# C1002 — Domain covariance under translation

## Question

Does translating both pivots and the whole field preserve the domain?

## Inputs

```json
{
  "translation": 215
}
```

## Sources

```json
[
  "C998",
  "C1001"
]
```

## Opened utc

2026-09-28T03:55:48.267438+00:00

## Results

```json
{
  "translated_residue": 283,
  "unchanged_period": 529,
  "first_output_shift": 215,
  "second_output_shift": 215,
  "general_rule": "Translate points and both pivots together; residue shifts by t."
}
```

## Finding

Whole-frame translation preserves the domain obstruction and period while shifting the residue. This is the same structural covariance seen in the SP cap capacities.

## Reassessment

Represent a nonintegral source field exactly instead of changing its values.

## Checks

```json
{
  "stage_shifts": true,
  "same_pivot_difference": true
}
```

## Closed utc

2026-09-28T03:55:48.267775+00:00

## Record provenance

Exact record hash: `d7a873e98ab35ec6e9ae9f4c8fdbcf8e79677c5bad44640e49c90d50a5525ebc`

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