# C667 — Explain retained-part conversion algebraically

## Question

What term distinguishes converting a core from converting its entire bracket?

## Inputs

```json
{
  "core": 690,
  "flanks": [
    30,
    30
  ],
  "operators": [
    "P",
    "E"
  ]
}
```

## Sources

```json
[
  "C666",
  "C631 retained-part synthesis"
]
```

## Opened utc

2026-09-28T01:08:54.665397+00:00

## Results

```json
{
  "rows": [
    {
      "retained": 760,
      "whole": "17500/23",
      "excess": "20/23",
      "flank_gain": "20/23"
    },
    {
      "retained": 810,
      "whole": "18750/23",
      "excess": "120/23",
      "flank_gain": "120/23"
    }
  ],
  "general_difference": "2(k−1)f"
}
```

## Finding

Whole conversion adds exactly2(k−1)f more than core conversion with retained flanks. This is the same retained-subdivision distinction that explains the earlier partial-E and whole-J routes: matching totals never establish identical internal paths.

## Reassessment

Reconstruct the native alternating ladder from its two rail parameters.

## Checks

```json
{
  "P_excess": true,
  "E_excess": true,
  "identity": true
}
```

## Closed utc

2026-09-28T01:08:54.665748+00:00

## Record provenance

Exact record hash: `65ceaa2a2d6692224d37483637d748381cb2e0a3bee2146ee8594dc7d403a603`

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