{
  "generated_at_utc": "2026-10-09T00:57:19.912783+00:00",
  "sympy": "1.13.1",
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        [
          "1",
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      ],
      [
        [
          "1",
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        [
          "1",
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      ]
    ]
  ],
  "cases": [
    {
      "case": "ordered_commutator",
      "known_nonzero_from_expansion_or_explicit_words": true,
      "scalar_result": "0",
      "scalar_ns": 35292,
      "probes": [
        {
          "preset": 0,
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            [
              "1",
              "0"
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            [
              "0",
              "-1"
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          ],
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        {
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            [
              "1",
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            [
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              "-1"
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          ],
          "nonzero_witness": true,
          "elapsed_ns": 34416
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      ],
      "prefix_32": {
        "variables": 2,
        "gates": 9,
        "source_dimension": 324,
        "requested_dimension": 32,
        "completed_dimension": 32,
        "requested_full_source_dimension": false,
        "model": {
          "field": "rational_characteristic_zero",
          "algebra": "free_associative",
          "gates": "division_free_tree"
        },
        "independent_identity_conclusion": "not_an_identity_in_the_declared_free_rational_algebra",
        "source_universal_hitting_acceptance": "not_independently_verified",
        "implementation": "Visible formula tree; structured rational matrix action, not an opaque black-box oracle",
        "budget_scope": "Counts selected rational add/multiply/divide calls only; excludes parsing, negation, bit arithmetic, allocation and wall time",
        "status": "nonzero_polynomial_witness",
        "witness": {
          "row": 3,
          "column": 0,
          "value": "-1/24",
          "dimension": 32,
          "generator": "[1] for n=s=1; otherwise entry (-1)^(r-q-1)/(r*i^(r-q)) for q<r, zero otherwise"
        },
        "arithmetic_charges": 436
      }
    },
    {
      "case": "distributive_identity",
      "known_nonzero_from_expansion_or_explicit_words": false,
      "scalar_result": "0",
      "scalar_ns": 8209,
      "probes": [
        {
          "preset": 0,
          "entries": [
            [
              "0",
              "0"
            ],
            [
              "0",
              "0"
            ]
          ],
          "nonzero_witness": false,
          "elapsed_ns": 51708
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        {
          "preset": 1,
          "entries": [
            [
              "0",
              "0"
            ],
            [
              "0",
              "0"
            ]
          ],
          "nonzero_witness": false,
          "elapsed_ns": 43792
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      ],
      "prefix_32": {
        "variables": 3,
        "gates": 15,
        "source_dimension": 1350,
        "requested_dimension": 32,
        "completed_dimension": 32,
        "requested_full_source_dimension": false,
        "model": {
          "field": "rational_characteristic_zero",
          "algebra": "free_associative",
          "gates": "division_free_tree"
        },
        "independent_identity_conclusion": "unknown",
        "source_universal_hitting_acceptance": "not_independently_verified",
        "implementation": "Visible formula tree; structured rational matrix action, not an opaque black-box oracle",
        "budget_scope": "Counts selected rational add/multiply/divide calls only; excludes parsing, negation, bit arithmetic, allocation and wall time",
        "status": "zero_at_truncation_unknown",
        "conditional_interpretation": "A full-dimension zero first column implies identity only under the source paper detection argument and correct implementation; neither is formally accepted here",
        "arithmetic_charges": 779
      }
    },
    {
      "case": "incorrect_square_rewrite",
      "known_nonzero_from_expansion_or_explicit_words": true,
      "scalar_result": "0",
      "scalar_ns": 15750,
      "probes": [
        {
          "preset": 0,
          "entries": [
            [
              "-1",
              "0"
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            [
              "0",
              "1"
            ]
          ],
          "nonzero_witness": true,
          "elapsed_ns": 87417
        },
        {
          "preset": 1,
          "entries": [
            [
              "-1",
              "0"
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            [
              "1",
              "1"
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          ],
          "nonzero_witness": true,
          "elapsed_ns": 67417
        }
      ],
      "prefix_32": {
        "variables": 2,
        "gates": 23,
        "source_dimension": 2116,
        "requested_dimension": 32,
        "completed_dimension": 32,
        "requested_full_source_dimension": false,
        "model": {
          "field": "rational_characteristic_zero",
          "algebra": "free_associative",
          "gates": "division_free_tree"
        },
        "independent_identity_conclusion": "not_an_identity_in_the_declared_free_rational_algebra",
        "source_universal_hitting_acceptance": "not_independently_verified",
        "implementation": "Visible formula tree; structured rational matrix action, not an opaque black-box oracle",
        "budget_scope": "Counts selected rational add/multiply/divide calls only; excludes parsing, negation, bit arithmetic, allocation and wall time",
        "status": "nonzero_polynomial_witness",
        "witness": {
          "row": 3,
          "column": 0,
          "value": "1/24",
          "dimension": 32,
          "generator": "[1] for n=s=1; otherwise entry (-1)^(r-q-1)/(r*i^(r-q)) for q<r, zero otherwise"
        },
        "arithmetic_charges": 1154
      }
    },
    {
      "case": "rational_scalar",
      "known_nonzero_from_expansion_or_explicit_words": true,
      "scalar_result": "8/21",
      "scalar_ns": 23083,
      "probes": [
        {
          "preset": 0,
          "entries": [
            [
              "2/3",
              "-1/7"
            ],
            [
              "0",
              "2/3"
            ]
          ],
          "nonzero_witness": true,
          "elapsed_ns": 75500
        },
        {
          "preset": 1,
          "entries": [
            [
              "11/21",
              "-1/7"
            ],
            [
              "0",
              "2/3"
            ]
          ],
          "nonzero_witness": true,
          "elapsed_ns": 48208
        }
      ],
      "prefix_32": {
        "variables": 2,
        "gates": 5,
        "source_dimension": 100,
        "requested_dimension": 32,
        "completed_dimension": 32,
        "requested_full_source_dimension": false,
        "model": {
          "field": "rational_characteristic_zero",
          "algebra": "free_associative",
          "gates": "division_free_tree"
        },
        "independent_identity_conclusion": "not_an_identity_in_the_declared_free_rational_algebra",
        "source_universal_hitting_acceptance": "not_independently_verified",
        "implementation": "Visible formula tree; structured rational matrix action, not an opaque black-box oracle",
        "budget_scope": "Counts selected rational add/multiply/divide calls only; excludes parsing, negation, bit arithmetic, allocation and wall time",
        "status": "nonzero_polynomial_witness",
        "witness": {
          "row": 0,
          "column": 0,
          "value": "2/3",
          "dimension": 32,
          "generator": "[1] for n=s=1; otherwise entry (-1)^(r-q-1)/(r*i^(r-q)) for q<r, zero otherwise"
        },
        "arithmetic_charges": 189
      }
    },
    {
      "case": "exceptional_variable",
      "known_nonzero_from_expansion_or_explicit_words": true,
      "scalar_result": "2",
      "scalar_ns": 1041,
      "probes": [
        {
          "preset": 0,
          "entries": [
            [
              "0",
              "1"
            ],
            [
              "0",
              "0"
            ]
          ],
          "nonzero_witness": true,
          "elapsed_ns": 4458
        },
        {
          "preset": 1,
          "entries": [
            [
              "1",
              "1"
            ],
            [
              "0",
              "0"
            ]
          ],
          "nonzero_witness": true,
          "elapsed_ns": 3292
        }
      ],
      "prefix_32": {
        "variables": 1,
        "gates": 1,
        "source_dimension": 1,
        "requested_dimension": 1,
        "completed_dimension": 1,
        "requested_full_source_dimension": true,
        "model": {
          "field": "rational_characteristic_zero",
          "algebra": "free_associative",
          "gates": "division_free_tree"
        },
        "independent_identity_conclusion": "not_an_identity_in_the_declared_free_rational_algebra",
        "source_universal_hitting_acceptance": "not_independently_verified",
        "implementation": "Visible formula tree; structured rational matrix action, not an opaque black-box oracle",
        "budget_scope": "Counts selected rational add/multiply/divide calls only; excludes parsing, negation, bit arithmetic, allocation and wall time",
        "status": "nonzero_polynomial_witness",
        "witness": {
          "row": 0,
          "column": 0,
          "value": "1",
          "dimension": 1,
          "generator": "[1] for n=s=1; otherwise entry (-1)^(r-q-1)/(r*i^(r-q)) for q<r, zero otherwise"
        },
        "arithmetic_charges": 0
      }
    },
    {
      "case": "product_of_sums_2",
      "known_nonzero_from_expansion_or_explicit_words": true,
      "scalar_result": "25",
      "scalar_ns": 2958,
      "probes": [
        {
          "preset": 0,
          "entries": [
            [
              "1",
              "0"
            ],
            [
              "0",
              "1"
            ]
          ],
          "nonzero_witness": true,
          "elapsed_ns": 20292
        },
        {
          "preset": 1,
          "entries": [
            [
              "2",
              "1"
            ],
            [
              "1",
              "1"
            ]
          ],
          "nonzero_witness": true,
          "elapsed_ns": 18500
        }
      ],
      "prefix_32": {
        "variables": 2,
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        "source_dimension": 196,
        "requested_dimension": 32,
        "completed_dimension": 32,
        "requested_full_source_dimension": false,
        "model": {
          "field": "rational_characteristic_zero",
          "algebra": "free_associative",
          "gates": "division_free_tree"
        },
        "independent_identity_conclusion": "not_an_identity_in_the_declared_free_rational_algebra",
        "source_universal_hitting_acceptance": "not_independently_verified",
        "implementation": "Visible formula tree; structured rational matrix action, not an opaque black-box oracle",
        "budget_scope": "Counts selected rational add/multiply/divide calls only; excludes parsing, negation, bit arithmetic, allocation and wall time",
        "status": "nonzero_polynomial_witness",
        "witness": {
          "row": 2,
          "column": 0,
          "value": "9/8",
          "dimension": 32,
          "generator": "[1] for n=s=1; otherwise entry (-1)^(r-q-1)/(r*i^(r-q)) for q<r, zero otherwise"
        },
        "arithmetic_charges": 436
      }
    },
    {
      "case": "product_of_sums_4",
      "known_nonzero_from_expansion_or_explicit_words": true,
      "scalar_result": "625",
      "scalar_ns": 5917,
      "probes": [
        {
          "preset": 0,
          "entries": [
            [
              "1",
              "0"
            ],
            [
              "0",
              "1"
            ]
          ],
          "nonzero_witness": true,
          "elapsed_ns": 39083
        },
        {
          "preset": 1,
          "entries": [
            [
              "5",
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            [
              "3",
              "2"
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          ],
          "nonzero_witness": true,
          "elapsed_ns": 36958
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      ],
      "prefix_32": {
        "variables": 2,
        "gates": 15,
        "source_dimension": 900,
        "requested_dimension": 32,
        "completed_dimension": 32,
        "requested_full_source_dimension": false,
        "model": {
          "field": "rational_characteristic_zero",
          "algebra": "free_associative",
          "gates": "division_free_tree"
        },
        "independent_identity_conclusion": "not_an_identity_in_the_declared_free_rational_algebra",
        "source_universal_hitting_acceptance": "not_independently_verified",
        "implementation": "Visible formula tree; structured rational matrix action, not an opaque black-box oracle",
        "budget_scope": "Counts selected rational add/multiply/divide calls only; excludes parsing, negation, bit arithmetic, allocation and wall time",
        "status": "nonzero_polynomial_witness",
        "witness": {
          "row": 4,
          "column": 0,
          "value": "27/128",
          "dimension": 32,
          "generator": "[1] for n=s=1; otherwise entry (-1)^(r-q-1)/(r*i^(r-q)) for q<r, zero otherwise"
        },
        "arithmetic_charges": 872
      }
    },
    {
      "case": "product_of_sums_8",
      "known_nonzero_from_expansion_or_explicit_words": true,
      "scalar_result": "390625",
      "scalar_ns": 10083,
      "probes": [
        {
          "preset": 0,
          "entries": [
            [
              "1",
              "0"
            ],
            [
              "0",
              "1"
            ]
          ],
          "nonzero_witness": true,
          "elapsed_ns": 72959
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        {
          "preset": 1,
          "entries": [
            [
              "34",
              "21"
            ],
            [
              "21",
              "13"
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          ],
          "nonzero_witness": true,
          "elapsed_ns": 72000
        }
      ],
      "prefix_32": {
        "variables": 2,
        "gates": 31,
        "source_dimension": 3844,
        "requested_dimension": 32,
        "completed_dimension": 32,
        "requested_full_source_dimension": false,
        "model": {
          "field": "rational_characteristic_zero",
          "algebra": "free_associative",
          "gates": "division_free_tree"
        },
        "independent_identity_conclusion": "not_an_identity_in_the_declared_free_rational_algebra",
        "source_universal_hitting_acceptance": "not_independently_verified",
        "implementation": "Visible formula tree; structured rational matrix action, not an opaque black-box oracle",
        "budget_scope": "Counts selected rational add/multiply/divide calls only; excludes parsing, negation, bit arithmetic, allocation and wall time",
        "status": "nonzero_polynomial_witness",
        "witness": {
          "row": 8,
          "column": 0,
          "value": "729/1146880",
          "dimension": 32,
          "generator": "[1] for n=s=1; otherwise entry (-1)^(r-q-1)/(r*i^(r-q)) for q<r, zero otherwise"
        },
        "arithmetic_charges": 1744
      }
    },
    {
      "case": "product_of_sums_16",
      "known_nonzero_from_expansion_or_explicit_words": true,
      "scalar_result": "152587890625",
      "scalar_ns": 20750,
      "probes": [
        {
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          "entries": [
            [
              "1",
              "0"
            ],
            [
              "0",
              "1"
            ]
          ],
          "nonzero_witness": true,
          "elapsed_ns": 156625
        },
        {
          "preset": 1,
          "entries": [
            [
              "1597",
              "987"
            ],
            [
              "987",
              "610"
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          ],
          "nonzero_witness": true,
          "elapsed_ns": 157458
        }
      ],
      "prefix_32": {
        "variables": 2,
        "gates": 63,
        "source_dimension": 15876,
        "requested_dimension": 32,
        "completed_dimension": 32,
        "requested_full_source_dimension": false,
        "model": {
          "field": "rational_characteristic_zero",
          "algebra": "free_associative",
          "gates": "division_free_tree"
        },
        "independent_identity_conclusion": "not_an_identity_in_the_declared_free_rational_algebra",
        "source_universal_hitting_acceptance": "not_independently_verified",
        "implementation": "Visible formula tree; structured rational matrix action, not an opaque black-box oracle",
        "budget_scope": "Counts selected rational add/multiply/divide calls only; excludes parsing, negation, bit arithmetic, allocation and wall time",
        "status": "nonzero_polynomial_witness",
        "witness": {
          "row": 16,
          "column": 0,
          "value": "59049/1880927240192000",
          "dimension": 32,
          "generator": "[1] for n=s=1; otherwise entry (-1)^(r-q-1)/(r*i^(r-q)) for q<r, zero otherwise"
        },
        "arithmetic_charges": 3488
      }
    },
    {
      "case": "nilpotent_probe_miss",
      "known_nonzero_from_expansion_or_explicit_words": true,
      "scalar_result": "0",
      "scalar_ns": 6042,
      "probes": [
        {
          "preset": 0,
          "entries": [
            [
              "0",
              "0"
            ],
            [
              "0",
              "0"
            ]
          ],
          "nonzero_witness": false,
          "elapsed_ns": 39875
        },
        {
          "preset": 1,
          "entries": [
            [
              "1",
              "0"
            ],
            [
              "-1",
              "-1"
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          ],
          "nonzero_witness": true,
          "elapsed_ns": 38167
        }
      ],
      "prefix_32": {
        "variables": 2,
        "gates": 13,
        "source_dimension": 676,
        "requested_dimension": 32,
        "completed_dimension": 32,
        "requested_full_source_dimension": false,
        "model": {
          "field": "rational_characteristic_zero",
          "algebra": "free_associative",
          "gates": "division_free_tree"
        },
        "independent_identity_conclusion": "not_an_identity_in_the_declared_free_rational_algebra",
        "source_universal_hitting_acceptance": "not_independently_verified",
        "implementation": "Visible formula tree; structured rational matrix action, not an opaque black-box oracle",
        "budget_scope": "Counts selected rational add/multiply/divide calls only; excludes parsing, negation, bit arithmetic, allocation and wall time",
        "status": "nonzero_polynomial_witness",
        "witness": {
          "row": 4,
          "column": 0,
          "value": "-1/48",
          "dimension": 32,
          "generator": "[1] for n=s=1; otherwise entry (-1)^(r-q-1)/(r*i^(r-q)) for q<r, zero otherwise"
        },
        "arithmetic_charges": 622
      }
    }
  ],
  "interpretation": "Exact scalar and two fixed 2-by-2 matrix probes. Nonzero is a witness; zero alone is unknown. Neither probe family is claimed universal. Timings are one local sample and do not benchmark optimal white-box PIT."
}