{
  "utc": "2026-10-13T23:22:57.844392+00:00",
  "arithmetic": "fractions.Fraction; integer determinants bound; exact residual shortest paths and direct best-response checks",
  "source_code_sha256": "78b06285de064c7c9725621768881de8f0ceac3da707c429e361a413f0efb44d",
  "whole_games": [
    {
      "weights": [
        1
      ],
      "partition_yes": false,
      "dimensions": [
        17,
        38
      ],
      "M_bits": 1538,
      "ssp_segments": 4,
      "exact_auxiliary_optimum_checks": 12,
      "negative_vertex_checks": 48,
      "additional_pair_parameters": [],
      "wrong_B_sign_rejected": "not_applicable",
      "wrong_default_column_rejected": true,
      "M_zero_infeasible_improvement": true
    },
    {
      "weights": [
        1,
        1
      ],
      "partition_yes": true,
      "dimensions": [
        25,
        54
      ],
      "M_bits": 3240,
      "ssp_segments": 8,
      "exact_auxiliary_optimum_checks": 24,
      "negative_vertex_checks": 72,
      "additional_pair_parameters": [
        "11/2",
        "5/2",
        "7/2",
        "9/2"
      ],
      "wrong_B_sign_rejected": true,
      "wrong_default_column_rejected": true,
      "M_zero_infeasible_improvement": true
    },
    {
      "weights": [
        1,
        2
      ],
      "partition_yes": false,
      "dimensions": [
        25,
        54
      ],
      "M_bits": 3272,
      "ssp_segments": 8,
      "exact_auxiliary_optimum_checks": 24,
      "negative_vertex_checks": 72,
      "additional_pair_parameters": [],
      "wrong_B_sign_rejected": "not_applicable",
      "wrong_default_column_rejected": true,
      "M_zero_infeasible_improvement": true
    },
    {
      "weights": [
        1,
        2,
        3
      ],
      "partition_yes": true,
      "dimensions": [
        33,
        70
      ],
      "M_bits": 5348,
      "ssp_segments": 16,
      "exact_auxiliary_optimum_checks": 48,
      "negative_vertex_checks": 96,
      "additional_pair_parameters": [
        "13/2",
        "15/2",
        "17/2",
        "19/2"
      ],
      "wrong_B_sign_rejected": true,
      "wrong_default_column_rejected": true,
      "M_zero_infeasible_improvement": true
    },
    {
      "weights": [
        1,
        2,
        4
      ],
      "partition_yes": false,
      "dimensions": [
        33,
        70
      ],
      "M_bits": 5364,
      "ssp_segments": 16,
      "exact_auxiliary_optimum_checks": 48,
      "negative_vertex_checks": 96,
      "additional_pair_parameters": [],
      "wrong_B_sign_rejected": "not_applicable",
      "wrong_default_column_rejected": true,
      "M_zero_infeasible_improvement": true
    },
    {
      "weights": [
        2,
        2,
        2
      ],
      "partition_yes": false,
      "dimensions": [
        33,
        70
      ],
      "M_bits": 5278,
      "ssp_segments": 16,
      "exact_auxiliary_optimum_checks": 48,
      "negative_vertex_checks": 96,
      "additional_pair_parameters": [],
      "wrong_B_sign_rejected": "not_applicable",
      "wrong_default_column_rejected": true,
      "M_zero_infeasible_improvement": true
    }
  ],
  "certificates": {
    "three_equilibrium_example": true,
    "two_pair_certificate_accepted": true,
    "duplicate_pair_certificate_rejected": true,
    "rank_zero_and_rank_two_invalid_recognized": true,
    "non_equilibrium_midpoint_rejected": true,
    "non_unit_and_negative_vectors_rejected": true,
    "degenerate_rank_one_continuum_boundary_witnesses": true,
    "one_by_two_continuum": true
  },
  "toy_penalty": {
    "exact_checks": 24,
    "M_equals_dual_norm_infeasible_tie": true
  },
  "limits": "Finite game checks certify each displayed pair and auxiliary optimum by exact inequalities. They do not prove complete equilibrium enumeration or the source-flow interface for arbitrary instances. Submitted old six-game and 39-network checks were not accessed or replayed."
}
