# C1209 — State the exact cap-round commutation condition

## Question

When does rounding commute with the cap?

## Inputs

```json
{}
```

## Sources

```json
[
  "C1208",
  "C1131 inherited cap-round theorem"
]
```

## Opened utc

2026-09-28T05:27:11.159718+00:00

## Results

```json
{
  "commutation_scope": {
    "path": "model/cap_round_commutation_scope.json",
    "sha256": "3a28ca3976b2b3e0c3db3acc855fbab20e1484a32e4cf271b3787b19eb686a3f",
    "bytes": 450
  }
}
```

## Finding

Cap and rounding commute when both quantities are rounded by the same monotone rule. Keeping the capacity exact on only one route changes the operation; it is not a counterexample to that theorem.

## Reassessment

Test whether rounding preserves the strict fact that a life exceeded capacity.

## Checks

```json
{
  "monotone_checks": true,
  "witness_values": true
}
```

## Closed utc

2026-09-28T05:27:11.160246+00:00

## Record provenance

Exact record hash: `74a19343aab6121ea572fd8ed73239378aaa8634bdb5fe33ead4fdee495f8336`

Source journal: `Research_Cycles/C1132_C1431_Recovered/journal.json`