{
  "schema": "bounded-fold-density-check/v1",
  "passed": true,
  "checks": [
    {
      "name": "full normal constraint determinants and exact constraint transformation",
      "passed": true,
      "bounded_evidence": "n=2,3,4: raw determinant (-1)^n, homogeneous determinant (-1)^n/rho; every constraint identity and the second tangential radial Jacobian are retained."
    },
    {
      "name": "exact degree-one hyperbolic elimination and reduced phase",
      "passed": true,
      "bounded_evidence": "n=2,3,4: full phase identity, auxiliary determinant one, exact Hessian with one positive and one negative eigenvalue, reduced critical determinant and all output/input covector signs."
    },
    {
      "name": "symbol-chain identities in positive and negative fractional orders",
      "passed": true,
      "bounded_evidence": "Three exact homogeneous amplitudes: fixed-tau radial derivatives, inverse amplitude change, s derivatives and critical substitution all agree; finite examples do not replace the higher-derivative proof."
    },
    {
      "name": "two normalizations and common half-density order",
      "passed": true,
      "bounded_evidence": "Three dimensions/orders: amplitude shifts plus/minus 1/2, prefactor exponent difference one, and symbol degree m+n/2; all arithmetic is exact."
    },
    {
      "name": "first-frequency Fourier inversion by a Gaussian approximate identity",
      "passed": true,
      "bounded_evidence": {
        "rho": 1.4,
        "signed_sheet_parameter": 0.7,
        "regularization": [
          0.01,
          0.001,
          0.0001
        ],
        "absolute_errors": [
          0.020331242662391502,
          0.0019780743751968404,
          0.0001971888379277542
        ],
        "expected_constant": "2pi",
        "expected_cubic_sign": "negative",
        "guard": "One Schwartz s amplitude and fixed radial/sheet parameters; convergence proof is Fourier inversion, not this numerical sample."
      }
    },
    {
      "name": "bounded first-frequency integration-by-parts model",
      "passed": true,
      "bounded_evidence": {
        "amplitude": "(1-s^2)^6 on [-1,1], extended by zero; finite boundary-vanishing model",
        "phase_derivative": "rho(t-s^2)",
        "first_frequency_ratio": "t>=3",
        "exact_iterates": [
          {
            "steps": 1,
            "radial_degree": "-2/3"
          },
          {
            "steps": 2,
            "radial_degree": "-5/3"
          },
          {
            "steps": 3,
            "radial_degree": "-8/3"
          },
          {
            "steps": 4,
            "radial_degree": "-11/3"
          }
        ],
        "guard": "Four integrations only; the arbitrary smooth compact-support all-derivative tail proof is analytic."
      }
    }
  ],
  "lesson": "fold-amplitudes-and-critical-densities.md",
  "lesson_sha256": "6639aaa65967c3b76b3636d839b554b06e970021b488aa95e429872f2e7816da",
  "script_sha256": "cdb08f87fedd9d47f40e63cbb7a0714100a121dcd2b9f25f3f6b11e9c08304d0",
  "independent_review": false,
  "full_course_complete": false,
  "guard": "Exact finite dimensions, selected amplitude identities and one Fourier numerical example support the written proof; no arbitrary-symbol representation or Airy continuity theorem is computationally certified."
}