{
  "title": "Lifted orbit flow: basepoint trace and invariant coordinate",
  "license": "CC0-1.0 to the extent of rights held; font terms separate",
  "circle": {
    "space": "R/Z",
    "measure": "normalized Lebesgue measure",
    "alpha": "sqrt(2)-1",
    "action": "(n,epsilon).x=x+n*alpha mod 1",
    "stabilizer": "{0} x C2",
    "delta": 1,
    "lift": "(n,epsilon).(x,lambda)=(x+n*alpha,lambda)",
    "basis": "distinct points R^n x, one per integer n",
    "samples": [
      {
        "n": 0,
        "exact_x": "fractional_part(0*(sqrt(2)-1))",
        "integer_part": 0,
        "in_B": true
      },
      {
        "n": 1,
        "exact_x": "fractional_part(1*(sqrt(2)-1))",
        "integer_part": 0,
        "in_B": false
      },
      {
        "n": 2,
        "exact_x": "fractional_part(2*(sqrt(2)-1))",
        "integer_part": 0,
        "in_B": false
      },
      {
        "n": 3,
        "exact_x": "fractional_part(3*(sqrt(2)-1))",
        "integer_part": 1,
        "in_B": true
      },
      {
        "n": 4,
        "exact_x": "fractional_part(4*(sqrt(2)-1))",
        "integer_part": 1,
        "in_B": false
      },
      {
        "n": 5,
        "exact_x": "fractional_part(5*(sqrt(2)-1))",
        "integer_part": 2,
        "in_B": true
      },
      {
        "n": 6,
        "exact_x": "fractional_part(6*(sqrt(2)-1))",
        "integer_part": 2,
        "in_B": false
      },
      {
        "n": 7,
        "exact_x": "fractional_part(7*(sqrt(2)-1))",
        "integer_part": 2,
        "in_B": false
      },
      {
        "n": 8,
        "exact_x": "fractional_part(8*(sqrt(2)-1))",
        "integer_part": 3,
        "in_B": true
      }
    ],
    "B": "[0,1/3)",
    "sampling_notice": "Only n=0,...,8 is displayed. Pigeonhole density, L2 averaging ergodicity, and the full almost-everywhere divergence proof are in LFT7.b-d and LFT7.k-m.",
    "projection": "p=1_(1,2)(lambda) I",
    "basepoint_trace": "1",
    "canonical_core_trace": "1/(2*pi)",
    "full_fiber_trace": "infinity",
    "full_obstruction": "W(T)=infinity for every nonzero positive T in N",
    "center": "L-infinity((0,infinity),d lambda)",
    "center_action": "c(lambda) -> c(exp(s)*lambda)"
  },
  "finite": {
    "space": [
      1,
      2,
      3
    ],
    "masses": [
      1,
      2,
      4
    ],
    "group": "S3",
    "invariant": "w=mu_x*lambda",
    "w_sample": 2,
    "complete_lifted_orbit": [
      [
        1,
        2
      ],
      [
        2,
        1
      ],
      [
        3,
        "1/2"
      ]
    ],
    "unitary": "V eta(y,x,w)=eta(y,x,w/mu_x)",
    "coordinate_measure": "dw in each of the three base columns",
    "algebra": "M3 tensor L-infinity(dw)",
    "density": "diag(1,2,4)/w",
    "core_group": "diag(1,2,4)^it * w^-it",
    "basepoint_trace": "integral Tr(A(w)) dw",
    "canonical_core_trace": "(1/(2*pi))*integral Tr(A(w)) dw",
    "dual_weight": "(1/(2*pi))*integral Tr(diag(1,2,4)*A(w)) dw/w",
    "projection": "P=I3*1_(1,2)(w)",
    "basepoint_projection_trace": 3,
    "canonical_projection_trace": "3/(2*pi)",
    "dual_projection_weight": "7*log(2)/(2*pi)",
    "s": "log(2)",
    "translated_projection_support": [
      "1/2",
      1
    ],
    "translated_canonical_trace": "3/(4*pi)"
  },
  "proof_locators": [
    "LFT7.b-d",
    "LFT7.g-o",
    "LFT7.p-v",
    "LFT8.b",
    "LFT8.e-f"
  ]
}