{
  "arithmetic": "exact integers and rational matrices; SymPy 1.14.0",
  "scope": "Finite checks and negative controls; all-modulus proof supplied separately. No spectral or headline-theorem certificate.",
  "lifting_cases": [
    {
      "case": 0,
      "modulus": 1125899906842624,
      "row_height": 8,
      "determinant_bound": 491520,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 1,
      "modulus": 150094635296999121,
      "row_height": 11,
      "determinant_bound": 1756920,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 2,
      "modulus": 250000000000000000000000,
      "row_height": 23,
      "determinant_bound": 33580920,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 3,
      "modulus": 1125899906842624,
      "row_height": 5,
      "determinant_bound": 75000,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 4,
      "modulus": 150094635296999121,
      "row_height": 33,
      "determinant_bound": 142310520,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 5,
      "modulus": 250000000000000000000000,
      "row_height": 9,
      "determinant_bound": 787320,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 6,
      "modulus": 1125899906842624,
      "row_height": 9,
      "determinant_bound": 787320,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 7,
      "modulus": 150094635296999121,
      "row_height": 6,
      "determinant_bound": 155520,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 8,
      "modulus": 250000000000000000000000,
      "row_height": 6,
      "determinant_bound": 155520,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 9,
      "modulus": 1125899906842624,
      "row_height": 11,
      "determinant_bound": 1756920,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 10,
      "modulus": 150094635296999121,
      "row_height": 38,
      "determinant_bound": 250216320,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 11,
      "modulus": 250000000000000000000000,
      "row_height": 17,
      "determinant_bound": 10022520,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 12,
      "modulus": 1125899906842624,
      "row_height": 8,
      "determinant_bound": 491520,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 13,
      "modulus": 150094635296999121,
      "row_height": 7,
      "determinant_bound": 288120,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 14,
      "modulus": 250000000000000000000000,
      "row_height": 18,
      "determinant_bound": 12597120,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 15,
      "modulus": 1125899906842624,
      "row_height": 10,
      "determinant_bound": 1200000,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 16,
      "modulus": 150094635296999121,
      "row_height": 26,
      "determinant_bound": 54837120,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 17,
      "modulus": 250000000000000000000000,
      "row_height": 15,
      "determinant_bound": 6075000,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 18,
      "modulus": 1125899906842624,
      "row_height": 10,
      "determinant_bound": 1200000,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 19,
      "modulus": 150094635296999121,
      "row_height": 22,
      "determinant_bound": 28110720,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 20,
      "modulus": 250000000000000000000000,
      "row_height": 9,
      "determinant_bound": 787320,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 21,
      "modulus": 1125899906842624,
      "row_height": 9,
      "determinant_bound": 787320,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 22,
      "modulus": 150094635296999121,
      "row_height": 7,
      "determinant_bound": 288120,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 23,
      "modulus": 250000000000000000000000,
      "row_height": 45,
      "determinant_bound": 492075000,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 24,
      "modulus": 1125899906842624,
      "row_height": 3,
      "determinant_bound": 9720,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 25,
      "modulus": 150094635296999121,
      "row_height": 6,
      "determinant_bound": 155520,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 26,
      "modulus": 250000000000000000000000,
      "row_height": 7,
      "determinant_bound": 288120,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 27,
      "modulus": 1125899906842624,
      "row_height": 11,
      "determinant_bound": 1756920,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 28,
      "modulus": 150094635296999121,
      "row_height": 8,
      "determinant_bound": 491520,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 29,
      "modulus": 250000000000000000000000,
      "row_height": 19,
      "determinant_bound": 15638520,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 30,
      "modulus": 1125899906842624,
      "row_height": 3,
      "determinant_bound": 9720,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 31,
      "modulus": 150094635296999121,
      "row_height": 6,
      "determinant_bound": 155520,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 32,
      "modulus": 250000000000000000000000,
      "row_height": 23,
      "determinant_bound": 33580920,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 33,
      "modulus": 1125899906842624,
      "row_height": 11,
      "determinant_bound": 1756920,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 34,
      "modulus": 150094635296999121,
      "row_height": 12,
      "determinant_bound": 2488320,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    },
    {
      "case": 35,
      "modulus": 250000000000000000000000,
      "row_height": 17,
      "determinant_bound": 10022520,
      "rank": 4,
      "primitive_lift": [
        1,
        0,
        0,
        0,
        1
      ]
    }
  ],
  "edge_cases": [
    "empty S",
    "all row vectors zero"
  ],
  "negative_controls": {
    "small_modulus_full_rank_determinant": 2,
    "nonprimitive_modular_vector": "full rank remains possible"
  },
  "prime_polynomial_checks": 3024,
  "composite_shear_checks": [
    {
      "modulus": 343,
      "distinct_prime_divisors": [
        7
      ],
      "cases": 50
    },
    {
      "modulus": 539,
      "distinct_prime_divisors": [
        7,
        11
      ],
      "cases": 50
    },
    {
      "modulus": 31603,
      "distinct_prime_divisors": [
        11,
        13,
        17
      ],
      "cases": 50
    }
  ],
  "all_assertions_passed": true
}