{
  "authorship": "Original mathematics and figures;GPT-6.1 Sol (OpenAI),Ultra;CC0 1.0",
  "harmonic_rectangle": {
    "domain": [
      [
        -1,
        1
      ],
      [
        0,
        1
      ]
    ],
    "formula": "sin(pi*(t+1)/2)*sinh(pi*(1-s)/2)/sinh(pi/2)",
    "boundary": "zero on top and vertical sides;between0 and1 on bottom",
    "marked_point": [
      0,
      0.5
    ],
    "marked_value": 0.37746985435706565,
    "proof_locators": [
      "Formal Lemma2.1",
      "Learner Exercise4"
    ]
  },
  "quadrant_barrier": {
    "angles": [
      0,
      1.5707963267948966
    ],
    "power": 1.5,
    "angular_formula": "cos(3*(theta-pi/4)/2)",
    "positive_minimum": 0.38268343236508984,
    "exact_minimum": "cos(3*pi/8)",
    "proof_locators": [
      "Formal Lemma3.1",
      "Learner Exercise5"
    ]
  },
  "windows": {
    "center": "2*pi*10^6",
    "radius_formula": "a*log(2+c)",
    "coefficients": [
      0.1,
      0.25
    ],
    "radii": [
      1.5653387942683457,
      3.913346985670864
    ],
    "exact_real_modulus": "2*abs(sin(t/2))",
    "zero_at_center": true,
    "peaks_offsets": [
      -3.141592653589793,
      3.141592653589793
    ],
    "proof_locators": [
      "Learner Example2",
      "Formal Theorem1.1"
    ]
  },
  "atomic_division": {
    "divisor": {
      "locations": [
        0,
        1
      ],
      "weights": [
        1,
        -1
      ]
    },
    "quotient": {
      "locations": [
        0,
        1,
        2
      ],
      "weights": [
        1,
        1,
        1
      ]
    },
    "product": {
      "locations": [
        0,
        3
      ],
      "weights": [
        1,
        -1
      ]
    },
    "formula": "(1-exp(-i*z))*(1+exp(-i*z)+exp(-2i*z))=1-exp(-3i*z)",
    "markers_are_point_masses_not_densities": true
  },
  "references": [
    "Terence Tao,246B Notes2 and245B Notes9",
    "Lars H\u00f6rmander,Analysis of Linear Partial Differential Operators I and II"
  ]
}