coefficient"
    },
    {
      "id": "CX:M1",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "### 12.2.",
      "dependencies": [
        "QF:N0",
        "G2:G0",
        "CX:N4",
        "CX:N5"
      ],
      "scope": "Whole positive complex plane, radial real nullspace, permitted complex one-sided vector and actual conic tangent"
    },
    {
      "id": "CX:M2",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "For an explicit **quadratic** operator,",
      "dependencies": [
        "CX:M1",
        "P3:L2",
        "U001:Q2",
        "P3:M4",
        "CT:T1"
      ],
      "scope": "All Fourier/delta constants, sharp Gaussian tensor norm and actual unbounded normalized evaluation packets"
    },
    {
      "id": "CX:M3",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "### 12.3.",
      "dependencies": [
        "CX:D0",
        "CX:D1",
        "U001:F0-DIFF",
        "U001:P15.1"
      ],
      "scope": "Every Gaussian recurrence, quartic degeneracy and exact widths, and failure for the real cubic"
    },
    {
      "id": "CX:X1",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "**Exercise 1 ",
      "dependencies": [
        "CX:T1"
      ],
      "scope": "Original Exercise 1 and its complete retained solution"
    },
    {
      "id": "CX:X2",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "**Exercise 2 ",
      "dependencies": [
        "CX:C11",
        "CX:C12"
      ],
      "scope": "Original Exercise 2 and its complete retained solution"
    },
    {
      "id": "CX:X3",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "**Exercise 3 ",
      "dependencies": [
        "CX:C3"
      ],
      "scope": "Original Exercise 3 and its complete retained solution"
    },
    {
      "id": "CX:X4",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "**Exercise 4 ",
      "dependencies": [
        "CX:M0"
      ],
      "scope": "Original Exercise 4 and its complete retained solution"
    },
    {
      "id": "CX:X5",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "**Exercise 5 ",
      "dependencies": [
        "CX:M0",
        "CX:C0"
      ],
      "scope": "Original Exercise 5 and its complete retained solution"
    },
    {
      "id": "CX:X6",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "**Exercise 6 ",
      "dependencies": [
        "CX:C5"
      ],
      "scope": "Original Exercise 6 and its complete retained solution"
    },
    {
      "id": "CX:X7",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "**Exercise 7 ",
      "dependencies": [
        "CX:C6",
        "CX:C7"
      ],
      "scope": "Original Exercise 7 and its complete retained solution"
    },
    {
      "id": "CX:X8",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "**Exercise 8 ",
      "dependencies": [
        "CX:C10"
      ],
      "scope": "Original Exercise 8 and its complete retained solution"
    },
    {
      "id": "CX:X9",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "**Exercise 9 ",
      "dependencies": [
        "CX:N1",
        "CX:N2"
      ],
      "scope": "Original Exercise 9 and its complete retained solution"
    },
    {
      "id": "CX:X10",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "**Exercise 10 ",
      "dependencies": [
        "CX:N5"
      ],
      "scope": "Original Exercise 10 and its complete retained solution"
    },
    {
      "id": "CX:X11",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "**Exercise 11 ",
      "dependencies": [
        "CX:M1"
      ],
      "scope": "Original Exercise 11 and its complete retained solution"
    },
    {
      "id": "CX:X12",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "**Exercise 12 ",
      "dependencies": [
        "CX:M2"
      ],
      "scope": "Original Exercise 12 and its complete retained solution"
    },
    {
      "id": "CX:X13",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "**Exercise 13 ",
      "dependencies": [
        "CX:S2"
      ],
      "scope": "Original Exercise 13 and its complete retained solution"
    },
    {
      "id": "CX:X14",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "**Exercise 14 ",
      "dependencies": [
        "CX:B1",
        "CX:B2",
        "CX:E4"
      ],
      "scope": "Original Exercise 14 and its complete retained solution"
    },
    {
      "id": "CX:X15",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "**Exercise 15 ",
      "dependencies": [
        "CX:M3"
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    {
      "id": "CX:X16",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "**Exercise 16 ",
      "dependencies": [
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    {
      "id": "CX:X17",
      "source": "complex-phase-fourier-integral-operators.md",
      "source_sha256": "c3a4e71cd6a364c51df99d145e22a27b2aa3cead86d8e55c25c1e0dbf376cab9",
      "proof_locator": "**Exercise 17 ",
      "dependencies": [
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        "CX:E4"
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      "source": "complex-phase-fourier-integral-operators.md",
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      "proof_locator": "**Figure 12.1.",
      "dependencies": [
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      "proof_locator": "The first panel gives the exact frozen scale",
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  ],
  "bounded_lesson_P514_closed": true,
  "human_review_complete": false,
  "full_course_complete": false,
  "public_release_authorized": false
}