{
  "title": "Two return partitions and the exact section-pairing square",
  "proof_source": "NEXT_INDUCED_CORNERS.md",
  "proof_locators": [
    "ic-returns/IC7",
    "ic-corner/IC11-12",
    "ic-pairing/IC41-46",
    "ic-full-transport/IC50-60"
  ],
  "central_partition": {
    "source": [
      "e_1",
      "e_2",
      "e_3",
      "e_n, n>=4"
    ],
    "range": [
      "alpha(e_1)",
      "alpha^2(e_2)",
      "alpha^3(e_3)",
      "alpha^n(e_n), n>=4"
    ],
    "arrow": "u^n e_n",
    "qualification": "Symbolic orthogonal central bands. Equal drawing widths encode no measure, probability or trace."
  },
  "hit_sample": {
    "first_section": [
      -4,
      0,
      4,
      8,
      12
    ],
    "second_section": [
      -3,
      1,
      7,
      10,
      13
    ],
    "displayed_pairs": [
      [
        0,
        1
      ],
      [
        4,
        7
      ],
      [
        8,
        10
      ]
    ],
    "first_pair": [
      0,
      1
    ],
    "next_pair": [
      4,
      7
    ],
    "a": 1,
    "a_next": 3,
    "R1": 4,
    "R2": 6,
    "identity": "R2=R1-a+a_next=4-1+3=7-1=6",
    "qualification": "Finite orbit samples of locally finite bi-infinite hit sets; no finite orbit or periodic closure is asserted."
  },
  "coincident_sample": {
    "hits": [
      0,
      4,
      8
    ],
    "a": 0,
    "b": 0,
    "pairing": "identity"
  },
  "deck_square": {
    "s": 9,
    "source_top": [
      9,
      "x"
    ],
    "target_top": [
      8,
      "W x"
    ],
    "source_bottom": [
      5,
      "x'"
    ],
    "target_bottom": [
      2,
      "W x'"
    ],
    "top_displacement": 1,
    "bottom_displacement": 3,
    "left_roof": 4,
    "right_roof": 6,
    "scalar_identity": "9-4-3=9-1-6=2",
    "fiber_identity": "B(5,x') A1_x = A2_Wx B(9,x)",
    "map": "unsheared product map J",
    "qualification": "J intertwines the variable-roof decks. D=H J aligns heights and restricts to Phi on constants; no whole-deck intertwining of D is asserted."
  },
  "shear": {
    "H": "(H G)(s,y)=G(s-a(V y),y)",
    "H_inverse": "(H^-1 G)(t,y)=G(t+a(V y),y)",
    "coefficient": "D(1 tensor x)=1 tensor Phi(x)"
  }
}