# C505 — The529 family is first a duration grammar

## Question

What does two-stage E expansion guarantee without any pivot choice?

## Inputs

```json
{
  "E": "25/23",
  "small_seed": 1058,
  "macro_seed": 10580
}
```

## Sources

```json
[
  "C480 checkpoint",
  "File52c §§3.8–3.9,6",
  "C481"
]
```

## Opened utc

2026-09-27T22:45:10.631228+00:00

## Results

```json
{
  "small": [
    1058,
    1150,
    1250
  ],
  "macro": [
    10580,
    11500,
    12500
  ],
  "general_integer_width_rule": "k*529 -> k*575 -> k*625",
  "proof": "D_k,a(y)-D_k,a(x)=k*(y-x); second pivot cancels from the width too"
}
```

## Finding

Every supplied529-multiple width follows this ladder under two E expansions. Tenfold scaling commutes with it. Width agreement therefore connects C480 and the rounded macro family while leaving their source roles and pivots distinct.

## Reassessment

Derive the additional congruence condition needed for integral endpoints.

## Checks

```json
{
  "small": true,
  "large": true,
  "scale": true
}
```

## Closed utc

2026-09-27T22:45:10.631494+00:00

## Record provenance

Exact record hash: `aaca498f1807d3744a9f95aa62a07a6b8de9dd53b097979c73df677fb1b93bc7`

Source journal: `Research_Cycles/C0482_C0531/journal.json`