# C881 — Classify all nine two-Key grid domains

## Question

Which ordered pairs differ between final and stagewise membership?

## Inputs

```json
{
  "inventory": "E,P,J each followed by E,P,J"
}
```

## Sources

```json
[
  "C880"
]
```

## Opened utc

2026-09-28T02:48:29.014031+00:00

## Results

```json
{
  "matrix": {
    "path": "model/two_key_domains.json",
    "sha256": "2dc016baa528ed002f1d353d53f91d2e5c971f0e0bb4222bf778e008349f1a58",
    "bytes": 1578
  },
  "rows": [
    {
      "execution": "E→E",
      "coefficient": "625/529",
      "final_integer_domain": 529,
      "all_stage_integer_domain": 529,
      "all_stage_5Z_domain": 2645
    },
    {
      "execution": "E→P",
      "coefficient": "1750/1587",
      "final_integer_domain": 1587,
      "all_stage_integer_domain": 1587,
      "all_stage_5Z_domain": 7935
    },
    {
      "execution": "E→J",
      "coefficient": "7500/6877",
      "final_integer_domain": 6877,
      "all_stage_integer_domain": 6877,
      "all_stage_5Z_domain": 34385
    },
    {
      "execution": "P→E",
      "coefficient": "1750/1587",
      "final_integer_domain": 1587,
      "all_stage_integer_domain": 1587,
      "all_stage_5Z_domain": 7935
    },
    {
      "execution": "P→P",
      "coefficient": "4900/4761",
      "final_integer_domain": 4761,
      "all_stage_integer_domain": 4761,
      "all_stage_5Z_domain": 23805
    },
    {
      "execution": "P→J",
      "coefficient": "7000/6877",
      "final_integer_domain": 6877,
      "all_stage_integer_domain": 20631,
      "all_stage_5Z_domain": 103155
    },
    {
      "execution": "J→E",
      "coefficient": "7500/6877",
      "final_integer_domain": 6877,
      "all_stage_integer_domain": 6877,
      "all_stage_5Z_domain": 34385
    },
    {
      "execution": "J→P",
      "coefficient": "7000/6877",
      "final_integer_domain": 6877,
      "all_stage_integer_domain": 6877,
      "all_stage_5Z_domain": 34385
    },
    {
      "execution": "J→J",
      "coefficient": "90000/89401",
      "final_integer_domain": 89401,
      "all_stage_integer_domain": 89401,
      "all_stage_5Z_domain": 447005
    }
  ]
}
```

## Finding

Eight of nine ordered pairs have the same final and stagewise integer domain. P→J is the exception: its final domain is6877Z, but its intermediate stage requires20631Z. Scalar products commute while partial integer-grid actions can differ.

## Reassessment

Expose the P/J asymmetry with the denominator-forced minimal example.

## Checks

```json
{
  "complete": true,
  "unique_exception": true
}
```

## Closed utc

2026-09-28T02:48:29.014485+00:00

## Record provenance

Exact record hash: `3233b76aea2010c86d72c2f7d7b9c7c56417bcef7d2297d864c262db6783b029`

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