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    "global_sheet_translates_per_valid_support": 101,
    "logical_witnesses_from_affine_and_sheet_translates": 1919,
    "tested_base_affine_maps": 171,
    "unique_valid_affine_supports_without_sheet_translates": 19,
    "valid_affine_logicals": 19,
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      {
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      {
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      },
      {
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      {
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      },
      {
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        "weight": 310
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      {
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        "weight": 310
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      {
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        "weight": 310
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        "weight": 310
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  "base_projection": {
    "branch_histogram": {
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      "1": 141
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    "distinct_base_columns": 221,
    "h_index_histogram": {
      "0": 44,
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      "2": 28,
      "3": 33,
      "4": 23,
      "5": 33,
      "6": 39,
      "7": 32,
      "8": 38
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    "multiplicity_histogram": {
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      "2": 46,
      "3": 17,
      "4": 3
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    "odd_projection_base_Hx_syndrome_weight": 0,
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    "t_histogram": {
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      "1": 16,
      "2": 14,
      "3": 6,
      "4": 20,
      "5": 16,
      "6": 10,
      "7": 16,
      "8": 18,
      "9": 18,
      "10": 20,
      "11": 18,
      "12": 18,
      "13": 14,
      "14": 18,
      "15": 18,
      "16": 14,
      "17": 14,
      "18": 16
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  "checks": {
    "hx_syndrome_weight": 0,
    "in_rowspace_Hz": false,
    "logical_nontrivial": true,
    "weight": 310
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  "conclusion": "The weight-310 support is connected and has one-dimensional kernel when H_X is restricted to it, so it contains no smaller nonempty H_X-syndrome-zero sub-support. The finite-field affine structure with the CPM sheet offsets gives an orbit of weight-310 logical witnesses, but no lower-weight witness was found by this structure search. Together with the existing stabilizer-coset tightening search, the best explicit upper bound remains d <= 310.",
  "input_witness": "/Users/kasaikenta/Library/CloudStorage/Dropbox/00_work/Projects/tmp/heisenberg_j3_l18_q9_girth6/arxiv2605_23894_3_18_P101_round4_kill_lowlog/results_fer_detail/low_weight_logical_upper_bound_search_20260625_153759.json",
  "restricted_Hx_structure": {
    "all_ones_vector_is_only_kernel_generator": true,
    "component_count": 1,
    "components": [
      {
        "checks": 407,
        "variables": 310
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    ],
    "hit_row_degree_histogram": {
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      "4": 33,
      "6": 7,
      "8": 1,
      "10": 2
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    "hit_rows": 407,
    "kernel_dimension_on_support": 1,
    "rank_on_support": 309
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  "sheet_projection": {
    "distinct_sheets": 100,
    "most_common_sheets": [
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        10
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      [
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      ],
      [
        48,
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      [
        33,
        6
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      [
        76,
        6
      ],
      [
        93,
        5
      ],
      [
        27,
        5
      ],
      [
        4,
        5
      ],
      [
        50,
        5
      ],
      [
        86,
        5
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      [
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        5
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      [
        55,
        5
      ]
    ],
    "sheet_multiplicity_max": 10,
    "sheet_multiplicity_min": 1
  },
  "stabilizer_coset_tightening_reference": "/Users/kasaikenta/Library/CloudStorage/Dropbox/00_work/Projects/tmp/heisenberg_j3_l18_q9_girth6/arxiv2605_23894_3_18_P101_round4_kill_lowlog/results_fer_detail/low_weight_logical_upper_bound_search_20260625_153759.json",
  "type": "structure search from the weight-310 Z logical witness"
}