# C557 — Derive Joseph overlap geometry

## Question

Why do all six paired states share27|83|27?

## Inputs

```json
{
  "trunk": 137,
  "collateral": 110
}
```

## Sources

```json
[
  "File61 §11; File62 §§5–6"
]
```

## Opened utc

2026-09-28T00:13:33.225665+00:00

## Results

```json
{
  "partition": [
    27,
    83,
    27
  ],
  "overlap": 83,
  "translation_commutes_with_collateral_construction": [
    true,
    true,
    true,
    true,
    true,
    true
  ]
}
```

## Finding

The27|83|27 geometry follows from lengths137 and110. Translation commutes with fixed-offset collateral construction, so twelve intervals reduce to one shape in six placements.

## Reassessment

Compare the moving collateral lives with fixed regular Joseph.

## Checks

```json
{
  "shape": true,
  "all_states": true
}
```

## Closed utc

2026-09-28T00:13:33.226047+00:00

## Record provenance

Exact record hash: `2f5aeccbea999cf71652786151b395b5347b29412bab2e753ad347c31b974f08`

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