{
  "schema": "canonical-coordinates-finite-check/v1",
  "passed": true,
  "lesson": "prescribed-canonical-coordinates-and-isotropic-fibers.md",
  "lesson_sha256": "0822251b59c9279212f165831dce586f34d6a32844659fa76da0ecb490e2b14b",
  "script_sha256": "b15c407933bef55e22539ea3e6a2c0d9955fcc88caac38001f204566c0e8f5ae",
  "checks": [
    {
      "name": "alternating dual correction and opposite sign",
      "passed": true,
      "models": [
        {
          "dimension": 4,
          "initial_skew_matrix": "Matrix([[0, 3], [-3, 0]])",
          "corrected_pairings_zero": true,
          "wrong_sign_doubles": true
        },
        {
          "dimension": 6,
          "initial_skew_matrix": "Matrix([[0, 2, -1], [-2, 0, 5], [1, -5, 0]])",
          "corrected_pairings_zero": true,
          "wrong_sign_doubles": true
        }
      ],
      "scope": "Explicit canonical dual vectors with nonzero initial pairings; general radical/complement argument remains in the proof."
    },
    {
      "name": "ordinary nonlinear canonical completion",
      "passed": true,
      "exact_primitive_difference": "d(x1*x2^2)",
      "scope": "Full two-dimensional nonlinear map, including all brackets."
    },
    {
      "name": "semidirect flow signs and full Euler equations",
      "passed": true,
      "radial_chart": "d/dell+tau1*d/dtau1+tau2*d/dtau2",
      "position": "z2+w^2",
      "momentum": "-tau1+exp(ell)*(b+w^2)",
      "scope": "Exact nonconstant section data, both Hamilton commutators and all stated field equations."
    },
    {
      "name": "exceptional nonzero-momentum position correction",
      "passed": true,
      "b_J": 2,
      "h": "(p-2)^2",
      "exact_correction": "p**2/2 - 4*p + 4*log(p) - 4*log(2) + 6",
      "scope": "Nonzero variable h, including logarithmic integral, marked differential and exact Euler equation."
    },
    {
      "name": "homogeneous primitive-preserving rotation including empty sums",
      "passed": true,
      "models": [
        {
          "n": 1,
          "k": 1,
          "primitive_and_form_preserved": true,
          "inverse_verified": true,
          "degrees": [
            0,
            1
          ],
          "auxiliary_model_sent_to_fiber": true
        },
        {
          "n": 2,
          "k": 1,
          "primitive_and_form_preserved": true,
          "inverse_verified": true,
          "degrees": [
            0,
            1
          ],
          "auxiliary_model_sent_to_fiber": true
        },
        {
          "n": 2,
          "k": 2,
          "primitive_and_form_preserved": true,
          "inverse_verified": true,
          "degrees": [
            0,
            1
          ],
          "auxiliary_model_sent_to_fiber": true
        },
        {
          "n": 3,
          "k": 2,
          "primitive_and_form_preserved": true,
          "inverse_verified": true,
          "degrees": [
            0,
            1
          ],
          "auxiliary_model_sent_to_fiber": true
        },
        {
          "n": 4,
          "k": 4,
          "primitive_and_form_preserved": true,
          "inverse_verified": true,
          "degrees": [
            0,
            1
          ],
          "auxiliary_model_sent_to_fiber": true
        }
      ],
      "scope": "Five exact dimensions/index counts, all covectors/base coordinates, and the full inverse."
    },
    {
      "name": "deformed cusp primitive, projection ranks and covector immersion",
      "passed": true,
      "base_ranks": [
        1,
        0,
        1
      ],
      "full_rank": 2,
      "lower_tangent_at_zero": [
        1,
        0
      ],
      "scope": "Full exact two-dimensional conic Lagrangian in the specified r range, not a global conormal claim."
    }
  ],
  "independent_mathematical_review": false,
  "scope": "Six bounded exact alternating, nonlinear symplectic, semidirect-flow, marked-coordinate, homogeneous rotation and cusp models. General smooth existence, isotropic induction and germ recognition are proved in the lesson, not certified by models."
}