{
  "origin": "coordinator_exact_symbolic_and_rational_checks_not_formal_verification",
  "source_sha256": "e6f3ee5af078764b1750ea6a8cdbc8c1d9577ef6d14d0bc638830f385d839db8",
  "checker_sha256": "f579ff99695b7fda60842ac337cc919457b5463cc8151ee77945126e9c5e2f21",
  "sympy_version": "1.14.0",
  "lemma_2_1_bracket_checks": 34,
  "stabilizer_counterexample": {
    "matrix": [
      [
        "-1",
        "0",
        "0"
      ],
      [
        "0",
        "-1",
        "0"
      ],
      [
        "0",
        "0",
        "1"
      ]
    ],
    "action_on_H_basis": [
      1,
      -1,
      -1
    ],
    "exterior_cube_multiplier": 1,
    "excluded_from_A0H": "first entry -1; A0H first entry is positive"
  },
  "unipotent_assertion_counterexample": {
    "matrix": [
      [
        "1",
        "0",
        "0"
      ],
      [
        "0",
        "-1",
        "1"
      ],
      [
        "0",
        "0",
        "-1"
      ]
    ],
    "semisimple_part": [
      [
        "1",
        "0",
        "0"
      ],
      [
        "0",
        "-1",
        "0"
      ],
      [
        "0",
        "0",
        "-1"
      ]
    ],
    "unipotent_part": [
      [
        "1",
        "0",
        "0"
      ],
      [
        "0",
        "1",
        "-1"
      ],
      [
        "0",
        "0",
        "1"
      ]
    ],
    "trace": -1,
    "square": [
      [
        "1",
        "0",
        "0"
      ],
      [
        "0",
        "1",
        "-2"
      ],
      [
        "0",
        "0",
        "1"
      ]
    ]
  },
  "positive_conjugation_weights": {
    "r1": "exp(-ell/2)",
    "r2": "exp(ell/2)",
    "thin_r1_after_positive_conjugation": "beta exp(-3ell/2)",
    "r2_after_positive_conjugation": "beta exp(ell/2)",
    "negative_conjugation_balances": "beta exp(-ell/2) in both"
  },
  "hyperbola_family": {
    "w": "beta q/2",
    "w_prime": "beta q/2+1/(beta q)",
    "k": 1,
    "k_prime": 1,
    "ell": 0,
    "A": 1,
    "q": "exp(s_prime)",
    "sufficient_conditions": [
      "0<beta<=1/10",
      "q>=10",
      "q^2>=2/beta^2",
      "q^(-10 C1 epsilon1)<=beta/2"
    ],
    "graph_distance_upper_bound": "(36/25) beta q^-3 < q^-3",
    "parameter_gap_over_claimed_bound": "beta^-1",
    "source_scales_compatible": true
  },
  "separation_example": {
    "ell": 1,
    "beta": "16^-4",
    "x": "Gamma",
    "r1": "beta/4",
    "r2": 0,
    "r3": 0,
    "v": "e beta u/4",
    "norm_qualification": "coordinate/Frobenius; e<3 gives ||v||<beta",
    "scope": "refutes the stated all-ell>=1 lemma; does not refute a corrected large-ell application"
  },
  "conditioning_example": {
    "mass_P": "1/4",
    "mass_A": "1/8",
    "conditional_mass": "1/2",
    "scope": "counterexample to the claimed inference from conditioning alone, not to the dynamical lemma"
  },
  "limitations": [
    "All-integer and analytic arguments are in accompanying notes.",
    "No repaired main theorem is proved.",
    "Finite exact algebra is not a formal proof certificate."
  ]
}