{
  "balanced_value": "0, using the proved normal bridge and published GHHR Remark 8.3",
  "checks": {
    "GF2 support solver rejects empty affine coset": "pass",
    "GF2 support solver returns exact witness": "pass",
    "all-r normal summand degree": "pass",
    "boundary vector after reindexing is its opposite": "pass",
    "complete character has valuation 2E-S": "pass",
    "complete character phase is NOT just B^(2E-S)": "pass",
    "dropping the Weyl exponent introduces an error": "pass",
    "dropping the central longitude scalar introduces an error": "pass",
    "half-integral raw representatives are kept exactly": "pass",
    "integer normal witness reconstructs exact lambda+rb": "pass",
    "lattice translation scalar is A^(-omega(v,w)/2)": "pass",
    "left seed translate and shifted average have identical Weyl phase": "pass",
    "native all-orders A exponent matches normal weight": "pass",
    "native i phase matches modulo four for integer meridian": "pass",
    "normal class is invariant under fractional common shifts": "pass",
    "other-cusp i phase is exact modulo four": "pass",
    "other-cusp native scalar matches the same normal weight": "pass",
    "other-cusp reindexing and simultaneous inversion leave a partial reversal": "pass",
    "outer A exponent after r inversion and n reindexing": "pass",
    "outer parity phase is unchanged modulo four": "pass",
    "root parity coset, not even lambda, is required": "pass",
    "same complete character requires exact orthogonality": "pass",
    "seed inversion shortcut fails before any global summation": "pass",
    "seed is not invariant under root inversion": "pass",
    "surjectivity reconstruction uses only an integer edge witness": "pass",
    "two balanced inputs have identical support modulo two": "pass"
  },
  "exact_checks": 26,
  "external_mathematical_dependencies": [
    "GHHR Theorem 7.1: integral kernel and image of the normal-homology map",
    "GHHR Section 7.1: peripheral raw rows have mod-2 class zero",
    "GHHR Definition 8.1 and Remark 8.3: extended index and boundary inversion",
    "retained global all-input convergence and correct-sided translate theorem"
  ],
  "local_coordinate_fixture": {
    "R": [
      -3,
      0,
      -4,
      2,
      1,
      -3
    ],
    "U": [
      "-3/2",
      "1/2",
      "-5/2",
      "3/2",
      "1/2",
      "-1/2"
    ],
    "X": [
      2,
      0,
      3,
      0,
      1,
      2
    ],
    "longitude": [
      0,
      1,
      -1,
      1,
      0,
      2
    ],
    "maxima": [
      "1/2",
      "3/2"
    ],
    "topological_realization_asserted": false
  },
  "normal_bridge": "F0(l^r)=sum_n (-1)^n A^(r*n) Z(-r*l-n*s)",
  "other_cusp_test": "(c_l/c_t)*sum_n (-1)^n A^n [Z(t-l-ns)-Z(-t-l-ns)], with nonvanishing unresolved",
  "same_complete_character_one_cusp": "all monomial translates kill the vacuum",
  "scope": "universal identities and local algebra; no topological instances searched",
  "seed_series_calibration": {
    "I_Delta_0_1": {
      "0": 1,
      "8": -1,
      "12": -2,
      "16": -3
    },
    "I_Delta_0_minus1": {
      "4": -1,
      "8": -1,
      "16": 1
    },
    "cutoff_A_degree": 16,
    "used_to_prove_global_boundary_inversion": false
  }
}
