{
  "schema": "AN02-original-L139-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), Lemmas16.2.1-16.2.2, printed314-315; original illustrations",
  "support_set_and_envelope": {
    "dimension": 2,
    "vertices": [
      [
        -1,
        0
      ],
      [
        1,
        0
      ],
      [
        0,
        2
      ]
    ],
    "intercepts": [
      0,
      0,
      0
    ],
    "support_set": "conv{(-1,0),(1,0),(0,2)}",
    "direction": [
      1,
      1
    ],
    "support_value": 2,
    "support_line": "xi1+xi2=2",
    "metric": "equal Euclidean coordinate scales in slope panel",
    "envelope": "0.5*log(exp(-2*y1)+exp(2*y1)+exp(4*y2))",
    "support_function": "max(-y1,y1,2*y2)",
    "section": "y=t*(1,1), -3<=t<=3",
    "section_envelope": "0.5*log(exp(-2*t)+exp(2*t)+exp(4*t))",
    "section_support": "max(-t,t,2*t)",
    "proved_gap_bound": "0<=M(y)-H(y)<=0.5*log(3)",
    "proof_locators": "Theorem E1, sections3-4; Example C, (E28)-(E30)",
    "sampling": "1601 evenly spaced section samples; no numerical inference of theorem"
  },
  "annulus_liouville": {
    "dimension": 2,
    "center": [
      0,
      0
    ],
    "inner_radius": 1,
    "outer_radius_geometry": 6,
    "alpha": -0.5,
    "C": 1,
    "observation_point": [
      3,
      0
    ],
    "observation_radius": 3,
    "barrier": "alpha+(C-alpha)*log(|z-c|/delta)/log(R/delta)",
    "fixed_point_curve": "-0.5+1.5*log(3)/log(R)",
    "R_domain": [
      3.2,
      100000000
    ],
    "horizontal_plot_axis": "log10(R)",
    "geometric_panel_metric": "equal Euclidean coordinate scales",
    "fixed_observation_point": true,
    "object_type": "harmonic comparison barrier, not an unspecified subharmonic function",
    "proved_limit": -0.5,
    "proof_locators": "Lemma C1, (E26); Theorem C2",
    "sampling": "1601 evenly spaced samples in log10(R)"
  },
  "output": {
    "svg": "vector illustration",
    "pdf": "original one-page vector illustration",
    "png": "native render from own PDF at scale1.7"
  }
}