# C665 — Derive the native core from carrier and flank

## Question

What minimal source inputs generate the720/30 family?

## Inputs

```json
{
  "carrier": 720,
  "flank": 30
}
```

## Sources

```json
[
  "Supplement A §§1.1–1.2"
]
```

## Opened utc

2026-09-28T01:08:30.974042+00:00

## Results

```json
{
  "core": 690,
  "flank_coefficients": {
    "core": 23,
    "carrier": 24,
    "native_bracket": 25,
    "expanded_core": 25
  }
}
```

## Finding

The crossed core690 is23 flanks of30. The carrier is24 flanks, and both the native two-flank bracket and E-expanded core are25 flanks. Their shared750 is forced by core=23×flank, while their component roles differ.

## Reassessment

Generate all native, P and E core/flank/bracket states together.

## Checks

```json
{
  "core": true,
  "forced_equality": true
}
```

## Closed utc

2026-09-28T01:08:30.976261+00:00

## Record provenance

Exact record hash: `1aeb72d2ecd8dc9dd6ded191817336a2590dc2700aed7b90d530052f84cb6a71`

Source journal: `Research_Cycles/C0632_C0731/journal.json`