data/theta_gap_certificate.json

{
  "rho": 0.8,
  "gap_lower_bound": 0.2737964150527131,
  "threshold": 9.869604401089358,
  "next_tree_gap": 5.551652475612764,
  "residue": [
    [
      0.0,
      0.0
    ],
    [
      0.0,
      2.0000000000000004
    ]
  ],
  "path_gram": [
    [
      2.0,
      -1.0
    ],
    [
      -1.0,
      3.0
    ]
  ],
  "k": 5,
  "lengths": [
    1.0,
    1.0,
    2.0
  ],
  "exact_excess_expression": "4*(pi-atan(sqrt(5)))**2-pi**2",
  "exact_excess_decimal": 5.991986821852391,
  "numerical_finest_excess": 5.99344283639317
}