{
  "schema": "AN02-L140-original-geometry134/v1",
  "dimension": "Figures use one complex dimension, with its ordinary Euclidean metric.",
  "logarithmic-tail-envelopes": {
    "proof_locators": [
      "UE3",
      "UE5",
      "UE8",
      "UE26",
      "UE27"
    ],
    "original_formula": "v_j(r)=log(r)/j, v_j(0)=-infinity",
    "tail_formula": "f_k(0)=-infinity; f_k(r)=0 for0<r<=1; f_k(r)=log(r)/k forr>1",
    "regularized_formula": "U_k(r)=max(0,log(r))/k forr>0; U_k(0)=0",
    "indices": [
      1,
      2,
      4,
      8
    ],
    "original_sample_radius": [
      0.001,
      3
    ],
    "original_sample_points": 1200,
    "regularized_sample_radius": [
      0,
      3
    ],
    "regularized_sample_points": 1000,
    "singularity_policy": "The original singular value is stated as minus infinity; it is never replaced by a finite plotted value."
  },
  "scaled-translation-averages": {
    "proof_locators": [
      "UE7",
      "UE8",
      "UE23",
      "UE28",
      "UE29"
    ],
    "input": "q(zeta)=abs(Im zeta)",
    "direction_y": 1,
    "K": {
      "real_interval": [
        -1,
        1
      ],
      "imaginary_interval": [
        0,
        1
      ],
      "area": 2
    },
    "support_function": "H(b)=abs(b)",
    "target_L": 1,
    "average_formula": "a_t(s)=2*integral_0^1 abs(b+s/t) db",
    "piecewise_variable": "c=s/t",
    "piecewise_branches": {
      "c>=0": "1+2c",
      "-1<=c<=0": "1+2c+2c^2",
      "c<=-1": "-1-2c"
    },
    "parameters_t": [
      1,
      2,
      4,
      8
    ],
    "profile_fixed_s": -1,
    "profile_b_interval": [
      0,
      1
    ],
    "center_sample_s_interval": [
      -2,
      2
    ],
    "center_sample_points": 1601,
    "values_at_s_minus_one": {
      "1": 1,
      "2": 0.5,
      "4": 0.625,
      "8": 0.78125
    },
    "proved_upper_bound": "a_t(s)<=1+2*abs(s)/t",
    "limit": "For each fixed s, a_t(s)->1."
  },
  "rendering": {
    "PNG_dpi": 160,
    "SVG_text": "Editable text with DejaVu Sans font notice."
  }
}