{
  "schema": "AN04-restored-oscillatory-cauchy-proof-map/v1",
  "proofs": [
    {
      "id": "EC:H0",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "Let \\(J\\) be a compact",
      "dependencies": [
        "SY:H1",
        "SY:H2",
        "SY:H7",
        "SY:H8",
        "SY:H10",
        "SY:R1"
      ],
      "scope": "Full scalar homogeneous principal direction and arbitrary finite-matrix ordinary lower symbols; actual global all-real Sobolev domains"
    },
    {
      "id": "EC:H1",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "The Hermitian real parts of",
      "dependencies": [
        "EC:H0",
        "SY:R0",
        "SY:R1",
        "SY:U0",
        "SY:U1"
      ],
      "scope": "Exact two-sided energy criterion, inverse evolution and all-real finite-time bound"
    },
    {
      "id": "EC:H2",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "We need the initial-time derivative",
      "dependencies": [
        "EC:H1",
        "SY:U0",
        "SY:U1",
        "SY:H15",
        "CE:H7"
      ],
      "scope": "Oriented initial derivative, correct sign and ordered left/right endpoint equations"
    },
    {
      "id": "EC:H3",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "These formulas imply smooth dependence",
      "dependencies": [
        "EC:H2",
        "SY:H10",
        "SY:H15",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "All finite joint strong parameter derivatives with the precise finite spatial-order loss"
    },
    {
      "id": "EC:H4",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "**Extra spatial orders give norm control.**",
      "dependencies": [
        "EC:H2",
        "EC:H3",
        "SY:H10",
        "CE:H7",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Coefficient norm continuity and actual evolution Taylor remainder with extra spatial orders, repeated endpoint products and smoothing norm regularity"
    },
    {
      "id": "EC:F0",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "Make the principal-symbol hypothesis precise.",
      "dependencies": [
        "EC:H0",
        "U001:P14.3"
      ],
      "scope": "Homogeneous principal extension, full ordinary lower coefficient and uniform high-frequency cutoff"
    },
    {
      "id": "EC:F1",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "The smooth flow and parameter theorem",
      "dependencies": [
        "EC:F0",
        "G2:NF0",
        "G2:NF1",
        "G2:NF5",
        "G2:NF6",
        "G2:NF7",
        "U001:F0-COMP"
      ],
      "scope": "Finite-time global Hamilton flow, two-sided covector comparison, continuation and positive homogeneity"
    },
    {
      "id": "EC:F2",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "The time-dependent flow is canonical as well.",
      "dependencies": [
        "EC:F1",
        "G2:G1",
        "G2:F2",
        "U001:P2"
      ],
      "scope": "Actual time-dependent variational symplectic identity, inverse flow and exact fixed-endpoint primitive"
    },
    {
      "id": "EC:F3",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "For clarity, the uniform derivative bounds needed next",
      "dependencies": [
        "EC:F1",
        "G2:NF5",
        "G2:NF6",
        "U001:F0-COMP",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Normalized all-derivative flow and inverse bounds, global base uniformity and every frequency weight"
    },
    {
      "id": "EC:F4",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "The variational equation at",
      "dependencies": [
        "EC:F3",
        "G2:NF0",
        "U001:P3"
      ],
      "scope": "Uniform short-time interval, actual global mixed base inverse, differentiated inverse and positive determinant"
    },
    {
      "id": "EC:F5",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "Define the action with its moving initial point,",
      "dependencies": [
        "EC:F2",
        "EC:F4",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full action variation with moving endpoints, Euler identity and both Hamilton-Jacobi signs"
    },
    {
      "id": "EC:F6",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "The mixed Hessian satisfies",
      "dependencies": [
        "EC:F4",
        "EC:F5",
        "GE:D1",
        "GE:D2"
      ],
      "scope": "Uniform mixed phase, nondegenerate critical equations, exact reflected graph and all phase bounds"
    },
    {
      "id": "EC:A0",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "Choose a matrix symbol",
      "dependencies": [
        "SY:H7",
        "SY:H8",
        "SY:H10",
        "P2:OP4",
        "U001:P14.3"
      ],
      "scope": "Exact compact-base proper initial operator and both bounded-frequency and localization defects"
    },
    {
      "id": "EC:A1",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "**The full amplitude action.**",
      "dependencies": [
        "EC:F6",
        "EC:A0",
        "EC:H0",
        "U001:U001-5.1",
        "U001:U001-6.1"
      ],
      "scope": "Exact eikonal cancellation and full order-preserving amplitude action with order-lowering residual"
    },
    {
      "id": "EC:A2",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "Here is the precise stationary-phase application",
      "dependencies": [
        "EC:F4",
        "EC:F6",
        "U001:P2",
        "U001:U001-5.1",
        "U001:U001-6.1"
      ],
      "scope": "Unique normalized critical point, unit absolute Hessian determinant, zero signature and full ordinary symbol remainder"
    },
    {
      "id": "EC:A3",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "For the coefficient signs in (CE16),",
      "dependencies": [
        "EC:A2",
        "P2:OP3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Actual bilinear Fourier coefficient pairing, positive Hessian and gradient signs and complete lower-order counting"
    },
    {
      "id": "EC:A4",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "Here are uniform bounds at the two noncompact ends",
      "dependencies": [
        "EC:A2",
        "U001:U001-2.1",
        "U001:U001-6.1",
        "P3:M3",
        "P3:M4"
      ],
      "scope": "Small physical frequency, large normalized frequency and compact annular tails with all derivative and integrability losses charged"
    },
    {
      "id": "EC:A5",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "In using such representatives, keep",
      "dependencies": [
        "EC:A1",
        "EC:A2",
        "EC:A3",
        "EC:A4",
        "K:K1",
        "GE:D17"
      ],
      "scope": "Full support-preserving representative, actual transform remainder and no continuity assumption on arbitrary summation choices"
    },
    {
      "id": "EC:D0",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "**The leading matrix and density.**",
      "dependencies": [
        "EC:F3",
        "EC:F4",
        "EC:A3",
        "GE:D2",
        "SY:H16"
      ],
      "scope": "Positive half-density and direct Jacobi row proof with the exact negative half-trace correction"
    },
    {
      "id": "EC:D1",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "The ordered finite-matrix existence, inverse and parameter proofs",
      "dependencies": [
        "EC:D0",
        "GE:D13",
        "GE:D14",
        "GE:D15",
        "EC:F3"
      ],
      "scope": "Actual ordered transport, every ordinary derivative and uniform inverse bound, including both time orientations"
    },
    {
      "id": "EC:D2",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "**Correct the full ordinary residual.**",
      "dependencies": [
        "EC:D1",
        "EC:A5",
        "GE:D16",
        "CE:H7"
      ],
      "scope": "Full lower residual, oriented variation of constants, zero initial value and one-order improvement without a classical expansion"
    },
    {
      "id": "EC:D3",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "Use parameter-aware support-preserving summation:",
      "dependencies": [
        "EC:D2",
        "EC:A5",
        "K:K1",
        "GE:D17"
      ],
      "scope": "One family of radii for all parameters, every differentiated asymptotic remainder and exact vanishing initial corrections"
    },
    {
      "id": "EC:D4",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "**Proper localization and the global output estimate.**",
      "dependencies": [
        "EC:D3",
        "EC:F6",
        "EC:A0",
        "GE:D3",
        "GE:D4",
        "GE:D7",
        "RC:C1"
      ],
      "scope": "Actual compact input/output supports, all-real graph estimates, strong continuity and exact initial smooth defect"
    },
    {
      "id": "EC:D5",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "Inside a slightly larger compact output set,",
      "dependencies": [
        "EC:D3",
        "EC:D4",
        "EC:A4",
        "P2:OP4",
        "P3:L1",
        "P3:M3",
        "P3:M4"
      ],
      "scope": "Every exterior PDO kernel derivative, additional z oscillatory gains and full global-output Schwartz estimates"
    },
    {
      "id": "EC:D6",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "These are genuine global output estimates.",
      "dependencies": [
        "EC:D5",
        "SY:H1",
        "SY:H2",
        "CE:H2",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Two-frequency rapid decay and norm-smooth input-to-arbitrary-global-Sobolev smoothing"
    },
    {
      "id": "EC:D7",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "The reduced graph symbol in",
      "dependencies": [
        "EC:D0",
        "EC:D1",
        "GE:D2",
        "GS:Z10",
        "GS:Z11"
      ],
      "scope": "Exact density normalization and every inverse ordinary graph-symbol bound"
    },
    {
      "id": "EC:C0",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "Fix an ordinary order-zero matrix operator",
      "dependencies": [
        "EC:H1",
        "EC:D4",
        "EC:D6"
      ],
      "scope": "Precise initial and residual correction contract, including all parameter operator-norm smoothing bounds"
    },
    {
      "id": "EC:C1",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "For compactly supported \\(\\phi\\in",
      "dependencies": [
        "EC:C0",
        "EC:H1",
        "SY:U2",
        "SY:C6",
        "CE:H3",
        "CE:H4"
      ],
      "scope": "Actual oriented Bochner correction of both errors and exact initial and equation identities"
    },
    {
      "id": "EC:C2",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "By uniqueness in the systems theorem,",
      "dependencies": [
        "EC:C1",
        "SY:C6",
        "SY:H2",
        "P3:L2",
        "P3:L3"
      ],
      "scope": "Full compact distribution domain, finite negative Sobolev order and exact correction identity"
    },
    {
      "id": "EC:C3",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "Here is the full kernel regularity conclusion.",
      "dependencies": [
        "EC:C2",
        "EC:D6",
        "EC:H1",
        "CE:H4"
      ],
      "scope": "All input-negative to global-positive Sobolev smoothing norms with one fixed datum order"
    },
    {
      "id": "EC:C4",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "To see that these bounds give an actual jointly smooth",
      "dependencies": [
        "EC:C3",
        "SY:H1",
        "SY:H2",
        "P3:M3",
        "CE:H4"
      ],
      "scope": "Actual delta columns, all input derivatives, arbitrary output evaluation and jointly smooth spatial kernel"
    },
    {
      "id": "EC:C5",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "This is the distribution kernel of the operator,",
      "dependencies": [
        "EC:C4",
        "CE:H4",
        "P3:L1",
        "P3:L2",
        "RC:K1",
        "RC:K2"
      ],
      "scope": "Bochner delta reconstruction and identification of the actual distribution kernel on compact data"
    },
    {
      "id": "EC:C6",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "The kernel is smooth in \\(t,r\\) as well.",
      "dependencies": [
        "EC:C3",
        "EC:C4",
        "EC:C5",
        "EC:H3",
        "EC:H4",
        "CE:H7",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Every endpoint derivative and oriented integral term, full jointly smooth parameter kernel"
    },
    {
      "id": "EC:C7",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "All the differentiations in this last paragraph also",
      "dependencies": [
        "EC:C6",
        "EC:H4",
        "CE:H4",
        "CE:H7"
      ],
      "scope": "Uniform input-unit-ball difference quotients and norm-integrable majorants after finite intermediate spatial losses"
    },
    {
      "id": "EC:G0",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "The family \\(V(t,r)\\) constructed",
      "dependencies": [
        "EC:D7",
        "EC:C5",
        "EC:C6",
        "EC:C7",
        "GE:D2"
      ],
      "scope": "Actual exact localized graph class and elliptic reduced matrix symbol with reflected kernel convention"
    },
    {
      "id": "EC:G1",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "We explain finite-time assembly on full compact input sets,",
      "dependencies": [
        "EC:G0",
        "EC:F1",
        "EC:F3",
        "U001:P14.3",
        "U001:F0-COMP"
      ],
      "scope": "Full-sphere compact initial localization and a finite time division on fixed parameter neighborhoods"
    },
    {
      "id": "EC:G2",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "Choose a compact multiplication cutoff",
      "dependencies": [
        "EC:G1",
        "EC:D4",
        "U001:P14.3"
      ],
      "scope": "Exact physical intermediate cutoff insertion and ordered evolution recursion"
    },
    {
      "id": "EC:G3",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "Set \\(F_{j+1}=V_jF_j\\).",
      "dependencies": [
        "EC:G2",
        "EC:F2",
        "RC:C3",
        "RC:C4",
        "RC:C5",
        "RC:C6",
        "RC:C7",
        "RC:C8",
        "RC:C9"
      ],
      "scope": "Actual zero-excess proper graph composition, unique matching covector and unchanged order"
    },
    {
      "id": "EC:G4",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "The other two terms have the asserted full smoothing bounds.",
      "dependencies": [
        "EC:G2",
        "EC:G3",
        "GE:D7",
        "EC:C3",
        "EC:C4",
        "EC:C5",
        "EC:C7",
        "EC:H3",
        "EC:H4"
      ],
      "scope": "Both accumulated smoothing errors, global Sobolev output bounds and every finite parameter loss"
    },
    {
      "id": "EC:G5",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "At each composition the principal symbols are multiplied",
      "dependencies": [
        "EC:G3",
        "EC:D7",
        "RC:C8",
        "GS:Z6",
        "GS:Z7",
        "GS:Z8",
        "GS:Z9",
        "GS:Z10",
        "GS:Z11"
      ],
      "scope": "Actual ordered bundle products, compatible Maslov frames and long-time assembly through caustics"
    },
    {
      "id": "EC:G6",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "All intermediate supports are compact:",
      "dependencies": [
        "EC:G1",
        "EC:G2",
        "EC:G3",
        "EC:F1",
        "EC:F3",
        "U001:F0-COMP"
      ],
      "scope": "Closed full punctured-product graph and quantitative exclusion of both zero-covector axes"
    },
    {
      "id": "EC:G7",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "We now prove every graph wavefront direction,",
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        "EC:G5",
        "EC:G6",
        "EC:D1",
        "EC:D7",
        "GE:D9",
        "RC:K4",
        "CH:W1",
        "EC:G8"
      ],
      "scope": "Full ordinary inverse-symbol ellipticity and microlocal inverse contradiction for every matrix-kernel graph direction"
    },
    {
      "id": "EC:G8",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "For completeness, the identity-kernel statement also",
      "dependencies": [
        "P3:L1",
        "P3:L2",
        "CH:W1",
        "CH:W3"
      ],
      "scope": "Arbitrary-cutoff diagonal wavefront proof, fixed tangential frequency and exact reflected conormal coordinates"
    },
    {
      "id": "EC:G9",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "For compactly supported vector data this also",
      "dependencies": [
        "EC:G7",
        "EC:G4",
        "EC:H1",
        "EC:H3",
        "RC:C1"
      ],
      "scope": "Exact spatial propagation by actual reversed evolution on a compact graph term and global smooth tail"
    },
    {
      "id": "EC:X1",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "### 7.1.",
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        "EC:C1",
        "EC:C2",
        "EC:D6"
      ],
      "scope": "Complete original zero-residual but nonzero initial-defect solution"
    },
    {
      "id": "EC:X2",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "### 7.2.",
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        "EC:H2",
        "EC:D1",
        "SY:H16",
        "U001:P2"
      ],
      "scope": "Complete original noncommuting pulse solution, both inverses and correct endpoint factor order"
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    {
      "id": "EC:X3",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "### 7.3.",
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        "EC:D0",
        "EC:D1",
        "EC:G9",
        "SY:H2",
        "P3:L2",
        "RP:F3",
        "EC:V1"
      ],
      "scope": "Complete original variable-speed flow, exact ordered solution, density, nonvanishing delta weight and sharp Sobolev threshold"
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      "id": "EC:V0",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "The two panels show (CE39)",
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        "EC:X3",
        "EC:V1",
        "U001:P2"
      ],
      "scope": "Exact sampled base/cotangent panels, kernel reflection, transported delta weight and coincident outer covector curves"
    },
    {
      "id": "EC:V1",
      "source": "oscillatory-cauchy-kernels-and-exact-evolution.md",
      "source_sha256": "af6859f6c2cbae615a1b9f39a364d1055b85041d75fd22859bdc5630a3de5d28",
      "proof_locator": "The elementary functions in the last example can be fixed",
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        "U001:P15.1",
        "U001:P15.2",
        "U001:P3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Complete hyperbolic-function inverse and derivatives, exact positive covector bounds and symmetry underlying the samples"
    }
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