{
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  "finite_groups": 18,
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      "name": "Complete two-root quotient and companion forms",
      "passed": true,
      "scope": "Both full inverses, all entries of the positive diagonal form and its exact companion symmetrization. The quotient and eigenvector inverses are explicitly distinct."
    },
    {
      "name": "Entire cubic polynomial and normalized matrix reconstruction",
      "passed": true,
      "scope": "All three Lagrange rows and both complete normalized matrix products, not a single selected coefficient."
    },
    {
      "name": "Nonautonomous complete operator division",
      "passed": true,
      "scope": "Exact action on arbitrary F(t,x), including every time/spatial derivative and the residual sign. Strictness is only asserted on a time interval separated from zero."
    },
    {
      "name": "Full homogeneous and forced wave solution",
      "passed": true,
      "scope": "Entire equation and both Cauchy jets for both terms, with both zero-frequency limits."
    },
    {
      "name": "Complete differential commutator sign and degree",
      "passed": true,
      "cases": [
        {
          "m": 1,
          "all_lower_normal_terms_verified": 1
        },
        {
          "m": 2,
          "all_lower_normal_terms_verified": 2
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        {
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          "all_lower_normal_terms_verified": 3
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        {
          "m": 4,
          "all_lower_normal_terms_verified": 4
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        {
          "m": 5,
          "all_lower_normal_terms_verified": 5
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      ],
      "scope": "All coefficients and all normal degrees for m1 through5, acting on arbitrary F. Leading normal coefficient contributes zero; total orderm and normal order at mostm-1 are kept."
    },
    {
      "name": "Scalar transport and commutant sign",
      "passed": true,
      "scope": "Full exact commutator for arbitrary multiplier q(x+ct) and arbitrary F(t,x), verifying partial_t-H_(c xi) and negative spatial branch velocity."
    },
    {
      "name": "Entire incoming-edge model and exit location",
      "passed": true,
      "scope": "The full operator identity holds for arbitrary one-variable G and hence its distributional specialization. Exact points locate a characteristic exiting the spatial domain before the initial surface; the local vanishing argument is written for every boundary point."
    },
    {
      "name": "Entire restricted extension weight and decomposition identities",
      "passed": true,
      "scope": "The adjoint complex multiplier has the exact full mixed weight in modulus. Both terms of the full-space decomposition sum to identity; both exact target/input ratios are computed. Backward support and quotient isometry are the separately source-bound B.2.4 input, not established by algebra."
    },
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      "name": "Full compressed normal commutator models",
      "passed": true,
      "cases": [
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        {
          "compressed_polynomial_degree": 1,
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        {
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        {
          "compressed_polynomial_degree": 5,
          "all_terms_verified": true
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        {
          "compressed_polynomial_degree": 6,
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      "scope": "For b(zeta)=zeta^k, the complete operator B=t^k D_t^k satisfies the normal commutator, including B_(D_zeta b)D_t and its sign, on arbitrary F. These models supplement, rather than replace, the source-bound full smooth-symbol identity and intrinsic dual-topology argument."
    },
    {
      "name": "Both full signed wave branch projections",
      "passed": true,
      "scope": "Both entire projected solutions, their sum and both first-order branch equations. Division is used only on cones separated from zero; the previous group checks the combined removable zero-frequency continuation."
    },
    {
      "name": "Forcing boundary model and nonextendibility growth",
      "passed": true,
      "scope": "Exact normal operator identity for arbitrary spatial distributions by specialization, with vanishing initial value. The exponential exceeds every positive polynomial order; the full compact-test distribution-order contradiction is written in Exercise19. Wavefront and N membership use the source-bound calculus, not a finite symbolic assertion."
    },
    {
      "name": "Weighted-forcing exponential kernel and all endpoints",
      "passed": true,
      "scope": "Exact Duhamel factorization retaining e^(-lambda r). Its normalized p-power integral equals lambda/[p(lambda-beta)] and is bounded by2 for lambda at least2beta and p at least1; p-infinity uses supremum1. The Volterra and Minkowski proofs are in Section3."
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      "name": "Complete initial left-division coefficient recursion",
      "passed": true,
      "cases": [
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          "m": 1,
          "complete_variable_coefficient_operator_identity": true
        },
        {
          "m": 2,
          "complete_variable_coefficient_operator_identity": true
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        {
          "m": 3,
          "complete_variable_coefficient_operator_identity": true
        },
        {
          "m": 4,
          "complete_variable_coefficient_operator_identity": true
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      ],
      "scope": "Full exact differential identities on arbitrary F for m1 through4, with arbitrary time-and-space dependent coefficients and both first-order and zeroth-order terms in A. The coefficient derivative is the complete [D_t,Q_k], including all coefficient derivatives; all positive normal powers cancel."
    },
    {
      "name": "Exact minimal extension and sharp normal-step bound",
      "passed": true
    },
    {
      "name": "Positive-real beta/Gamma trace constant",
      "passed": true,
      "cases": [
        {
          "j": 0,
          "r": "1",
          "relative_error": "0.0"
        },
        {
          "j": 0,
          "r": "0.8",
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          "j": 1,
          "r": "2",
          "relative_error": "0.0"
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        {
          "j": 1,
          "r": "2.3",
          "relative_error": "0.0"
        },
        {
          "j": 3,
          "r": "5.7",
          "relative_error": "0.0"
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      ]
    },
    {
      "name": "Full Green differential identity with every coefficient derivative",
      "passed": true,
      "orders": [
        1,
        2,
        3,
        4,
        5
      ]
    },
    {
      "name": "All order-three boundary coordinate terms and H2 counterexample exponents",
      "passed": true
    },
    {
      "name": "Bounded moment weights, exact derivative cancellation and local limit",
      "passed": true
    }
  ],
  "checked_utc": "2026-10-05T19:58:51.320598+00:00",
  "script_sha256": "7bedd8b5f4546bd11b891fec9feba4386a6e27f6f0103d1b1fd8bef7783af024",
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  "finite_checks_replace_proofs": false,
  "human_review_claimed": false,
  "full_course_complete": false
}