{
  "schema": "AN04-restored-first-order-systems-proof-map/v1",
  "proofs": [
    {
      "id": "SY:H1",
      "source": "hilbert-coefficient-calculus-and-positivity.md",
      "source_sha256": "3320652328eede9d258da38f97d97d88e4e8f93640e113915564c22652dd9787",
      "proof_locator": "Let \\(K\\) be a separable",
      "dependencies": [
        "CE:H3",
        "CE:H4",
        "CE:H5",
        "P3:M4",
        "P3:M6",
        "P3:M7",
        "P3:L1",
        "P3:L2",
        "P3:L3",
        "U001:P2"
      ],
      "scope": "Actual separable Hilbert basis, norm-valued Schwartz Fourier inversion, full Plancherel and onto L2 extension"
    },
    {
      "id": "SY:H2",
      "source": "hilbert-coefficient-calculus-and-positivity.md",
      "source_sha256": "3320652328eede9d258da38f97d97d88e4e8f93640e113915564c22652dd9787",
      "proof_locator": "Define \\(H^s(K)\\)",
      "dependencies": [
        "SY:H1",
        "CE:H11",
        "CE:H12",
        "P3:M7",
        "U001:P14.2-powers"
      ],
      "scope": "Every-real Hilbert Sobolev scale, onto weight isometry, simultaneous density, separability and exact dual pairing"
    },
    {
      "id": "SY:H3",
      "source": "hilbert-coefficient-calculus-and-positivity.md",
      "source_sha256": "3320652328eede9d258da38f97d97d88e4e8f93640e113915564c22652dd9787",
      "proof_locator": "We will also use finite-coordinate",
      "dependencies": [
        "SY:H1",
        "U001:F0-COMP",
        "U001:P14.3",
        "P3:M7"
      ],
      "scope": "Full Schwartz finite-coordinate approximation from compact vector images and uniform contraction convergence"
    },
    {
      "id": "SY:H4",
      "source": "hilbert-coefficient-calculus-and-positivity.md",
      "source_sha256": "3320652328eede9d258da38f97d97d88e4e8f93640e113915564c22652dd9787",
      "proof_locator": "The space \\(\\mathcal L(K)\\)",
      "dependencies": [
        "CE:H4",
        "CE:H5"
      ],
      "scope": "Completeness of operator norm spaces from uniform Cauchy estimates on all vectors"
    },
    {
      "id": "SY:H5",
      "source": "hilbert-coefficient-calculus-and-positivity.md",
      "source_sha256": "3320652328eede9d258da38f97d97d88e4e8f93640e113915564c22652dd9787",
      "proof_locator": "Left quantization on",
      "dependencies": [
        "SY:H1",
        "SY:H4",
        "CE:H4",
        "P2:OP1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Norm-convergent left Fourier action and all weighted differentiated Schwartz bounds"
    },
    {
      "id": "SY:H6",
      "source": "hilbert-coefficient-calculus-and-positivity.md",
      "source_sha256": "3320652328eede9d258da38f97d97d88e4e8f93640e113915564c22652dd9787",
      "proof_locator": "Here is the norm-valued extension",
      "dependencies": [
        "SY:H4",
        "CE:H1",
        "CE:H3",
        "CE:H4",
        "P2:OP1",
        "P2:OP2",
        "P2:OP3"
      ],
      "scope": "Actual Banach-valued quadratic multiplier limits using uniform norm separation, summable cutoff tails and differentiated completeness"
    },
    {
      "id": "SY:H7",
      "source": "hilbert-coefficient-calculus-and-positivity.md",
      "source_sha256": "3320652328eede9d258da38f97d97d88e4e8f93640e113915564c22652dd9787",
      "proof_locator": "Take \\(E=\\mathcal L(K)\\)",
      "dependencies": [
        "SY:H6",
        "P2:OP3"
      ],
      "scope": "Ordered product and adjoint expansions with every pre-diagonal, derivative and finite-seminorm remainder in operator norm"
    },
    {
      "id": "SY:H8",
      "source": "hilbert-coefficient-calculus-and-positivity.md",
      "source_sha256": "3320652328eede9d258da38f97d97d88e4e8f93640e113915564c22652dd9787",
      "proof_locator": "For compact smooth symbols, norm Fubini",
      "dependencies": [
        "SY:H5",
        "SY:H7",
        "CE:H4",
        "P2:OP3"
      ],
      "scope": "Actual Schwartz composition/adjoint identities, bounded local-norm convergence and no finite-rank symbol approximation"
    },
    {
      "id": "SY:H9",
      "source": "hilbert-coefficient-calculus-and-positivity.md",
      "source_sha256": "3320652328eede9d258da38f97d97d88e4e8f93640e113915564c22652dd9787",
      "proof_locator": "Choose the fixed scalar normalized Schwartz",
      "dependencies": [
        "SY:H1",
        "SY:H3",
        "SY:H5",
        "CE:H4",
        "CE:H5",
        "AM:B3",
        "P3:M4"
      ],
      "scope": "Vector packet isometry, actual synthesis, norm-valued packet matrix, all derivative estimates and dimension-independent Schur reconstruction"
    },
    {
      "id": "SY:H10",
      "source": "hilbert-coefficient-calculus-and-positivity.md",
      "source_sha256": "3320652328eede9d258da38f97d97d88e4e8f93640e113915564c22652dd9787",
      "proof_locator": "For arbitrary real \\(s,r\\)",
      "dependencies": [
        "SY:H2",
        "SY:H8",
        "SY:H9",
        "U001:P14.2-powers"
      ],
      "scope": "Complete global Hilbert Sobolev estimates at every real order and exact quadratic-form domain"
    },
    {
      "id": "SY:H11",
      "source": "hilbert-coefficient-calculus-and-positivity.md",
      "source_sha256": "3320652328eede9d258da38f97d97d88e4e8f93640e113915564c22652dd9787",
      "proof_locator": "Retain the scalar \\(B,\\psi",
      "dependencies": [
        "SY:H4",
        "SY:H5",
        "CE:G2",
        "CE:G3",
        "CE:G4",
        "CE:H3",
        "CE:H4"
      ],
      "scope": "Actual operator-valued moving-probe integral with all locally uniform differentiated absolute majorants"
    },
    {
      "id": "SY:H12",
      "source": "hilbert-coefficient-calculus-and-positivity.md",
      "source_sha256": "3320652328eede9d258da38f97d97d88e4e8f93640e113915564c22652dd9787",
      "proof_locator": "For completeness, the full cancellation estimate",
      "dependencies": [
        "SY:H11",
        "CE:H1",
        "CE:G5",
        "CE:G6",
        "CE:G7",
        "CE:G8"
      ],
      "scope": "Dimension-independent full one-order moving-scale cancellation and exact repeated derivative identities"
    },
    {
      "id": "SY:H13",
      "source": "hilbert-coefficient-calculus-and-positivity.md",
      "source_sha256": "3320652328eede9d258da38f97d97d88e4e8f93640e113915564c22652dd9787",
      "proof_locator": "If \\(b=b^*\\ge0\\)",
      "dependencies": [
        "SY:H3",
        "SY:H5",
        "SY:H11",
        "CE:G3",
        "CE:G4",
        "CE:H4",
        "P3:M3",
        "P3:M4"
      ],
      "scope": "Actual positive Hilbert quadratic form, norm-rapid probe bounds, finite input approximation and all unbounded-parameter limits"
    },
    {
      "id": "SY:H14",
      "source": "hilbert-coefficient-calculus-and-positivity.md",
      "source_sha256": "3320652328eede9d258da38f97d97d88e4e8f93640e113915564c22652dd9787",
      "proof_locator": "Let \\(a\\in S^{2m+1}",
      "dependencies": [
        "SY:H7",
        "SY:H10",
        "SY:H12",
        "SY:H13",
        "SY:H2"
      ],
      "scope": "Full dimension-independent Hermitian sharp lower bound with adjoint error and exact H1 energy domain"
    },
    {
      "id": "SY:H15",
      "source": "hilbert-coefficient-calculus-and-positivity.md",
      "source_sha256": "3320652328eede9d258da38f97d97d88e4e8f93640e113915564c22652dd9787",
      "proof_locator": "Suppose \\(a(t)\\) is bounded",
      "dependencies": [
        "SY:H5",
        "SY:H8",
        "SY:H10",
        "SY:H2",
        "P3:M3"
      ],
      "scope": "Strong parameter action, actual adjoint and product continuity through uniform Schwartz tails and density"
    },
    {
      "id": "SY:H16",
      "source": "hilbert-coefficient-calculus-and-positivity.md",
      "source_sha256": "3320652328eede9d258da38f97d97d88e4e8f93640e113915564c22652dd9787",
      "proof_locator": "A Hermitian matrix has a unitary",
      "dependencies": [
        "U001:P2",
        "U001:F0-COMP",
        "U001:F0-CALC",
        "U001:P15.1",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Full finite Hermitian spectral proof, positive square root, norm equivalence, matrix exponential and determinant identity without an imported triangularization theorem"
    },
    {
      "id": "SY:A0",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "Let \\(K\\) be a separable",
      "dependencies": [
        "SY:H1",
        "SY:H2"
      ],
      "scope": "Exact Hilbert Sobolev conventions and E2s dual isometry"
    },
    {
      "id": "SY:N0",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "Left quantization uses",
      "dependencies": [
        "SY:H5",
        "SY:H14",
        "SY:H15"
      ],
      "scope": "Full uniform operator-norm ordinary symbol, local norm time continuity and Hermitian hypotheses"
    },
    {
      "id": "SY:N1",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "For a finite system, distributional",
      "dependencies": [
        "SY:N0",
        "GL:V0",
        "GL:V2",
        "U001:F0-COMP",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "Finite-entry distributional/local-smooth continuity equivalence via full finite grids, diagonal subsequences and derivative identification"
    },
    {
      "id": "SY:N2",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "The entrywise calculus is not",
      "dependencies": [
        "SY:A0",
        "SY:N0",
        "SY:H14",
        "SY:H10",
        "FC:X16"
      ],
      "scope": "Dimension-independent Hilbert sharp lower bound on the exact H1 domain and exclusion of an eigenvalue-only replacement"
    },
    {
      "id": "SY:N3",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "Here is the time-continuity step.",
      "dependencies": [
        "SY:H15",
        "SY:N0"
      ],
      "scope": "Strong all-real parameter action for fixed and varying vectors with no operator-norm continuity hypothesis"
    },
    {
      "id": "SY:E0",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "For any real \\(s\\)",
      "dependencies": [
        "SY:H7",
        "SY:H8",
        "SY:H10",
        "SY:N2"
      ],
      "scope": "Ordered all-real Sobolev conjugation and every ordinary order-zero error"
    },
    {
      "id": "SY:E1",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "Let \\(u\\in C^1",
      "dependencies": [
        "SY:E0",
        "SY:H2",
        "U001:F0-CALC",
        "CE:H2"
      ],
      "scope": "Energy differential inequality, compact maximum and exact nonnegative-root estimate"
    },
    {
      "id": "SY:E2",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "For \\(1\\leq p<\\infty\\)",
      "dependencies": [
        "SY:E1",
        "CE:H2",
        "U001:P15.1"
      ],
      "scope": "Exact weighted kernel and lambda threshold, all-p norm coefficient and maximum endpoint"
    },
    {
      "id": "SY:E3",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "The formal Hilbert adjoint",
      "dependencies": [
        "SY:H7",
        "SY:H8",
        "SY:H15",
        "SY:N2",
        "SY:E0",
        "SY:E1",
        "SY:E2"
      ],
      "scope": "Actual adjoint symbol/form and correct reversed dual equation with the same energy hypothesis"
    },
    {
      "id": "SY:C0",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "Choose smooth tests \\(v\\)",
      "dependencies": [
        "SY:E3",
        "SY:H3",
        "CE:H1",
        "CE:H6"
      ],
      "scope": "Injective test range, conjugate-linear functional and complete complex Hahn-Banach extension"
    },
    {
      "id": "SY:C1",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "The finite time interval and separability",
      "dependencies": [
        "SY:C0",
        "SY:H1",
        "SY:H2",
        "CE:H5",
        "CE:H11",
        "CE:H12"
      ],
      "scope": "Actual separable Hilbert L2 representation with the exact Sobolev isometry and norm bound"
    },
    {
      "id": "SY:C2",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "The \\(L^1\\) bound implies",
      "dependencies": [
        "SY:C1",
        "SY:H2",
        "CE:H3",
        "CE:H4",
        "CE:H5"
      ],
      "scope": "Normalized measurable dual test, L-infinity bound and extension of the actual representation to all L1"
    },
    {
      "id": "SY:C3",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "Interior tests imply",
      "dependencies": [
        "SY:C2",
        "SY:N3",
        "CE:H6",
        "CE:H7",
        "CE:H8",
        "CE:H9",
        "CE:H10",
        "SY:H3"
      ],
      "scope": "Full distributional equation, strong measurability, Banach primitive, countable separation and actual initial trace"
    },
    {
      "id": "SY:C4",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "For smooth time-dependent Schwartz data,",
      "dependencies": [
        "SY:C3",
        "SY:N3",
        "SY:E1",
        "SY:E2"
      ],
      "scope": "High-order existence bootstrap obtains the exact energy domain before using uniqueness"
    },
    {
      "id": "SY:C5",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "Approximate arbitrary \\(\\phi\\)",
      "dependencies": [
        "SY:C4",
        "SY:H2",
        "CE:H3",
        "CE:H6",
        "SY:E2"
      ],
      "scope": "Full density of data, Cauchy convergence in C Hs, integrated equation and every-p energy passage"
    },
    {
      "id": "SY:C6",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "If \\(u\\) is the difference",
      "dependencies": [
        "SY:C5",
        "SY:N3",
        "SY:E1",
        "SY:E2"
      ],
      "scope": "Uniqueness on the justified one-order-lower energy domain"
    },
    {
      "id": "SY:C7",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "For homogeneous forcing and data",
      "dependencies": [
        "SY:C6",
        "SY:C5",
        "SY:N3",
        "U001:FTC-TAYLOR-COMPACT-PARAMETERS"
      ],
      "scope": "All-order spatial regularity and correctly conditioned higher time regularity"
    },
    {
      "id": "SY:U0",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "For \\(0\\leq r\\leq t\\leq T\\)",
      "dependencies": [
        "SY:C5",
        "SY:C6",
        "SY:E2",
        "SY:N3"
      ],
      "scope": "Uniform ordered evolution bounds, composition by uniqueness and final-time strong derivative in Hs-1"
    },
    {
      "id": "SY:U1",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "We need joint strong continuity",
      "dependencies": [
        "SY:U0",
        "SY:H2",
        "SY:H10",
        "CE:H7"
      ],
      "scope": "Joint strong continuity on the closed time triangle using dense smooth vectors and uniform near-diagonal estimates"
    },
    {
      "id": "SY:U2",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "The integral is Bochner",
      "dependencies": [
        "SY:U1",
        "SY:U0",
        "SY:C6",
        "CE:H3",
        "CE:H4",
        "CE:H7",
        "CE:H8",
        "P3:M3"
      ],
      "scope": "Actual forcing integral, common majorants, full Banach Fubini identity and zero trace"
    },
    {
      "id": "SY:R0",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "Forward accretivity does not supply",
      "dependencies": [
        "SY:C5",
        "SY:C6",
        "SY:U0",
        "SY:U1",
        "SY:E3"
      ],
      "scope": "Correct reverse coefficient, two-sided accretivity and full inverse/composition identities on the square"
    },
    {
      "id": "SY:R1",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "A principal Hermitian system",
      "dependencies": [
        "SY:R0",
        "SY:N0"
      ],
      "scope": "Reversible Hermitian principal system with arbitrary bounded complex ordinary lower terms"
    },
    {
      "id": "SY:R2",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "A fixed positive symmetrizer also",
      "dependencies": [
        "SY:H16",
        "SY:N0",
        "SY:C5",
        "SY:C6",
        "SY:R0"
      ],
      "scope": "Exact fixed matrix congruence, all-real norm equivalence and reversible symmetrized evolution"
    },
    {
      "id": "SY:X1",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "### 5.1.",
      "dependencies": [
        "SY:H16",
        "SY:H2",
        "SY:N2",
        "SY:C6",
        "SY:U0",
        "SY:R0",
        "U001:P14.3"
      ],
      "scope": "Complete original noncommuting accretive model, forward contraction, determinant growth and localized-frequency unbounded inverse"
    },
    {
      "id": "SY:X2",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "### 5.2.",
      "dependencies": [
        "SY:H16",
        "SY:R2",
        "SY:H1",
        "U001:P2"
      ],
      "scope": "Complete original Euclidean form failure, exact symmetrizer, uniform norm and both polarized spatial speeds"
    },
    {
      "id": "SY:X3",
      "source": "first-order-systems-and-ordered-evolution.md",
      "source_sha256": "33b311a5bb673fb1680cd1dcadddf3240230df3a046398a5984a4c5df0d32eb5",
      "proof_locator": "### 5.3.",
      "dependencies": [
        "SY:H16",
        "SY:R1",
        "SY:U0",
        "SY:H1",
        "SY:H2",
        "P3:L2",
        "CH:W1",
        "RP:F2",
        "RP:F3",
        "U001:P16.1",
        "U001:P16.2"
      ],
      "scope": "Complete original ordered triangular evolution, continuous zero frequency, exact delta/interval kernels, both wavefronts and sharp Sobolev thresholds"
    },
    {
      "id": "SY:V0",
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      "proof_locator": "For a Hermitian principal system,",
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