{
  "criterion": "A translation descends to all nearest-five cells iff its integer shift is divisible by5.",
  "proof": "Multiples of5 translate entire cells. For a nonzero residue1..4, a five-point cell straddles a new cell boundary after translation; adding multiples of5 changes neither fact.",
  "witnesses": [
    {
      "shift_residue": 1,
      "same_cell_inputs": [
        98,
        99,
        100,
        101,
        102
      ],
      "shifted_rounds": [
        100,
        100,
        100,
        100,
        105
      ]
    },
    {
      "shift_residue": 2,
      "same_cell_inputs": [
        98,
        99,
        100,
        101,
        102
      ],
      "shifted_rounds": [
        100,
        100,
        100,
        105,
        105
      ]
    },
    {
      "shift_residue": 3,
      "same_cell_inputs": [
        98,
        99,
        100,
        101,
        102
      ],
      "shifted_rounds": [
        100,
        100,
        105,
        105,
        105
      ]
    },
    {
      "shift_residue": 4,
      "same_cell_inputs": [
        98,
        99,
        100,
        101,
        102
      ],
      "shifted_rounds": [
        100,
        105,
        105,
        105,
        105
      ]
    }
  ],
  "scope": "Restricted finite source domains can avoid a split accidentally; universal necessity does not forbid that."
}