{
  "schema_version": "2.2.0",
  "dataset_version": "m1-m48-v1",
  "id": "fibonacci-m26-p84",
  "name": "Fibonacci M26P84",
  "modulus": 26,
  "pisano_period": 84,
  "conventions": {
    "index_origin": 0,
    "clock_labels": [
      84,
      1,
      2,
      3,
      4,
      5,
      6,
      7,
      8,
      9,
      10,
      11,
      12,
      13,
      14,
      15,
      16,
      17,
      18,
      19,
      20,
      21,
      22,
      23,
      24,
      25,
      26,
      27,
      28,
      29,
      30,
      31,
      32,
      33,
      34,
      35,
      36,
      37,
      38,
      39,
      40,
      41,
      42,
      43,
      44,
      45,
      46,
      47,
      48,
      49,
      50,
      51,
      52,
      53,
      54,
      55,
      56,
      57,
      58,
      59,
      60,
      61,
      62,
      63,
      64,
      65,
      66,
      67,
      68,
      69,
      70,
      71,
      72,
      73,
      74,
      75,
      76,
      77,
      78,
      79,
      80,
      81,
      82,
      83
    ],
    "orientation": "clockwise from north",
    "zero_display": 26,
    "alpha_circle_positions": 84,
    "beta_circle_positions": 26,
    "node_rule": "Native residues divisible by 3, including zero; preserves the approved M6/M9/M12 reference selection. Additional regular formations of all native values are catalogued separately.",
    "m1_convention": "Modulo 1 has period 1 and residue 0."
  },
  "alpha": {
    "residues": [
      0,
      1,
      1,
      2,
      3,
      5,
      8,
      13,
      21,
      8,
      3,
      11,
      14,
      25,
      13,
      12,
      25,
      11,
      10,
      21,
      5,
      0,
      5,
      5,
      10,
      15,
      25,
      14,
      13,
      1,
      14,
      15,
      3,
      18,
      21,
      13,
      8,
      21,
      3,
      24,
      1,
      25,
      0,
      25,
      25,
      24,
      23,
      21,
      18,
      13,
      5,
      18,
      23,
      15,
      12,
      1,
      13,
      14,
      1,
      15,
      16,
      5,
      21,
      0,
      21,
      21,
      16,
      11,
      1,
      12,
      13,
      25,
      12,
      11,
      23,
      8,
      5,
      13,
      18,
      5,
      23,
      2,
      25,
      1
    ],
    "display_values": [
      26,
      1,
      1,
      2,
      3,
      5,
      8,
      13,
      21,
      8,
      3,
      11,
      14,
      25,
      13,
      12,
      25,
      11,
      10,
      21,
      5,
      26,
      5,
      5,
      10,
      15,
      25,
      14,
      13,
      1,
      14,
      15,
      3,
      18,
      21,
      13,
      8,
      21,
      3,
      24,
      1,
      25,
      26,
      25,
      25,
      24,
      23,
      21,
      18,
      13,
      5,
      18,
      23,
      15,
      12,
      1,
      13,
      14,
      1,
      15,
      16,
      5,
      21,
      26,
      21,
      21,
      16,
      11,
      1,
      12,
      13,
      25,
      12,
      11,
      23,
      8,
      5,
      13,
      18,
      5,
      23,
      2,
      25,
      1
    ]
  },
  "gamma": {
    "alpha_indices": [
      0,
      4,
      8,
      10,
      15,
      19,
      21,
      25,
      31,
      32,
      33,
      34,
      37,
      38,
      39,
      42,
      45,
      47,
      48,
      51,
      53,
      54,
      59,
      62,
      63,
      64,
      65,
      69,
      72,
      78
    ],
    "clock_positions": [
      84,
      4,
      8,
      10,
      15,
      19,
      21,
      25,
      31,
      32,
      33,
      34,
      37,
      38,
      39,
      42,
      45,
      47,
      48,
      51,
      53,
      54,
      59,
      62,
      63,
      64,
      65,
      69,
      72,
      78
    ],
    "clockwise_values": [
      26,
      3,
      21,
      3,
      12,
      21,
      26,
      15,
      15,
      3,
      18,
      21,
      21,
      3,
      24,
      26,
      24,
      21,
      18,
      18,
      15,
      12,
      15,
      21,
      26,
      21,
      21,
      12,
      12,
      18
    ],
    "triangle_paths": []
  },
  "beta": {
    "interval_sums": [
      4,
      26,
      8,
      63,
      46,
      5,
      20,
      67,
      0,
      0,
      0,
      21,
      0,
      0,
      26,
      50,
      23,
      0,
      18,
      23,
      0,
      29,
      21,
      0,
      0,
      0,
      28,
      38,
      60,
      56
    ],
    "interval_residues": [
      4,
      0,
      8,
      11,
      20,
      5,
      20,
      15,
      0,
      0,
      0,
      21,
      0,
      0,
      0,
      24,
      23,
      0,
      18,
      23,
      0,
      3,
      21,
      0,
      0,
      0,
      2,
      12,
      8,
      4
    ],
    "display_cycle": [
      4,
      26,
      8,
      11,
      20,
      5,
      20,
      15,
      26,
      26,
      26,
      21,
      26,
      26,
      26,
      24,
      23,
      26,
      18,
      23,
      26,
      3,
      21,
      26,
      26,
      26,
      2,
      12,
      8,
      4
    ],
    "intervals": [
      {
        "from_index": 0,
        "to_index": 4,
        "interior_indices": [
          1,
          2,
          3
        ],
        "interior_values": [
          1,
          1,
          2
        ],
        "sum": 4,
        "residue": 4
      },
      {
        "from_index": 4,
        "to_index": 8,
        "interior_indices": [
          5,
          6,
          7
        ],
        "interior_values": [
          5,
          8,
          13
        ],
        "sum": 26,
        "residue": 0
      },
      {
        "from_index": 8,
        "to_index": 10,
        "interior_indices": [
          9
        ],
        "interior_values": [
          8
        ],
        "sum": 8,
        "residue": 8
      },
      {
        "from_index": 10,
        "to_index": 15,
        "interior_indices": [
          11,
          12,
          13,
          14
        ],
        "interior_values": [
          11,
          14,
          25,
          13
        ],
        "sum": 63,
        "residue": 11
      },
      {
        "from_index": 15,
        "to_index": 19,
        "interior_indices": [
          16,
          17,
          18
        ],
        "interior_values": [
          25,
          11,
          10
        ],
        "sum": 46,
        "residue": 20
      },
      {
        "from_index": 19,
        "to_index": 21,
        "interior_indices": [
          20
        ],
        "interior_values": [
          5
        ],
        "sum": 5,
        "residue": 5
      },
      {
        "from_index": 21,
        "to_index": 25,
        "interior_indices": [
          22,
          23,
          24
        ],
        "interior_values": [
          5,
          5,
          10
        ],
        "sum": 20,
        "residue": 20
      },
      {
        "from_index": 25,
        "to_index": 31,
        "interior_indices": [
          26,
          27,
          28,
          29,
          30
        ],
        "interior_values": [
          25,
          14,
          13,
          1,
          14
        ],
        "sum": 67,
        "residue": 15
      },
      {
        "from_index": 31,
        "to_index": 32,
        "interior_indices": [],
        "interior_values": [],
        "sum": 0,
        "residue": 0
      },
      {
        "from_index": 32,
        "to_index": 33,
        "interior_indices": [],
        "interior_values": [],
        "sum": 0,
        "residue": 0
      },
      {
        "from_index": 33,
        "to_index": 34,
        "interior_indices": [],
        "interior_values": [],
        "sum": 0,
        "residue": 0
      },
      {
        "from_index": 34,
        "to_index": 37,
        "interior_indices": [
          35,
          36
        ],
        "interior_values": [
          13,
          8
        ],
        "sum": 21,
        "residue": 21
      },
      {
        "from_index": 37,
        "to_index": 38,
        "interior_indices": [],
        "interior_values": [],
        "sum": 0,
        "residue": 0
      },
      {
        "from_index": 38,
        "to_index": 39,
        "interior_indices": [],
        "interior_values": [],
        "sum": 0,
        "residue": 0
      },
      {
        "from_index": 39,
        "to_index": 42,
        "interior_indices": [
          40,
          41
        ],
        "interior_values": [
          1,
          25
        ],
        "sum": 26,
        "residue": 0
      },
      {
        "from_index": 42,
        "to_index": 45,
        "interior_indices": [
          43,
          44
        ],
        "interior_values": [
          25,
          25
        ],
        "sum": 50,
        "residue": 24
      },
      {
        "from_index": 45,
        "to_index": 47,
        "interior_indices": [
          46
        ],
        "interior_values": [
          23
        ],
        "sum": 23,
        "residue": 23
      },
      {
        "from_index": 47,
        "to_index": 48,
        "interior_indices": [],
        "interior_values": [],
        "sum": 0,
        "residue": 0
      },
      {
        "from_index": 48,
        "to_index": 51,
        "interior_indices": [
          49,
          50
        ],
        "interior_values": [
          13,
          5
        ],
        "sum": 18,
        "residue": 18
      },
      {
        "from_index": 51,
        "to_index": 53,
        "interior_indices": [
          52
        ],
        "interior_values": [
          23
        ],
        "sum": 23,
        "residue": 23
      },
      {
        "from_index": 53,
        "to_index": 54,
        "interior_indices": [],
        "interior_values": [],
        "sum": 0,
        "residue": 0
      },
      {
        "from_index": 54,
        "to_index": 59,
        "interior_indices": [
          55,
          56,
          57,
          58
        ],
        "interior_values": [
          1,
          13,
          14,
          1
        ],
        "sum": 29,
        "residue": 3
      },
      {
        "from_index": 59,
        "to_index": 62,
        "interior_indices": [
          60,
          61
        ],
        "interior_values": [
          16,
          5
        ],
        "sum": 21,
        "residue": 21
      },
      {
        "from_index": 62,
        "to_index": 63,
        "interior_indices": [],
        "interior_values": [],
        "sum": 0,
        "residue": 0
      },
      {
        "from_index": 63,
        "to_index": 64,
        "interior_indices": [],
        "interior_values": [],
        "sum": 0,
        "residue": 0
      },
      {
        "from_index": 64,
        "to_index": 65,
        "interior_indices": [],
        "interior_values": [],
        "sum": 0,
        "residue": 0
      },
      {
        "from_index": 65,
        "to_index": 69,
        "interior_indices": [
          66,
          67,
          68
        ],
        "interior_values": [
          16,
          11,
          1
        ],
        "sum": 28,
        "residue": 2
      },
      {
        "from_index": 69,
        "to_index": 72,
        "interior_indices": [
          70,
          71
        ],
        "interior_values": [
          13,
          25
        ],
        "sum": 38,
        "residue": 12
      },
      {
        "from_index": 72,
        "to_index": 78,
        "interior_indices": [
          73,
          74,
          75,
          76,
          77
        ],
        "interior_values": [
          11,
          23,
          8,
          5,
          13
        ],
        "sum": 60,
        "residue": 8
      },
      {
        "from_index": 78,
        "to_index": 0,
        "interior_indices": [
          79,
          80,
          81,
          82,
          83
        ],
        "interior_values": [
          5,
          23,
          2,
          25,
          1
        ],
        "sum": 56,
        "residue": 4
      }
    ],
    "minimal_cycle": [
      4,
      0,
      8,
      11,
      20,
      5,
      20,
      15,
      0,
      0,
      0,
      21,
      0,
      0,
      0,
      24,
      23,
      0,
      18,
      23,
      0,
      3,
      21,
      0,
      0,
      0,
      2,
      12,
      8,
      4
    ],
    "native_doubling": false,
    "recognised_doubling_cycle": null,
    "rotation_to_recognised": null,
    "m9_doubling_cycle_present": false,
    "start_value": 4,
    "start_interval_clock_positions": [
      84,
      4
    ],
    "start_interval_values": [
      1,
      1,
      2
    ],
    "circle_positions": 26,
    "cycle_length": 30,
    "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
  },
  "epsilon": null,
  "epsilon_routes": [],
  "epsilon_status": "Null: no tested regular six-node reference traversal contains only 3, 6 and 9 with all three present.",
  "geometry_summary": {
    "rule_version": "native-pilot-v1",
    "scope": "Shapes in the primary Gamma node set (native residues divisible by 3, including zero). All regular polygons on the period circle are catalogued separately with their native values; their existence alone is a property of the plotting grid.",
    "circle_role": "Pisano-period plotting framework",
    "shapes": [
      {
        "name": "Equilateral triangle",
        "status": "Absent",
        "count": 0,
        "orientation": null,
        "rotation_degrees": null,
        "rotations_degrees": [],
        "alpha_paths": []
      },
      {
        "name": "Hexagon",
        "status": "Absent",
        "count": 0,
        "orientation": null,
        "rotation_degrees": null,
        "rotations_degrees": [],
        "alpha_paths": []
      },
      {
        "name": "2D stellated octahedron",
        "status": "Absent",
        "count": 0,
        "orientation": null,
        "rotation_degrees": null,
        "rotations_degrees": [],
        "alpha_paths": []
      },
      {
        "name": "Chevron",
        "status": "Absent",
        "count": 0,
        "orientation": null,
        "rotation_degrees": null,
        "rotations_degrees": [],
        "alpha_paths": []
      },
      {
        "name": "Unicursal hexagram",
        "status": "Absent",
        "count": 0,
        "orientation": null,
        "rotation_degrees": null,
        "rotations_degrees": [],
        "alpha_paths": []
      }
    ],
    "other_orientations_status": "Absent",
    "epsilon_rule": "A regular six-node formation must support the reference unicursal route through only 3, 6 and 9, containing all three. Rotations and reflections of the route are tested; reversed or cyclically shifted traversals are deduplicated."
  },
  "formation_catalogue": [
    {
      "id": "3-nodes-offset-0",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        0,
        28,
        56
      ],
      "clock_positions": [
        84,
        28,
        56
      ],
      "native_values": [
        26,
        13,
        13
      ],
      "rotation_degrees": 0.0,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        26,
        13,
        13
      ],
      "beta": {
        "interval_sums": [
          286,
          390,
          338
        ],
        "interval_residues": [
          0,
          0,
          0
        ],
        "display_cycle": [
          26,
          26,
          26
        ],
        "intervals": [
          {
            "from_index": 0,
            "to_index": 28,
            "interior_indices": [
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27
            ],
            "interior_values": [
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14
            ],
            "sum": 286,
            "residue": 0
          },
          {
            "from_index": 28,
            "to_index": 56,
            "interior_indices": [
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55
            ],
            "interior_values": [
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1
            ],
            "sum": 390,
            "residue": 0
          },
          {
            "from_index": 56,
            "to_index": 0,
            "interior_indices": [
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83
            ],
            "interior_values": [
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1
            ],
            "sum": 338,
            "residue": 0
          }
        ],
        "minimal_cycle": [
          0
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 26,
        "start_interval_clock_positions": [
          84,
          28
        ],
        "start_interval_values": [
          1,
          1,
          2,
          3,
          5,
          8,
          13,
          21,
          8,
          3,
          11,
          14,
          25,
          13,
          12,
          25,
          11,
          10,
          21,
          5,
          0,
          5,
          5,
          10,
          15,
          25,
          14
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-1",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        1,
        29,
        57
      ],
      "clock_positions": [
        1,
        29,
        57
      ],
      "native_values": [
        1,
        1,
        14
      ],
      "rotation_degrees": 4.285714285714286,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        1,
        1,
        14
      ],
      "beta": {
        "interval_sums": [
          298,
          402,
          324
        ],
        "interval_residues": [
          12,
          12,
          12
        ],
        "display_cycle": [
          12,
          12,
          12
        ],
        "intervals": [
          {
            "from_index": 1,
            "to_index": 29,
            "interior_indices": [
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28
            ],
            "interior_values": [
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13
            ],
            "sum": 298,
            "residue": 12
          },
          {
            "from_index": 29,
            "to_index": 57,
            "interior_indices": [
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56
            ],
            "interior_values": [
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13
            ],
            "sum": 402,
            "residue": 12
          },
          {
            "from_index": 57,
            "to_index": 1,
            "interior_indices": [
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0
            ],
            "interior_values": [
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0
            ],
            "sum": 324,
            "residue": 12
          }
        ],
        "minimal_cycle": [
          12
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 12,
        "start_interval_clock_positions": [
          1,
          29
        ],
        "start_interval_values": [
          1,
          2,
          3,
          5,
          8,
          13,
          21,
          8,
          3,
          11,
          14,
          25,
          13,
          12,
          25,
          11,
          10,
          21,
          5,
          0,
          5,
          5,
          10,
          15,
          25,
          14,
          13
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-2",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        2,
        30,
        58
      ],
      "clock_positions": [
        2,
        30,
        58
      ],
      "native_values": [
        1,
        14,
        1
      ],
      "rotation_degrees": 8.571428571428571,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        1,
        14,
        1
      ],
      "beta": {
        "interval_sums": [
          298,
          402,
          324
        ],
        "interval_residues": [
          12,
          12,
          12
        ],
        "display_cycle": [
          12,
          12,
          12
        ],
        "intervals": [
          {
            "from_index": 2,
            "to_index": 30,
            "interior_indices": [
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29
            ],
            "interior_values": [
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1
            ],
            "sum": 298,
            "residue": 12
          },
          {
            "from_index": 30,
            "to_index": 58,
            "interior_indices": [
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57
            ],
            "interior_values": [
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14
            ],
            "sum": 402,
            "residue": 12
          },
          {
            "from_index": 58,
            "to_index": 2,
            "interior_indices": [
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1
            ],
            "interior_values": [
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1
            ],
            "sum": 324,
            "residue": 12
          }
        ],
        "minimal_cycle": [
          12
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 12,
        "start_interval_clock_positions": [
          2,
          30
        ],
        "start_interval_values": [
          2,
          3,
          5,
          8,
          13,
          21,
          8,
          3,
          11,
          14,
          25,
          13,
          12,
          25,
          11,
          10,
          21,
          5,
          0,
          5,
          5,
          10,
          15,
          25,
          14,
          13,
          1
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-3",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        3,
        31,
        59
      ],
      "clock_positions": [
        3,
        31,
        59
      ],
      "native_values": [
        2,
        15,
        15
      ],
      "rotation_degrees": 12.857142857142858,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        2,
        15,
        15
      ],
      "beta": {
        "interval_sums": [
          310,
          388,
          310
        ],
        "interval_residues": [
          24,
          24,
          24
        ],
        "display_cycle": [
          24,
          24,
          24
        ],
        "intervals": [
          {
            "from_index": 3,
            "to_index": 31,
            "interior_indices": [
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30
            ],
            "interior_values": [
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14
            ],
            "sum": 310,
            "residue": 24
          },
          {
            "from_index": 31,
            "to_index": 59,
            "interior_indices": [
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58
            ],
            "interior_values": [
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1
            ],
            "sum": 388,
            "residue": 24
          },
          {
            "from_index": 59,
            "to_index": 3,
            "interior_indices": [
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2
            ],
            "interior_values": [
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1
            ],
            "sum": 310,
            "residue": 24
          }
        ],
        "minimal_cycle": [
          24
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 24,
        "start_interval_clock_positions": [
          3,
          31
        ],
        "start_interval_values": [
          3,
          5,
          8,
          13,
          21,
          8,
          3,
          11,
          14,
          25,
          13,
          12,
          25,
          11,
          10,
          21,
          5,
          0,
          5,
          5,
          10,
          15,
          25,
          14,
          13,
          1,
          14
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-4",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        4,
        32,
        60
      ],
      "clock_positions": [
        4,
        32,
        60
      ],
      "native_values": [
        3,
        3,
        16
      ],
      "rotation_degrees": 17.142857142857142,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        3,
        3,
        16
      ],
      "beta": {
        "interval_sums": [
          322,
          400,
          296
        ],
        "interval_residues": [
          10,
          10,
          10
        ],
        "display_cycle": [
          10,
          10,
          10
        ],
        "intervals": [
          {
            "from_index": 4,
            "to_index": 32,
            "interior_indices": [
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31
            ],
            "interior_values": [
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15
            ],
            "sum": 322,
            "residue": 10
          },
          {
            "from_index": 32,
            "to_index": 60,
            "interior_indices": [
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59
            ],
            "interior_values": [
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15
            ],
            "sum": 400,
            "residue": 10
          },
          {
            "from_index": 60,
            "to_index": 4,
            "interior_indices": [
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3
            ],
            "interior_values": [
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2
            ],
            "sum": 296,
            "residue": 10
          }
        ],
        "minimal_cycle": [
          10
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 10,
        "start_interval_clock_positions": [
          4,
          32
        ],
        "start_interval_values": [
          5,
          8,
          13,
          21,
          8,
          3,
          11,
          14,
          25,
          13,
          12,
          25,
          11,
          10,
          21,
          5,
          0,
          5,
          5,
          10,
          15,
          25,
          14,
          13,
          1,
          14,
          15
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-5",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        5,
        33,
        61
      ],
      "clock_positions": [
        5,
        33,
        61
      ],
      "native_values": [
        5,
        18,
        5
      ],
      "rotation_degrees": 21.428571428571427,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        5,
        18,
        5
      ],
      "beta": {
        "interval_sums": [
          320,
          398,
          294
        ],
        "interval_residues": [
          8,
          8,
          8
        ],
        "display_cycle": [
          8,
          8,
          8
        ],
        "intervals": [
          {
            "from_index": 5,
            "to_index": 33,
            "interior_indices": [
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32
            ],
            "interior_values": [
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3
            ],
            "sum": 320,
            "residue": 8
          },
          {
            "from_index": 33,
            "to_index": 61,
            "interior_indices": [
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60
            ],
            "interior_values": [
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16
            ],
            "sum": 398,
            "residue": 8
          },
          {
            "from_index": 61,
            "to_index": 5,
            "interior_indices": [
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4
            ],
            "interior_values": [
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3
            ],
            "sum": 294,
            "residue": 8
          }
        ],
        "minimal_cycle": [
          8
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 8,
        "start_interval_clock_positions": [
          5,
          33
        ],
        "start_interval_values": [
          8,
          13,
          21,
          8,
          3,
          11,
          14,
          25,
          13,
          12,
          25,
          11,
          10,
          21,
          5,
          0,
          5,
          5,
          10,
          15,
          25,
          14,
          13,
          1,
          14,
          15,
          3
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-6",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        6,
        34,
        62
      ],
      "clock_positions": [
        6,
        34,
        62
      ],
      "native_values": [
        8,
        21,
        21
      ],
      "rotation_degrees": 25.714285714285715,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        8,
        21,
        21
      ],
      "beta": {
        "interval_sums": [
          330,
          382,
          278
        ],
        "interval_residues": [
          18,
          18,
          18
        ],
        "display_cycle": [
          18,
          18,
          18
        ],
        "intervals": [
          {
            "from_index": 6,
            "to_index": 34,
            "interior_indices": [
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33
            ],
            "interior_values": [
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18
            ],
            "sum": 330,
            "residue": 18
          },
          {
            "from_index": 34,
            "to_index": 62,
            "interior_indices": [
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61
            ],
            "interior_values": [
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5
            ],
            "sum": 382,
            "residue": 18
          },
          {
            "from_index": 62,
            "to_index": 6,
            "interior_indices": [
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5
            ],
            "interior_values": [
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5
            ],
            "sum": 278,
            "residue": 18
          }
        ],
        "minimal_cycle": [
          18
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 18,
        "start_interval_clock_positions": [
          6,
          34
        ],
        "start_interval_values": [
          13,
          21,
          8,
          3,
          11,
          14,
          25,
          13,
          12,
          25,
          11,
          10,
          21,
          5,
          0,
          5,
          5,
          10,
          15,
          25,
          14,
          13,
          1,
          14,
          15,
          3,
          18
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-7",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        7,
        35,
        63
      ],
      "clock_positions": [
        7,
        35,
        63
      ],
      "native_values": [
        13,
        13,
        26
      ],
      "rotation_degrees": 30.0,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        13,
        13,
        26
      ],
      "beta": {
        "interval_sums": [
          338,
          390,
          286
        ],
        "interval_residues": [
          0,
          0,
          0
        ],
        "display_cycle": [
          26,
          26,
          26
        ],
        "intervals": [
          {
            "from_index": 7,
            "to_index": 35,
            "interior_indices": [
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34
            ],
            "interior_values": [
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21
            ],
            "sum": 338,
            "residue": 0
          },
          {
            "from_index": 35,
            "to_index": 63,
            "interior_indices": [
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62
            ],
            "interior_values": [
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21
            ],
            "sum": 390,
            "residue": 0
          },
          {
            "from_index": 63,
            "to_index": 7,
            "interior_indices": [
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6
            ],
            "interior_values": [
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8
            ],
            "sum": 286,
            "residue": 0
          }
        ],
        "minimal_cycle": [
          0
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 26,
        "start_interval_clock_positions": [
          7,
          35
        ],
        "start_interval_values": [
          21,
          8,
          3,
          11,
          14,
          25,
          13,
          12,
          25,
          11,
          10,
          21,
          5,
          0,
          5,
          5,
          10,
          15,
          25,
          14,
          13,
          1,
          14,
          15,
          3,
          18,
          21
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-8",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        8,
        36,
        64
      ],
      "clock_positions": [
        8,
        36,
        64
      ],
      "native_values": [
        21,
        8,
        21
      ],
      "rotation_degrees": 34.285714285714285,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        21,
        8,
        21
      ],
      "beta": {
        "interval_sums": [
          330,
          382,
          278
        ],
        "interval_residues": [
          18,
          18,
          18
        ],
        "display_cycle": [
          18,
          18,
          18
        ],
        "intervals": [
          {
            "from_index": 8,
            "to_index": 36,
            "interior_indices": [
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35
            ],
            "interior_values": [
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13
            ],
            "sum": 330,
            "residue": 18
          },
          {
            "from_index": 36,
            "to_index": 64,
            "interior_indices": [
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63
            ],
            "interior_values": [
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0
            ],
            "sum": 382,
            "residue": 18
          },
          {
            "from_index": 64,
            "to_index": 8,
            "interior_indices": [
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7
            ],
            "interior_values": [
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13
            ],
            "sum": 278,
            "residue": 18
          }
        ],
        "minimal_cycle": [
          18
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 18,
        "start_interval_clock_positions": [
          8,
          36
        ],
        "start_interval_values": [
          8,
          3,
          11,
          14,
          25,
          13,
          12,
          25,
          11,
          10,
          21,
          5,
          0,
          5,
          5,
          10,
          15,
          25,
          14,
          13,
          1,
          14,
          15,
          3,
          18,
          21,
          13
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-9",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        9,
        37,
        65
      ],
      "clock_positions": [
        9,
        37,
        65
      ],
      "native_values": [
        8,
        21,
        21
      ],
      "rotation_degrees": 38.57142857142857,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        8,
        21,
        21
      ],
      "beta": {
        "interval_sums": [
          330,
          382,
          278
        ],
        "interval_residues": [
          18,
          18,
          18
        ],
        "display_cycle": [
          18,
          18,
          18
        ],
        "intervals": [
          {
            "from_index": 9,
            "to_index": 37,
            "interior_indices": [
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36
            ],
            "interior_values": [
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8
            ],
            "sum": 330,
            "residue": 18
          },
          {
            "from_index": 37,
            "to_index": 65,
            "interior_indices": [
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64
            ],
            "interior_values": [
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21
            ],
            "sum": 382,
            "residue": 18
          },
          {
            "from_index": 65,
            "to_index": 9,
            "interior_indices": [
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8
            ],
            "interior_values": [
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21
            ],
            "sum": 278,
            "residue": 18
          }
        ],
        "minimal_cycle": [
          18
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 18,
        "start_interval_clock_positions": [
          9,
          37
        ],
        "start_interval_values": [
          3,
          11,
          14,
          25,
          13,
          12,
          25,
          11,
          10,
          21,
          5,
          0,
          5,
          5,
          10,
          15,
          25,
          14,
          13,
          1,
          14,
          15,
          3,
          18,
          21,
          13,
          8
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-10",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        10,
        38,
        66
      ],
      "clock_positions": [
        10,
        38,
        66
      ],
      "native_values": [
        3,
        3,
        16
      ],
      "rotation_degrees": 42.857142857142854,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        3,
        3,
        16
      ],
      "beta": {
        "interval_sums": [
          348,
          400,
          270
        ],
        "interval_residues": [
          10,
          10,
          10
        ],
        "display_cycle": [
          10,
          10,
          10
        ],
        "intervals": [
          {
            "from_index": 10,
            "to_index": 38,
            "interior_indices": [
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37
            ],
            "interior_values": [
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21
            ],
            "sum": 348,
            "residue": 10
          },
          {
            "from_index": 38,
            "to_index": 66,
            "interior_indices": [
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65
            ],
            "interior_values": [
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21
            ],
            "sum": 400,
            "residue": 10
          },
          {
            "from_index": 66,
            "to_index": 10,
            "interior_indices": [
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9
            ],
            "interior_values": [
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8
            ],
            "sum": 270,
            "residue": 10
          }
        ],
        "minimal_cycle": [
          10
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 10,
        "start_interval_clock_positions": [
          10,
          38
        ],
        "start_interval_values": [
          11,
          14,
          25,
          13,
          12,
          25,
          11,
          10,
          21,
          5,
          0,
          5,
          5,
          10,
          15,
          25,
          14,
          13,
          1,
          14,
          15,
          3,
          18,
          21,
          13,
          8,
          21
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-11",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        11,
        39,
        67
      ],
      "clock_positions": [
        11,
        39,
        67
      ],
      "native_values": [
        11,
        24,
        11
      ],
      "rotation_degrees": 47.142857142857146,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        11,
        24,
        11
      ],
      "beta": {
        "interval_sums": [
          340,
          392,
          262
        ],
        "interval_residues": [
          2,
          2,
          2
        ],
        "display_cycle": [
          2,
          2,
          2
        ],
        "intervals": [
          {
            "from_index": 11,
            "to_index": 39,
            "interior_indices": [
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38
            ],
            "interior_values": [
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3
            ],
            "sum": 340,
            "residue": 2
          },
          {
            "from_index": 39,
            "to_index": 67,
            "interior_indices": [
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66
            ],
            "interior_values": [
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16
            ],
            "sum": 392,
            "residue": 2
          },
          {
            "from_index": 67,
            "to_index": 11,
            "interior_indices": [
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10
            ],
            "interior_values": [
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3
            ],
            "sum": 262,
            "residue": 2
          }
        ],
        "minimal_cycle": [
          2
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 2,
        "start_interval_clock_positions": [
          11,
          39
        ],
        "start_interval_values": [
          14,
          25,
          13,
          12,
          25,
          11,
          10,
          21,
          5,
          0,
          5,
          5,
          10,
          15,
          25,
          14,
          13,
          1,
          14,
          15,
          3,
          18,
          21,
          13,
          8,
          21,
          3
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-12",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        12,
        40,
        68
      ],
      "clock_positions": [
        12,
        40,
        68
      ],
      "native_values": [
        14,
        1,
        1
      ],
      "rotation_degrees": 51.42857142857143,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        14,
        1,
        1
      ],
      "beta": {
        "interval_sums": [
          350,
          402,
          272
        ],
        "interval_residues": [
          12,
          12,
          12
        ],
        "display_cycle": [
          12,
          12,
          12
        ],
        "intervals": [
          {
            "from_index": 12,
            "to_index": 40,
            "interior_indices": [
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39
            ],
            "interior_values": [
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24
            ],
            "sum": 350,
            "residue": 12
          },
          {
            "from_index": 40,
            "to_index": 68,
            "interior_indices": [
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67
            ],
            "interior_values": [
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11
            ],
            "sum": 402,
            "residue": 12
          },
          {
            "from_index": 68,
            "to_index": 12,
            "interior_indices": [
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11
            ],
            "interior_values": [
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11
            ],
            "sum": 272,
            "residue": 12
          }
        ],
        "minimal_cycle": [
          12
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 12,
        "start_interval_clock_positions": [
          12,
          40
        ],
        "start_interval_values": [
          25,
          13,
          12,
          25,
          11,
          10,
          21,
          5,
          0,
          5,
          5,
          10,
          15,
          25,
          14,
          13,
          1,
          14,
          15,
          3,
          18,
          21,
          13,
          8,
          21,
          3,
          24
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-13",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        13,
        41,
        69
      ],
      "clock_positions": [
        13,
        41,
        69
      ],
      "native_values": [
        25,
        25,
        12
      ],
      "rotation_degrees": 55.714285714285715,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        25,
        25,
        12
      ],
      "beta": {
        "interval_sums": [
          326,
          378,
          274
        ],
        "interval_residues": [
          14,
          14,
          14
        ],
        "display_cycle": [
          14,
          14,
          14
        ],
        "intervals": [
          {
            "from_index": 13,
            "to_index": 41,
            "interior_indices": [
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40
            ],
            "interior_values": [
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1
            ],
            "sum": 326,
            "residue": 14
          },
          {
            "from_index": 41,
            "to_index": 69,
            "interior_indices": [
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68
            ],
            "interior_values": [
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1
            ],
            "sum": 378,
            "residue": 14
          },
          {
            "from_index": 69,
            "to_index": 13,
            "interior_indices": [
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12
            ],
            "interior_values": [
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14
            ],
            "sum": 274,
            "residue": 14
          }
        ],
        "minimal_cycle": [
          14
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 14,
        "start_interval_clock_positions": [
          13,
          41
        ],
        "start_interval_values": [
          13,
          12,
          25,
          11,
          10,
          21,
          5,
          0,
          5,
          5,
          10,
          15,
          25,
          14,
          13,
          1,
          14,
          15,
          3,
          18,
          21,
          13,
          8,
          21,
          3,
          24,
          1
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-14",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        14,
        42,
        70
      ],
      "clock_positions": [
        14,
        42,
        70
      ],
      "native_values": [
        13,
        26,
        13
      ],
      "rotation_degrees": 60.0,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        13,
        26,
        13
      ],
      "beta": {
        "interval_sums": [
          338,
          390,
          286
        ],
        "interval_residues": [
          0,
          0,
          0
        ],
        "display_cycle": [
          26,
          26,
          26
        ],
        "intervals": [
          {
            "from_index": 14,
            "to_index": 42,
            "interior_indices": [
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41
            ],
            "interior_values": [
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25
            ],
            "sum": 338,
            "residue": 0
          },
          {
            "from_index": 42,
            "to_index": 70,
            "interior_indices": [
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69
            ],
            "interior_values": [
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12
            ],
            "sum": 390,
            "residue": 0
          },
          {
            "from_index": 70,
            "to_index": 14,
            "interior_indices": [
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13
            ],
            "interior_values": [
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25
            ],
            "sum": 286,
            "residue": 0
          }
        ],
        "minimal_cycle": [
          0
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 26,
        "start_interval_clock_positions": [
          14,
          42
        ],
        "start_interval_values": [
          12,
          25,
          11,
          10,
          21,
          5,
          0,
          5,
          5,
          10,
          15,
          25,
          14,
          13,
          1,
          14,
          15,
          3,
          18,
          21,
          13,
          8,
          21,
          3,
          24,
          1,
          25
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-15",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        15,
        43,
        71
      ],
      "clock_positions": [
        15,
        43,
        71
      ],
      "native_values": [
        12,
        25,
        25
      ],
      "rotation_degrees": 64.28571428571429,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        12,
        25,
        25
      ],
      "beta": {
        "interval_sums": [
          326,
          378,
          274
        ],
        "interval_residues": [
          14,
          14,
          14
        ],
        "display_cycle": [
          14,
          14,
          14
        ],
        "intervals": [
          {
            "from_index": 15,
            "to_index": 43,
            "interior_indices": [
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42
            ],
            "interior_values": [
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0
            ],
            "sum": 326,
            "residue": 14
          },
          {
            "from_index": 43,
            "to_index": 71,
            "interior_indices": [
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70
            ],
            "interior_values": [
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13
            ],
            "sum": 378,
            "residue": 14
          },
          {
            "from_index": 71,
            "to_index": 15,
            "interior_indices": [
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14
            ],
            "interior_values": [
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13
            ],
            "sum": 274,
            "residue": 14
          }
        ],
        "minimal_cycle": [
          14
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 14,
        "start_interval_clock_positions": [
          15,
          43
        ],
        "start_interval_values": [
          25,
          11,
          10,
          21,
          5,
          0,
          5,
          5,
          10,
          15,
          25,
          14,
          13,
          1,
          14,
          15,
          3,
          18,
          21,
          13,
          8,
          21,
          3,
          24,
          1,
          25,
          0
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-16",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        16,
        44,
        72
      ],
      "clock_positions": [
        16,
        44,
        72
      ],
      "native_values": [
        25,
        25,
        12
      ],
      "rotation_degrees": 68.57142857142857,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        25,
        25,
        12
      ],
      "beta": {
        "interval_sums": [
          326,
          378,
          274
        ],
        "interval_residues": [
          14,
          14,
          14
        ],
        "display_cycle": [
          14,
          14,
          14
        ],
        "intervals": [
          {
            "from_index": 16,
            "to_index": 44,
            "interior_indices": [
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43
            ],
            "interior_values": [
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25
            ],
            "sum": 326,
            "residue": 14
          },
          {
            "from_index": 44,
            "to_index": 72,
            "interior_indices": [
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71
            ],
            "interior_values": [
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25
            ],
            "sum": 378,
            "residue": 14
          },
          {
            "from_index": 72,
            "to_index": 16,
            "interior_indices": [
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15
            ],
            "interior_values": [
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12
            ],
            "sum": 274,
            "residue": 14
          }
        ],
        "minimal_cycle": [
          14
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 14,
        "start_interval_clock_positions": [
          16,
          44
        ],
        "start_interval_values": [
          11,
          10,
          21,
          5,
          0,
          5,
          5,
          10,
          15,
          25,
          14,
          13,
          1,
          14,
          15,
          3,
          18,
          21,
          13,
          8,
          21,
          3,
          24,
          1,
          25,
          0,
          25
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-17",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        17,
        45,
        73
      ],
      "clock_positions": [
        17,
        45,
        73
      ],
      "native_values": [
        11,
        24,
        11
      ],
      "rotation_degrees": 72.85714285714286,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        11,
        24,
        11
      ],
      "beta": {
        "interval_sums": [
          340,
          366,
          288
        ],
        "interval_residues": [
          2,
          2,
          2
        ],
        "display_cycle": [
          2,
          2,
          2
        ],
        "intervals": [
          {
            "from_index": 17,
            "to_index": 45,
            "interior_indices": [
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44
            ],
            "interior_values": [
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25
            ],
            "sum": 340,
            "residue": 2
          },
          {
            "from_index": 45,
            "to_index": 73,
            "interior_indices": [
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72
            ],
            "interior_values": [
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12
            ],
            "sum": 366,
            "residue": 2
          },
          {
            "from_index": 73,
            "to_index": 17,
            "interior_indices": [
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16
            ],
            "interior_values": [
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25
            ],
            "sum": 288,
            "residue": 2
          }
        ],
        "minimal_cycle": [
          2
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 2,
        "start_interval_clock_positions": [
          17,
          45
        ],
        "start_interval_values": [
          10,
          21,
          5,
          0,
          5,
          5,
          10,
          15,
          25,
          14,
          13,
          1,
          14,
          15,
          3,
          18,
          21,
          13,
          8,
          21,
          3,
          24,
          1,
          25,
          0,
          25,
          25
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-18",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        18,
        46,
        74
      ],
      "clock_positions": [
        18,
        46,
        74
      ],
      "native_values": [
        10,
        23,
        23
      ],
      "rotation_degrees": 77.14285714285714,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        10,
        23,
        23
      ],
      "beta": {
        "interval_sums": [
          354,
          354,
          276
        ],
        "interval_residues": [
          16,
          16,
          16
        ],
        "display_cycle": [
          16,
          16,
          16
        ],
        "intervals": [
          {
            "from_index": 18,
            "to_index": 46,
            "interior_indices": [
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45
            ],
            "interior_values": [
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24
            ],
            "sum": 354,
            "residue": 16
          },
          {
            "from_index": 46,
            "to_index": 74,
            "interior_indices": [
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73
            ],
            "interior_values": [
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11
            ],
            "sum": 354,
            "residue": 16
          },
          {
            "from_index": 74,
            "to_index": 18,
            "interior_indices": [
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17
            ],
            "interior_values": [
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11
            ],
            "sum": 276,
            "residue": 16
          }
        ],
        "minimal_cycle": [
          16
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 16,
        "start_interval_clock_positions": [
          18,
          46
        ],
        "start_interval_values": [
          21,
          5,
          0,
          5,
          5,
          10,
          15,
          25,
          14,
          13,
          1,
          14,
          15,
          3,
          18,
          21,
          13,
          8,
          21,
          3,
          24,
          1,
          25,
          0,
          25,
          25,
          24
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-19",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        19,
        47,
        75
      ],
      "clock_positions": [
        19,
        47,
        75
      ],
      "native_values": [
        21,
        21,
        8
      ],
      "rotation_degrees": 81.42857142857143,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        21,
        21,
        8
      ],
      "beta": {
        "interval_sums": [
          356,
          356,
          278
        ],
        "interval_residues": [
          18,
          18,
          18
        ],
        "display_cycle": [
          18,
          18,
          18
        ],
        "intervals": [
          {
            "from_index": 19,
            "to_index": 47,
            "interior_indices": [
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46
            ],
            "interior_values": [
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23
            ],
            "sum": 356,
            "residue": 18
          },
          {
            "from_index": 47,
            "to_index": 75,
            "interior_indices": [
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74
            ],
            "interior_values": [
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23
            ],
            "sum": 356,
            "residue": 18
          },
          {
            "from_index": 75,
            "to_index": 19,
            "interior_indices": [
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18
            ],
            "interior_values": [
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10
            ],
            "sum": 278,
            "residue": 18
          }
        ],
        "minimal_cycle": [
          18
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 18,
        "start_interval_clock_positions": [
          19,
          47
        ],
        "start_interval_values": [
          5,
          0,
          5,
          5,
          10,
          15,
          25,
          14,
          13,
          1,
          14,
          15,
          3,
          18,
          21,
          13,
          8,
          21,
          3,
          24,
          1,
          25,
          0,
          25,
          25,
          24,
          23
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-20",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        20,
        48,
        76
      ],
      "clock_positions": [
        20,
        48,
        76
      ],
      "native_values": [
        5,
        18,
        5
      ],
      "rotation_degrees": 85.71428571428571,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        5,
        18,
        5
      ],
      "beta": {
        "interval_sums": [
          372,
          346,
          294
        ],
        "interval_residues": [
          8,
          8,
          8
        ],
        "display_cycle": [
          8,
          8,
          8
        ],
        "intervals": [
          {
            "from_index": 20,
            "to_index": 48,
            "interior_indices": [
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47
            ],
            "interior_values": [
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21
            ],
            "sum": 372,
            "residue": 8
          },
          {
            "from_index": 48,
            "to_index": 76,
            "interior_indices": [
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75
            ],
            "interior_values": [
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8
            ],
            "sum": 346,
            "residue": 8
          },
          {
            "from_index": 76,
            "to_index": 20,
            "interior_indices": [
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19
            ],
            "interior_values": [
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21
            ],
            "sum": 294,
            "residue": 8
          }
        ],
        "minimal_cycle": [
          8
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 8,
        "start_interval_clock_positions": [
          20,
          48
        ],
        "start_interval_values": [
          0,
          5,
          5,
          10,
          15,
          25,
          14,
          13,
          1,
          14,
          15,
          3,
          18,
          21,
          13,
          8,
          21,
          3,
          24,
          1,
          25,
          0,
          25,
          25,
          24,
          23,
          21
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-21",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        21,
        49,
        77
      ],
      "clock_positions": [
        21,
        49,
        77
      ],
      "native_values": [
        26,
        13,
        13
      ],
      "rotation_degrees": 90.0,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        26,
        13,
        13
      ],
      "beta": {
        "interval_sums": [
          390,
          338,
          286
        ],
        "interval_residues": [
          0,
          0,
          0
        ],
        "display_cycle": [
          26,
          26,
          26
        ],
        "intervals": [
          {
            "from_index": 21,
            "to_index": 49,
            "interior_indices": [
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48
            ],
            "interior_values": [
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18
            ],
            "sum": 390,
            "residue": 0
          },
          {
            "from_index": 49,
            "to_index": 77,
            "interior_indices": [
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76
            ],
            "interior_values": [
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5
            ],
            "sum": 338,
            "residue": 0
          },
          {
            "from_index": 77,
            "to_index": 21,
            "interior_indices": [
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20
            ],
            "interior_values": [
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5
            ],
            "sum": 286,
            "residue": 0
          }
        ],
        "minimal_cycle": [
          0
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 26,
        "start_interval_clock_positions": [
          21,
          49
        ],
        "start_interval_values": [
          5,
          5,
          10,
          15,
          25,
          14,
          13,
          1,
          14,
          15,
          3,
          18,
          21,
          13,
          8,
          21,
          3,
          24,
          1,
          25,
          0,
          25,
          25,
          24,
          23,
          21,
          18
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-22",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        22,
        50,
        78
      ],
      "clock_positions": [
        22,
        50,
        78
      ],
      "native_values": [
        5,
        5,
        18
      ],
      "rotation_degrees": 94.28571428571429,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        5,
        5,
        18
      ],
      "beta": {
        "interval_sums": [
          398,
          346,
          268
        ],
        "interval_residues": [
          8,
          8,
          8
        ],
        "display_cycle": [
          8,
          8,
          8
        ],
        "intervals": [
          {
            "from_index": 22,
            "to_index": 50,
            "interior_indices": [
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49
            ],
            "interior_values": [
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13
            ],
            "sum": 398,
            "residue": 8
          },
          {
            "from_index": 50,
            "to_index": 78,
            "interior_indices": [
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77
            ],
            "interior_values": [
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13
            ],
            "sum": 346,
            "residue": 8
          },
          {
            "from_index": 78,
            "to_index": 22,
            "interior_indices": [
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21
            ],
            "interior_values": [
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0
            ],
            "sum": 268,
            "residue": 8
          }
        ],
        "minimal_cycle": [
          8
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 8,
        "start_interval_clock_positions": [
          22,
          50
        ],
        "start_interval_values": [
          5,
          10,
          15,
          25,
          14,
          13,
          1,
          14,
          15,
          3,
          18,
          21,
          13,
          8,
          21,
          3,
          24,
          1,
          25,
          0,
          25,
          25,
          24,
          23,
          21,
          18,
          13
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-23",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        23,
        51,
        79
      ],
      "clock_positions": [
        23,
        51,
        79
      ],
      "native_values": [
        5,
        18,
        5
      ],
      "rotation_degrees": 98.57142857142857,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        5,
        18,
        5
      ],
      "beta": {
        "interval_sums": [
          398,
          346,
          268
        ],
        "interval_residues": [
          8,
          8,
          8
        ],
        "display_cycle": [
          8,
          8,
          8
        ],
        "intervals": [
          {
            "from_index": 23,
            "to_index": 51,
            "interior_indices": [
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50
            ],
            "interior_values": [
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5
            ],
            "sum": 398,
            "residue": 8
          },
          {
            "from_index": 51,
            "to_index": 79,
            "interior_indices": [
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78
            ],
            "interior_values": [
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18
            ],
            "sum": 346,
            "residue": 8
          },
          {
            "from_index": 79,
            "to_index": 23,
            "interior_indices": [
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22
            ],
            "interior_values": [
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5
            ],
            "sum": 268,
            "residue": 8
          }
        ],
        "minimal_cycle": [
          8
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 8,
        "start_interval_clock_positions": [
          23,
          51
        ],
        "start_interval_values": [
          10,
          15,
          25,
          14,
          13,
          1,
          14,
          15,
          3,
          18,
          21,
          13,
          8,
          21,
          3,
          24,
          1,
          25,
          0,
          25,
          25,
          24,
          23,
          21,
          18,
          13,
          5
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-24",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        24,
        52,
        80
      ],
      "clock_positions": [
        24,
        52,
        80
      ],
      "native_values": [
        10,
        23,
        23
      ],
      "rotation_degrees": 102.85714285714286,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        10,
        23,
        23
      ],
      "beta": {
        "interval_sums": [
          406,
          328,
          250
        ],
        "interval_residues": [
          16,
          16,
          16
        ],
        "display_cycle": [
          16,
          16,
          16
        ],
        "intervals": [
          {
            "from_index": 24,
            "to_index": 52,
            "interior_indices": [
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51
            ],
            "interior_values": [
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18
            ],
            "sum": 406,
            "residue": 16
          },
          {
            "from_index": 52,
            "to_index": 80,
            "interior_indices": [
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79
            ],
            "interior_values": [
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5
            ],
            "sum": 328,
            "residue": 16
          },
          {
            "from_index": 80,
            "to_index": 24,
            "interior_indices": [
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23
            ],
            "interior_values": [
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5
            ],
            "sum": 250,
            "residue": 16
          }
        ],
        "minimal_cycle": [
          16
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 16,
        "start_interval_clock_positions": [
          24,
          52
        ],
        "start_interval_values": [
          15,
          25,
          14,
          13,
          1,
          14,
          15,
          3,
          18,
          21,
          13,
          8,
          21,
          3,
          24,
          1,
          25,
          0,
          25,
          25,
          24,
          23,
          21,
          18,
          13,
          5,
          18
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-25",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        25,
        53,
        81
      ],
      "clock_positions": [
        25,
        53,
        81
      ],
      "native_values": [
        15,
        15,
        2
      ],
      "rotation_degrees": 107.14285714285714,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        15,
        15,
        2
      ],
      "beta": {
        "interval_sums": [
          414,
          336,
          258
        ],
        "interval_residues": [
          24,
          24,
          24
        ],
        "display_cycle": [
          24,
          24,
          24
        ],
        "intervals": [
          {
            "from_index": 25,
            "to_index": 53,
            "interior_indices": [
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52
            ],
            "interior_values": [
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23
            ],
            "sum": 414,
            "residue": 24
          },
          {
            "from_index": 53,
            "to_index": 81,
            "interior_indices": [
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80
            ],
            "interior_values": [
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23
            ],
            "sum": 336,
            "residue": 24
          },
          {
            "from_index": 81,
            "to_index": 25,
            "interior_indices": [
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24
            ],
            "interior_values": [
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10
            ],
            "sum": 258,
            "residue": 24
          }
        ],
        "minimal_cycle": [
          24
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 24,
        "start_interval_clock_positions": [
          25,
          53
        ],
        "start_interval_values": [
          25,
          14,
          13,
          1,
          14,
          15,
          3,
          18,
          21,
          13,
          8,
          21,
          3,
          24,
          1,
          25,
          0,
          25,
          25,
          24,
          23,
          21,
          18,
          13,
          5,
          18,
          23
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-26",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        26,
        54,
        82
      ],
      "clock_positions": [
        26,
        54,
        82
      ],
      "native_values": [
        25,
        12,
        25
      ],
      "rotation_degrees": 111.42857142857143,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        25,
        12,
        25
      ],
      "beta": {
        "interval_sums": [
          404,
          326,
          248
        ],
        "interval_residues": [
          14,
          14,
          14
        ],
        "display_cycle": [
          14,
          14,
          14
        ],
        "intervals": [
          {
            "from_index": 26,
            "to_index": 54,
            "interior_indices": [
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53
            ],
            "interior_values": [
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15
            ],
            "sum": 404,
            "residue": 14
          },
          {
            "from_index": 54,
            "to_index": 82,
            "interior_indices": [
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81
            ],
            "interior_values": [
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2
            ],
            "sum": 326,
            "residue": 14
          },
          {
            "from_index": 82,
            "to_index": 26,
            "interior_indices": [
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25
            ],
            "interior_values": [
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15
            ],
            "sum": 248,
            "residue": 14
          }
        ],
        "minimal_cycle": [
          14
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 14,
        "start_interval_clock_positions": [
          26,
          54
        ],
        "start_interval_values": [
          14,
          13,
          1,
          14,
          15,
          3,
          18,
          21,
          13,
          8,
          21,
          3,
          24,
          1,
          25,
          0,
          25,
          25,
          24,
          23,
          21,
          18,
          13,
          5,
          18,
          23,
          15
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "3-nodes-offset-27",
      "shape": "Equilateral triangle",
      "alpha_indices": [
        27,
        55,
        83
      ],
      "clock_positions": [
        27,
        55,
        83
      ],
      "native_values": [
        14,
        1,
        1
      ],
      "rotation_degrees": 115.71428571428571,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        14,
        1,
        1
      ],
      "beta": {
        "interval_sums": [
          402,
          350,
          272
        ],
        "interval_residues": [
          12,
          12,
          12
        ],
        "display_cycle": [
          12,
          12,
          12
        ],
        "intervals": [
          {
            "from_index": 27,
            "to_index": 55,
            "interior_indices": [
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54
            ],
            "interior_values": [
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12
            ],
            "sum": 402,
            "residue": 12
          },
          {
            "from_index": 55,
            "to_index": 83,
            "interior_indices": [
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82
            ],
            "interior_values": [
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25
            ],
            "sum": 350,
            "residue": 12
          },
          {
            "from_index": 83,
            "to_index": 27,
            "interior_indices": [
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26
            ],
            "interior_values": [
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25
            ],
            "sum": 272,
            "residue": 12
          }
        ],
        "minimal_cycle": [
          12
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 12,
        "start_interval_clock_positions": [
          27,
          55
        ],
        "start_interval_values": [
          13,
          1,
          14,
          15,
          3,
          18,
          21,
          13,
          8,
          21,
          3,
          24,
          1,
          25,
          0,
          25,
          25,
          24,
          23,
          21,
          18,
          13,
          5,
          18,
          23,
          15,
          12
        ],
        "circle_positions": 26,
        "cycle_length": 3,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      }
    },
    {
      "id": "6-nodes-offset-0",
      "shape": "Hexagon",
      "alpha_indices": [
        0,
        14,
        28,
        42,
        56,
        70
      ],
      "clock_positions": [
        84,
        14,
        28,
        42,
        56,
        70
      ],
      "native_values": [
        26,
        13,
        13,
        26,
        13,
        13
      ],
      "rotation_degrees": 0.0,
      "north_south": true,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        26,
        13,
        13
      ],
      "beta": {
        "interval_sums": [
          115,
          158,
          167,
          223,
          154,
          171
        ],
        "interval_residues": [
          11,
          2,
          11,
          15,
          24,
          15
        ],
        "display_cycle": [
          11,
          2,
          11,
          15,
          24,
          15
        ],
        "intervals": [
          {
            "from_index": 0,
            "to_index": 14,
            "interior_indices": [
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13
            ],
            "interior_values": [
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25
            ],
            "sum": 115,
            "residue": 11
          },
          {
            "from_index": 14,
            "to_index": 28,
            "interior_indices": [
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27
            ],
            "interior_values": [
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14
            ],
            "sum": 158,
            "residue": 2
          },
          {
            "from_index": 28,
            "to_index": 42,
            "interior_indices": [
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41
            ],
            "interior_values": [
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25
            ],
            "sum": 167,
            "residue": 11
          },
          {
            "from_index": 42,
            "to_index": 56,
            "interior_indices": [
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55
            ],
            "interior_values": [
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1
            ],
            "sum": 223,
            "residue": 15
          },
          {
            "from_index": 56,
            "to_index": 70,
            "interior_indices": [
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69
            ],
            "interior_values": [
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12
            ],
            "sum": 154,
            "residue": 24
          },
          {
            "from_index": 70,
            "to_index": 0,
            "interior_indices": [
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83
            ],
            "interior_values": [
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1
            ],
            "sum": 171,
            "residue": 15
          }
        ],
        "minimal_cycle": [
          11,
          2,
          11,
          15,
          24,
          15
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 11,
        "start_interval_clock_positions": [
          84,
          14
        ],
        "start_interval_values": [
          1,
          1,
          2,
          3,
          5,
          8,
          13,
          21,
          8,
          3,
          11,
          14,
          25
        ],
        "circle_positions": 26,
        "cycle_length": 6,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      },
      "epsilon_routes": []
    },
    {
      "id": "6-nodes-offset-1",
      "shape": "Hexagon",
      "alpha_indices": [
        1,
        15,
        29,
        43,
        57,
        71
      ],
      "clock_positions": [
        1,
        15,
        29,
        43,
        57,
        71
      ],
      "native_values": [
        1,
        12,
        1,
        25,
        14,
        25
      ],
      "rotation_degrees": 4.285714285714286,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        1,
        12,
        1,
        25,
        14,
        25
      ],
      "beta": {
        "interval_sums": [
          127,
          159,
          166,
          211,
          153,
          146
        ],
        "interval_residues": [
          23,
          3,
          10,
          3,
          23,
          16
        ],
        "display_cycle": [
          23,
          3,
          10,
          3,
          23,
          16
        ],
        "intervals": [
          {
            "from_index": 1,
            "to_index": 15,
            "interior_indices": [
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14
            ],
            "interior_values": [
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13
            ],
            "sum": 127,
            "residue": 23
          },
          {
            "from_index": 15,
            "to_index": 29,
            "interior_indices": [
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28
            ],
            "interior_values": [
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13
            ],
            "sum": 159,
            "residue": 3
          },
          {
            "from_index": 29,
            "to_index": 43,
            "interior_indices": [
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42
            ],
            "interior_values": [
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0
            ],
            "sum": 166,
            "residue": 10
          },
          {
            "from_index": 43,
            "to_index": 57,
            "interior_indices": [
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56
            ],
            "interior_values": [
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13
            ],
            "sum": 211,
            "residue": 3
          },
          {
            "from_index": 57,
            "to_index": 71,
            "interior_indices": [
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70
            ],
            "interior_values": [
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13
            ],
            "sum": 153,
            "residue": 23
          },
          {
            "from_index": 71,
            "to_index": 1,
            "interior_indices": [
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0
            ],
            "interior_values": [
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0
            ],
            "sum": 146,
            "residue": 16
          }
        ],
        "minimal_cycle": [
          23,
          3,
          10,
          3,
          23,
          16
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 23,
        "start_interval_clock_positions": [
          1,
          15
        ],
        "start_interval_values": [
          1,
          2,
          3,
          5,
          8,
          13,
          21,
          8,
          3,
          11,
          14,
          25,
          13
        ],
        "circle_positions": 26,
        "cycle_length": 6,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      },
      "epsilon_routes": []
    },
    {
      "id": "6-nodes-offset-2",
      "shape": "Hexagon",
      "alpha_indices": [
        2,
        16,
        30,
        44,
        58,
        72
      ],
      "clock_positions": [
        2,
        16,
        30,
        44,
        58,
        72
      ],
      "native_values": [
        1,
        25,
        14,
        25,
        1,
        12
      ],
      "rotation_degrees": 8.571428571428571,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        1,
        25,
        14,
        25,
        1,
        12
      ],
      "beta": {
        "interval_sums": [
          138,
          135,
          177,
          200,
          177,
          135
        ],
        "interval_residues": [
          8,
          5,
          21,
          18,
          21,
          5
        ],
        "display_cycle": [
          8,
          5,
          21,
          18,
          21,
          5
        ],
        "intervals": [
          {
            "from_index": 2,
            "to_index": 16,
            "interior_indices": [
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15
            ],
            "interior_values": [
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12
            ],
            "sum": 138,
            "residue": 8
          },
          {
            "from_index": 16,
            "to_index": 30,
            "interior_indices": [
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29
            ],
            "interior_values": [
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1
            ],
            "sum": 135,
            "residue": 5
          },
          {
            "from_index": 30,
            "to_index": 44,
            "interior_indices": [
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43
            ],
            "interior_values": [
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25
            ],
            "sum": 177,
            "residue": 21
          },
          {
            "from_index": 44,
            "to_index": 58,
            "interior_indices": [
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57
            ],
            "interior_values": [
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14
            ],
            "sum": 200,
            "residue": 18
          },
          {
            "from_index": 58,
            "to_index": 72,
            "interior_indices": [
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71
            ],
            "interior_values": [
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25
            ],
            "sum": 177,
            "residue": 21
          },
          {
            "from_index": 72,
            "to_index": 2,
            "interior_indices": [
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1
            ],
            "interior_values": [
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1
            ],
            "sum": 135,
            "residue": 5
          }
        ],
        "minimal_cycle": [
          8,
          5,
          21,
          18,
          21,
          5
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 8,
        "start_interval_clock_positions": [
          2,
          16
        ],
        "start_interval_values": [
          2,
          3,
          5,
          8,
          13,
          21,
          8,
          3,
          11,
          14,
          25,
          13,
          12
        ],
        "circle_positions": 26,
        "cycle_length": 6,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      },
      "epsilon_routes": []
    },
    {
      "id": "6-nodes-offset-3",
      "shape": "Hexagon",
      "alpha_indices": [
        3,
        17,
        31,
        45,
        59,
        73
      ],
      "clock_positions": [
        3,
        17,
        31,
        45,
        59,
        73
      ],
      "native_values": [
        2,
        11,
        15,
        24,
        15,
        11
      ],
      "rotation_degrees": 12.857142857142858,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        2,
        11,
        15,
        24,
        15,
        11
      ],
      "beta": {
        "interval_sums": [
          161,
          138,
          187,
          177,
          174,
          125
        ],
        "interval_residues": [
          5,
          8,
          5,
          21,
          18,
          21
        ],
        "display_cycle": [
          5,
          8,
          5,
          21,
          18,
          21
        ],
        "intervals": [
          {
            "from_index": 3,
            "to_index": 17,
            "interior_indices": [
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16
            ],
            "interior_values": [
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25
            ],
            "sum": 161,
            "residue": 5
          },
          {
            "from_index": 17,
            "to_index": 31,
            "interior_indices": [
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30
            ],
            "interior_values": [
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14
            ],
            "sum": 138,
            "residue": 8
          },
          {
            "from_index": 31,
            "to_index": 45,
            "interior_indices": [
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44
            ],
            "interior_values": [
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25
            ],
            "sum": 187,
            "residue": 5
          },
          {
            "from_index": 45,
            "to_index": 59,
            "interior_indices": [
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58
            ],
            "interior_values": [
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1
            ],
            "sum": 177,
            "residue": 21
          },
          {
            "from_index": 59,
            "to_index": 73,
            "interior_indices": [
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72
            ],
            "interior_values": [
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12
            ],
            "sum": 174,
            "residue": 18
          },
          {
            "from_index": 73,
            "to_index": 3,
            "interior_indices": [
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2
            ],
            "interior_values": [
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1
            ],
            "sum": 125,
            "residue": 21
          }
        ],
        "minimal_cycle": [
          5,
          8,
          5,
          21,
          18,
          21
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 5,
        "start_interval_clock_positions": [
          3,
          17
        ],
        "start_interval_values": [
          3,
          5,
          8,
          13,
          21,
          8,
          3,
          11,
          14,
          25,
          13,
          12,
          25
        ],
        "circle_positions": 26,
        "cycle_length": 6,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      },
      "epsilon_routes": []
    },
    {
      "id": "6-nodes-offset-4",
      "shape": "Hexagon",
      "alpha_indices": [
        4,
        18,
        32,
        46,
        60,
        74
      ],
      "clock_positions": [
        4,
        18,
        32,
        46,
        60,
        74
      ],
      "native_values": [
        3,
        10,
        3,
        23,
        16,
        23
      ],
      "rotation_degrees": 17.142857142857142,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        3,
        10,
        3,
        23,
        16,
        23
      ],
      "beta": {
        "interval_sums": [
          169,
          143,
          208,
          169,
          169,
          104
        ],
        "interval_residues": [
          13,
          13,
          0,
          13,
          13,
          0
        ],
        "display_cycle": [
          13,
          13,
          26,
          13,
          13,
          26
        ],
        "intervals": [
          {
            "from_index": 4,
            "to_index": 18,
            "interior_indices": [
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17
            ],
            "interior_values": [
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11
            ],
            "sum": 169,
            "residue": 13
          },
          {
            "from_index": 18,
            "to_index": 32,
            "interior_indices": [
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31
            ],
            "interior_values": [
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15
            ],
            "sum": 143,
            "residue": 13
          },
          {
            "from_index": 32,
            "to_index": 46,
            "interior_indices": [
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45
            ],
            "interior_values": [
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24
            ],
            "sum": 208,
            "residue": 0
          },
          {
            "from_index": 46,
            "to_index": 60,
            "interior_indices": [
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59
            ],
            "interior_values": [
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15
            ],
            "sum": 169,
            "residue": 13
          },
          {
            "from_index": 60,
            "to_index": 74,
            "interior_indices": [
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73
            ],
            "interior_values": [
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11
            ],
            "sum": 169,
            "residue": 13
          },
          {
            "from_index": 74,
            "to_index": 4,
            "interior_indices": [
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3
            ],
            "interior_values": [
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2
            ],
            "sum": 104,
            "residue": 0
          }
        ],
        "minimal_cycle": [
          13,
          13,
          0
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 13,
        "start_interval_clock_positions": [
          4,
          18
        ],
        "start_interval_values": [
          5,
          8,
          13,
          21,
          8,
          3,
          11,
          14,
          25,
          13,
          12,
          25,
          11
        ],
        "circle_positions": 26,
        "cycle_length": 6,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      },
      "epsilon_routes": []
    },
    {
      "id": "6-nodes-offset-5",
      "shape": "Hexagon",
      "alpha_indices": [
        5,
        19,
        33,
        47,
        61,
        75
      ],
      "clock_positions": [
        5,
        19,
        33,
        47,
        61,
        75
      ],
      "native_values": [
        5,
        21,
        18,
        21,
        5,
        8
      ],
      "rotation_degrees": 21.428571428571427,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        5,
        21,
        18,
        21,
        5,
        8
      ],
      "beta": {
        "interval_sums": [
          174,
          125,
          213,
          164,
          187,
          99
        ],
        "interval_residues": [
          18,
          21,
          5,
          8,
          5,
          21
        ],
        "display_cycle": [
          18,
          21,
          5,
          8,
          5,
          21
        ],
        "intervals": [
          {
            "from_index": 5,
            "to_index": 19,
            "interior_indices": [
              6,
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18
            ],
            "interior_values": [
              8,
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10
            ],
            "sum": 174,
            "residue": 18
          },
          {
            "from_index": 19,
            "to_index": 33,
            "interior_indices": [
              20,
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32
            ],
            "interior_values": [
              5,
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3
            ],
            "sum": 125,
            "residue": 21
          },
          {
            "from_index": 33,
            "to_index": 47,
            "interior_indices": [
              34,
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46
            ],
            "interior_values": [
              21,
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23
            ],
            "sum": 213,
            "residue": 5
          },
          {
            "from_index": 47,
            "to_index": 61,
            "interior_indices": [
              48,
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60
            ],
            "interior_values": [
              18,
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16
            ],
            "sum": 164,
            "residue": 8
          },
          {
            "from_index": 61,
            "to_index": 75,
            "interior_indices": [
              62,
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74
            ],
            "interior_values": [
              21,
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23
            ],
            "sum": 187,
            "residue": 5
          },
          {
            "from_index": 75,
            "to_index": 5,
            "interior_indices": [
              76,
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4
            ],
            "interior_values": [
              5,
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3
            ],
            "sum": 99,
            "residue": 21
          }
        ],
        "minimal_cycle": [
          18,
          21,
          5,
          8,
          5,
          21
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 18,
        "start_interval_clock_positions": [
          5,
          19
        ],
        "start_interval_values": [
          8,
          13,
          21,
          8,
          3,
          11,
          14,
          25,
          13,
          12,
          25,
          11,
          10
        ],
        "circle_positions": 26,
        "cycle_length": 6,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      },
      "epsilon_routes": []
    },
    {
      "id": "6-nodes-offset-6",
      "shape": "Hexagon",
      "alpha_indices": [
        6,
        20,
        34,
        48,
        62,
        76
      ],
      "clock_positions": [
        6,
        20,
        34,
        48,
        62,
        76
      ],
      "native_values": [
        8,
        5,
        21,
        18,
        21,
        5
      ],
      "rotation_degrees": 25.714285714285715,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        8,
        5,
        21,
        18,
        21,
        5
      ],
      "beta": {
        "interval_sums": [
          187,
          138,
          213,
          151,
          174,
          99
        ],
        "interval_residues": [
          5,
          8,
          5,
          21,
          18,
          21
        ],
        "display_cycle": [
          5,
          8,
          5,
          21,
          18,
          21
        ],
        "intervals": [
          {
            "from_index": 6,
            "to_index": 20,
            "interior_indices": [
              7,
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19
            ],
            "interior_values": [
              13,
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21
            ],
            "sum": 187,
            "residue": 5
          },
          {
            "from_index": 20,
            "to_index": 34,
            "interior_indices": [
              21,
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33
            ],
            "interior_values": [
              0,
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18
            ],
            "sum": 138,
            "residue": 8
          },
          {
            "from_index": 34,
            "to_index": 48,
            "interior_indices": [
              35,
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47
            ],
            "interior_values": [
              13,
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21
            ],
            "sum": 213,
            "residue": 5
          },
          {
            "from_index": 48,
            "to_index": 62,
            "interior_indices": [
              49,
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61
            ],
            "interior_values": [
              13,
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5
            ],
            "sum": 151,
            "residue": 21
          },
          {
            "from_index": 62,
            "to_index": 76,
            "interior_indices": [
              63,
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75
            ],
            "interior_values": [
              0,
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8
            ],
            "sum": 174,
            "residue": 18
          },
          {
            "from_index": 76,
            "to_index": 6,
            "interior_indices": [
              77,
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5
            ],
            "interior_values": [
              13,
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5
            ],
            "sum": 99,
            "residue": 21
          }
        ],
        "minimal_cycle": [
          5,
          8,
          5,
          21,
          18,
          21
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 5,
        "start_interval_clock_positions": [
          6,
          20
        ],
        "start_interval_values": [
          13,
          21,
          8,
          3,
          11,
          14,
          25,
          13,
          12,
          25,
          11,
          10,
          21
        ],
        "circle_positions": 26,
        "cycle_length": 6,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      },
      "epsilon_routes": []
    },
    {
      "id": "6-nodes-offset-7",
      "shape": "Hexagon",
      "alpha_indices": [
        7,
        21,
        35,
        49,
        63,
        77
      ],
      "clock_positions": [
        7,
        21,
        35,
        49,
        63,
        77
      ],
      "native_values": [
        13,
        26,
        13,
        13,
        26,
        13
      ],
      "rotation_degrees": 30.0,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        13,
        26,
        13
      ],
      "beta": {
        "interval_sums": [
          179,
          159,
          218,
          159,
          179,
          94
        ],
        "interval_residues": [
          23,
          3,
          10,
          3,
          23,
          16
        ],
        "display_cycle": [
          23,
          3,
          10,
          3,
          23,
          16
        ],
        "intervals": [
          {
            "from_index": 7,
            "to_index": 21,
            "interior_indices": [
              8,
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20
            ],
            "interior_values": [
              21,
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5
            ],
            "sum": 179,
            "residue": 23
          },
          {
            "from_index": 21,
            "to_index": 35,
            "interior_indices": [
              22,
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34
            ],
            "interior_values": [
              5,
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21
            ],
            "sum": 159,
            "residue": 3
          },
          {
            "from_index": 35,
            "to_index": 49,
            "interior_indices": [
              36,
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48
            ],
            "interior_values": [
              8,
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18
            ],
            "sum": 218,
            "residue": 10
          },
          {
            "from_index": 49,
            "to_index": 63,
            "interior_indices": [
              50,
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62
            ],
            "interior_values": [
              5,
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21
            ],
            "sum": 159,
            "residue": 3
          },
          {
            "from_index": 63,
            "to_index": 77,
            "interior_indices": [
              64,
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76
            ],
            "interior_values": [
              21,
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5
            ],
            "sum": 179,
            "residue": 23
          },
          {
            "from_index": 77,
            "to_index": 7,
            "interior_indices": [
              78,
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6
            ],
            "interior_values": [
              18,
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8
            ],
            "sum": 94,
            "residue": 16
          }
        ],
        "minimal_cycle": [
          23,
          3,
          10,
          3,
          23,
          16
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 23,
        "start_interval_clock_positions": [
          7,
          21
        ],
        "start_interval_values": [
          21,
          8,
          3,
          11,
          14,
          25,
          13,
          12,
          25,
          11,
          10,
          21,
          5
        ],
        "circle_positions": 26,
        "cycle_length": 6,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      },
      "epsilon_routes": []
    },
    {
      "id": "6-nodes-offset-8",
      "shape": "Hexagon",
      "alpha_indices": [
        8,
        22,
        36,
        50,
        64,
        78
      ],
      "clock_positions": [
        8,
        22,
        36,
        50,
        64,
        78
      ],
      "native_values": [
        21,
        5,
        8,
        5,
        21,
        18
      ],
      "rotation_degrees": 34.285714285714285,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        21,
        5,
        8,
        5,
        21,
        18
      ],
      "beta": {
        "interval_sums": [
          158,
          167,
          223,
          154,
          171,
          89
        ],
        "interval_residues": [
          2,
          11,
          15,
          24,
          15,
          11
        ],
        "display_cycle": [
          2,
          11,
          15,
          24,
          15,
          11
        ],
        "intervals": [
          {
            "from_index": 8,
            "to_index": 22,
            "interior_indices": [
              9,
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21
            ],
            "interior_values": [
              8,
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0
            ],
            "sum": 158,
            "residue": 2
          },
          {
            "from_index": 22,
            "to_index": 36,
            "interior_indices": [
              23,
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35
            ],
            "interior_values": [
              5,
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13
            ],
            "sum": 167,
            "residue": 11
          },
          {
            "from_index": 36,
            "to_index": 50,
            "interior_indices": [
              37,
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49
            ],
            "interior_values": [
              21,
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13
            ],
            "sum": 223,
            "residue": 15
          },
          {
            "from_index": 50,
            "to_index": 64,
            "interior_indices": [
              51,
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63
            ],
            "interior_values": [
              18,
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0
            ],
            "sum": 154,
            "residue": 24
          },
          {
            "from_index": 64,
            "to_index": 78,
            "interior_indices": [
              65,
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77
            ],
            "interior_values": [
              21,
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13
            ],
            "sum": 171,
            "residue": 15
          },
          {
            "from_index": 78,
            "to_index": 8,
            "interior_indices": [
              79,
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7
            ],
            "interior_values": [
              5,
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13
            ],
            "sum": 89,
            "residue": 11
          }
        ],
        "minimal_cycle": [
          2,
          11,
          15,
          24,
          15,
          11
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 2,
        "start_interval_clock_positions": [
          8,
          22
        ],
        "start_interval_values": [
          8,
          3,
          11,
          14,
          25,
          13,
          12,
          25,
          11,
          10,
          21,
          5,
          0
        ],
        "circle_positions": 26,
        "cycle_length": 6,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      },
      "epsilon_routes": []
    },
    {
      "id": "6-nodes-offset-9",
      "shape": "Hexagon",
      "alpha_indices": [
        9,
        23,
        37,
        51,
        65,
        79
      ],
      "clock_positions": [
        9,
        23,
        37,
        51,
        65,
        79
      ],
      "native_values": [
        8,
        5,
        21,
        18,
        21,
        5
      ],
      "rotation_degrees": 38.57142857142857,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        8,
        5,
        21,
        18,
        21,
        5
      ],
      "beta": {
        "interval_sums": [
          155,
          170,
          207,
          157,
          168,
          105
        ],
        "interval_residues": [
          25,
          14,
          25,
          1,
          12,
          1
        ],
        "display_cycle": [
          25,
          14,
          25,
          1,
          12,
          1
        ],
        "intervals": [
          {
            "from_index": 9,
            "to_index": 23,
            "interior_indices": [
              10,
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22
            ],
            "interior_values": [
              3,
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5
            ],
            "sum": 155,
            "residue": 25
          },
          {
            "from_index": 23,
            "to_index": 37,
            "interior_indices": [
              24,
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36
            ],
            "interior_values": [
              10,
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8
            ],
            "sum": 170,
            "residue": 14
          },
          {
            "from_index": 37,
            "to_index": 51,
            "interior_indices": [
              38,
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50
            ],
            "interior_values": [
              3,
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5
            ],
            "sum": 207,
            "residue": 25
          },
          {
            "from_index": 51,
            "to_index": 65,
            "interior_indices": [
              52,
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64
            ],
            "interior_values": [
              23,
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21
            ],
            "sum": 157,
            "residue": 1
          },
          {
            "from_index": 65,
            "to_index": 79,
            "interior_indices": [
              66,
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78
            ],
            "interior_values": [
              16,
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18
            ],
            "sum": 168,
            "residue": 12
          },
          {
            "from_index": 79,
            "to_index": 9,
            "interior_indices": [
              80,
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8
            ],
            "interior_values": [
              23,
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21
            ],
            "sum": 105,
            "residue": 1
          }
        ],
        "minimal_cycle": [
          25,
          14,
          25,
          1,
          12,
          1
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 25,
        "start_interval_clock_positions": [
          9,
          23
        ],
        "start_interval_values": [
          3,
          11,
          14,
          25,
          13,
          12,
          25,
          11,
          10,
          21,
          5,
          0,
          5
        ],
        "circle_positions": 26,
        "cycle_length": 6,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      },
      "epsilon_routes": []
    },
    {
      "id": "6-nodes-offset-10",
      "shape": "Hexagon",
      "alpha_indices": [
        10,
        24,
        38,
        52,
        66,
        80
      ],
      "clock_positions": [
        10,
        24,
        38,
        52,
        66,
        80
      ],
      "native_values": [
        3,
        10,
        3,
        23,
        16,
        23
      ],
      "rotation_degrees": 42.857142857142854,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        3,
        10,
        3,
        23,
        16,
        23
      ],
      "beta": {
        "interval_sums": [
          157,
          181,
          222,
          155,
          157,
          90
        ],
        "interval_residues": [
          1,
          25,
          14,
          25,
          1,
          12
        ],
        "display_cycle": [
          1,
          25,
          14,
          25,
          1,
          12
        ],
        "intervals": [
          {
            "from_index": 10,
            "to_index": 24,
            "interior_indices": [
              11,
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23
            ],
            "interior_values": [
              11,
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5
            ],
            "sum": 157,
            "residue": 1
          },
          {
            "from_index": 24,
            "to_index": 38,
            "interior_indices": [
              25,
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37
            ],
            "interior_values": [
              15,
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21
            ],
            "sum": 181,
            "residue": 25
          },
          {
            "from_index": 38,
            "to_index": 52,
            "interior_indices": [
              39,
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51
            ],
            "interior_values": [
              24,
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18
            ],
            "sum": 222,
            "residue": 14
          },
          {
            "from_index": 52,
            "to_index": 66,
            "interior_indices": [
              53,
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65
            ],
            "interior_values": [
              15,
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21
            ],
            "sum": 155,
            "residue": 25
          },
          {
            "from_index": 66,
            "to_index": 80,
            "interior_indices": [
              67,
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79
            ],
            "interior_values": [
              11,
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5
            ],
            "sum": 157,
            "residue": 1
          },
          {
            "from_index": 80,
            "to_index": 10,
            "interior_indices": [
              81,
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9
            ],
            "interior_values": [
              2,
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8
            ],
            "sum": 90,
            "residue": 12
          }
        ],
        "minimal_cycle": [
          1,
          25,
          14,
          25,
          1,
          12
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 1,
        "start_interval_clock_positions": [
          10,
          24
        ],
        "start_interval_values": [
          11,
          14,
          25,
          13,
          12,
          25,
          11,
          10,
          21,
          5,
          0,
          5,
          5
        ],
        "circle_positions": 26,
        "cycle_length": 6,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      },
      "epsilon_routes": []
    },
    {
      "id": "6-nodes-offset-11",
      "shape": "Hexagon",
      "alpha_indices": [
        11,
        25,
        39,
        53,
        67,
        81
      ],
      "clock_positions": [
        11,
        25,
        39,
        53,
        67,
        81
      ],
      "native_values": [
        11,
        15,
        24,
        15,
        11,
        2
      ],
      "rotation_degrees": 47.142857142857146,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        11,
        15,
        24,
        15,
        11,
        2
      ],
      "beta": {
        "interval_sums": [
          156,
          169,
          221,
          156,
          169,
          91
        ],
        "interval_residues": [
          0,
          13,
          13,
          0,
          13,
          13
        ],
        "display_cycle": [
          26,
          13,
          13,
          26,
          13,
          13
        ],
        "intervals": [
          {
            "from_index": 11,
            "to_index": 25,
            "interior_indices": [
              12,
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24
            ],
            "interior_values": [
              14,
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10
            ],
            "sum": 156,
            "residue": 0
          },
          {
            "from_index": 25,
            "to_index": 39,
            "interior_indices": [
              26,
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38
            ],
            "interior_values": [
              25,
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3
            ],
            "sum": 169,
            "residue": 13
          },
          {
            "from_index": 39,
            "to_index": 53,
            "interior_indices": [
              40,
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52
            ],
            "interior_values": [
              1,
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23
            ],
            "sum": 221,
            "residue": 13
          },
          {
            "from_index": 53,
            "to_index": 67,
            "interior_indices": [
              54,
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66
            ],
            "interior_values": [
              12,
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16
            ],
            "sum": 156,
            "residue": 0
          },
          {
            "from_index": 67,
            "to_index": 81,
            "interior_indices": [
              68,
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80
            ],
            "interior_values": [
              1,
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23
            ],
            "sum": 169,
            "residue": 13
          },
          {
            "from_index": 81,
            "to_index": 11,
            "interior_indices": [
              82,
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10
            ],
            "interior_values": [
              25,
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3
            ],
            "sum": 91,
            "residue": 13
          }
        ],
        "minimal_cycle": [
          0,
          13,
          13
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 26,
        "start_interval_clock_positions": [
          11,
          25
        ],
        "start_interval_values": [
          14,
          25,
          13,
          12,
          25,
          11,
          10,
          21,
          5,
          0,
          5,
          5,
          10
        ],
        "circle_positions": 26,
        "cycle_length": 6,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      },
      "epsilon_routes": []
    },
    {
      "id": "6-nodes-offset-12",
      "shape": "Hexagon",
      "alpha_indices": [
        12,
        26,
        40,
        54,
        68,
        82
      ],
      "clock_positions": [
        12,
        26,
        40,
        54,
        68,
        82
      ],
      "native_values": [
        14,
        25,
        1,
        12,
        1,
        25
      ],
      "rotation_degrees": 51.42857142857143,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        14,
        25,
        1,
        12,
        1,
        25
      ],
      "beta": {
        "interval_sums": [
          157,
          168,
          235,
          155,
          170,
          77
        ],
        "interval_residues": [
          1,
          12,
          1,
          25,
          14,
          25
        ],
        "display_cycle": [
          1,
          12,
          1,
          25,
          14,
          25
        ],
        "intervals": [
          {
            "from_index": 12,
            "to_index": 26,
            "interior_indices": [
              13,
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25
            ],
            "interior_values": [
              25,
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15
            ],
            "sum": 157,
            "residue": 1
          },
          {
            "from_index": 26,
            "to_index": 40,
            "interior_indices": [
              27,
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39
            ],
            "interior_values": [
              14,
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24
            ],
            "sum": 168,
            "residue": 12
          },
          {
            "from_index": 40,
            "to_index": 54,
            "interior_indices": [
              41,
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53
            ],
            "interior_values": [
              25,
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15
            ],
            "sum": 235,
            "residue": 1
          },
          {
            "from_index": 54,
            "to_index": 68,
            "interior_indices": [
              55,
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67
            ],
            "interior_values": [
              1,
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11
            ],
            "sum": 155,
            "residue": 25
          },
          {
            "from_index": 68,
            "to_index": 82,
            "interior_indices": [
              69,
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81
            ],
            "interior_values": [
              12,
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2
            ],
            "sum": 170,
            "residue": 14
          },
          {
            "from_index": 82,
            "to_index": 12,
            "interior_indices": [
              83,
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11
            ],
            "interior_values": [
              1,
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11
            ],
            "sum": 77,
            "residue": 25
          }
        ],
        "minimal_cycle": [
          1,
          12,
          1,
          25,
          14,
          25
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 1,
        "start_interval_clock_positions": [
          12,
          26
        ],
        "start_interval_values": [
          25,
          13,
          12,
          25,
          11,
          10,
          21,
          5,
          0,
          5,
          5,
          10,
          15
        ],
        "circle_positions": 26,
        "cycle_length": 6,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      },
      "epsilon_routes": []
    },
    {
      "id": "6-nodes-offset-13",
      "shape": "Hexagon",
      "alpha_indices": [
        13,
        27,
        41,
        55,
        69,
        83
      ],
      "clock_positions": [
        13,
        27,
        41,
        55,
        69,
        83
      ],
      "native_values": [
        25,
        14,
        25,
        1,
        12,
        1
      ],
      "rotation_degrees": 55.714285714285715,
      "north_south": false,
      "in_primary_gamma": false,
      "uniform_values": false,
      "minimal_value_cycle": [
        25,
        14,
        25,
        1,
        12,
        1
      ],
      "beta": {
        "interval_sums": [
          157,
          155,
          222,
          155,
          183,
          90
        ],
        "interval_residues": [
          1,
          25,
          14,
          25,
          1,
          12
        ],
        "display_cycle": [
          1,
          25,
          14,
          25,
          1,
          12
        ],
        "intervals": [
          {
            "from_index": 13,
            "to_index": 27,
            "interior_indices": [
              14,
              15,
              16,
              17,
              18,
              19,
              20,
              21,
              22,
              23,
              24,
              25,
              26
            ],
            "interior_values": [
              13,
              12,
              25,
              11,
              10,
              21,
              5,
              0,
              5,
              5,
              10,
              15,
              25
            ],
            "sum": 157,
            "residue": 1
          },
          {
            "from_index": 27,
            "to_index": 41,
            "interior_indices": [
              28,
              29,
              30,
              31,
              32,
              33,
              34,
              35,
              36,
              37,
              38,
              39,
              40
            ],
            "interior_values": [
              13,
              1,
              14,
              15,
              3,
              18,
              21,
              13,
              8,
              21,
              3,
              24,
              1
            ],
            "sum": 155,
            "residue": 25
          },
          {
            "from_index": 41,
            "to_index": 55,
            "interior_indices": [
              42,
              43,
              44,
              45,
              46,
              47,
              48,
              49,
              50,
              51,
              52,
              53,
              54
            ],
            "interior_values": [
              0,
              25,
              25,
              24,
              23,
              21,
              18,
              13,
              5,
              18,
              23,
              15,
              12
            ],
            "sum": 222,
            "residue": 14
          },
          {
            "from_index": 55,
            "to_index": 69,
            "interior_indices": [
              56,
              57,
              58,
              59,
              60,
              61,
              62,
              63,
              64,
              65,
              66,
              67,
              68
            ],
            "interior_values": [
              13,
              14,
              1,
              15,
              16,
              5,
              21,
              0,
              21,
              21,
              16,
              11,
              1
            ],
            "sum": 155,
            "residue": 25
          },
          {
            "from_index": 69,
            "to_index": 83,
            "interior_indices": [
              70,
              71,
              72,
              73,
              74,
              75,
              76,
              77,
              78,
              79,
              80,
              81,
              82
            ],
            "interior_values": [
              13,
              25,
              12,
              11,
              23,
              8,
              5,
              13,
              18,
              5,
              23,
              2,
              25
            ],
            "sum": 183,
            "residue": 1
          },
          {
            "from_index": 83,
            "to_index": 13,
            "interior_indices": [
              0,
              1,
              2,
              3,
              4,
              5,
              6,
              7,
              8,
              9,
              10,
              11,
              12
            ],
            "interior_values": [
              0,
              1,
              1,
              2,
              3,
              5,
              8,
              13,
              21,
              8,
              3,
              11,
              14
            ],
            "sum": 90,
            "residue": 12
          }
        ],
        "minimal_cycle": [
          1,
          25,
          14,
          25,
          1,
          12
        ],
        "native_doubling": false,
        "recognised_doubling_cycle": null,
        "rotation_to_recognised": null,
        "m9_doubling_cycle_present": false,
        "start_value": 1,
        "start_interval_clock_positions": [
          13,
          27
        ],
        "start_interval_values": [
          13,
          12,
          25,
          11,
          10,
          21,
          5,
          0,
          5,
          5,
          10,
          15,
          25
        ],
        "circle_positions": 26,
        "cycle_length": 6,
        "endpoint_rule": "Exclude endpoints, include wraparound; raw residues used for sums. Empty interiors sum to zero."
      },
      "epsilon_routes": []
    }
  ],
  "notable_gamma_patterns": [
    {
      "formation_id": "6-nodes-offset-0",
      "shape": "Hexagon",
      "values": [
        26,
        13,
        13,
        26,
        13,
        13
      ],
      "rotation_degrees": 0.0,
      "reasons": [
        "Repeating native-value pattern"
      ],
      "python_status": "Passed",
      "wolfram_status": "Pending",
      "significance": "Not yet assessed"
    },
    {
      "formation_id": "6-nodes-offset-7",
      "shape": "Hexagon",
      "values": [
        13,
        26,
        13,
        13,
        26,
        13
      ],
      "rotation_degrees": 30.0,
      "reasons": [
        "Repeating native-value pattern"
      ],
      "python_status": "Passed",
      "wolfram_status": "Pending",
      "significance": "Not yet assessed"
    }
  ],
  "research": {
    "focus": "OXTR gene expression",
    "biological_significance_status": "Not yet assessed",
    "statistical_significance_status": "Not yet assessed",
    "quantum_annealing_status": "Not yet performed",
    "tiers": [
      {
        "name": "Creates cellular life",
        "range": [
          8,
          10
        ],
        "wss_score": null,
        "evidence": []
      },
      {
        "name": "Supports cellular life",
        "range": [
          5,
          7
        ],
        "wss_score": null,
        "evidence": []
      },
      {
        "name": "Creates elemental life",
        "range": [
          1,
          4
        ],
        "wss_score": null,
        "evidence": []
      }
    ],
    "background_literature": {
      "register_id": "oxtr-phase1-literature",
      "register_version": "2026-10-18.1",
      "register_file": "research-register.json",
      "relationship": "background_context",
      "scope": "Shared background literature for the initial OXTR research programme across M1–M1001. Studies of related mechanisms in other systems provide context for investigation.",
      "publication_ids": [
        "E01",
        "E02",
        "E03",
        "E04",
        "E05",
        "E06",
        "E07",
        "E08",
        "E09",
        "E10",
        "E11",
        "E12",
        "E13",
        "E14"
      ],
      "hero_publication_ids": [
        "E03",
        "E06",
        "E08",
        "E11"
      ],
      "modulus_relevance_status": "Not yet assessed"
    }
  },
  "verification": {
    "python": "Passed",
    "wolfram": "Passed",
    "scope": "Native period, sequences, regular polygon index spacing and reference Epsilon routes; no physical or biological validation.",
    "wolfram_scope": "Native period, complete Alpha, primary Gamma indices, Beta sums/residues, primary triangles/hexagons and exact north-oriented M9 reference path comparison. Extended catalogue, native Epsilon search and Chevron remain Python-checked.",
    "evidence": "wolfram-batch-result.json"
  },
  "reduction_to_m9": {
    "version": "m9-reduction-v1",
    "target_modulus": 9,
    "status": "Outside divisible-by-nine family",
    "applicable": false,
    "research_question": "Does the shared M9 pattern have shared biological relevance, or does native-modulus complexity change it? Unassessed: requires peer-reviewed evidence, in silico modelling (including proposed CUDA/CUDA-Q work), and appropriate experimental validation. No spatial-dimension claim or WSS score follows from this arithmetic match.",
    "biological_significance": "Not yet assessed",
    "native_data_preserved": true
  },
  "epsilon_m9_comparison": {
    "rule_version": "epsilon-m9-exact-v2",
    "geometry_identical": true,
    "reduced_sequence_identical": false,
    "combined_eligible": false,
    "alpha_indices": [
      0,
      28,
      70,
      42,
      14,
      56,
      0
    ],
    "clock_positions": [
      84,
      28,
      70,
      42,
      14,
      56,
      84
    ],
    "native_values": [
      26,
      13,
      13,
      26,
      13,
      13,
      26
    ],
    "reduced_m9_values": [
      9,
      4,
      4,
      9,
      4,
      4,
      9
    ],
    "reference_values": [
      9,
      3,
      6,
      9,
      3,
      6,
      9
    ],
    "normalized_route_sixths": [
      0,
      2,
      5,
      3,
      1,
      4,
      0
    ],
    "orientation": "clockwise from north; fixed M9 route and starting phase",
    "reason": "Reference geometry exists, but the reduced traversal does not equal 9–3–6–9–3–6–9.",
    "scope": "Tests the fixed north-oriented M9 path on the full native period circle; does not fold the period or rotate/reverse the values to force a match."
  },
  "epsilon_assessment": {
    "rule_version": "native-epsilon-diagnostics-v1",
    "native_369_present": false,
    "candidate_count": 14,
    "candidates": [
      {
        "formation_id": "6-nodes-offset-0",
        "alpha_indices": [
          0,
          14,
          28,
          42,
          56,
          70
        ],
        "clock_positions": [
          84,
          14,
          28,
          42,
          56,
          70
        ],
        "native_values": [
          26,
          13,
          13,
          26,
          13,
          13
        ],
        "rotation_degrees": 0.0,
        "missing_required_values": [
          3,
          6,
          9
        ],
        "failing_nodes": [
          {
            "alpha_index": 0,
            "clock_position": 84,
            "value": 26
          },
          {
            "alpha_index": 14,
            "clock_position": 14,
            "value": 13
          },
          {
            "alpha_index": 28,
            "clock_position": 28,
            "value": 13
          },
          {
            "alpha_index": 42,
            "clock_position": 42,
            "value": 26
          },
          {
            "alpha_index": 56,
            "clock_position": 56,
            "value": 13
          },
          {
            "alpha_index": 70,
            "clock_position": 70,
            "value": 13
          }
        ],
        "native_369_qualifies": false
      },
      {
        "formation_id": "6-nodes-offset-1",
        "alpha_indices": [
          1,
          15,
          29,
          43,
          57,
          71
        ],
        "clock_positions": [
          1,
          15,
          29,
          43,
          57,
          71
        ],
        "native_values": [
          1,
          12,
          1,
          25,
          14,
          25
        ],
        "rotation_degrees": 4.285714285714286,
        "missing_required_values": [
          3,
          6,
          9
        ],
        "failing_nodes": [
          {
            "alpha_index": 1,
            "clock_position": 1,
            "value": 1
          },
          {
            "alpha_index": 15,
            "clock_position": 15,
            "value": 12
          },
          {
            "alpha_index": 29,
            "clock_position": 29,
            "value": 1
          },
          {
            "alpha_index": 43,
            "clock_position": 43,
            "value": 25
          },
          {
            "alpha_index": 57,
            "clock_position": 57,
            "value": 14
          },
          {
            "alpha_index": 71,
            "clock_position": 71,
            "value": 25
          }
        ],
        "native_369_qualifies": false
      },
      {
        "formation_id": "6-nodes-offset-2",
        "alpha_indices": [
          2,
          16,
          30,
          44,
          58,
          72
        ],
        "clock_positions": [
          2,
          16,
          30,
          44,
          58,
          72
        ],
        "native_values": [
          1,
          25,
          14,
          25,
          1,
          12
        ],
        "rotation_degrees": 8.571428571428571,
        "missing_required_values": [
          3,
          6,
          9
        ],
        "failing_nodes": [
          {
            "alpha_index": 2,
            "clock_position": 2,
            "value": 1
          },
          {
            "alpha_index": 16,
            "clock_position": 16,
            "value": 25
          },
          {
            "alpha_index": 30,
            "clock_position": 30,
            "value": 14
          },
          {
            "alpha_index": 44,
            "clock_position": 44,
            "value": 25
          },
          {
            "alpha_index": 58,
            "clock_position": 58,
            "value": 1
          },
          {
            "alpha_index": 72,
            "clock_position": 72,
            "value": 12
          }
        ],
        "native_369_qualifies": false
      },
      {
        "formation_id": "6-nodes-offset-3",
        "alpha_indices": [
          3,
          17,
          31,
          45,
          59,
          73
        ],
        "clock_positions": [
          3,
          17,
          31,
          45,
          59,
          73
        ],
        "native_values": [
          2,
          11,
          15,
          24,
          15,
          11
        ],
        "rotation_degrees": 12.857142857142858,
        "missing_required_values": [
          3,
          6,
          9
        ],
        "failing_nodes": [
          {
            "alpha_index": 3,
            "clock_position": 3,
            "value": 2
          },
          {
            "alpha_index": 17,
            "clock_position": 17,
            "value": 11
          },
          {
            "alpha_index": 31,
            "clock_position": 31,
            "value": 15
          },
          {
            "alpha_index": 45,
            "clock_position": 45,
            "value": 24
          },
          {
            "alpha_index": 59,
            "clock_position": 59,
            "value": 15
          },
          {
            "alpha_index": 73,
            "clock_position": 73,
            "value": 11
          }
        ],
        "native_369_qualifies": false
      },
      {
        "formation_id": "6-nodes-offset-4",
        "alpha_indices": [
          4,
          18,
          32,
          46,
          60,
          74
        ],
        "clock_positions": [
          4,
          18,
          32,
          46,
          60,
          74
        ],
        "native_values": [
          3,
          10,
          3,
          23,
          16,
          23
        ],
        "rotation_degrees": 17.142857142857142,
        "missing_required_values": [
          6,
          9
        ],
        "failing_nodes": [
          {
            "alpha_index": 18,
            "clock_position": 18,
            "value": 10
          },
          {
            "alpha_index": 46,
            "clock_position": 46,
            "value": 23
          },
          {
            "alpha_index": 60,
            "clock_position": 60,
            "value": 16
          },
          {
            "alpha_index": 74,
            "clock_position": 74,
            "value": 23
          }
        ],
        "native_369_qualifies": false
      },
      {
        "formation_id": "6-nodes-offset-5",
        "alpha_indices": [
          5,
          19,
          33,
          47,
          61,
          75
        ],
        "clock_positions": [
          5,
          19,
          33,
          47,
          61,
          75
        ],
        "native_values": [
          5,
          21,
          18,
          21,
          5,
          8
        ],
        "rotation_degrees": 21.428571428571427,
        "missing_required_values": [
          3,
          6,
          9
        ],
        "failing_nodes": [
          {
            "alpha_index": 5,
            "clock_position": 5,
            "value": 5
          },
          {
            "alpha_index": 19,
            "clock_position": 19,
            "value": 21
          },
          {
            "alpha_index": 33,
            "clock_position": 33,
            "value": 18
          },
          {
            "alpha_index": 47,
            "clock_position": 47,
            "value": 21
          },
          {
            "alpha_index": 61,
            "clock_position": 61,
            "value": 5
          },
          {
            "alpha_index": 75,
            "clock_position": 75,
            "value": 8
          }
        ],
        "native_369_qualifies": false
      },
      {
        "formation_id": "6-nodes-offset-6",
        "alpha_indices": [
          6,
          20,
          34,
          48,
          62,
          76
        ],
        "clock_positions": [
          6,
          20,
          34,
          48,
          62,
          76
        ],
        "native_values": [
          8,
          5,
          21,
          18,
          21,
          5
        ],
        "rotation_degrees": 25.714285714285715,
        "missing_required_values": [
          3,
          6,
          9
        ],
        "failing_nodes": [
          {
            "alpha_index": 6,
            "clock_position": 6,
            "value": 8
          },
          {
            "alpha_index": 20,
            "clock_position": 20,
            "value": 5
          },
          {
            "alpha_index": 34,
            "clock_position": 34,
            "value": 21
          },
          {
            "alpha_index": 48,
            "clock_position": 48,
            "value": 18
          },
          {
            "alpha_index": 62,
            "clock_position": 62,
            "value": 21
          },
          {
            "alpha_index": 76,
            "clock_position": 76,
            "value": 5
          }
        ],
        "native_369_qualifies": false
      },
      {
        "formation_id": "6-nodes-offset-7",
        "alpha_indices": [
          7,
          21,
          35,
          49,
          63,
          77
        ],
        "clock_positions": [
          7,
          21,
          35,
          49,
          63,
          77
        ],
        "native_values": [
          13,
          26,
          13,
          13,
          26,
          13
        ],
        "rotation_degrees": 30.0,
        "missing_required_values": [
          3,
          6,
          9
        ],
        "failing_nodes": [
          {
            "alpha_index": 7,
            "clock_position": 7,
            "value": 13
          },
          {
            "alpha_index": 21,
            "clock_position": 21,
            "value": 26
          },
          {
            "alpha_index": 35,
            "clock_position": 35,
            "value": 13
          },
          {
            "alpha_index": 49,
            "clock_position": 49,
            "value": 13
          },
          {
            "alpha_index": 63,
            "clock_position": 63,
            "value": 26
          },
          {
            "alpha_index": 77,
            "clock_position": 77,
            "value": 13
          }
        ],
        "native_369_qualifies": false
      },
      {
        "formation_id": "6-nodes-offset-8",
        "alpha_indices": [
          8,
          22,
          36,
          50,
          64,
          78
        ],
        "clock_positions": [
          8,
          22,
          36,
          50,
          64,
          78
        ],
        "native_values": [
          21,
          5,
          8,
          5,
          21,
          18
        ],
        "rotation_degrees": 34.285714285714285,
        "missing_required_values": [
          3,
          6,
          9
        ],
        "failing_nodes": [
          {
            "alpha_index": 8,
            "clock_position": 8,
            "value": 21
          },
          {
            "alpha_index": 22,
            "clock_position": 22,
            "value": 5
          },
          {
            "alpha_index": 36,
            "clock_position": 36,
            "value": 8
          },
          {
            "alpha_index": 50,
            "clock_position": 50,
            "value": 5
          },
          {
            "alpha_index": 64,
            "clock_position": 64,
            "value": 21
          },
          {
            "alpha_index": 78,
            "clock_position": 78,
            "value": 18
          }
        ],
        "native_369_qualifies": false
      },
      {
        "formation_id": "6-nodes-offset-9",
        "alpha_indices": [
          9,
          23,
          37,
          51,
          65,
          79
        ],
        "clock_positions": [
          9,
          23,
          37,
          51,
          65,
          79
        ],
        "native_values": [
          8,
          5,
          21,
          18,
          21,
          5
        ],
        "rotation_degrees": 38.57142857142857,
        "missing_required_values": [
          3,
          6,
          9
        ],
        "failing_nodes": [
          {
            "alpha_index": 9,
            "clock_position": 9,
            "value": 8
          },
          {
            "alpha_index": 23,
            "clock_position": 23,
            "value": 5
          },
          {
            "alpha_index": 37,
            "clock_position": 37,
            "value": 21
          },
          {
            "alpha_index": 51,
            "clock_position": 51,
            "value": 18
          },
          {
            "alpha_index": 65,
            "clock_position": 65,
            "value": 21
          },
          {
            "alpha_index": 79,
            "clock_position": 79,
            "value": 5
          }
        ],
        "native_369_qualifies": false
      },
      {
        "formation_id": "6-nodes-offset-10",
        "alpha_indices": [
          10,
          24,
          38,
          52,
          66,
          80
        ],
        "clock_positions": [
          10,
          24,
          38,
          52,
          66,
          80
        ],
        "native_values": [
          3,
          10,
          3,
          23,
          16,
          23
        ],
        "rotation_degrees": 42.857142857142854,
        "missing_required_values": [
          6,
          9
        ],
        "failing_nodes": [
          {
            "alpha_index": 24,
            "clock_position": 24,
            "value": 10
          },
          {
            "alpha_index": 52,
            "clock_position": 52,
            "value": 23
          },
          {
            "alpha_index": 66,
            "clock_position": 66,
            "value": 16
          },
          {
            "alpha_index": 80,
            "clock_position": 80,
            "value": 23
          }
        ],
        "native_369_qualifies": false
      },
      {
        "formation_id": "6-nodes-offset-11",
        "alpha_indices": [
          11,
          25,
          39,
          53,
          67,
          81
        ],
        "clock_positions": [
          11,
          25,
          39,
          53,
          67,
          81
        ],
        "native_values": [
          11,
          15,
          24,
          15,
          11,
          2
        ],
        "rotation_degrees": 47.142857142857146,
        "missing_required_values": [
          3,
          6,
          9
        ],
        "failing_nodes": [
          {
            "alpha_index": 11,
            "clock_position": 11,
            "value": 11
          },
          {
            "alpha_index": 25,
            "clock_position": 25,
            "value": 15
          },
          {
            "alpha_index": 39,
            "clock_position": 39,
            "value": 24
          },
          {
            "alpha_index": 53,
            "clock_position": 53,
            "value": 15
          },
          {
            "alpha_index": 67,
            "clock_position": 67,
            "value": 11
          },
          {
            "alpha_index": 81,
            "clock_position": 81,
            "value": 2
          }
        ],
        "native_369_qualifies": false
      },
      {
        "formation_id": "6-nodes-offset-12",
        "alpha_indices": [
          12,
          26,
          40,
          54,
          68,
          82
        ],
        "clock_positions": [
          12,
          26,
          40,
          54,
          68,
          82
        ],
        "native_values": [
          14,
          25,
          1,
          12,
          1,
          25
        ],
        "rotation_degrees": 51.42857142857143,
        "missing_required_values": [
          3,
          6,
          9
        ],
        "failing_nodes": [
          {
            "alpha_index": 12,
            "clock_position": 12,
            "value": 14
          },
          {
            "alpha_index": 26,
            "clock_position": 26,
            "value": 25
          },
          {
            "alpha_index": 40,
            "clock_position": 40,
            "value": 1
          },
          {
            "alpha_index": 54,
            "clock_position": 54,
            "value": 12
          },
          {
            "alpha_index": 68,
            "clock_position": 68,
            "value": 1
          },
          {
            "alpha_index": 82,
            "clock_position": 82,
            "value": 25
          }
        ],
        "native_369_qualifies": false
      },
      {
        "formation_id": "6-nodes-offset-13",
        "alpha_indices": [
          13,
          27,
          41,
          55,
          69,
          83
        ],
        "clock_positions": [
          13,
          27,
          41,
          55,
          69,
          83
        ],
        "native_values": [
          25,
          14,
          25,
          1,
          12,
          1
        ],
        "rotation_degrees": 55.714285714285715,
        "missing_required_values": [
          3,
          6,
          9
        ],
        "failing_nodes": [
          {
            "alpha_index": 13,
            "clock_position": 13,
            "value": 25
          },
          {
            "alpha_index": 27,
            "clock_position": 27,
            "value": 14
          },
          {
            "alpha_index": 41,
            "clock_position": 41,
            "value": 25
          },
          {
            "alpha_index": 55,
            "clock_position": 55,
            "value": 1
          },
          {
            "alpha_index": 69,
            "clock_position": 69,
            "value": 12
          },
          {
            "alpha_index": 83,
            "clock_position": 83,
            "value": 1
          }
        ],
        "native_369_qualifies": false
      }
    ],
    "absence_reason": "No tested six-node formation contains only 3, 6 and 9 with all three present.",
    "reference_path_available": false
  },
  "views": [
    {
      "id": "alpha",
      "name": "Alpha",
      "svg": "pilot-v1/views/m26/alpha.svg",
      "png": "pilot-v1/views/m26/alpha.png"
    },
    {
      "id": "beta",
      "name": "Beta",
      "svg": "pilot-v1/views/m26/beta.svg",
      "png": "pilot-v1/views/m26/beta.png"
    },
    {
      "id": "gamma",
      "name": "Gamma",
      "svg": "pilot-v1/views/m26/gamma.svg",
      "png": "pilot-v1/views/m26/gamma.png"
    },
    {
      "id": "fibonacci-m26p84",
      "name": "Fibonacci M26P84",
      "svg": "pilot-v1/views/m26/fibonacci-m26p84.svg",
      "png": "pilot-v1/views/m26/fibonacci-m26p84.png"
    }
  ],
  "research_register": {
    "schema_version": "1.0.0",
    "id": "oxtr-phase1-literature",
    "version": "2026-10-18.1",
    "title": "First Principles Genomics — research register",
    "initial_biological_focus": "OXTR",
    "scope": "Shared background literature for the initial OXTR research programme across M1–M1001. Studies of related mechanisms in other systems provide context for investigation.",
    "relationship_to_atlas": "background_context",
    "modulus_relevance_status": "Not yet assessed",
    "hypothesis_reference_record_id": "fibonacci-m9-p24",
    "hypothesis": "The Alpha, Beta, Gamma and Epsilon sequences of M9P24 may offer useful descriptions of biological timing, cellular division and molecular organisation in the context of OXTR. Bioinformatics and in silico modelling provide the planned means of investigation.",
    "interpretation": "Publication findings describe their original experimental systems. Links from Atlas records identify shared reading; a sequence-to-biology correspondence remains open for assessment.",
    "publication_count": 14,
    "hero_publication_ids": [
      "E03",
      "E06",
      "E08",
      "E11"
    ],
    "hero_selection": "Four lead references within the fourteen unique publications, one for each theme.",
    "wss": {
      "status": "Awaiting assessment",
      "score": null,
      "meaning": "Future assessment of relevance to individual Atlas records. Publication trust scores remain separate from WSS."
    },
    "appraisal_method": {
      "id": "provisional-editorial-v1",
      "label": "Provisional editorial trust",
      "range": [
        0,
        10
      ],
      "purpose": "Prioritise closer reading of reported findings. This is an editorial assessment, without calibration or independent peer review.",
      "components": {
        "design": {
          "range": [
            0,
            3
          ],
          "criteria": {
            "3": "Controlled mechanistic test or quantitative prediction tested by manipulation",
            "2": "Structured measurement or comparative design",
            "1": "Descriptive observation",
            "0": "Insufficient information"
          }
        },
        "validation": {
          "range": [
            0,
            3
          ],
          "criteria": {
            "3": "Independent replication appraised",
            "2": "Complementary evidence within the paper",
            "1": "Limited or abstract-level corroboration",
            "0": "Unassessed"
          }
        },
        "review": {
          "range": [
            0,
            3
          ],
          "criteria": {
            "3": "Main text, relevant methods/results and captions reviewed",
            "2": "Selected full-text sections",
            "1": "Abstract, metadata or figure descriptions",
            "0": "Unreviewed"
          }
        },
        "provenance": {
          "range": [
            0,
            1
          ],
          "criteria": {
            "1": "Verified primary peer-reviewed publication",
            "0": "Unverified"
          }
        }
      },
      "independent_replication_appraisal": "Not yet completed for this collection",
      "citation_policy": "Crossref cited-by counts are a dated contextual snapshot and contribute no points. Coverage depends on deposited references. Unavailable counts remain null.",
      "journal_policy": "Journal reputation and impact factor contribute no points.",
      "jadad_policy": "Jadad concerns reporting of randomised clinical trials. It is not applicable to these fourteen study designs; no numeric Jadad score is assigned."
    },
    "publications": [
      {
        "id": "E01",
        "title": "Reconstitution of circadian oscillation of cyanobacterial KaiC phosphorylation in vitro",
        "author": "Nakajima et al.",
        "year": "2005",
        "journal": "Science",
        "doi": "10.1126/science.1108451",
        "theme": "Alpha · circadian timing",
        "hero": null,
        "methods": "Purified cyanobacterial clock proteins; biochemical reconstitution",
        "finding": "KaiA, KaiB, KaiC and ATP supported self-sustaining phosphorylation oscillation. Temperature compensation and agreement between mutant periods in vitro and in vivo strengthened the biological interpretation.",
        "relevance": "direct evidence for a molecular timing mechanism; a broad Alpha reference independent of OXTR.",
        "limit": "biochemical reconstruction is more informative about mechanism than visual analogy. It is a cyanobacterial system, not a mammalian receptor study, and does not support a universal exact 24-hour period or M9 correspondence.",
        "access": "primary abstract and publication metadata. Replication and detailed methods require full-text appraisal before a formal quality rating.",
        "scoreParts": {
          "design": 3,
          "validation": 1,
          "review": 1,
          "provenance": 1
        },
        "score": 6,
        "scoreReason": "Molecular reconstitution tests a timing mechanism; appraisal currently covers the abstract and metadata.",
        "jadad": "Not applicable",
        "citationCount": 1020,
        "citationSource": "Crossref cited-by count",
        "citationChecked": "17 October 2026",
        "crossrefTitle": "Reconstitution of Circadian Oscillation of Cyanobacterial KaiC Phosphorylation in Vitro"
      },
      {
        "id": "E02",
        "title": "Phase locking and multiple oscillating attractors for the coupled mammalian clock and cell cycle",
        "author": "Feillet et al.",
        "year": "2014",
        "journal": "PNAS",
        "doi": "10.1073/pnas.1320474111",
        "theme": "Alpha / Beta · cycle coupling",
        "hero": null,
        "methods": "Mouse fibroblasts; live-cell imaging and modelling",
        "finding": "live imaging of proliferating mouse fibroblasts, combined with modelling, identified phase locking and additional clock states under different conditions.",
        "relevance": "direct support for studying relationships between biological cycles; connects the Alpha and Beta research themes without making them equivalent.",
        "limit": "single-cell observations support the dynamical interpretation. Culture conditions and synchronisation matter, and related cells are not automatically independent replicates. No modular doubling or Atlas geometry was tested.",
        "access": "primary abstract and selected results, including period distributions. A phase-space representation is not evidence of a physical toroidal cell structure.",
        "scoreParts": {
          "design": 2,
          "validation": 2,
          "review": 2,
          "provenance": 1
        },
        "score": 7,
        "scoreReason": "Live-cell observations and modelling provide complementary evidence for phase locking; selected results appraised.",
        "jadad": "Not applicable",
        "citationCount": 216,
        "citationSource": "Crossref cited-by count",
        "citationChecked": "17 October 2026",
        "crossrefTitle": "Phase locking and multiple oscillating attractors for the coupled mammalian clock and cell cycle"
      },
      {
        "id": "E03",
        "title": "Interaction between time-of-day and oxytocin efficacy in mice and humans with and without gestational diabetes",
        "author": "Duong et al.",
        "year": "2026 (online 2025)",
        "journal": "Molecular Metabolism",
        "doi": "10.1016/j.molmet.2025.102269",
        "theme": "Alpha · circadian regulation",
        "hero": "alpha",
        "methods": "Mouse genetics; human cells; retrospective human records",
        "finding": "mouse experiments support BMAL1 regulation of Oxtr and time-dependent oxytocin responsiveness. Human myometrial-cell experiments and records from 2,367 patients add distinct translational observations.",
        "relevance": "a direct OXTR connection to the Alpha theme.",
        "limit": "genetic perturbation and promoter experiments strengthen mechanism. Animal, cell and retrospective components need separate appraisal; the human observations can be confounded. Uterine findings do not establish effects in brain tissue.",
        "access": "final publication abstract and selected full-text results. The 2024 preprint is superseded for this register and is not counted as another study.",
        "scoreParts": {
          "design": 3,
          "validation": 2,
          "review": 2,
          "provenance": 1
        },
        "score": 8,
        "scoreReason": "Genetic and promoter experiments support mechanism; the human observational component has separate confounding limits.",
        "jadad": "Not applicable",
        "citationCount": 3,
        "citationSource": "Crossref cited-by count",
        "citationChecked": "17 October 2026",
        "crossrefTitle": "Interaction between time-of-day and oxytocin efficacy in mice and humans with and without gestational diabetes"
      },
      {
        "id": "E04",
        "title": "Cryptochromes mediate rhythmic repression of the glucocorticoid receptor",
        "author": "Lamia et al.",
        "year": "2011",
        "journal": "Nature",
        "doi": "10.1038/nature10700",
        "theme": "Alpha · receptor regulation",
        "hero": null,
        "methods": "Mouse cells and mice; genetic and biochemical experiments",
        "finding": "Cry1/Cry2 interacted with the glucocorticoid receptor and altered glucocorticoid-responsive transcription in mouse cells, with related metabolic phenotypes in mice.",
        "relevance": "direct circadian–receptor evidence provides a natural NR3C1 example beyond oxytocin.",
        "limit": "molecular interactions, genetic perturbations and physiological observations complement one another. Multiple pathways change in knockout animals; the results do not isolate a universal geometric explanation or establish human clinical effects.",
        "access": "primary abstract and selected experimental/results text. NR3C1 identifies the receptor target; the principal genetic perturbations here involved cryptochromes.",
        "scoreParts": {
          "design": 3,
          "validation": 2,
          "review": 2,
          "provenance": 1
        },
        "score": 8,
        "scoreReason": "Genetic perturbations, molecular interactions and physiological measurements support the reported mechanism.",
        "jadad": "Not applicable",
        "citationCount": 550,
        "citationSource": "Crossref cited-by count",
        "citationChecked": "17 October 2026",
        "crossrefTitle": "Cryptochromes mediate rhythmic repression of the glucocorticoid receptor"
      },
      {
        "id": "E05",
        "title": "Experimental and theoretical study of mitotic spindle orientation",
        "author": "Théry et al.",
        "year": "2007",
        "journal": "Nature",
        "doi": "10.1038/nature05786",
        "theme": "Beta · mitotic geometry",
        "hero": null,
        "methods": "Human HeLa cells; controlled adhesion geometry and modelling",
        "finding": "controlled adhesion patterns in human HeLa cells, combined with a physical model, accounted for spindle-orientation distributions across different microenvironment geometries.",
        "relevance": "direct evidence that geometry is a meaningful biological variable; supports interest in the Beta/division theme and related shape families.",
        "limit": "manipulation of the cell environment offers more than an observational resemblance. The model uses specific force assumptions and a transformed cell line. Division orientation is distinct from division rate, population doubling or a Fibonacci sequence.",
        "access": "publisher abstract and figure descriptions. Full methods and sample accounting remain to be appraised.",
        "scoreParts": {
          "design": 3,
          "validation": 1,
          "review": 1,
          "provenance": 1
        },
        "score": 6,
        "scoreReason": "Controlled adhesion geometry tests a quantitative model; full methods and sample accounting remain open for appraisal.",
        "jadad": "Not applicable",
        "citationCount": 424,
        "citationSource": "Crossref cited-by count",
        "citationChecked": "17 October 2026",
        "crossrefTitle": "Experimental and theoretical study of mitotic spindle orientation"
      },
      {
        "id": "E06",
        "title": "Oxytocin promotes epicardial cell activation and heart regeneration after cardiac injury",
        "author": "Wasserman et al.",
        "year": "2022",
        "journal": "Frontiers in Cell and Developmental Biology",
        "doi": "10.3389/fcell.2022.985298",
        "theme": "Beta · proliferation",
        "hero": "beta",
        "methods": "Human stem-cell-derived epicardial cells and zebrafish",
        "finding": "OXT promoted activation and proliferation-related outcomes in human stem-cell-derived epicardial cells; OXTR knockdown removed the response. Zebrafish experiments supplied a separate regeneration context.",
        "relevance": "a direct OXTR-related example for the Beta research theme.",
        "limit": "receptor knockdown strengthens causal interpretation within the model. Check independent cell lines and differentiation batches; images are subsamples, not donors. Cultured epicardial cells and fish regeneration do not establish regeneration of adult human hearts.",
        "access": "primary full-text methods/results and relevant figure captions, including OXTR knockdown and proliferation measurements.",
        "scoreParts": {
          "design": 3,
          "validation": 2,
          "review": 3,
          "provenance": 1
        },
        "score": 9,
        "scoreReason": "Receptor knockdown provides a mechanistic test alongside cell and zebrafish experiments; methods and results appraised.",
        "jadad": "Not applicable",
        "citationCount": 25,
        "citationSource": "Crossref cited-by count",
        "citationChecked": "17 October 2026",
        "crossrefTitle": "Oxytocin promotes epicardial cell activation and heart regeneration after cardiac injury"
      },
      {
        "id": "E07",
        "title": "Oxytocin stimulates hippocampal neurogenesis via oxytocin receptor expressed in CA3 pyramidal neurons",
        "author": "Lin et al.",
        "year": "2017",
        "journal": "Nature Communications",
        "doi": "10.1038/s41467-017-00675-5",
        "theme": "Beta · survival / proliferation",
        "hero": null,
        "methods": "Mice; conditional Oxtr deletion",
        "finding": "conditional Oxtr deletion in mouse hippocampal excitatory neurons affected survival and maturation of newly generated neurons, while measured progenitor proliferation and early differentiation were unchanged.",
        "relevance": "direct evidence that the biological outcome must be named precisely. It bounds generalisations from E06 without contradicting a different tissue-specific finding.",
        "limit": "conditional genetics help localise the process. Appraise cell-type specificity, blinding and precision; an unchanged measured outcome does not prove absence of every possible effect.",
        "access": "primary results/discussion. Retained deliberately to avoid presenting all OXTR growth-related findings as evidence of duplication.",
        "scoreParts": {
          "design": 3,
          "validation": 2,
          "review": 2,
          "provenance": 1
        },
        "score": 8,
        "scoreReason": "Conditional genetics and cell-outcome measurements distinguish survival from proliferation in this mouse system.",
        "jadad": "Not applicable",
        "citationCount": 149,
        "citationSource": "Crossref cited-by count",
        "citationChecked": "17 October 2026",
        "crossrefTitle": "Oxytocin stimulates hippocampal neurogenesis via oxytocin receptor expressed in CA3 pyramidal neurons"
      },
      {
        "id": "E08",
        "title": "Structural basis for the activation and ligand recognition of the human oxytocin receptor",
        "author": "Waltenspühl et al.",
        "year": "2022",
        "journal": "Nature Communications",
        "doi": "10.1038/s41467-022-31325-0",
        "theme": "Gamma · molecular recognition",
        "hero": "gamma",
        "methods": "Engineered human receptor complex; cryo-EM and functional assays",
        "finding": "cryo-EM at 3.2 Å and functional experiments characterised OXT recognition and receptor activation. All nine peptide residues participate in binding.",
        "relevance": "a direct structural foundation for Gamma's molecular-recognition theme; useful context, but not confirmation, for the OXT-specific Epsilon proposal.",
        "limit": "structure–function comparisons strengthen interpretation. Receptor stabilisation and an engineered G protein affect generalisation. The six-residue disulfide loop is not itself an aromatic hexagon; the reported contacts do not establish an Epsilon trajectory.",
        "access": "full primary article main text, receptor-engineering description and functional-assay captions. No new molecular analysis performed.",
        "scoreParts": {
          "design": 3,
          "validation": 2,
          "review": 3,
          "provenance": 1
        },
        "score": 9,
        "scoreReason": "Structural observations and functional residue tests provide complementary evidence; main text and assay descriptions appraised.",
        "jadad": "Not applicable",
        "citationCount": 46,
        "citationSource": "Crossref cited-by count",
        "citationChecked": "17 October 2026",
        "crossrefTitle": "Structural basis for the activation and ligand recognition of the human oxytocin receptor"
      },
      {
        "id": "E09",
        "title": "X-ray structures and mechanism of the human serotonin transporter",
        "author": "Coleman, Green & Gouaux",
        "year": "2016",
        "journal": "Nature",
        "doi": "10.1038/nature17629",
        "theme": "Gamma · molecular recognition",
        "hero": null,
        "methods": "Engineered human serotonin transporter; crystallography and assays",
        "finding": "structures of human SERT bound to antidepressants identified central and allosteric binding sites and explained aspects of inhibitor recognition.",
        "relevance": "direct structural evidence broadens Gamma beyond OXTR to SLC6A4.",
        "limit": "structural and binding/transport measurements are complementary. Thermostabilised constructs were used; the ts3 construct had no detectable transport activity. Results describe particular inhibitor-bound conformations rather than every physiological state.",
        "access": "publisher abstract and selected primary methods/results. These data concern the protein product and ligand recognition, not regulation of SLC6A4 gene expression.",
        "scoreParts": {
          "design": 2,
          "validation": 2,
          "review": 2,
          "provenance": 1
        },
        "score": 7,
        "scoreReason": "Structures and functional measurements complement one another; engineered constructs constrain interpretation.",
        "jadad": "Not applicable",
        "citationCount": 674,
        "citationSource": "Crossref cited-by count",
        "citationChecked": "17 October 2026",
        "crossrefTitle": "X-ray structures and mechanism of the human serotonin transporter"
      },
      {
        "id": "E10",
        "title": "Structure of a D2 dopamine receptor–G-protein complex in a lipid membrane",
        "author": "Yin et al.",
        "year": "2020",
        "journal": "Nature",
        "doi": "10.1038/s41586-020-2379-5",
        "theme": "Gamma · molecular recognition",
        "hero": null,
        "methods": "Engineered human DRD2 complex in a lipid membrane; cryo-EM",
        "finding": "cryo-EM characterised an agonist-bound DRD2–Gi complex within a reconstituted phospholipid membrane, identifying receptor, G-protein and lipid interactions.",
        "relevance": "direct evidence for the structural context of molecular recognition; another Gamma hero example.",
        "limit": "including a lipid environment improves contextual relevance relative to isolated detergent structures. The receptor complex is engineered and ligand-bound; it does not represent every receptor state or establish clinical outcomes.",
        "access": "primary abstract and selected full-text structural results, with deposited structural metadata checked. No geometric correspondence to the Atlas was analysed.",
        "scoreParts": {
          "design": 2,
          "validation": 1,
          "review": 2,
          "provenance": 1
        },
        "score": 6,
        "scoreReason": "A membrane-reconstituted structural observation provides a defined molecular state; independent functional validation needs fuller appraisal.",
        "jadad": "Not applicable",
        "citationCount": 199,
        "citationSource": "Crossref cited-by count",
        "citationChecked": "17 October 2026",
        "crossrefTitle": "Structure of a D2 dopamine receptor–G-protein complex in a lipid membrane"
      },
      {
        "id": "E11",
        "title": "Assembly of a π–π stack of ligands in the binding site of an acetylcholine-binding protein",
        "author": "Stornaiuolo et al.",
        "year": "2013",
        "journal": "Nature Communications",
        "doi": "10.1038/ncomms2900",
        "theme": "Epsilon · aromatic stacking",
        "hero": "epsilon",
        "methods": "Molluscan binding protein; crystallography and solution assays",
        "finding": "an Aplysia acetylcholine-binding protein accommodated a stack of three aromatic ligands. Solution binding measurements supported the structural observation.",
        "relevance": "direct experimental support for π-stacking as a molecular-recognition phenomenon; an Epsilon reference that does not depend on OXTR.",
        "limit": "crystallography, mutagenesis and solution assays help address crystallographic-artifact concerns. This soluble molluscan protein is a model for receptor ligand-binding domains, not a complete human receptor. Neither the three-ligand count nor the stacked geometry establishes the Atlas traversal.",
        "access": "full primary main text, including solution validation and scope of the receptor model.",
        "scoreParts": {
          "design": 3,
          "validation": 2,
          "review": 3,
          "provenance": 1
        },
        "score": 9,
        "scoreReason": "Structures, mutagenesis and solution assays support the binding interpretation; full main text appraised.",
        "jadad": "Not applicable",
        "citationCount": 66,
        "citationSource": "Crossref cited-by count",
        "citationChecked": "17 October 2026",
        "crossrefTitle": "Assembly of a π–π stack of ligands in the binding site of an acetylcholine-binding protein"
      },
      {
        "id": "E12",
        "title": "Targeted disruption of pi–pi stacking in Malaysian banana lectin reduces mitogenicity while preserving antiviral activity",
        "author": "Covés-Datson et al.",
        "year": "2021",
        "journal": "Scientific Reports",
        "doi": "10.1038/s41598-020-80577-7",
        "theme": "Epsilon · aromatic stacking",
        "hero": null,
        "methods": "Engineered plant lectin; structures and cellular assays",
        "finding": "an F84T mutation disrupted aromatic stacking in Malay BanLec, reduced mitogenic activity and retained antiviral activity in the studied assays.",
        "relevance": "a direct structure–function example for Epsilon; also connects molecular interactions with proliferation as a biological outcome.",
        "limit": "structural checks and functional comparisons strengthen the rationale. Mutation can change several properties simultaneously, so this is not a pure isolation of one energetic contribution. Plant-lectin effects do not establish OXTR behaviour or clinical benefit.",
        "access": "primary abstract and main structural/results sections. Published in 2021 despite “2020” appearing within the DOI.",
        "scoreParts": {
          "design": 3,
          "validation": 2,
          "review": 2,
          "provenance": 1
        },
        "score": 8,
        "scoreReason": "Targeted mutation and structural / cellular comparisons connect aromatic organisation with biological outcomes.",
        "jadad": "Not applicable",
        "citationCount": 24,
        "citationSource": "Crossref cited-by count",
        "citationChecked": "17 October 2026",
        "crossrefTitle": "Targeted disruption of pi–pi stacking in Malaysian banana lectin reduces mitogenicity while preserving antiviral activity"
      },
      {
        "id": "E13",
        "title": "Reconstruction of molecular orbital densities from photoemission data",
        "author": "Puschnig et al.",
        "year": "2009",
        "journal": "Science",
        "doi": "10.1126/science.1176105",
        "theme": "Valence electron orbitals",
        "hero": null,
        "methods": "Organic molecular films; photoemission and calculations",
        "finding": "photoemission measurements and Fourier reconstruction yielded orbital-density distributions for pentacene films and p-sexiphenyl monolayers, compared with density-functional calculations.",
        "relevance": "direct physics evidence for studying spatial electronic structure; contextual to the valence-electron theme and more distant from biological significance.",
        "limit": "comparison of measurement and theory is valuable. Reconstruction is model-dependent and the samples are non-living materials. Orbital densities are not classical electron orbits or evidence of unicursal electron motion.",
        "access": "primary abstract and metadata. No biological role, creation-of-life claim or numeric WSS assignment follows from this paper.",
        "scoreParts": {
          "design": 2,
          "validation": 1,
          "review": 1,
          "provenance": 1
        },
        "score": 5,
        "scoreReason": "Measurement and reconstruction are compared with calculations; current appraisal is abstract-level.",
        "jadad": "Not applicable",
        "citationCount": 334,
        "citationSource": "Crossref cited-by count",
        "citationChecked": "17 October 2026",
        "crossrefTitle": "Reconstruction of Molecular Orbital Densities from Photoemission Data"
      },
      {
        "id": "E14",
        "title": "CD38 is critical for social behaviour by regulating oxytocin secretion",
        "author": "Jin et al.",
        "year": "2007",
        "journal": "Nature",
        "doi": "10.1038/nature05526",
        "theme": "Supporting OXT context",
        "hero": null,
        "methods": "Mice; Cd38 knockout and rescue experiments",
        "finding": "Cd38-knockout mice showed altered oxytocin secretion and behavioural phenotypes. Oxytocin replacement or hypothalamic delivery of human CD38 rescued selected outcomes.",
        "relevance": "direct support for CD38 involvement in the oxytocin system. Retained as supporting hero context without forcing it into a particular sequence family.",
        "limit": "knockout plus rescue strengthens causal interpretation in mice. Global deletion, behavioural measurement and human translation require careful appraisal. This concerns secretion, not proof of OXTR-expression regulation or geometric correspondence.",
        "access": "primary abstract and figure descriptions. Full methods are needed before formal risk-of-bias grading.",
        "scoreParts": {
          "design": 3,
          "validation": 1,
          "review": 1,
          "provenance": 1
        },
        "score": 6,
        "scoreReason": "Knockout and rescue experiments test the secretion mechanism; full methods remain open for appraisal.",
        "jadad": "Not applicable",
        "citationCount": 585,
        "citationSource": "Crossref cited-by count",
        "citationChecked": "17 October 2026",
        "crossrefTitle": "CD38 is critical for social behaviour by regulating oxytocin secretion"
      }
    ]
  },
  "family": {
    "m9": false,
    "label": null
  }
}
