# C906 — Derive the retained-subdivision criterion

## Question

When can two constructions share an outer span but disagree at an internal biography?

## Inputs

```json
{
  "S": 144900,
  "l": 147,
  "k": "70/69"
}
```

## Sources

```json
[
  "C898",
  "C831 complete-path criterion"
]
```

## Opened utc

2026-09-28T02:55:38.238978+00:00

## Results

```json
{
  "criterion": {
    "path": "model/subdivision_retention_criterion.json",
    "sha256": "f99e9c904c85e66da56fa237f9701db460678c4fd4ceb1744d71bd1a91e452fb",
    "bytes": 325
  },
  "partitions": {
    "uniform": [
      "3377570/23",
      "3430/23"
    ],
    "retained": [
      146853,
      147
    ]
  },
  "cut_difference": "49/23"
}
```

## Finding

Uniform expansion and retained biography have the same outer span but internal cuts differing by(k−1)l. They coincide only for unit scale or zero-length biography. The complete path, rather than its total, distinguishes these constructions.

## Reassessment

Compare the distinct mechanisms behind apparent operation-order effects.

## Checks

```json
{
  "outer_equal": true,
  "cut_delta": true,
  "nontrivial": true
}
```

## Closed utc

2026-09-28T02:55:38.239426+00:00

## Record provenance

Exact record hash: `65893c56d5a9dbd7a122e42927d3ef837f86d3468b95bf5e11dd55fb5a71160b`

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