{
  "schema": "original-singular-hull-geometry/v1",
  "fourier_convention": "F_u(z)=<u,exp(-i x dot z)>;D=(1/i)partial",
  "first_figure": {
    "spatial_dimension": 2,
    "carrier": "K={0}x[-1,1]",
    "observation_point": [
      2,
      0
    ],
    "closest_point": [
      0,
      0
    ],
    "open_neighborhood_radius": 0.5,
    "unit_separator": [
      1,
      0
    ],
    "uniform_gap": 1.5,
    "equal_spatial_axes": true,
    "complex_contour_dimension": 2,
    "displayed_slice": "xi2=0,zeta2=0; real xi1 in[-4,4]",
    "R": 2,
    "m": 4,
    "contour_height": "2log(2+xi1^2)",
    "strip_upper_boundary": "positive root of eta=4log(1+sqrt(xi1^2+eta^2))",
    "shaded_set": "Upper eta>=0 part of the actual implicit strip,not a bound using real frequency alone",
    "root_solver": "brentq on[1e-8,100];601 horizontal samples",
    "full_proof_uses_all_real_frequency_coordinates": true,
    "proof_locators": [
      "Theorem1.1,sufficiency;equations3.1–3.8",
      "Exercise5"
    ]
  },
  "second_figure": {
    "function": "(1-abs(x))_+",
    "singular_points": [
      -1,
      0,
      1
    ],
    "singular_hull": [
      -1,
      1
    ],
    "operator": "D^2=-partial_x^2",
    "point_mass_weights": [
      -1,
      2,
      -1
    ],
    "weights_are_distributional_coefficients": true,
    "entire_transform": "2(1-cos z)/z^2 with value1 at0",
    "real_centers": "R_j=2pi j",
    "indices": [
      1,
      10,
      100
    ],
    "eta_range": [
      -3,
      3
    ],
    "exact_profile": "[log4+2log(abs(sinh(eta logR/2)))-log(R^2+(eta logR)^2)]/logR",
    "eta0": "exact minus-infinite value excluded from finite curves",
    "proper_limit": "abs(eta)-2",
    "recession": "abs(eta)",
    "proof_locators": [
      "Theorems4.1,5.1",
      "Worked example4;equations9.2–9.3",
      "Exercise10"
    ]
  },
  "human_sources": [
    "Terence Tao,Some connections with the Fourier transform,2021",
    "Lars Hormander,The Analysis of Linear Partial Differential Operators I and II"
  ],
  "authorship": "GPT-6.1 Sol(OpenAI),Ultra;original CC0 1.0"
}