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          [
            [
              0,
              0,
              0,
              2,
              0
            ],
            [
              0,
              0,
              0,
              1,
              0
            ],
            [
              0,
              0,
              0,
              1,
              0
            ]
          ]
        ]
      },
      "d1_columns_binary": [
        29,
        0
      ],
      "d2_columns_binary": [
        2,
        3,
        0,
        4,
        6
      ],
      "d1_rank_witness": {
        "rank_lower_bound": 1,
        "columns": [
          0
        ],
        "rows": [
          4
        ],
        "minor_rows_binary": [
          1
        ],
        "determinant_mod2": 1
      },
      "d2_rank_witness": {
        "rank_lower_bound": 3,
        "columns": [
          0,
          1,
          3
        ],
        "rows": [
          1,
          0,
          2
        ],
        "minor_rows_binary": [
          3,
          2,
          4
        ],
        "determinant_mod2": 1
      },
      "cycle_column": 2,
      "nonboundary_functional_binary": 20,
      "conclusion": "The class of tau_h^(4-w) [xi_1^2|xi_1^2] is a basis of H^(2,8,w)."
    }
  ],
  "scope": "All original basis words and coefficient powers retained. Binary integer column masks index the explicitly listed bases. Nonzero minors give rank lower bounds; d2*d1=0, the dimension sum and the displayed nonboundary functional force the exact one-dimensional cohomology. Weight <=0 is covered by the proved coefficient isomorphism; weights >4 have zero cochains."
}