# C873 — Derive the exact lattice-preservation criterion

## Question

When does a reduced Key p/q preserve hZ?

## Inputs

```json
{
  "keys": {
    "E": "25/23",
    "P": "70/69",
    "J": "300/299"
  }
}
```

## Sources

```json
[
  "C872 declared domains"
]
```

## Opened utc

2026-09-28T02:46:40.097441+00:00

## Results

```json
{
  "theorem": {
    "path": "model/one_key_grid_theorem.json",
    "sha256": "99ef73648f71a47a6a4c02ada19ea51bdd1db7db3bd5264862b0864719ee7a34",
    "bytes": 232
  },
  "reduced_denominators": {
    "E": 23,
    "P": 69,
    "J": 299
  }
}
```

## Finding

A Key preserves a declared grid exactly on hqZ. This is an input-domain condition, not a demand to round rational outputs or a license to apply a Key to every chronology.

## Reassessment

Classify integer and Rounded domains for all three Keys and their intersection.

## Checks

```json
{
  "coprime": true,
  "grid_cancels": true
}
```

## Closed utc

2026-09-28T02:46:40.097734+00:00

## Record provenance

Exact record hash: `7874513c4f183894460c3b9ad772dfe9222ba95d4ef5223cbbcc59d2b79f937e`

Source journal: `Research_Cycles/C0832_C0931/journal.json`