# C760 — Prove rounding equivariance for admitted shifts

## Question

Which state changes commute with componentwise rounding?

## Inputs

```json
{
  "admitted_amounts": [
    60,
    130,
    215,
    460
  ]
}
```

## Sources

```json
[
  "File51a variants; C752"
]
```

## Opened utc

2026-09-28T01:55:50.707706+00:00

## Results

```json
{
  "identity": "For k=5m, Q(n+k)=5 floor((n+5m+2)/5)=Q(n)+5m. Thus row residuals and corresponding boundary residuals persist under compatible admitted changes.",
  "checked_amounts": {
    "60": true,
    "130": true,
    "215": true,
    "460": true
  },
  "scope": "compatible coordinates or rows; insertion retains topology; finite source states only"
}
```

## Finding

Rounding compatibility is an equivariance under multiples of five. It explains why several variants preserve residual profiles without making all chronology operations interchangeable.

## Reassessment

Test counted-versus-completed input order where the shift is not a multiple of five.

## Checks

```json
{
  "all_shifts": true,
  "one_unit_counterexample": true
}
```

## Closed utc

2026-09-28T01:55:50.708572+00:00

## Record provenance

Exact record hash: `56187441824de6877048d05176f46bedcef35df34d274dc6aa93645d11bbc0b3`

Source journal: `Research_Cycles/C0732_C0831/journal.json`