{
  "schema": "AN02-projective-cover-tube-exact-geometry/v1",
  "license": "CC0-1.0",
  "double_cover": {
    "F": "Z0*Z1",
    "degree": 2,
    "affine_level": "z*w=1",
    "projective_coordinate": "u=Z0/Z1=z^2",
    "base_cycle": "u=exp(2*pi*i*t),0<=t<=1",
    "lifts": [
      "z=exp(pi*i*t)",
      "z=-exp(pi*i*t)"
    ],
    "w": "1/z",
    "lift_endpoints": [
      [
        "+1",
        "-1"
      ],
      [
        "-1",
        "+1"
      ]
    ],
    "transfer_boundary": "([-1]-[+1])+([+1]-[-1])=0",
    "upstairs_normalized_period": 1,
    "pulled_back_base_normalized_period": 2,
    "scope": "Base complex coordinate and upstairs z-projection; not all real coordinates of X.",
    "proof_locator": "Example2;formal Proposition4.1"
  },
  "normal_tube": {
    "map": "T(theta,phi)=(exp(i*phi),(1/2)*exp(i*theta))",
    "domain": [
      [
        "0",
        "2*pi"
      ],
      [
        "0",
        "2*pi"
      ]
    ],
    "orientation": "dtheta wedge dphi",
    "normal_angle": "theta",
    "base_angle": "phi",
    "local_boundary_coefficient": 1,
    "form": "db/b wedge da/a",
    "pullback": "-dtheta wedge dphi",
    "period": "-4*pi^2=(2*pi*i)^2",
    "scope": "Parameter rectangle with identified edges; not an ambient embedding.",
    "proof_locator": "Example4;formal Theorem5.1 and product boundary(29)"
  }
}