{
  "contraction_validation": {
    "asymptotic_projector_residual": 9.841760953537459e-11,
    "carrier_dimension": 125,
    "complement_dimension": 78,
    "complement_spectral_radius": 0.9437500000000003,
    "epsilon": 4.822530864197531e-06,
    "epsilon_max": 1.0716735253772278e-05,
    "errors": [],
    "gamma_max_eigenvalue": 1.0000000000000002,
    "gamma_min_eigenvalue": 0.09999999999999887,
    "hermitian_residual": 4.5614838681767503e-14,
    "iterations": 398,
    "k2_max_eigenvalue": 186624.00000000023,
    "k2_min_eigenvalue": -3.023683698931091e-11,
    "kernel_dimension": 47,
    "kernel_fixed_residual": 5.2903539071262844e-15,
    "monotone_epsilon_max": 5.358367626886139e-06,
    "projector_commutator_residual": 5.302322383667821e-15,
    "spectral_gap": 11663.999999999933,
    "tolerance": 1e-10,
    "valid": true
  },
  "kernel_validation": {
    "checks": {
      "carrier_dimension": {
        "computed": 125,
        "description": "dim(V) = dim(V\u2082 \u2297 V\u2082 \u2297 V\u2082) = 5\u00b3 = 125.",
        "expected": 125,
        "pass": true
      },
      "casimir_eigenvalue_spectrum": {
        "computed": {
          "0": 1,
          "12": 28,
          "2": 9,
          "20": 27,
          "30": 22,
          "42": 13,
          "6": 25
        },
        "description": "C has eigenvalue multiplicities consistent with V\u2082 \u2297 V\u2082 \u2297 V\u2082 decomposed into SU(2) irreps.",
        "expected": {
          "0": 1,
          "12": 28,
          "2": 9,
          "20": 27,
          "30": 22,
          "42": 13,
          "6": 25
        },
        "pass": true
      },
      "casimir_hermiticity": {
        "computed": 0.0,
        "description": "C is Hermitian.",
        "expected": 0.0,
        "pass": true
      },
      "coherence_fraction": {
        "computed": 0.376,
        "description": "dim(E\u2084\u2087) / dim(V) = 47/125.",
        "expected": 0.376,
        "pass": true
      },
      "k2_max_eigenvalue": {
        "computed": 186624,
        "description": "Largest eigenvalue of K\u00b2 is 186624.",
        "expected": 186624,
        "pass": true
      },
      "k2_positive_semidefinite": {
        "computed": -3.8513545013291515e-11,
        "description": "K\u00b2 is positive semidefinite.",
        "expected": 0.0,
        "pass": true
      },
      "k2_spectral_gap": {
        "computed": 11664,
        "description": "Smallest positive eigenvalue of K\u00b2 is 11664.",
        "expected": 11664,
        "pass": true
      },
      "k2_spectrum": {
        "computed": [
          0,
          11664,
          12544,
          19600,
          32400,
          186624
        ],
        "description": "K\u00b2 spectrum equals {0, 11664, 12544, 19600, 32400, 186624}.",
        "expected": [
          0,
          11664,
          12544,
          19600,
          32400,
          186624
        ],
        "pass": true
      },
      "kernel_annihilation": {
        "computed": 3.66356143212542e-13,
        "description": "K annihilates every vector in ker(K\u00b2).",
        "expected": 0.0,
        "pass": true
      },
      "kernel_dimension": {
        "computed": 47,
        "description": "dim(ker K) = 47.",
        "expected": 47,
        "pass": true
      },
      "kernel_hermiticity": {
        "computed": 0.0,
        "description": "K = (C \u2212 6I)(C \u2212 30I) is Hermitian.",
        "expected": 0.0,
        "pass": true
      },
      "kernel_squared_hermiticity": {
        "computed": 0.0,
        "description": "K\u00b2 is Hermitian.",
        "expected": 0.0,
        "pass": true
      }
    },
    "results": {
      "carrier_dimension": 125,
      "casimir_hermitian_residual": 0.0,
      "coherence_fraction": 0.376,
      "k2_hermitian_residual": 0.0,
      "k2_max_eigenvalue": 186624,
      "k2_spectral_gap": 11664,
      "k2_unique_eigenvalues": [
        0,
        11664,
        12544,
        19600,
        32400,
        186624
      ],
      "kernel_dimension": 47,
      "kernel_hermitian_residual": 0.0,
      "max_kernel_annihilation_error": 3.66356143212542e-13
    },
    "status": "pass"
  },
  "projector_validation": {
    "checks": {
      "complement_idempotence": {
        "computed": 4.9771532071534406e-14,
        "description": "I - P\u2084\u2087 is also an orthogonal projector.",
        "expected": 0.0,
        "pass": true
      },
      "kernel_annihilation": {
        "computed": 2.306547804400579e-11,
        "description": "K annihilates the range of P\u2084\u2087: K P\u2084\u2087 = 0.",
        "expected": 0.0,
        "pass": true
      },
      "kernel_basis_dimension": {
        "computed": 47,
        "description": "The extracted zero eigenspace of K\u00b2 contains exactly 47 orthonormal vectors.",
        "expected": 47,
        "pass": true
      },
      "kernel_basis_orthonormality": {
        "computed": 8.019108826791589e-15,
        "description": "The extracted kernel eigenvectors form an orthonormal basis.",
        "expected": 0.0,
        "pass": true
      },
      "kernel_squared_annihilation": {
        "computed": 3.0778843423254525e-09,
        "description": "K\u00b2 annihilates the range of P\u2084\u2087.",
        "expected": 0.0,
        "pass": true
      },
      "orthogonal_decomposition": {
        "computed": 4.813039516702142e-14,
        "description": "The projector and its complement have orthogonal ranges.",
        "expected": 0.0,
        "pass": true
      },
      "projector_commutes_with_casimir": {
        "computed": 5.1880980981082575e-12,
        "description": "P\u2084\u2087 is a spectral projector of C and therefore commutes with C.",
        "expected": 0.0,
        "pass": true
      },
      "projector_commutes_with_kernel": {
        "computed": 4.4871719659351695e-11,
        "description": "P\u2084\u2087 and K commute.",
        "expected": 0.0,
        "pass": true
      },
      "projector_hermiticity": {
        "computed": 0.0,
        "description": "P\u2084\u2087 is Hermitian: P\u2084\u2087\u2020 = P\u2084\u2087.",
        "eigenvalue_imaginary_max": 0.0,
        "expected": 0.0,
        "pass": true
      },
      "projector_idempotence": {
        "computed": 4.906691467917368e-14,
        "description": "P\u2084\u2087 is idempotent: P\u2084\u2087\u00b2 = P\u2084\u2087.",
        "expected": 0.0,
        "pass": true
      },
      "projector_identity_on_basis": {
        "computed": 2.7641683914309598e-15,
        "description": "P\u2084\u2087 acts as the identity on every kernel-basis vector.",
        "expected": 0.0,
        "pass": true
      },
      "projector_rank": {
        "computed": 47,
        "description": "The numerical rank of P\u2084\u2087 is 47.",
        "expected": 47,
        "pass": true
      },
      "projector_trace": {
        "absolute_error": 7.105427357601002e-15,
        "computed": 46.99999999999999,
        "description": "The trace of an orthogonal projector equals its rank.",
        "expected": 47,
        "imaginary_component_error": 0.0,
        "pass": true
      }
    },
    "inputs": {
      "carrier_dimension": 125,
      "expected_rank": 47,
      "kernel_operator": "K = (C - 6I)(C - 30I)",
      "projector_definition": "P47 = sum_i |e_i><e_i| over the zero eigenspace of K^2"
    },
    "results": {
      "basis_orthonormality_error": 8.019108826791589e-15,
      "carrier_dimension": 125,
      "complement_idempotence_error": 4.9771532071534406e-14,
      "construction_tolerance": 1e-08,
      "kernel_annihilation_error": 2.306547804400579e-11,
      "kernel_basis_dimension": 47,
      "kernel_squared_annihilation_error": 3.0778843423254525e-09,
      "maximum_basis_action_error": 2.7641683914309598e-15,
      "orthogonal_decomposition_error": 4.813039516702142e-14,
      "projector_casimir_commutator_error": 5.1880980981082575e-12,
      "projector_hermiticity_error": 0.0,
      "projector_idempotence_error": 4.906691467917368e-14,
      "projector_kernel_commutator_error": 4.4871719659351695e-11,
      "projector_rank": 47,
      "projector_trace": 46.99999999999999
    },
    "status": "pass",
    "tolerances": {
      "annihilation_tolerance": 1e-08,
      "basis_tolerance": 1e-10,
      "hermiticity_tolerance": 1e-10,
      "idempotence_tolerance": 1e-10,
      "trace_tolerance": 1e-08
    }
  },
  "qutip_validation": {
    "carrier_dimension": 125,
    "coherence_fraction": 0.376,
    "errors": [],
    "k2_spectral_gap": 11664,
    "kernel_dimension": 47,
    "valid": true
  },
  "semigroup_validation": {
    "asymptotic_projector_residual": 8.597836122919425e-06,
    "carrier_dimension": 125,
    "complement_dimension": 78,
    "contraction_violation": 6.372680161348399e-14,
    "errors": [],
    "expected_decay_bound": 8.597836122737877e-06,
    "hermitian_residual": 1.589393196129677e-16,
    "identity_residual": 2.608735010101322e-15,
    "k2_max_eigenvalue": 186624.0,
    "kernel_dimension": 47,
    "kernel_fixed_residual": 1.0097045299853382e-13,
    "positivity_violation": 4.630113888715213e-16,
    "projector_commutator_residual": 9.517276153749086e-14,
    "second_time": 0.00075,
    "semigroup_law_residual": 2.401747409270404e-15,
    "semigroup_max_eigenvalue": 1.0000000000000637,
    "semigroup_min_eigenvalue": -4.630113888715213e-16,
    "spectral_gap": 11663.999999999964,
    "time": 0.001,
    "tolerance": 1e-10,
    "valid": true
  },
  "valid": true
}