# C1464 — Explain the LXX paired-Key join by its mode gap

## Question

Why do LXX cumulative-to-Exodus and regular-to-Conquest Priestly routes meet at386?

## Sources

```json
[
  "Comparative Study §4.3",
  "LXX mode difference500"
]
```

## Opened utc

2026-09-29T01:19:39.207475+00:00

## Results

```json
{
  "head_gap": 500,
  "target_gap": 40,
  "head_gap_required": 500,
  "regular_output": 386,
  "cumulative_output": 386,
  "output_difference_formula": 0
}
```

## Finding

The500-year LXX head gap is exactly(25/2)×40, the condition for the two head-held Priestly routes to share a target. The known386 join is an algebraic expression of that compatibility.

## Reassessment

Apply the same fixed Exodus/Conquest comparison to the other two manuscripts, retaining every result.

## Checks

```json
{
  "join386": true,
  "criterion_exact": true
}
```

## Closed utc

2026-09-29T01:19:39.208000+00:00

## Record provenance

Exact record hash: `1c8a743eadabf9a6b202e7e68b557c234c8bdefba8afa10963f98bd9df4dbbd8`

Source journal: `Research_Cycles/C1433_C1482/journal.json`