{
  "release": "0.7.0",
  "protocol_frozen_before_execution": true,
  "constants": {
    "phi_exact": {
      "a": "1/2",
      "b": "1/2",
      "decimal": "1.6180339887498948482045868343656381177203091798057628621354486227052604628189024"
    },
    "phi_decimal": "1.6180339887498948482045868343656381177203091798057628621354486227052604628189024",
    "golden_turn_fraction_exact": {
      "a": "3/2",
      "b": "-1/2",
      "decimal": "0.3819660112501051517954131656343618822796908201942371378645513772947395371810976"
    },
    "golden_turn_fraction_decimal": "0.3819660112501051517954131656343618822796908201942371378645513772947395371810976",
    "golden_angle_degrees": 137.50776405003785,
    "golden_angle_radians": 2.399963229728653
  },
  "exact_identity_checks": {
    "phi_squared_equals_phi_plus_one": true,
    "phi_times_phi_minus_one_equals_one": true,
    "golden_turn_equals_one_minus_inverse_phi": true,
    "fibonacci_companion_characteristic": "lambda^2-lambda-1",
    "dominant_root_is_phi": true,
    "E8_identity_2cos_pi_over_5_equals_phi_error": 0.0
  },
  "fibonacci_convergence": [
    {
      "index": 2,
      "numerator": 2,
      "denominator": 1,
      "fraction": "2/1",
      "value": 2.0,
      "absolute_error": 0.3819660112501051
    },
    {
      "index": 3,
      "numerator": 3,
      "denominator": 2,
      "fraction": "3/2",
      "value": 1.5,
      "absolute_error": 0.1180339887498949
    },
    {
      "index": 4,
      "numerator": 5,
      "denominator": 3,
      "fraction": "5/3",
      "value": 1.6666666666666667,
      "absolute_error": 0.04863267791677184
    },
    {
      "index": 5,
      "numerator": 8,
      "denominator": 5,
      "fraction": "8/5",
      "value": 1.6,
      "absolute_error": 0.018033988749894814
    },
    {
      "index": 6,
      "numerator": 13,
      "denominator": 8,
      "fraction": "13/8",
      "value": 1.625,
      "absolute_error": 0.0069660112501050975
    },
    {
      "index": 7,
      "numerator": 21,
      "denominator": 13,
      "fraction": "21/13",
      "value": 1.6153846153846154,
      "absolute_error": 0.0026493733652794837
    },
    {
      "index": 8,
      "numerator": 34,
      "denominator": 21,
      "fraction": "34/21",
      "value": 1.619047619047619,
      "absolute_error": 0.0010136302977241662
    },
    {
      "index": 9,
      "numerator": 55,
      "denominator": 34,
      "fraction": "55/34",
      "value": 1.6176470588235294,
      "absolute_error": 0.00038692992636546464
    },
    {
      "index": 10,
      "numerator": 89,
      "denominator": 55,
      "fraction": "89/55",
      "value": 1.6181818181818182,
      "absolute_error": 0.00014782943192326314
    },
    {
      "index": 11,
      "numerator": 144,
      "denominator": 89,
      "fraction": "144/89",
      "value": 1.6179775280898876,
      "absolute_error": 5.6460660007306984e-05
    },
    {
      "index": 12,
      "numerator": 233,
      "denominator": 144,
      "fraction": "233/144",
      "value": 1.6180555555555556,
      "absolute_error": 2.15668056606777e-05
    },
    {
      "index": 13,
      "numerator": 377,
      "denominator": 233,
      "fraction": "377/233",
      "value": 1.6180257510729614,
      "absolute_error": 8.237676933475768e-06
    },
    {
      "index": 14,
      "numerator": 610,
      "denominator": 377,
      "fraction": "610/377",
      "value": 1.6180371352785146,
      "absolute_error": 3.1465286196574738e-06
    },
    {
      "index": 15,
      "numerator": 987,
      "denominator": 610,
      "fraction": "987/610",
      "value": 1.618032786885246,
      "absolute_error": 1.2018646489142526e-06
    },
    {
      "index": 16,
      "numerator": 1597,
      "denominator": 987,
      "fraction": "1597/987",
      "value": 1.618034447821682,
      "absolute_error": 4.590717870289751e-07
    },
    {
      "index": 17,
      "numerator": 2584,
      "denominator": 1597,
      "fraction": "2584/1597",
      "value": 1.6180338134001253,
      "absolute_error": 1.7534976959332482e-07
    },
    {
      "index": 18,
      "numerator": 4181,
      "denominator": 2584,
      "fraction": "4181/2584",
      "value": 1.618034055727554,
      "absolute_error": 6.697765919660981e-08
    }
  ],
  "fixed_grid_golden_turn": {
    "exact_finite_grid_representation": false,
    "proof": "A finite angular grid step is k/N, which is rational. The golden-turn fraction (3-sqrt(5))/2 is irrational.",
    "convergents": [
      {
        "index": 3,
        "ticks_advanced": 1,
        "grid_size": 2,
        "fraction": "1/2",
        "turn_error": 0.11803398874989485,
        "angle_error_degrees": 42.492235949962144
      },
      {
        "index": 4,
        "ticks_advanced": 1,
        "grid_size": 3,
        "fraction": "1/3",
        "turn_error": 0.04863267791677184,
        "angle_error_degrees": 17.507764050037864
      },
      {
        "index": 5,
        "ticks_advanced": 2,
        "grid_size": 5,
        "fraction": "2/5",
        "turn_error": 0.01803398874989487,
        "angle_error_degrees": 6.4922359499621525
      },
      {
        "index": 6,
        "ticks_advanced": 3,
        "grid_size": 8,
        "fraction": "3/8",
        "turn_error": 0.006966011250105153,
        "angle_error_degrees": 2.507764050037855
      },
      {
        "index": 7,
        "ticks_advanced": 5,
        "grid_size": 13,
        "fraction": "5/13",
        "turn_error": 0.0026493733652794837,
        "angle_error_degrees": 0.9537744115006141
      },
      {
        "index": 8,
        "ticks_advanced": 8,
        "grid_size": 21,
        "fraction": "8/21",
        "turn_error": 0.0010136302977242218,
        "angle_error_degrees": 0.36490690718071983
      },
      {
        "index": 9,
        "ticks_advanced": 13,
        "grid_size": 34,
        "fraction": "13/34",
        "turn_error": 0.0003869299263654091,
        "angle_error_degrees": 0.13929477349154729
      },
      {
        "index": 10,
        "ticks_advanced": 21,
        "grid_size": 55,
        "fraction": "21/55",
        "turn_error": 0.00014782943192331865,
        "angle_error_degrees": 0.053218595492394716
      },
      {
        "index": 11,
        "ticks_advanced": 34,
        "grid_size": 89,
        "fraction": "34/89",
        "turn_error": 5.646066000719596e-05,
        "angle_error_degrees": 0.020325837602590546
      },
      {
        "index": 12,
        "ticks_advanced": 55,
        "grid_size": 144,
        "fraction": "55/144",
        "turn_error": 2.1566805660733213e-05,
        "angle_error_degrees": 0.0077640500378639565
      },
      {
        "index": 13,
        "ticks_advanced": 89,
        "grid_size": 233,
        "fraction": "89/233",
        "turn_error": 8.237676933475768e-06,
        "angle_error_degrees": 0.0029655636960512766
      },
      {
        "index": 14,
        "ticks_advanced": 144,
        "grid_size": 377,
        "fraction": "144/377",
        "turn_error": 3.146528619768496e-06,
        "angle_error_degrees": 0.0011327503031166586
      },
      {
        "index": 15,
        "ticks_advanced": 233,
        "grid_size": 610,
        "fraction": "233/610",
        "turn_error": 1.2018646489697637e-06,
        "angle_error_degrees": 0.00043267127362911495
      },
      {
        "index": 16,
        "ticks_advanced": 377,
        "grid_size": 987,
        "fraction": "377/987",
        "turn_error": 4.590717870289751e-07,
        "angle_error_degrees": 0.00016526584333043104
      },
      {
        "index": 17,
        "ticks_advanced": 610,
        "grid_size": 1597,
        "fraction": "610/1597",
        "turn_error": 1.7534976959332482e-07,
        "angle_error_degrees": 6.312591705359694e-05
      },
      {
        "index": 18,
        "ticks_advanced": 987,
        "grid_size": 2584,
        "fraction": "987/2584",
        "turn_error": 6.697765930763211e-08,
        "angle_error_degrees": 2.411195735074756e-05
      }
    ],
    "prefix_count": 144,
    "prefix_star_discrepancy": 0.006944444444444444,
    "exact_recurrence": false,
    "boundary": "Replacing the irrational turn by p/q creates a q-tick periodic approximation, not an exact golden rotation."
  },
  "ideal_hydrogen_gap_audit": {
    "n_max": 40,
    "transition_count": 780,
    "distinct_ratio_comparisons": 303809,
    "numerator_transition": [
      19,
      30
    ],
    "denominator_transition": [
      23,
      34
    ],
    "ratio_fraction": "7491169/4629825",
    "ratio_decimal": "1.6180242233777734579600740848736183333063344726852526823368053868126765050514868",
    "absolute_error_to_phi": "0.0000097653721213902445127494920197844139747071205101797986432358925839577674156",
    "relative_error_to_phi": "0.0000060353318838098351651409269242503367339374217477390224584031788796950220332929405",
    "exact_phi_equality": false,
    "exact_reason": "Every ideal Coulomb bound-state gap is rational in Hartree; a ratio of two nonzero gaps is rational, whereas phi is irrational."
  },
  "icosahedral_geometry": {
    "vertex_count": 12,
    "unique_pair_distances": [
      2.0,
      3.23606797749979,
      3.804226065180614
    ],
    "edge_length": 2.0,
    "next_separation": 3.23606797749979,
    "next_to_edge_ratio": 1.618033988749895,
    "absolute_error_to_phi": 0.0,
    "classification": "construction-imposed exact geometry"
  },
  "classification": [
    {
      "object": "Fibonacci companion matrix",
      "status": "exact intrinsic algebra",
      "reason": "its characteristic polynomial is x^2-x-1"
    },
    {
      "object": "E8 first two mass scales",
      "status": "exact within the E8 theory; experimentally tested elsewhere",
      "reason": "m2/m1 = 2 cos(pi/5) = phi"
    },
    {
      "object": "icosahedral coordinates / golden rectangles",
      "status": "exact when that geometry is declared",
      "reason": "the coordinates explicitly contain phi"
    },
    {
      "object": "golden-angle sampler",
      "status": "construction-imposed",
      "reason": "the engine chooses the irrational turn fraction"
    },
    {
      "object": "ideal hydrogen Coulomb gap ratios",
      "status": "no exact phi ratio",
      "reason": "all such ratios are rational"
    },
    {
      "object": "finite fixed-tick angular grid",
      "status": "cannot contain the exact golden turn",
      "reason": "every finite-grid step is rational"
    }
  ],
  "epistemic_status": {
    "written_math": "Q(sqrt(5)) identities, Fibonacci eigenvalue, hydrogen rationality, and finite-grid irrationality arguments",
    "finite_computation": "n<=40 hydrogen scan, Fibonacci convergence table, icosahedron distance check, and 144-point discrepancy",
    "measured_evidence_added": false,
    "hypothesis": "A phi-based hidden-dimensional mechanism remains unspecified and is not inferred by this audit."
  }
}