{
  "schema": "AN02-original-L132-figures/v1",
  "license": "CC0-1.0",
  "author": "GPT-6.1 Sol (OpenAI)",
  "human_source_credit": "Lars Hörmander, The Analysis of Linear Partial Differential Operators II (1983; second revised printing 1990; reprint 2005), section 16.1, Lemma 16.1.3 and Corollary 16.1.5; original illustrations and disk example",
  "poisson": {
    "dimension": 2,
    "domain": "open unit disk",
    "boundary": "(cos(theta),sin(theta))",
    "observation_point": "(t,0)",
    "radii": [
      0.0,
      0.6,
      0.9,
      0.98
    ],
    "exact_density": "(1-t^2)/(2*pi*(1-2*t*cos(theta)+t^2))",
    "exact_peak": "(1+t)/(2*pi*(1-t))",
    "t098_peak": "99/(2*pi)",
    "full_angle_domain": [
      -3.141592653589793,
      3.141592653589793
    ],
    "zoom_angle_domain": [
      -0.25,
      0.25
    ],
    "samples_per_curve": 12001,
    "proved_mass": 1,
    "proof_locators": [
      "Theorem P3",
      "P6",
      "P7",
      "P8"
    ],
    "interpretation": "Numerical samples of exact densities; total mass and concentration are proved in the lesson."
  },
  "usc": {
    "dimension": 2,
    "domain": "open unit disk",
    "section": "radial, not whole-disk",
    "exact_family": "max(log(r)/k,-1); value at r=0 is -1",
    "k_values": [
      1,
      2,
      5,
      10
    ],
    "exact_switch_radii": [
      "exp(-1)",
      "exp(-2)",
      "exp(-5)",
      "exp(-10)"
    ],
    "switch_radius_samples": [
      0.36787944117144233,
      0.1353352832366127,
      0.006737946999085467,
      4.5399929762484854e-05
    ],
    "raw_supremum": "0 for 0<r<1; -1 at r=0",
    "raw_filled_point": [
      0,
      -1
    ],
    "raw_open_point": [
      0,
      0
    ],
    "regularization": "constant0",
    "regularized_filled_point": [
      0,
      0
    ],
    "displayed_radius_one": "boundary limit; r=1 is not in the lesson's open disk",
    "proof_locators": [
      "Theorem S2",
      "S4",
      "S5",
      "Worked example3",
      "Exercise6"
    ]
  }
}