{
  "schema": "AN04-restored-higher-order-cauchy-proof-map/v1",
  "proofs": [
    {
      "id": "HC:R1",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "## 1.",
      "dependencies": [
        "U001:P3",
        "U001:U001-A4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS",
        "FC:A0",
        "CE:G1",
        "P2:OP3"
      ],
      "scope": "Uniform root bounds, all mixed derivatives, global ordering, low cutoff and m1; only one principal step assumed"
    },
    {
      "id": "HC:D1",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "## 2.",
      "dependencies": [
        "HC:R1",
        "P2:OP3"
      ],
      "scope": "Exact left division with every normal coefficient derivative and purely tangential one-order-lower residual"
    },
    {
      "id": "HC:D2",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "Lagrange interpolation,",
      "dependencies": [
        "HC:D1",
        "U001:F0-COMP",
        "P2:OP3",
        "CE:G1"
      ],
      "scope": "Entire polynomial interpolation, all column orders and exact lower-order reconstruction"
    },
    {
      "id": "HC:E1",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "## 3.",
      "dependencies": [
        "HC:D2",
        "FC:E3",
        "FC:C7",
        "CE:H7",
        "CE:H10",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Actual weak quotient energy and all lower jet primitives, full Gronwall, m1 and real s"
    },
    {
      "id": "HC:E2",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "The stronger estimate retains",
      "dependencies": [
        "HC:E1",
        "CE:H2",
        "P3:M4",
        "U001:P14.3"
      ],
      "scope": "Complete Volterra kernel and all weighted p endpoints with independent p/lambda/T constant"
    },
    {
      "id": "HC:M1",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "### Mixed spaces,",
      "dependencies": [
        "MH:B1",
        "MH:B2",
        "MH:B3",
        "MH:Q1",
        "MH:M1",
        "MH:J1",
        "MH:J2",
        "MH:J3",
        "MH:I1",
        "MH:I2",
        "PB:M1",
        "PB:G1",
        "PB:G2",
        "PB:K1",
        "PB:K2"
      ],
      "scope": "All real mixed spaces and minimal extension; one actual extension at all tangential orders"
    },
    {
      "id": "HC:M2",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "### Duality,",
      "dependencies": [
        "HC:M1",
        "MH:A1",
        "MH:A2",
        "MH:D1",
        "MH:E1",
        "MH:E2",
        "MH:N3"
      ],
      "scope": "Full antiduals, support/quotient distinction, density, exact embedding criterion and supported decomposition"
    },
    {
      "id": "HC:M3",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "### The normal step,",
      "dependencies": [
        "HC:M2",
        "MH:N1",
        "MH:N2",
        "MH:TR1",
        "MH:TR2",
        "MH:TR3"
      ],
      "scope": "Both complete normal-step estimates, full integer jets, strict trace threshold and continuous slices"
    },
    {
      "id": "HC:M4",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "### Boundary operators survive",
      "dependencies": [
        "HC:M3",
        "U001:P3",
        "CH:T0",
        "CNF:F1",
        "GB:H1",
        "GB:H2",
        "GB:H3",
        "PS:PS5",
        "MH:D1",
        "CE:G1"
      ],
      "scope": "Full boundary chart independence, flat-test annihilation, both normal supports, all-real coordinates and locally finite partitions"
    },
    {
      "id": "HC:M5",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "Let \\(E\\to X\\)",
      "dependencies": [
        "HC:M4",
        "HC:M3",
        "CE:G1",
        "MH:M1",
        "MH:D1",
        "U001:U001-A4"
      ],
      "scope": "Intrinsic transversal order, all finite bundle coefficients, full trace bound, density uniqueness and support"
    },
    {
      "id": "HC:M6",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "### Differential equations recover",
      "dependencies": [
        "HC:M2",
        "HC:M3",
        "MH:M1",
        "MH:R1",
        "MH:R2",
        "MH:T1",
        "MH:T2"
      ],
      "scope": "Every real recovery inequality, full commutators, finite nested fractional iteration and genuine tangential mapping"
    },
    {
      "id": "HC:G1",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "### Green's formula",
      "dependencies": [
        "HC:D1",
        "HC:E1",
        "P2:OP3",
        "CE:H1",
        "CE:H11",
        "CE:H12",
        "FC:C2",
        "FC:C3",
        "FC:C4"
      ],
      "scope": "Full signed Green form, reordered adjoint and dual estimate, actual Hahn-Banach L-infinity weak solution"
    },
    {
      "id": "HC:G2",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "We first take",
      "dependencies": [
        "HC:G1",
        "HC:M3",
        "HC:M6",
        "MH:T1",
        "MH:T2",
        "MH:E1"
      ],
      "scope": "Both interval endpoints, full negative-integer action and normal descent to actual provisional continuous jets"
    },
    {
      "id": "HC:G3",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "Compare the weak identity",
      "dependencies": [
        "HC:G2",
        "HC:G1",
        "U001:U001-A4",
        "CNF:J2",
        "CE:H7",
        "CE:H10"
      ],
      "scope": "Distributional Green identity and every triangular initial trace with arbitrary smooth test jets"
    },
    {
      "id": "HC:G4",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "Construct smooth-data solutions",
      "dependencies": [
        "HC:G3",
        "HC:E1",
        "CE:H5",
        "CE:H6",
        "CE:H10",
        "FC:M6"
      ],
      "scope": "Compatible all-Sobolev smooth construction, full temporal bootstrap and exact continuous-jet limit for arbitrary data"
    },
    {
      "id": "HC:L1",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "## 5.",
      "dependencies": [
        "HC:R1",
        "HC:G4",
        "G2:NF1",
        "G2:NF2",
        "G2:NF7",
        "U001:P3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Strict relative hyperbolicity, actual transverse flow and triangular noncommuting jet changes"
    },
    {
      "id": "HC:L2",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "On a compact coordinate patch",
      "dependencies": [
        "HC:L1",
        "HC:R1",
        "U001:U001-A4",
        "P2:OP3",
        "PS:PS5"
      ],
      "scope": "Full bounded root/coefficient extension with uniform convex gaps; differential agreement patch"
    },
    {
      "id": "HC:L3",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "**Local mixed Cauchy statement.**",
      "dependencies": [
        "HC:L2",
        "HC:G4",
        "HC:M3",
        "HC:M6",
        "MH:E1",
        "P3:M5"
      ],
      "scope": "Exact nonnegative mixed local Cauchy theorem and all recovery inequalities"
    },
    {
      "id": "HC:L4",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "**Companion local restricted-source statement.**",
      "dependencies": [
        "HC:L3",
        "HC:M1",
        "HC:M2",
        "HC:M6",
        "MH:M1",
        "HC:G4"
      ],
      "scope": "Real negative-order finite whole-space descent with exact Fourier multipliers, all commutator terms and nested cutoffs"
    },
    {
      "id": "HC:L5",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "For \\(r=+\\infty\\) and any fixed finite",
      "dependencies": [
        "HC:L4",
        "HC:M2",
        "HC:G4",
        "HC:M6"
      ],
      "scope": "One actual solution for infinite normal/tangential intersections by exact embedding and energy uniqueness"
    },
    {
      "id": "HC:L6",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "**Supported local solution.**",
      "dependencies": [
        "HC:L3",
        "HC:L4",
        "HC:L5",
        "HC:M2",
        "HC:M6",
        "HC:G4",
        "NE:J1",
        "NE:J2"
      ],
      "scope": "All supported real/infinite orders, boundary delta cancellation, one-sided decomposition and full correction"
    },
    {
      "id": "HC:L7",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "**One-sided local distribution uniqueness.**",
      "dependencies": [
        "HC:L6",
        "HC:L2",
        "U001:P3",
        "U001:U001-A4",
        "CH:T0",
        "P3:L3"
      ],
      "scope": "Curved shrinking boxes, persistent hyperbolicity, reverse adjoint support and actual compact test"
    },
    {
      "id": "HC:L8",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "**Semiglobal theorem.**",
      "dependencies": [
        "HC:L7",
        "HC:L6",
        "HC:L3",
        "HC:M4",
        "PS:PS5",
        "U001:F0-COMP",
        "U001:U001-A4",
        "CE:G1"
      ],
      "scope": "Full common-patch refinement, germ equality, uniform finite bands, supported residual orders and relatively compact Y"
    },
    {
      "id": "HC:F1",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "## 7.",
      "dependencies": [
        "HC:D1",
        "P2:OP3",
        "K:K2",
        "HT:H6"
      ],
      "scope": "Nonclassical all-order corrections with full signs/orders; finite exact recursion after final asymptotic sum"
    },
    {
      "id": "HC:P1",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "## 8.",
      "dependencies": [
        "HC:F1",
        "K:K3",
        "CH:W1",
        "CH:W4",
        "P2:OP4",
        "FC:M3",
        "FC:M4",
        "FC:M5",
        "FC:M6",
        "FC:S5",
        "FC:S8",
        "P3:M3",
        "P3:L1"
      ],
      "scope": "Actual nonnormal compact localization, all temporal Fourier bounds, genuine slices and spatial-to-full smoothing"
    },
    {
      "id": "HC:P2",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "The scalar source",
      "dependencies": [
        "HC:P1",
        "FC:M2",
        "FC:M3",
        "FC:M4",
        "FC:M5",
        "K:K3",
        "P2:OP3",
        "U001:U001-A4"
      ],
      "scope": "Remove source at all off-characteristic normal frequencies, full nested cutoffs and exact proper parametrix"
    },
    {
      "id": "HC:P3",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "Construct such a test",
      "dependencies": [
        "HC:P2",
        "FC:F2",
        "FC:F3",
        "FC:Q2",
        "FC:Q3",
        "HT:H6",
        "P2:OP3"
      ],
      "scope": "Scalar nonclassical commuting test by full-order recurrence, sign, support and parameter summation"
    },
    {
      "id": "HC:P4",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "Before passing to a slice,",
      "dependencies": [
        "HC:P3",
        "HC:P2",
        "HC:P1",
        "FC:C7",
        "FC:C1",
        "FC:M8",
        "FC:S3",
        "FC:S8",
        "K:K3"
      ],
      "scope": "All normal lifts regular before pullback, actual smooth evolution, reversibility and transported singular arcs"
    },
    {
      "id": "HC:P5",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "On the branch",
      "dependencies": [
        "HC:P4",
        "G2:G0",
        "HC:D1",
        "K:K3"
      ],
      "scope": "Exact Hamiltonian multiplier and both elliptic transfers; no unrestricted global propagation"
    },
    {
      "id": "HC:B1",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "## 9.",
      "dependencies": [
        "BW:N1",
        "BW:N2",
        "BW:N3",
        "BW:N4",
        "NE:E3",
        "NE:G1",
        "NE:G2",
        "NE:G3",
        "CNF:U2",
        "CNF:U3",
        "CNF:J2",
        "CNF:J3"
      ],
      "scope": "Actual N extension, intrinsic normal jets and exact supported delta correction"
    },
    {
      "id": "HC:B2",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "We need the intrinsic jets",
      "dependencies": [
        "HC:B1",
        "CNF:J3",
        "CNF:J4",
        "CNF:U3",
        "CNF:U4",
        "GB:D1",
        "BW:W9"
      ],
      "scope": "Complete intrinsic commutator by interior equality and actual dual injectivity, including normal symbol term"
    },
    {
      "id": "HC:B3",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "At a compressed covector",
      "dependencies": [
        "HC:B2",
        "BW:W0",
        "BW:W5",
        "BW:W6",
        "BW:W7",
        "BW:W8",
        "GB:E3",
        "BW:S1",
        "BW:S2",
        "CNF:S10"
      ],
      "scope": "Full intrinsic differential wavefront inclusion with base constancy modulo residual and compact receiving"
    },
    {
      "id": "HC:B4",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "**Boundary wavefront equality.**",
      "dependencies": [
        "HC:B3",
        "BW:S3",
        "BW:S4",
        "BW:S5",
        "BW:T6",
        "BW:T11",
        "BW:T12"
      ],
      "scope": "Forward inclusion with every intrinsic trace and forcing term"
    },
    {
      "id": "HC:B5",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "For the reverse inclusion,",
      "dependencies": [
        "HC:B4",
        "BW:T11",
        "BW:S1",
        "BW:S2",
        "BW:S5",
        "BW:W2",
        "U001:F0-COMP"
      ],
      "scope": "Actual compact collar smoothness for spatially residual N output, preserving pure-normal interior exception"
    },
    {
      "id": "HC:B6",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "Use the refined factors",
      "dependencies": [
        "HC:B5",
        "HC:F1",
        "HC:P3",
        "CNF:J2",
        "CNF:J4",
        "BW:T6",
        "BW:T11",
        "BW:T12",
        "P2:OP3"
      ],
      "scope": "All localized quotient equations and intrinsic initial data, full residuals stay in dual class before smoothness"
    },
    {
      "id": "HC:B7",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "The first-order smooth Cauchy solution",
      "dependencies": [
        "HC:B6",
        "HC:B1",
        "CNF:J3",
        "FC:C7",
        "CH:T0",
        "FC:M1"
      ],
      "scope": "Canonical zero-trace supported difference and exact compact-time Schwartz distribution domain"
    },
    {
      "id": "HC:B8",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "For a compact smooth test forcing",
      "dependencies": [
        "HC:B7",
        "FC:C7",
        "FC:M6",
        "P2:OP3",
        "P2:OP4",
        "U001:P14.2-powers",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full uniformly bounded-weight commutator, actual remainder with polynomial base ratios and all parameter derivatives"
    },
    {
      "id": "HC:B9",
      "source": "higher-order-cauchy-roots-jets-and-propagation.md",
      "source_sha256": "b1b8dad95d3f7f5d90811d7f482585028efd3b9ec124b747eeaf3bcf773c01ba",
      "proof_locator": "Assume all weights through",
      "dependencies": [
        "HC:B8",
        "HC:E1",
        "FC:E3",
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      "proof_locator": "**Exercise 20 —",
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      "proof_locator": "**Exercise 22 —",
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