{
  "gauge_example": {
    "gradient_dot_orbit": 0,
    "critical_hessian": "Matrix([[8, 0, 0], [0, 0, 0], [0, 0, 2]])",
    "gradient_at_critical_point": "Matrix([[0], [0], [0]])"
  },
  "decomposition_counterexample": {
    "S": "y²+yz+z²",
    "gauge": "x direction",
    "Hessian": "Matrix([[0, 0, 0], [0, 2, 1], [0, 1, 2]])",
    "mixed_yz": 1
  },
  "K3_exact": {
    "chern_series": "6*h**2 + 1",
    "integral_H2": 4,
    "chi": 24,
    "signature": -16,
    "b2": 22,
    "h11": 20,
    "unit_volume_metric_moduli": 57,
    "with_volume": 58,
    "scalar_a4": "4/15",
    "scalar_zeta0": "-11/15"
  },
  "laplace": {
    "potential": "(x²−1)²(x²+x+1)",
    "H_left": 8,
    "H_right": 24,
    "limit_left": 0.6339745962155613,
    "limit_right": 0.3660254037844387,
    "numerics": [
      {
        "epsilon": 0.1,
        "left_mass": 0.6407736506606446,
        "difference_from_limit": 0.00679905444508333
      },
      {
        "epsilon": 0.02,
        "left_mass": 0.6353885313916223,
        "difference_from_limit": 0.001413935176061032
      },
      {
        "epsilon": 0.005,
        "left_mass": 0.6343259874057366,
        "difference_from_limit": 0.00035139119017535414
      },
      {
        "epsilon": 0.001,
        "left_mass": 0.6340447321844239,
        "difference_from_limit": 7.013596886262796e-05
      }
    ]
  },
  "kraft": {
    "prefix_code": "0,10,110,...",
    "partial_sum_N30": "1073741823/1073741824",
    "unbounded_prefactor_example": "d_n=2^n, S_n=0: each weighted term is 1; sum diverges"
  },
  "coercivity_counterexample": {
    "S": "0.5*log(1+x²)",
    "Z_epsilon1": "diverges: integral of 1/sqrt(1+x²)",
    "finite_window_integral_R1000": 15.201805419083977,
    "regularity": "smooth, nonnegative, coercive; unique nondegenerate minimum"
  },
  "born_split": {
    "norm_preserving_split": "c -> (c/sqrt(2),c/sqrt(2))",
    "weight_multipliers": {
      "1": 1.4142135623730951,
      "2": 1.0,
      "4": 0.5
    },
    "condition": "positive power law and additive branch weights; quantum norm is input"
  },
  "modular_kernel": {
    "eta_i": "0.768225422326056659002594179576180644517866914464805014676703",
    "V_i": "0.348300982421419214796061565081342060664370304378904469743571",
    "V_rho": "0.355752762782858968975134335031959773752152420380914248523653",
    "max_modular_relative_error": 1.2446030555722283e-60,
    "minus_log_4Vrho2": "0.680744190805339655448696864909294127660700505928405450787951",
    "divided_by_4pi": "0.0541719014738810864726753758527732617466979556857884622505822",
    "scope": "kernel arithmetic only, no electromagnetic coupling matching"
  },
  "constant_j": {
    "free_type": 1,
    "j1728_fibre_types": 9,
    "j0_fibre_types": 46,
    "total_types": 56,
    "generic_j": "6912*a**3/(4*a**3 + 27*b**2)",
    "interpretation": "fibre multiplicity patterns, not 56 individual metrics or universes; positions and generic j remain variable"
  },
  "archived_spectrum_checks": [
    {
      "file": "k3_run2.npz",
      "k": 3,
      "N": 20,
      "modes": 400,
      "V_raw": 78.92306342436659,
      "mean_first_12": 44.193637836707886,
      "zero_mode": 0.007482344374582608,
      "zeta_prime_unadjusted_ts005": 1.0157729168961738,
      "old_integrator_difference": 3.307447795641849e-07,
      "fitted_a4_check": -3.5640257256595467
    },
    {
      "file": "k4_run1.npz",
      "k": 4,
      "N": 34,
      "modes": 1156,
      "V_raw": 79.00720442514546,
      "mean_first_12": 43.95780746896526,
      "zero_mode": 0.01001513021994984,
      "zeta_prime_unadjusted_ts005": 1.0238767815115102,
      "old_integrator_difference": 3.3545138578006117e-07,
      "fitted_a4_check": -2.1180248442731955
    },
    {
      "file": "k5_run1.npz",
      "k": 5,
      "N": 52,
      "modes": 2704,
      "V_raw": 79.05326653101423,
      "mean_first_12": 43.797472549045686,
      "zero_mode": 0.010152253152706462,
      "zeta_prime_unadjusted_ts005": 1.028623696743923,
      "old_integrator_difference": 3.376112207575943e-07,
      "fitted_a4_check": -1.7600420157577013
    },
    {
      "file": "k6_run1.npz",
      "k": 6,
      "N": 74,
      "modes": 5476,
      "V_raw": 79.01656403145859,
      "mean_first_12": 43.52348027842439,
      "zero_mode": 0.016314771265325923,
      "zeta_prime_unadjusted_ts005": 1.039934602372297,
      "old_integrator_difference": 3.4362037881763285e-07,
      "fitted_a4_check": -1.0248238048091045
    }
  ],
  "gauge_running": [
    {
      "loops": 1,
      "crossings_GeV": {
        "1-2": 10308931063427.56,
        "1-3": 242023640070928.66,
        "2-3": 9.596709683230115e+16
      },
      "predicted_inverse_alpha_at_MZ": 145.56633552643177,
      "relative_difference_percent": 13.76725115585793,
      "warning": "gauge-only two-loop equations; alpha at MZ is already used as input; not a derivation of alpha(0)"
    },
    {
      "loops": 2,
      "crossings_GeV": {
        "1-2": 10873454353789.703,
        "1-3": 171285378221123.5,
        "2-3": 2.8818256922810828e+16
      },
      "predicted_inverse_alpha_at_MZ": 143.1220293720984,
      "relative_difference_percent": 11.856905668653162,
      "warning": "gauge-only two-loop equations; alpha at MZ is already used as input; not a derivation of alpha(0)"
    }
  ],
  "gauge_coefficient_algebra": {
    "stated_coefficient": "f0*n*g²/(96*pi²) = 1/4",
    "solution": "24*pi**2/(f0*n)",
    "implemented": "12*pi**2/(f0*n)",
    "implemented_coefficient": "1/8",
    "scope": "factor-two inconsistency inside alpha_u_toets.py; physical trace convention still requires justification"
  },
  "environment": {
    "python": "3.14.4 (main, Aug 20 2026, 10:41:58) [GCC 15.2.0]",
    "numpy": "2.3.5",
    "sympy": "1.14.0",
    "mpmath": "1.3.0"
  }
}