{
  "license": "CC0-1.0 original code/data/diagram; font terms separate",
  "figure_kind": "Exact translation partition and exact real-exponent graphs",
  "coordinate": "r",
  "action": "theta_s(f)(r)=f(r+s)",
  "trace": "tau(f)=integral f(r) exp(r) dr",
  "scalar_density": "h(r)=exp(-r)",
  "T": 1,
  "m": 2,
  "partition_formula": "q_n=1_[-nT,(1-n)T)",
  "visible_intervals": [
    {
      "q_index": 3,
      "left": -3,
      "right": -2,
      "left_endpoint": "included",
      "right_endpoint": "excluded",
      "in_P2": false
    },
    {
      "q_index": 2,
      "left": -2,
      "right": -1,
      "left_endpoint": "included",
      "right_endpoint": "excluded",
      "in_P2": true
    },
    {
      "q_index": 1,
      "left": -1,
      "right": 0,
      "left_endpoint": "included",
      "right_endpoint": "excluded",
      "in_P2": true
    },
    {
      "q_index": 0,
      "left": 0,
      "right": 1,
      "left_endpoint": "included",
      "right_endpoint": "excluded",
      "in_P2": true
    },
    {
      "q_index": -1,
      "left": 1,
      "right": 2,
      "left_endpoint": "included",
      "right_endpoint": "excluded",
      "in_P2": true
    },
    {
      "q_index": -2,
      "left": 2,
      "right": 3,
      "left_endpoint": "included",
      "right_endpoint": "excluded",
      "in_P2": true
    },
    {
      "q_index": -3,
      "left": 3,
      "right": 4,
      "left_endpoint": "included",
      "right_endpoint": "excluded",
      "in_P2": false
    }
  ],
  "P2": {
    "left": -2,
    "right": 3,
    "right_endpoint": "excluded",
    "norm": 1,
    "positive_integral": 5
  },
  "support_translation": {
    "start_interval": [
      0,
      1
    ],
    "image_interval": [
      -1,
      0
    ],
    "time": 1,
    "direction": "left"
  },
  "continuous_unitary": "v_t(r)=exp(i*t*r)",
  "staircase_unitary": "w_t(r)=exp(i*T*t*floor(r/T))",
  "graphed_real_exponents": {
    "t": 1,
    "continuous": "r",
    "staircase": "floor(r)"
  },
  "half_step_check": {
    "r": "1/4",
    "s": "1/2",
    "t": 1,
    "r_plus_s": "3/4",
    "continuous_exponents": [
      "1/4",
      "3/4"
    ],
    "staircase_exponents": [
      0,
      0
    ],
    "continuous_ratio": "exp(i/2)",
    "staircase_ratio": "1",
    "required_ratio": "exp(i/2)",
    "failure_set": "0<r<1/2 (positive Lebesgue measure)"
  },
  "matrix_model": {
    "K": [
      [
        "1",
        "0"
      ],
      [
        "0",
        "-1"
      ]
    ],
    "B": [
      [
        "2",
        "1"
      ],
      [
        "1",
        "2"
      ]
    ],
    "B_eigenvalues": [
      1,
      3
    ],
    "P_plus": [
      [
        "1/2",
        "1/2"
      ],
      [
        "1/2",
        "1/2"
      ]
    ],
    "P_minus": [
      [
        "1/2",
        "-1/2"
      ],
      [
        "-1/2",
        "1/2"
      ]
    ],
    "A(r)": [
      [
        "2",
        "exp(-2*I*r)"
      ],
      [
        "exp(2*I*r)",
        "2"
      ]
    ],
    "h_covariance_checked": true,
    "A_transport_checked": true,
    "noncentrality_checked": true
  },
  "proof_locators": [
    "OA-FLOW-L19.md#l19-5",
    "L19.5.f",
    "L19.5.g",
    "L19.5.h",
    "OA-FLOW-L19.md#l19-7",
    "L19.7.c",
    "L19.7.d"
  ],
  "checks": {
    "exact_partition_endpoints": true,
    "sharp_bound": true,
    "integer_step": true,
    "half_step_failure": true,
    "matrix_covariance": true,
    "matrix_nonscalar_density": true
  },
  "native_png_pixels": [
    2400,
    2000
  ]
}