{
  "schema": "AN02-L155-geometry178/v1",
  "coordinates_and_metric": "Euclidean complex frequency space; z=xi+i*eta, real volume dxi*deta. Dot products in exponentials are bilinear.",
  "first_figure": {
    "proof_locators": [
      "learner worked example2, equationsL155.5-L155.8",
      "solutions7-8",
      "formalAR1"
    ],
    "actual_density": "U_(r,sigma)(z)=z^r*exp(-eta^2/sigma^2)/(2*pi^(3/2)*sigma)",
    "density_normalization": "g_sigma integrates to1; actual test integral represents D^r delta0, D=-i*d/dx.",
    "explicit_weight": "t=4096,a=2,R=1/64; phi=p_t(eta)-a*log(t^2+xi^2+eta^2)",
    "left_slice": "eta=0, q=xi/t; weighted energy divided by g_sigma(0)^2/(4*pi^2)*t^(2r-8) is q^(2r)/(1+q^2)^4",
    "left_orders": [
      0,
      1,
      3
    ],
    "q_samples": "900 samples over[-5,5]; exact function plotted",
    "right_profiles": [
      0.5,
      1,
      2
    ],
    "eta_samples": "900 samples over[-5,5]; entire Gaussian profiles exist beyond plotted interval",
    "full_surface_example": "For n=2,P=z1,N={z1=0},parameter z2=xi2+i*eta2, induced dS=dxi2*deta2. U=z2^3*g1(eta2)/(2*pi), producing1_x1 tensor D2^3 delta0."
  },
  "second_figure": {
    "proof_locators": [
      "learner examples1,4, equationsL155.3-L155.4,L155.11",
      "formalAR3"
    ],
    "left_domain": "Displayed x in[-1,1]; all three kernels solve(D-i)^3 u=0 globally.",
    "left_characteristic_measure": "N={i}, counting measure, tau=1, amplitudes a=0,1,2",
    "right_domain": "X=(-1,1)^2; only x2 coordinate and derivative order shown.",
    "right_objects": [
      {
        "l": 1,
        "a_l": 0.5,
        "coefficient": 0.5
      },
      {
        "l": 2,
        "a_l": 0.75,
        "coefficient": 0.25
      },
      {
        "l": 3,
        "a_l": 0.875,
        "coefficient": 0.125
      },
      {
        "l": 4,
        "a_l": 0.9375,
        "coefficient": 0.0625
      },
      {
        "l": 5,
        "a_l": 0.96875,
        "coefficient": 0.03125
      },
      {
        "l": 6,
        "a_l": 0.984375,
        "coefficient": 0.015625
      },
      {
        "l": 7,
        "a_l": 0.9921875,
        "coefficient": 0.0078125
      }
    ],
    "infinite_scope": "a_l=1-2^(-l), l>=1, locally finite on X; plotted seven terms are numerical samples, not the complete distribution. Boundary1 is outside X.",
    "homogeneous_operator": "P(z)=z1, P(D)=D1, all local orders preserved."
  },
  "authorship": "GPT-6.1 Sol (OpenAI), Ultra; original exact scientific plots; CC0 1.0",
  "human_source": "Lars Hormander, The Analysis of Linear Partial Differential Operators II, two unnumbered representation exercises on printedp300.",
  "Blender_assessment": "Exact scalar energy profiles, physical one-dimensional kernels and order-location coordinates are fully described by these plots. A3D embedding would add no mathematical information.",
  "source_body_or_source_images_used": false
}