# C877 — Derive the quarter-family divisibility law

## Question

What single condition explains each whole-family grid profile?

## Inputs

```json
{
  "seed": "161/4"
}
```

## Sources

```json
[
  "C873,876"
]
```

## Opened utc

2026-09-28T02:47:39.086772+00:00

## Results

```json
{
  "coefficients": {
    "E": "175/4",
    "P": "245/6",
    "J": "525/13"
  },
  "quarter_index_divisors": {
    "E": 1,
    "P": 3,
    "J": 13
  },
  "artifact": {
    "path": "model/quarter_multiplier_law.json",
    "sha256": "ca5bfffc2493bf2b6fdb41f0f8d6a5c203fc730d0372a1cf7cce21c17ab9afd2",
    "bytes": 229
  }
}
```

## Finding

For source161m/4, E gives175m/4, P gives245m/6 and J gives525m/13. Quarter preservation is automatic for E, requires3|m for P and13|m for J. This explains the entire finite ladder.

## Reassessment

Apply E to every head in File46 rather than only its outer width.

## Checks

```json
{
  "divisors": true,
  "whole_family": true
}
```

## Closed utc

2026-09-28T02:47:39.087292+00:00

## Record provenance

Exact record hash: `cf65c4496550ec684c3078f89c83c62e8ebcc84d085c61a1d9eda8c0482bdfd5`

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