{
  "schema": "original-joint-frequency-geometry/v1",
  "fourier_convention": "F_u(z)=<u,exp(-i x dot z)>; D=(1/i)partial",
  "metric": "Euclidean on real and complex spaces; observation integrals use 2n-dimensional Lebesgue measure",
  "first_figure": {
    "real_frequency_centers": "R_j=(2j+1)pi",
    "indices": [
      1,
      10,
      100
    ],
    "imaginary_coordinate_range": [
      -2,
      2
    ],
    "atom_profile": "log(abs(expm1(eta*log(R))))/log(R)",
    "zero_value_at_eta0": "exactly minus infinity; not represented as a finite ordinate",
    "proper_atom_limit": "max(eta,0)",
    "derivative_order": 2,
    "derivative_location": 0.5,
    "derivative_profile": "2+log(1+(eta*log(R)/R)^2)/log(R)+eta/2",
    "derivative_limit": "2+eta/2",
    "recession": "eta/2",
    "proof_locators": [
      "Theorem2.1",
      "Worked example2, equation9.2",
      "Worked example3, equation9.3"
    ]
  },
  "second_figure": {
    "normalized_real_frequency_circle_radius": 1,
    "equal_axis_scale": true,
    "dominant_coordinate_threshold": "1/sqrt(2)",
    "first_profile_collapses_when": "abs(xi1)>=R/sqrt(2)",
    "second_profile_collapses_when": "abs(xi2)>=R/sqrt(2)",
    "intersection": "four diagonal boundary directions",
    "distributions": "u1=phi(x1) tensor delta0(x2); u2=delta0(x1) tensor phi(x2)",
    "bump": "phi=exp(-1/(1-x^2)) on abs(x)<1, zero otherwise; exact support[-1,1]",
    "realized_routes": [
      "(R,0): empty,|eta2|",
      "(0,R): |eta1|,empty",
      "(R/sqrt2,R/sqrt2): empty,empty"
    ],
    "forbidden_joint_pair": "|eta1|,|eta2|",
    "not_a_classification_of_all_possible_proper_components": true,
    "proof_locators": [
      "Worked example4,equation9.4",
      "Exercise7 and its complete solution"
    ]
  },
  "sources": [
    "Lars Hormander,The Analysis of Linear Partial Differential Operators II,1983;1990;2005",
    "Exact linked original PSH compactness,smooth complex decay and endpoint-limit proofs"
  ],
  "authorship": "GPT-6.1 Sol (OpenAI), Ultra; original CC0 1.0 figures"
}