{
  "scope": "Original coordinate diagrams, not numerical proofs or ambient embeddings",
  "figure1": {
    "punctures": [
      -1,
      1
    ],
    "strip_delta": 0.65,
    "positive_circle_radius": 0.45,
    "rays": "x=a, y<=0 excluded from A; y<0 lies in B",
    "matrix": [
      [
        1,
        1,
        1,
        1
      ],
      [
        -1,
        -1,
        0,
        0
      ],
      [
        0,
        0,
        -1,
        -1
      ]
    ],
    "kernel_basis": [
      [
        1,
        -1,
        0,
        0
      ],
      [
        0,
        0,
        1,
        -1
      ]
    ],
    "proof": [
      "PR11",
      "PR12",
      "PR15"
    ]
  },
  "figure2": {
    "dimension_d": 2,
    "input_order": [
      "phi",
      "theta"
    ],
    "output_order": [
      "alpha1",
      "alpha2"
    ],
    "matrix": [
      [
        -1,
        1
      ],
      [
        -1,
        0
      ]
    ],
    "determinant": 1,
    "angle_identification": "modulo 2pi",
    "coordinate_formula": [
      "h=epsilon exp(i phi)",
      "w=exp(i theta)",
      "t1=L exp(i(theta-phi))",
      "t2=L exp(-i phi)"
    ],
    "proof": [
      "PR21",
      "PR23",
      "PR24"
    ]
  },
  "figure3": {
    "ordered_pairs": [
      [
        -1,
        0
      ],
      [
        -1,
        2
      ],
      [
        1,
        0
      ],
      [
        1,
        2
      ]
    ],
    "normalized_period_matrix": [
      [
        1,
        0,
        0,
        0
      ],
      [
        0,
        1,
        0,
        0
      ],
      [
        0,
        0,
        1,
        0
      ],
      [
        0,
        0,
        0,
        1
      ]
    ],
    "normalization": "(2pi i)^(-2)",
    "tube_vector": [
      1,
      1,
      1,
      1
    ],
    "sample_cycle_and_normalized_periods": [
      2,
      -1,
      0,
      3
    ],
    "proof": [
      "PR6",
      "PR17",
      "PR25",
      "PR26"
    ]
  },
  "source_credit": "Original diagrams from the written PR036 calculation; human context ABG I p187, ABG II p163 and p176; finite-chain context Hatcher Algebraic Topology Chapter 2",
  "blender_assessment": "Coordinate cover, phase map and exact matrix are explained directly by 2D vector diagrams; a 3D ambient scene would not represent these complex complements"
}