{
  "title": "Center flows: periods and eigenfrequencies",
  "license": "CC0-1.0 to the extent of rights held; font terms separate",
  "circle": {
    "a": "log(4)",
    "flow": "theta_s f(r)=f(r+s)",
    "character": "exp(2*pi*i*r/a)",
    "displayed_time": "a/4",
    "displayed_phase": "i",
    "K": "a Z",
    "E": "(2*pi/a) Z",
    "phase_circle_notice": "The rendered unit circle is the value of the character, not physical circumference a."
  },
  "scalar": {
    "algebra": "C",
    "flow": "identity",
    "K": "R",
    "E": "{0}"
  },
  "torus": {
    "flow": "(x,y)->(x+s,y+sqrt(2)*s) mod Z^2",
    "interval": [
      0,
      2
    ],
    "exact_segment_breaks": [
      "0",
      "1/sqrt(2)",
      "1",
      "sqrt(2)",
      "2"
    ],
    "rendered_segments": [
      {
        "s_start": 0.0,
        "s_end": 0.7071067811865475,
        "x_floor": 0,
        "y_floor": 0
      },
      {
        "s_start": 0.7071067811865475,
        "s_end": 1.0,
        "x_floor": 0,
        "y_floor": 1
      },
      {
        "s_start": 1.0,
        "s_end": 1.414213562373095,
        "x_floor": 1,
        "y_floor": 1
      },
      {
        "s_start": 1.414213562373095,
        "s_end": 2.0,
        "x_floor": 1,
        "y_floor": 2
      }
    ],
    "samples": [
      {
        "s": 0,
        "exact_x": "0",
        "exact_y": "0*sqrt(2)-floor(0*sqrt(2))"
      },
      {
        "s": 1,
        "exact_x": "0",
        "exact_y": "1*sqrt(2)-floor(1*sqrt(2))"
      },
      {
        "s": 2,
        "exact_x": "0",
        "exact_y": "2*sqrt(2)-floor(2*sqrt(2))"
      }
    ],
    "K": "{0}",
    "E": "2*pi*(Z+sqrt(2)*Z)",
    "fixed_algebra": "C1",
    "notice": "Finite orbit segment only; complete basis and ergodicity are proved in MIV6.c-m."
  },
  "frequencies": {
    "axis": "p/(2*pi)",
    "range_of_indices": [
      -3,
      3
    ],
    "sample": [
      {
        "m": -3,
        "n": -3,
        "exact_value": "-3+(-3)*sqrt(2)"
      },
      {
        "m": -2,
        "n": -3,
        "exact_value": "-2+(-3)*sqrt(2)"
      },
      {
        "m": -3,
        "n": -2,
        "exact_value": "-3+(-2)*sqrt(2)"
      },
      {
        "m": -1,
        "n": -3,
        "exact_value": "-1+(-3)*sqrt(2)"
      },
      {
        "m": -2,
        "n": -2,
        "exact_value": "-2+(-2)*sqrt(2)"
      },
      {
        "m": -3,
        "n": -1,
        "exact_value": "-3+(-1)*sqrt(2)"
      },
      {
        "m": 0,
        "n": -3,
        "exact_value": "0+(-3)*sqrt(2)"
      },
      {
        "m": -1,
        "n": -2,
        "exact_value": "-1+(-2)*sqrt(2)"
      },
      {
        "m": -2,
        "n": -1,
        "exact_value": "-2+(-1)*sqrt(2)"
      },
      {
        "m": 1,
        "n": -3,
        "exact_value": "1+(-3)*sqrt(2)"
      },
      {
        "m": -3,
        "n": 0,
        "exact_value": "-3+(0)*sqrt(2)"
      },
      {
        "m": 0,
        "n": -2,
        "exact_value": "0+(-2)*sqrt(2)"
      },
      {
        "m": -1,
        "n": -1,
        "exact_value": "-1+(-1)*sqrt(2)"
      },
      {
        "m": 2,
        "n": -3,
        "exact_value": "2+(-3)*sqrt(2)"
      },
      {
        "m": -2,
        "n": 0,
        "exact_value": "-2+(0)*sqrt(2)"
      },
      {
        "m": 1,
        "n": -2,
        "exact_value": "1+(-2)*sqrt(2)"
      },
      {
        "m": -3,
        "n": 1,
        "exact_value": "-3+(1)*sqrt(2)"
      },
      {
        "m": 0,
        "n": -1,
        "exact_value": "0+(-1)*sqrt(2)"
      },
      {
        "m": 3,
        "n": -3,
        "exact_value": "3+(-3)*sqrt(2)"
      },
      {
        "m": -1,
        "n": 0,
        "exact_value": "-1+(0)*sqrt(2)"
      },
      {
        "m": 2,
        "n": -2,
        "exact_value": "2+(-2)*sqrt(2)"
      },
      {
        "m": -2,
        "n": 1,
        "exact_value": "-2+(1)*sqrt(2)"
      },
      {
        "m": 1,
        "n": -1,
        "exact_value": "1+(-1)*sqrt(2)"
      },
      {
        "m": -3,
        "n": 2,
        "exact_value": "-3+(2)*sqrt(2)"
      },
      {
        "m": 0,
        "n": 0,
        "exact_value": "0+(0)*sqrt(2)"
      },
      {
        "m": 3,
        "n": -2,
        "exact_value": "3+(-2)*sqrt(2)"
      },
      {
        "m": -1,
        "n": 1,
        "exact_value": "-1+(1)*sqrt(2)"
      },
      {
        "m": 2,
        "n": -1,
        "exact_value": "2+(-1)*sqrt(2)"
      },
      {
        "m": -2,
        "n": 2,
        "exact_value": "-2+(2)*sqrt(2)"
      },
      {
        "m": 1,
        "n": 0,
        "exact_value": "1+(0)*sqrt(2)"
      },
      {
        "m": -3,
        "n": 3,
        "exact_value": "-3+(3)*sqrt(2)"
      },
      {
        "m": 0,
        "n": 1,
        "exact_value": "0+(1)*sqrt(2)"
      },
      {
        "m": 3,
        "n": -1,
        "exact_value": "3+(-1)*sqrt(2)"
      },
      {
        "m": -1,
        "n": 2,
        "exact_value": "-1+(2)*sqrt(2)"
      },
      {
        "m": 2,
        "n": 0,
        "exact_value": "2+(0)*sqrt(2)"
      },
      {
        "m": -2,
        "n": 3,
        "exact_value": "-2+(3)*sqrt(2)"
      },
      {
        "m": 1,
        "n": 1,
        "exact_value": "1+(1)*sqrt(2)"
      },
      {
        "m": 0,
        "n": 2,
        "exact_value": "0+(2)*sqrt(2)"
      },
      {
        "m": 3,
        "n": 0,
        "exact_value": "3+(0)*sqrt(2)"
      },
      {
        "m": -1,
        "n": 3,
        "exact_value": "-1+(3)*sqrt(2)"
      },
      {
        "m": 2,
        "n": 1,
        "exact_value": "2+(1)*sqrt(2)"
      },
      {
        "m": 1,
        "n": 2,
        "exact_value": "1+(2)*sqrt(2)"
      },
      {
        "m": 0,
        "n": 3,
        "exact_value": "0+(3)*sqrt(2)"
      },
      {
        "m": 3,
        "n": 1,
        "exact_value": "3+(1)*sqrt(2)"
      },
      {
        "m": 2,
        "n": 2,
        "exact_value": "2+(2)*sqrt(2)"
      },
      {
        "m": 1,
        "n": 3,
        "exact_value": "1+(3)*sqrt(2)"
      },
      {
        "m": 3,
        "n": 2,
        "exact_value": "3+(2)*sqrt(2)"
      },
      {
        "m": 2,
        "n": 3,
        "exact_value": "2+(3)*sqrt(2)"
      },
      {
        "m": 3,
        "n": 3,
        "exact_value": "3+(3)*sqrt(2)"
      }
    ],
    "omitted_point": "1/2 is not in Z+sqrt(2) Z",
    "notice": "Finite dots do not prove density; MIV6.n does."
  },
  "conditional_type_III": {
    "circle": {
      "lambda": "1/4",
      "S": "{0} union {4^n:n in Z}",
      "T": "(2*pi/log(4))*Z"
    },
    "scalar": {
      "S": "[0,infinity)",
      "T": "{0}"
    },
    "torus": {
      "S": "{0,1}",
      "T": "2*pi*(Z+sqrt(2)*Z)"
    },
    "condition": "Only when the model is the center flow of a system with type III factor crossing; no factor realization asserted."
  },
  "translation": {
    "N": "L-infinity(R)",
    "trace": "integral exp(x)*f(x) dx",
    "flow": "f(x)->f(x+s)",
    "trace_scale": "exp(-s)",
    "regular_coeff": "f(x-r)",
    "regular_group": "xi(r-s,x)",
    "coordinates": {
      "y": "x-r",
      "z": "x",
      "inverse_r": "z-y",
      "inverse_x": "z",
      "absolute_jacobian": 1
    },
    "W": "W xi(y,z)=xi(z-y,z)",
    "transformed_group": "eta(y+s,z)",
    "whole_crossing": "B(L2(R_y)) tensor I_(L2(R_z))",
    "K": "{0}",
    "S": "{1}",
    "T": "R",
    "projection": "W^*(|1_[0,1]><1_[0,1]| tensor I)W",
    "projection_factor_trace": 1
  },
  "proof_locators": [
    "MIV6.d-f",
    "MIV6.h-o",
    "MIV6.p-x",
    "MIV7.a-f"
  ]
}