# C556 — Generate all collateral Joseph intervals

## Question

Do two fixed-offset constructions account for every source Joseph rail?

## Inputs

```json
{
  "Levi": 1933,
  "Kohath": 1796,
  "Joseph_life": 110
}
```

## Sources

```json
[
  "File61 §§11.2–11.3A; File62 §5.2"
]
```

## Opened utc

2026-09-28T00:13:20.705602+00:00

## Results

```json
{
  "intervals": [
    [
      1933,
      1823
    ],
    [
      1906,
      1796
    ],
    [
      1926,
      1816
    ],
    [
      1899,
      1789
    ],
    [
      1919,
      1809
    ],
    [
      1892,
      1782
    ],
    [
      "3873/2",
      "3653/2"
    ],
    [
      "3819/2",
      "3599/2"
    ],
    [
      "3859/2",
      "3639/2"
    ],
    [
      "3805/2",
      "3585/2"
    ],
    [
      "3845/2",
      "3625/2"
    ],
    [
      "3791/2",
      "3571/2"
    ]
  ],
  "matches": 12
}
```

## Finding

Two collateral constructors, (L,L−110) and (K+110,K), generate all12 exact-phase Joseph intervals. Joseph remains a collateral interval, not an added trunk generation.

## Reassessment

Derive the invariant geometry shared by these interval pairs.

## Checks

```json
{
  "count": true,
  "lifespans": true
}
```

## Closed utc

2026-09-28T00:13:20.706343+00:00

## Record provenance

Exact record hash: `aa8061f481db035298b9c58aa226626ddcc53eb100ff7e64f2530e48549e1456`

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