# C608 — Derive covariance of anchored Keys

## Question

When does a translation commute with an anchored dilation?

## Inputs

```json
{
  "anchor": 14004,
  "point": 1446,
  "translation": 2,
  "Key": "300/299"
}
```

## Sources

```json
[
  "General anchored-Key grammar; C572"
]
```

## Opened utc

2026-09-28T00:27:14.184026+00:00

## Results

```json
{
  "translate_output": 1406,
  "move_anchor_and_point": 1406,
  "move_point_only": "420396/299",
  "point_only_residual": "2/299",
  "identity": "T_t D(a,k)=D(a+t,k) T_t"
}
```

## Finding

Translation commutes with dilation when the anchor moves with the frame. Holding the anchor while translating only the point leaves residual(k−1)t, here2/299. This formalizes why whole-pair and one-end changes behave differently.

## Reassessment

Determine how allocation equivalence changes when interior junctions are retained.

## Checks

```json
{
  "covariance": true,
  "residual": true
}
```

## Closed utc

2026-09-28T00:27:14.184353+00:00

## Record provenance

Exact record hash: `94c0c4984af8effb5bd24e52a28db64fa404a490e7f33f7148c45ef3db5eee21`

Source journal: `Research_Cycles/C0532_C0631/journal.json`