{
  "schema": "homogeneous-phase-equivalence-bounded-check/v1",
  "passed": true,
  "checks": [
    {
      "name": "Taylor integral factor and full nonlinear indefinite matrix change",
      "passed": true,
      "critical_point": [
        "a^2",
        "a*b"
      ],
      "marked_Hessian": [
        1,
        -2
      ],
      "parameter_coordinate_determinant": "sqrt(2)",
      "scope": "Actual polynomial with parameter-dependent critical shift, exact Taylor integral, nonconstant LDL congruence and local coordinate derivative; signs fixed on a neighborhood of v=0."
    },
    {
      "name": "actual phase equations, critical maps and fiber Hessian nullity",
      "passed": true,
      "models": [
        {
          "n": 2,
          "N": 1,
          "k": 1,
          "critical_equation_rank": 1,
          "critical_map_rank": 2,
          "base_projection_rank": 1,
          "fiber_Hessian_rank": 0,
          "signature": 0
        },
        {
          "n": 2,
          "N": 3,
          "k": 1,
          "critical_equation_rank": 3,
          "critical_map_rank": 2,
          "base_projection_rank": 1,
          "fiber_Hessian_rank": 2,
          "signature": 0
        },
        {
          "n": 2,
          "N": 4,
          "k": 2,
          "critical_equation_rank": 4,
          "critical_map_rank": 2,
          "base_projection_rank": 0,
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        },
        {
          "n": 3,
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          "k": 2,
          "critical_equation_rank": 5,
          "critical_map_rank": 3,
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          "fiber_Hessian_rank": 3,
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        },
        {
          "n": 3,
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          "critical_equation_rank": 4,
          "critical_map_rank": 3,
          "base_projection_rank": 0,
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        },
        {
          "n": 4,
          "N": 6,
          "k": 3,
          "critical_equation_rank": 6,
          "critical_map_rank": 4,
          "base_projection_rank": 1,
          "fiber_Hessian_rank": 3,
          "signature": 1
        }
      ],
      "scope": "Six complete phase/critical-map tangent models, including a curved base graph and empty quadratic block; not a general smooth rank certification."
    },
    {
      "name": "varying-rank minimal homotopy and independently differentiated exact radial flow",
      "passed": true,
      "cusp": "g=(-2v^3/3-3v^4/4,v^2+v^3), v=q/p",
      "scope": "Full rational C_t and vertical field, exact cancellation, Euler degree, critical fixation, covectors and nondegeneracy; nonzero nearby g_eta. A separate conormal flow is solved and differentiated exactly. Domains retain p>0 and invertible C_t."
    },
    {
      "name": "complete homogeneous map, two-sided inverse, radius defect and mixed amplitude degrees",
      "passed": true,
      "phase_variable_determinant": "D^(-2)",
      "unequal_signatures": [
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        2
      ],
      "scope": "Exact full three-variable map, wrong square-root coefficient D^2, opposite Hessian signature, and nonconstant amplitude with base/frequency mixed derivatives; D>0 and compact angular/base domains explicit."
    },
    {
      "name": "actual stable phases and full block permutations including empty blocks",
      "passed": true,
      "models": [
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          "k": 1,
          "original_N": [
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            1
          ],
          "stabilized_N": 1,
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          "full_phase_block_permutation": [
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          "full_phase_block_permutation": [
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            2,
            5,
            6,
            7,
            3,
            4
          ],
          "absolute_permutation_Jacobian": 1
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        {
          "k": 3,
          "original_N": [
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            4
          ],
          "stabilized_N": 6,
          "combined_signature": -1,
          "full_phase_block_permutation": [
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            2,
            3,
            6,
            4,
            5
          ],
          "absolute_permutation_Jacobian": 1
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        {
          "k": 2,
          "original_N": [
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          ],
          "stabilized_N": 4,
          "combined_signature": -2,
          "full_phase_block_permutation": [
            1,
            2,
            4,
            3
          ],
          "absolute_permutation_Jacobian": 1
        }
      ],
      "scope": "Five exact stabilized phases, full variable maps, nondegenerate equations and Hessian signs; not just numerical dimension/signature counts."
    },
    {
      "name": "independent bounded source/target integral comparison and omitted-Jacobian defect",
      "passed": true,
      "base_point": [
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        0.3333333333333333
      ],
      "D_exact": "1721/1350",
      "Gauss_nodes_per_axis": 32,
      "target_box": [
        [
          0.7,
          1.3
        ],
        [
          -0.2,
          0.2
        ],
        [
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          0.2
        ]
      ],
      "absolute_integral_difference": 1.3985787785349405e-17,
      "omitted_Jacobian_absolute_defect": 0.033967933260206286,
      "scope": "Absolutely integrable finite boxes, independently assembled source/target phases, amplitudes, bounds and weights; not verification of the unrestricted oscillatory cutoff limit."
    }
  ],
  "lesson_sha256": "d62be990dcf5e284b3942ed539653ed8631901281069715abfc80bec64b38ee2",
  "script_sha256": "8ac50e2b262daa039bca6bcdd09b56584519bbd2cc76495c5ebb4f87a164e826",
  "general_smooth_proof_independently_certified": false,
  "whole_source_parent_closed": false
}