# C1000 — Verify fixed-pivot domains

## Question

Do the declared residue classes preserve integrality at both stages?

## Inputs

```json
{}
```

## Sources

```json
[
  "C999"
]
```

## Opened utc

2026-09-28T03:55:11.412548+00:00

## Results

```json
{
  "nonempty_class_checks": 24,
  "pivot_difference_mod3": 1,
  "J_to_P_reason": "gcd(300,69)=3 does not divide−9170"
}
```

## Finding

All nonempty class formulas pass exact staged substitution. J→P fails because the two fixed pivots differ modulo3; no endpoint choice repairs that within the declared integer domain.

## Reassessment

Extend the congruence result from one point to an entire labelled field.

## Checks

```json
{
  "all_substitutions": true,
  "obstruction": true
}
```

## Closed utc

2026-09-28T03:55:11.413103+00:00

## Record provenance

Exact record hash: `b0ce4d9805172c518f2bdc73dfa1d9406777f1135d5baa0f8fc5008d4eb164e4`

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