# C1001 — Whole-field placement and difference conditions

## Question

What makes every point of a source field belong to one admissible domain?

## Inputs

```json
{}
```

## Sources

```json
[
  "C998",
  "C1000"
]
```

## Opened utc

2026-09-28T03:55:13.386588+00:00

## Results

```json
{
  "criterion": {
    "criterion": "All x_i lie in r+mZ iff x_0≡r mod m and every x_i−x_0 is divisible by m.",
    "difference_condition": "Controls field shape up to translation.",
    "placement_condition": "One retained point fixes the residue class."
  },
  "inherited_witnesses_reclassified": {
    "C480": {
      "points": [
        2148,
        3206
      ],
      "residue": 32,
      "period": 529,
      "width_ok": true,
      "placement_ok": true
    },
    "fixed12026": {
      "points": [
        4114,
        3056
      ],
      "residue": 388,
      "period": 529,
      "width_ok": true,
      "placement_ok": false
    }
  }
}
```

## Finding

A whole-field test separates the difference lattice from one placement residue. The inherited529-width examples share the width condition but differ in placement.

## Reassessment

Check how this domain transforms under a whole-frame translation.

## Checks

```json
{
  "both_widths": true,
  "placement_distinction": true
}
```

## Closed utc

2026-09-28T03:55:13.386862+00:00

## Record provenance

Exact record hash: `3c01eb998f1b0f440c676831c9ab5eab4e50200b7c7ea2de588c0cdc3d642b5b`

Source journal: `Research_Cycles/C0932_C1131/journal.json`