{
  "authorship": "Original mathematics and figures;GPT-6.1 Sol (OpenAI),Ultra;CC0 1.0",
  "rectangle_geometry": {
    "K": [
      [
        -3,
        -2
      ],
      [
        5,
        4
      ]
    ],
    "C": [
      [
        1,
        -1
      ],
      [
        2,
        1
      ]
    ],
    "all_admissible_translations": [
      [
        -4,
        -1
      ],
      [
        3,
        3
      ]
    ],
    "Minkowski_difference": [
      [
        -5,
        -3
      ],
      [
        4,
        5
      ]
    ],
    "good_translation": [
      2,
      1
    ],
    "good_translated_carrier": [
      [
        3,
        0
      ],
      [
        4,
        2
      ]
    ],
    "bad_translation": [
      4,
      4
    ],
    "bad_translated_carrier": [
      [
        5,
        3
      ],
      [
        6,
        5
      ]
    ],
    "geometric_example_does_not_prescribe_an_actual_profile_family": true,
    "proof_locators": [
      "Formal Theorem3.1",
      "Formal Corollary6.1",
      "Learner Exercise3"
    ]
  },
  "invertible_counterexample": {
    "distribution": "delta_(0,0)+(1/4)*(delta_0 tensor f)",
    "f": {
      "support": [
        1,
        2
      ],
      "positive_on_interior": true,
      "smooth": true,
      "mass": 1
    },
    "real_Fourier_lower_bound": 0.75,
    "input_singular_locus": {
      "isolated_point": [
        0,
        0
      ],
      "segment": [
        [
          0,
          1
        ],
        [
          0,
          2
        ]
      ]
    },
    "input_singular_hull": [
      [
        0,
        0
      ],
      [
        0,
        2
      ]
    ],
    "selected_profile_indicator": "zero",
    "selected_factor_singular_support": [
      [
        0,
        0
      ]
    ],
    "selected_factor_support_bound_radius": 0.2,
    "actual_selected_output_hull": [
      [
        0,
        0
      ]
    ],
    "hull_sum": [
      [
        0,
        0
      ],
      [
        0,
        2
      ]
    ],
    "diagram_shows_singular_loci_not_distribution_densities": true,
    "physical_axes_have_equal_scale": true,
    "selected_factor_has_no_closed_formula_asserted": true,
    "proof_locators": [
      "Learner Example3",
      "Formal Theorem2.1",
      "Formal Theorem5.1"
    ]
  },
  "references": [
    "Terence Tao,246B Notes2 and245B Notes9",
    "Lars H\u00f6rmander,Analysis of Linear Partial Differential Operators I and II"
  ]
}