{
  "schema": "AN04-restored-first-order-cauchy-proof-map/v1",
  "proofs": [
    {
      "id": "FC:M1",
      "source": "spacetime-symbol-composition.md",
      "source_sha256": "5aaaa67173d246ec0afa772ca8beb9527f6ff9d4b5b97e3d0d1b6daeca0a9701",
      "proof_locator": "## M1.",
      "dependencies": [
        "P2:OP1",
        "P2:OP3",
        "P3:L3"
      ],
      "scope": "Exact full and partial classes, Schwartz action and adjoint, all joint time parameter derivatives"
    },
    {
      "id": "FC:M2",
      "source": "spacetime-symbol-composition.md",
      "source_sha256": "5aaaa67173d246ec0afa772ca8beb9527f6ff9d4b5b97e3d0d1b6daeca0a9701",
      "proof_locator": "## M2.",
      "dependencies": [
        "FC:M1",
        "P2:OP3",
        "P3:L1",
        "P3:M3",
        "P3:M4"
      ],
      "scope": "Actual partial-times-full smoothing ideal in both orders, rapid independent parameter bounds and full operator identity"
    },
    {
      "id": "FC:M3",
      "source": "spacetime-symbol-composition.md",
      "source_sha256": "5aaaa67173d246ec0afa772ca8beb9527f6ff9d4b5b97e3d0d1b6daeca0a9701",
      "proof_locator": "## M3.",
      "dependencies": [
        "FC:M2",
        "U001:U001-A4",
        "P2:OP3",
        "U001:P14.2-powers"
      ],
      "scope": "Strict conic cutoff, exact multiplier product, BA full ordinary expansion with every finite remainder"
    },
    {
      "id": "FC:M4",
      "source": "spacetime-symbol-composition.md",
      "source_sha256": "5aaaa67173d246ec0afa772ca8beb9527f6ff9d4b5b97e3d0d1b6daeca0a9701",
      "proof_locator": "## M4.",
      "dependencies": [
        "FC:M3",
        "FC:M2",
        "P2:OP3"
      ],
      "scope": "Smoothing-cone adjoint split, AB full expansion, finite Leibniz/binomial cancellation and all remainders"
    },
    {
      "id": "FC:M5",
      "source": "spacetime-symbol-composition.md",
      "source_sha256": "5aaaa67173d246ec0afa772ca8beb9527f6ff9d4b5b97e3d0d1b6daeca0a9701",
      "proof_locator": "We record explicitly",
      "dependencies": [
        "P2:OP4",
        "CH:W4",
        "CH:T0",
        "K:K3"
      ],
      "scope": "Actual global tempered off-diagonal Schwartz tests, every derivative, global pseudolocality and elliptic regularity"
    },
    {
      "id": "FC:M6",
      "source": "spacetime-symbol-composition.md",
      "source_sha256": "5aaaa67173d246ec0afa772ca8beb9527f6ff9d4b5b97e3d0d1b6daeca0a9701",
      "proof_locator": "For use here and in U030",
      "dependencies": [
        "P3:L1",
        "P3:L3",
        "P3:M3",
        "P3:M4",
        "P3:M6"
      ],
      "scope": "Complete all-dimension Fourier Sobolev embedding with exact integrability condition and parameter bounds"
    },
    {
      "id": "FC:M7",
      "source": "spacetime-symbol-composition.md",
      "source_sha256": "5aaaa67173d246ec0afa772ca8beb9527f6ff9d4b5b97e3d0d1b6daeca0a9701",
      "proof_locator": "Now let \\(u\\in",
      "dependencies": [
        "FC:M4",
        "FC:M5",
        "CE:G1",
        "P3:M4"
      ],
      "scope": "Tempered compact-time solution, actual characteristic operator and separated forcing error"
    },
    {
      "id": "FC:M8",
      "source": "spacetime-symbol-composition.md",
      "source_sha256": "5aaaa67173d246ec0afa772ca8beb9527f6ff9d4b5b97e3d0d1b6daeca0a9701",
      "proof_locator": "For the transported spatial test",
      "dependencies": [
        "FC:M4",
        "FC:M5",
        "FC:M6",
        "CE:G1",
        "P3:L1",
        "P3:L3",
        "P3:M4"
      ],
      "scope": "Actual ordinary localized commuting test, joint all-Sobolev regularity and elliptic removal"
    },
    {
      "id": "FC:S0",
      "source": "smooth-pullback-and-time-slices.md",
      "source_sha256": "fe045cb0810aa59200a7e2310481a7a2835e7e54a5ff3e55f2d5a5ddc1d9eeb9",
      "proof_locator": "## Exact earlier proofs",
      "dependencies": [
        "U001:U001-A4",
        "U001:P3",
        "U001:F0-COMP"
      ],
      "scope": "Finite smooth localization construction and actual current retained source bindings"
    },
    {
      "id": "FC:S1",
      "source": "smooth-pullback-and-time-slices.md",
      "source_sha256": "fe045cb0810aa59200a7e2310481a7a2835e7e54a5ff3e55f2d5a5ddc1d9eeb9",
      "proof_locator": "### 18.5.",
      "dependencies": [
        "FC:S0",
        "CH:T0",
        "CH:W1",
        "CH:W2",
        "P3:L1",
        "P3:L3",
        "P3:M3",
        "P3:M4"
      ],
      "scope": "Full original smooth-map transverse pullback integral and every finite distribution estimate"
    },
    {
      "id": "FC:S2",
      "source": "smooth-pullback-and-time-slices.md",
      "source_sha256": "fe045cb0810aa59200a7e2310481a7a2835e7e54a5ff3e55f2d5a5ddc1d9eeb9",
      "proof_locator": "Here is the needed independence",
      "dependencies": [
        "FC:S1",
        "P3:L1",
        "P3:M3",
        "U001:U001-A4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full Gaussian approximation, all separated-support derivatives, cutoff independence and local gluing"
    },
    {
      "id": "FC:S3",
      "source": "smooth-pullback-and-time-slices.md",
      "source_sha256": "fe045cb0810aa59200a7e2310481a7a2835e7e54a5ff3e55f2d5a5ddc1d9eeb9",
      "proof_locator": "For completeness fix a nonzero",
      "dependencies": [
        "FC:S2",
        "CH:W1",
        "CH:W2",
        "U001:F0-COMP",
        "P3:M3"
      ],
      "scope": "Full exact pullback wavefront inclusion, both regular-frequency regions and all powers"
    },
    {
      "id": "FC:S4",
      "source": "smooth-pullback-and-time-slices.md",
      "source_sha256": "fe045cb0810aa59200a7e2310481a7a2835e7e54a5ff3e55f2d5a5ddc1d9eeb9",
      "proof_locator": "For a submersion",
      "dependencies": [
        "FC:S3",
        "RP:W2",
        "CH:W3",
        "U001:P3"
      ],
      "scope": "Exact submersion converse from actual inverse chart and constant-factor projection, all zero dimensions"
    },
    {
      "id": "FC:S5",
      "source": "smooth-pullback-and-time-slices.md",
      "source_sha256": "fe045cb0810aa59200a7e2310481a7a2835e7e54a5ff3e55f2d5a5ddc1d9eeb9",
      "proof_locator": "### 18.6.",
      "dependencies": [
        "FC:S1",
        "FC:S2",
        "FC:S3",
        "CH:W1",
        "P3:M3",
        "P3:M4",
        "P3:L1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full no-normal parameter family, every derivative in strong test topology and reconstruction of original distribution"
    },
    {
      "id": "FC:S6",
      "source": "smooth-pullback-and-time-slices.md",
      "source_sha256": "fe045cb0810aa59200a7e2310481a7a2835e7e54a5ff3e55f2d5a5ddc1d9eeb9",
      "proof_locator": "## S1.",
      "dependencies": [
        "FC:S1",
        "FC:S3",
        "TG:U1",
        "CH:W1",
        "U001:F0-COMP",
        "P3:M3"
      ],
      "scope": "Fixed-wavefront sequential continuity, uniform finite order and compact Fourier convergence, closed output cone"
    },
    {
      "id": "FC:S7",
      "source": "smooth-pullback-and-time-slices.md",
      "source_sha256": "fe045cb0810aa59200a7e2310481a7a2835e7e54a5ff3e55f2d5a5ddc1d9eeb9",
      "proof_locator": "The smooth Gaussian approximations",
      "dependencies": [
        "FC:S2",
        "FC:S6",
        "CH:W1",
        "P3:M3",
        "P3:L1"
      ],
      "scope": "Actual smooth fixed-wavefront approximation and unique continuous extension"
    },
    {
      "id": "FC:S8",
      "source": "smooth-pullback-and-time-slices.md",
      "source_sha256": "fe045cb0810aa59200a7e2310481a7a2835e7e54a5ff3e55f2d5a5ddc1d9eeb9",
      "proof_locator": "## S2.",
      "dependencies": [
        "FC:S5",
        "CE:G1",
        "U001:U001-A4",
        "P3:M4"
      ],
      "scope": "Actual equality of smooth pullback and continuous Sobolev trace by continuous scalar testing"
    },
    {
      "id": "FC:A0",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "## 1.",
      "dependencies": [
        "CE:G1",
        "CE:G9",
        "CE:G10",
        "CE:H6",
        "CE:H12",
        "P2:OP3"
      ],
      "scope": "Exact global scalar hypotheses, full strong continuity, dual pairing and source scope"
    },
    {
      "id": "FC:E1",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "Begin with",
      "dependencies": [
        "FC:A0",
        "CE:G9",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Complete scalar weighted squared-norm and maximum quadratic proof including zero norm"
    },
    {
      "id": "FC:E2",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "Extend the weighted forcing",
      "dependencies": [
        "FC:E1",
        "CE:H2",
        "P3:M4"
      ],
      "scope": "All p energy bound, exact factor2/lambda-half and infinity endpoint with common threshold"
    },
    {
      "id": "FC:E3",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "For general \\(s\\), put",
      "dependencies": [
        "FC:E2",
        "CE:G1",
        "CE:G9",
        "CE:G10",
        "P2:OP3"
      ],
      "scope": "All-real Sobolev conjugation, uniform zero-order remainder and actual lower form bound"
    },
    {
      "id": "FC:C1",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "**Uniqueness in the claimed class.**",
      "dependencies": [
        "FC:E3",
        "CE:G10",
        "CE:H10"
      ],
      "scope": "Correct lower-order energy uniqueness with actual C1 representative"
    },
    {
      "id": "FC:C2",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "**The reversed adjoint estimate.**",
      "dependencies": [
        "FC:E3",
        "P2:OP3",
        "CE:G9",
        "CE:G10"
      ],
      "scope": "Adjoint plus sign, exact zero terminal test domain and reverse dual-order estimate"
    },
    {
      "id": "FC:C3",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "Define on the range",
      "dependencies": [
        "FC:C2",
        "CE:H1",
        "CE:H2",
        "CE:H12"
      ],
      "scope": "Well-defined bounded conjugate-linear functional and full complex Hahn–Banach extension"
    },
    {
      "id": "FC:C4",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "**The Hilbert representation",
      "dependencies": [
        "FC:C3",
        "CE:H5",
        "CE:H11",
        "CE:H12",
        "CT:T1"
      ],
      "scope": "Actual Hilbert representation, L-infinity norm test and L1 density extension"
    },
    {
      "id": "FC:C5",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "The test identity is now",
      "dependencies": [
        "FC:C4",
        "CE:H6",
        "CE:H7",
        "CE:H9",
        "CE:H10"
      ],
      "scope": "Strong measurable action, actual vector primitive, unique continuous representative and exact initial trace"
    },
    {
      "id": "FC:C6",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "**Gain the regularity needed",
      "dependencies": [
        "FC:C5",
        "CE:G10",
        "CE:H10",
        "FC:E3"
      ],
      "scope": "Smooth-data construction two orders higher achieves actual energy domain"
    },
    {
      "id": "FC:C7",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "For arbitrary data,",
      "dependencies": [
        "FC:C6",
        "FC:C1",
        "CE:H6",
        "CE:H10",
        "CE:H5"
      ],
      "scope": "Full uniform solution limit and completeness, exact equation trace and every energy endpoint"
    },
    {
      "id": "FC:C8",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "For homogeneous forcing",
      "dependencies": [
        "FC:C7",
        "FC:C1",
        "FC:M6",
        "CE:G10"
      ],
      "scope": "All spatial orders, correct one-time-derivative scope and smooth-time bootstrap"
    },
    {
      "id": "FC:F1",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "## 4.",
      "dependencies": [
        "FC:A0",
        "FC:C7",
        "P2:OP3"
      ],
      "scope": "Exact uniform classical family and both time-direction lower bounds"
    },
    {
      "id": "FC:F2",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "Here is the needed time-continuous",
      "dependencies": [
        "FC:F1",
        "G2:NF1",
        "G2:NF2",
        "G2:NF7",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "U001:F0-COMP"
      ],
      "scope": "Full time-continuous state-smooth contraction and differentiated integral proof, complete finite-time frequency/base bounds"
    },
    {
      "id": "FC:F3",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "Homogeneity and uniqueness give",
      "dependencies": [
        "FC:F2",
        "G2:G0",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Homogeneous two-time flow, all normalized derivative estimates and full symplectic variational identity"
    },
    {
      "id": "FC:Q1",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "Choose a compactly based",
      "dependencies": [
        "CH:W1",
        "CH:W4",
        "FC:M5",
        "P2:OP4",
        "U001:U001-A4"
      ],
      "scope": "Actual initial regular compact-base classical conic test with H-infinity output"
    },
    {
      "id": "FC:Q2",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "Here \\(b=a/i\\)",
      "dependencies": [
        "FC:Q1",
        "FC:F3",
        "P2:OP3"
      ],
      "scope": "Full scalar commutator expansion, first sign, triangular recurrence and degree count"
    },
    {
      "id": "FC:Q3",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "The complete solution is",
      "dependencies": [
        "FC:Q2",
        "FC:F2",
        "FC:F3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Every transported homogeneous coefficient, actual integral and full compact-support/derivative estimates"
    },
    {
      "id": "FC:Q4",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "For a parameter-dependent asymptotic sum",
      "dependencies": [
        "FC:Q3",
        "HT:H6",
        "P2:OP3"
      ],
      "scope": "Actual first-time differentiated symbol sum and both complete uniformly smoothing errors"
    },
    {
      "id": "FC:Q5",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "For \\(u\\in C_tH^r\\) solving",
      "dependencies": [
        "FC:Q4",
        "CE:G1",
        "CE:G10",
        "FC:C1",
        "FC:C7",
        "FC:M6",
        "K:K3"
      ],
      "scope": "Distributional product rule, smooth forcing at every order and uniqueness identification of transported test"
    },
    {
      "id": "FC:W1",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "Since the preceding construction works",
      "dependencies": [
        "FC:Q5",
        "FC:F1",
        "FC:F3"
      ],
      "scope": "Both inclusions, restart/reversal, exact all-time equality and two-time evolution with correct initial Sobolev scope"
    },
    {
      "id": "FC:W2",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "With these proved contracts",
      "dependencies": [
        "FC:W1",
        "FC:M7",
        "FC:S8",
        "FC:S3"
      ],
      "scope": "Full characteristic containment with temporal exception and exact pullback trace inclusion"
    },
    {
      "id": "FC:W3",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "For the other inclusion,",
      "dependencies": [
        "FC:W2",
        "FC:Q3",
        "FC:Q4",
        "FC:Q5",
        "HT:H6",
        "FC:C8",
        "FC:M8"
      ],
      "scope": "All-time-derivative summation and forcing, actual spacetime elliptic removal and conditional slice equality"
    },
    {
      "id": "FC:W4",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "Finally \\(H_{\\tau+b_0}\\)",
      "dependencies": [
        "FC:W3",
        "FC:F3",
        "G2:G0"
      ],
      "scope": "All four spacetime Hamilton equations and actual characteristic constraint"
    },
    {
      "id": "FC:D1",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "## 7.",
      "dependencies": [
        "FC:C7",
        "FC:F3",
        "P3:L1",
        "P3:L3",
        "CE:G1",
        "CH:W1",
        "RP:C1",
        "U001:P14.3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "All seven exact models: signs, nonnorm continuity, tempered obstruction, damping, full tanh flow and half-density"
    },
    {
      "id": "FC:X1",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "**Exercise 1 (",
      "dependencies": [
        "FC:E2"
      ],
      "scope": "Original complete solved Exercise1"
    },
    {
      "id": "FC:X2",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "**Exercise 2 (",
      "dependencies": [
        "FC:C7",
        "CE:H9"
      ],
      "scope": "Original complete solved Exercise2"
    },
    {
      "id": "FC:X3",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "**Exercise 3 (",
      "dependencies": [
        "CE:G1",
        "CE:G10"
      ],
      "scope": "Original complete solved Exercise3"
    },
    {
      "id": "FC:X4",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "**Exercise 4 (",
      "dependencies": [
        "FC:C1"
      ],
      "scope": "Original complete solved Exercise4"
    },
    {
      "id": "FC:X5",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "**Exercise 5 (",
      "dependencies": [
        "FC:C4"
      ],
      "scope": "Original complete solved Exercise5"
    },
    {
      "id": "FC:X6",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "**Exercise 6 (",
      "dependencies": [
        "FC:E3"
      ],
      "scope": "Original complete solved Exercise6"
    },
    {
      "id": "FC:X7",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "**Exercise 7 (",
      "dependencies": [
        "FC:C2"
      ],
      "scope": "Original complete solved Exercise7"
    },
    {
      "id": "FC:X8",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "**Exercise 8 (",
      "dependencies": [
        "FC:F3",
        "FC:D1"
      ],
      "scope": "Original complete solved Exercise8"
    },
    {
      "id": "FC:X9",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "**Exercise 9 (",
      "dependencies": [
        "FC:F3",
        "FC:D1"
      ],
      "scope": "Original complete solved Exercise9"
    },
    {
      "id": "FC:X10",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "**Exercise 10 (",
      "dependencies": [
        "FC:D1"
      ],
      "scope": "Original complete solved Exercise10"
    },
    {
      "id": "FC:X11",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "**Exercise 11 (",
      "dependencies": [
        "FC:D1",
        "CH:W1"
      ],
      "scope": "Original complete solved Exercise11"
    },
    {
      "id": "FC:X12",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "**Exercise 12 (",
      "dependencies": [
        "FC:D1",
        "P3:L3"
      ],
      "scope": "Original complete solved Exercise12"
    },
    {
      "id": "FC:X13",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "**Exercise 13 (",
      "dependencies": [
        "FC:Q2"
      ],
      "scope": "Original complete solved Exercise13"
    },
    {
      "id": "FC:X14",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "**Exercise 14 (",
      "dependencies": [
        "FC:Q5"
      ],
      "scope": "Original complete solved Exercise14"
    },
    {
      "id": "FC:X15",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "**Exercise 15 (",
      "dependencies": [
        "FC:W2",
        "FC:W3",
        "FC:S3",
        "FC:S8"
      ],
      "scope": "Original complete solved Exercise15"
    },
    {
      "id": "FC:X16",
      "source": "first-order-cauchy-energy-and-wavefront.md",
      "source_sha256": "a084c9b4d1eb13e87df057a83a356555a7eef2723a11cfc2a6d64b35d571ad9a",
      "proof_locator": "**Exercise 16 (",
      "dependencies": [
        "CE:G1",
        "P1:B1"
      ],
      "scope": "Original complete solved Exercise16"
    }
  ],
  "earlier_maps": [
    "../20261004-free-stationary-phase/proof-map.json",
    "../20261004-free-intrinsic-graph/proof-map.json",
    "../20261004-free-tangent-zoom/proof-map.json",
    "../20261004-free-canonical-composition/proof-map.json",
    "../20261005-restored-oscillatory/proof-map.json",
    "../20261005-restored-phase-space/proof-map.json",
    "../20261005-restored-phase-equivalence/proof-map.json",
    "../20261005-restored-intrinsic-regularity/proof-map.json",
    "../20261005-restored-tangent-zoom/proof-map.json",
    "../20261005-restored-gaussian-symbols/proof-map.json",
    "../20261005-restored-linear-reduction/proof-map.json",
    "../20261005-restored-analytic-composition/proof-map.json",
    "../20261005-restored-subprincipal-transport/proof-map.json",
    "../20261005-restored-graph-egorov/proof-map.json",
    "../20261005-restored-corank-continuity/proof-map.json",
    "../20261005-restored-submanifolds/proof-map.json",
    "../20261005-restored-cubic-scaling/proof-map.json",
    "../20261005-restored-smooth-descent/proof-map.json",
    "../20261005-restored-simultaneous-reflections/proof-map.json",
    "../20261005-restored-folded-forms/proof-map.json",
    "../20261005-restored-symplectic-reflections/proof-map.json",
    "../20261005-restored-canonical-folds/proof-map.json",
    "../20261005-restored-fold-densities/proof-map.json",
    "../20261005-restored-airy-models/proof-map.json",
    "../20261005-restored-invariant-folds/proof-map.json",
    "../20261005-restored-prescribed-coordinates/proof-map.json",
    "../20261005-restored-clean-pairs/proof-map.json",
    "../20261005-restored-function-normal-forms/proof-map.json",
    "../20261005-restored-quadratic-forms/proof-map.json",
    "../20261005-restored-positive-ideals/proof-map.json",
    "../20261005-restored-complex-fios/proof-map.json",
    "../20261005-restored-maslov-topology/proof-map.json",
    "../20261005-restored-real-principal/proof-map.json",
    "../20261005-cauchy-foundations/proof-map.json"
  ],
  "component_proofs_complete": true,
  "U030_restored": true,
  "full_course_complete": false,
  "public_release_authorized": false
}