{
  "schema": "fundamental_force_pipeline_v1",
  "input": {
    "n": 8,
    "pairs": [
      [
        0,
        1
      ],
      [
        2,
        3
      ],
      [
        4,
        5
      ],
      [
        6,
        7
      ]
    ],
    "chiral": [
      "A",
      "A",
      "A",
      "A",
      "A",
      "A",
      "A",
      "A"
    ],
    "sidon": [
      1,
      2,
      4,
      8,
      16,
      32,
      64,
      128
    ],
    "phases": [
      0,
      0,
      0,
      0,
      0,
      0,
      0,
      0
    ]
  },
  "transform": {
    "character_matrix": [
      [
        1,
        0,
        0,
        0
      ],
      [
        -1,
        0,
        0,
        0
      ],
      [
        0,
        1,
        0,
        0
      ],
      [
        0,
        -1,
        0,
        0
      ],
      [
        0,
        0,
        1,
        0
      ],
      [
        0,
        0,
        -1,
        0
      ],
      [
        0,
        0,
        0,
        1
      ],
      [
        0,
        0,
        0,
        -1
      ]
    ],
    "gram_matrix": [
      [
        1,
        -1,
        0,
        0,
        0,
        0,
        0,
        0
      ],
      [
        -1,
        1,
        0,
        0,
        0,
        0,
        0,
        0
      ],
      [
        0,
        0,
        1,
        -1,
        0,
        0,
        0,
        0
      ],
      [
        0,
        0,
        -1,
        1,
        0,
        0,
        0,
        0
      ],
      [
        0,
        0,
        0,
        0,
        1,
        -1,
        0,
        0
      ],
      [
        0,
        0,
        0,
        0,
        -1,
        1,
        0,
        0
      ],
      [
        0,
        0,
        0,
        0,
        0,
        0,
        1,
        -1
      ],
      [
        0,
        0,
        0,
        0,
        0,
        0,
        -1,
        1
      ]
    ],
    "block_weights": [
      256,
      256,
      256,
      256
    ],
    "eigenvalues": {
      "lambda_max": 529,
      "lambda_min": 17
    }
  },
  "derived": {
    "cartan_diagonal": 273,
    "block_weight": 256,
    "lambda_max": 529,
    "lambda_min": 17,
    "denominator": 1792,
    "sigma": [
      0.15234375,
      "\u03c3 = 273/1792 = 39/256"
    ],
    "tau": [
      0.14285714285714285,
      "\u03c4 = 256/1792"
    ],
    "delta": [
      0.00948660714285715,
      "\u2206 = (273 \u2212 256)/1792 = 17/1792"
    ],
    "coupling_count": 28,
    "pair_count": 4,
    "sidon_doublings": 7,
    "gap_numerator": 17,
    "gap_equation": "\u2206 = \u03bb_min/D = (529 \u2212 2\u00d7256)/1792 = 17/1792"
  },
  "verification": [
    "\u2705 spectral gap positive",
    "\u2705 denominator D = 2\u2078 \u00d7 7 = 1792",
    "\u2705 coupling count C(8,2) = 28",
    "\u2705 gap numerator integer: 17"
  ],
  "note": "All values derived from input. Zero hardcoded constants."
}