{
  "title": "The active coordinate and the corner unit",
  "license": "CC0-1.0 to the extent of rights held; font terms retained separately",
  "model": {
    "N": "L-infinity(R,dr)",
    "action": "theta_s f(r)=f(r+s)",
    "trace": "tau(f)=integral exp(r)*f(r)*dr",
    "scaling": "tau theta_s=exp(-s)tau"
  },
  "coordinates": {
    "map": "x=r-s, y=r",
    "inverse": "s=y-x, r=y",
    "matrix": [
      [
        -1,
        1
      ],
      [
        0,
        1
      ]
    ],
    "determinant_exact": -1,
    "absolute_determinant_exact": 1,
    "unitary": "J xi(x,y)=xi(y-x,y)",
    "inverse_unitary": "J* eta(s,r)=eta(r-s,r)",
    "coefficient": "J pi(f) J*=M_f tensor 1",
    "group_operator": "J u_t J* eta(x,y)=eta(x+t,y)"
  },
  "support_window": {
    "test_vector": "1_[0,1)(r-s)*1_[-1,1)(r)",
    "time_exact": 1,
    "area_exact": 2,
    "original_sr_polygon_vertices": [
      [
        -1,
        -1
      ],
      [
        -2,
        -1
      ],
      [
        0,
        1
      ],
      [
        1,
        1
      ]
    ],
    "translated_sr_polygon_vertices": [
      [
        0,
        -1
      ],
      [
        -1,
        -1
      ],
      [
        1,
        1
      ],
      [
        2,
        1
      ]
    ],
    "original_xy_polygon_vertices": [
      [
        0,
        -1
      ],
      [
        1,
        -1
      ],
      [
        1,
        1
      ],
      [
        0,
        1
      ]
    ],
    "translated_xy_polygon_vertices": [
      [
        -1,
        -1
      ],
      [
        0,
        -1
      ],
      [
        0,
        1
      ],
      [
        -1,
        1
      ]
    ],
    "support_centers_sr": [
      [
        "-1/2",
        "0"
      ],
      [
        "1/2",
        "0"
      ]
    ],
    "support_centers_xy": [
      [
        "1/2",
        "0"
      ],
      [
        "-1/2",
        "0"
      ]
    ],
    "support_motion": "u_1 moves support by +1 in s and -1 in x; y unchanged",
    "argument_motion": "u_1 evaluates at s-1; J u_1 J* evaluates at x+1",
    "boundaries": "Half-open sets are specified by the test-vector formula. Polygon outlines draw boundaries of Lebesgue measure zero.",
    "scope": "finite window of one test vector; full representation on R^2"
  },
  "operator_algebra": {
    "regular_image": "B(L2(R_x)) tensor 1_L2(R_y)",
    "multiplicity_space": "L2(R_y)",
    "abstract_algebra": "B(L2(R))",
    "normal_inverse": "compression by zeta -> zeta tensor eta, for a fixed unit vector eta"
  },
  "direct_sum": {
    "N": "L-infinity(R) direct-sum L-infinity(R)",
    "M": "B(L2(R)) direct-sum B(L2(R))",
    "map": "iota_1(f)=(f,0)",
    "map_unit": "e_1=(1,0)",
    "complement_unit": "e_2=(0,1)",
    "detected_corner": "M e_1",
    "center": "C e_1 direct-sum C e_2",
    "each_corner_type": "I_infinity factor, semifinite and properly infinite",
    "whole_type": "semifinite, properly infinite, nonfactor",
    "qualification": "This complement is semifinite by its own copy; one map alone does not determine a general complementary corner."
  },
  "proof_locators": [
    "L18.9.c",
    "L18.9.d",
    "L18.9.e",
    "L18.9.f",
    "L18.9.g",
    "L18.9.h",
    "L18.9.i",
    "L18.9.j",
    "L18.10.g"
  ],
  "symbolic_checks": {
    "coordinate_inverse": true,
    "absolute_determinant": true,
    "support_polygons_and_areas": true,
    "group_argument_sign": true,
    "support_center_signs": true
  },
  "render": {
    "inches": [
      15,
      11
    ],
    "dpi": 160,
    "png_pixels": [
      2400,
      1760
    ],
    "font": "DejaVu Sans",
    "svg_fonttype": "path"
  },
  "runtime": {
    "matplotlib": "3.10.9",
    "numpy": "2.4.4",
    "sympy": "1.13.1"
  }
}