# C690 — Reconstruct the three cumulative completion paths

## Question

Can one retained2940 component generate the12600/12740/13440 family?

## Inputs

```json
{
  "source_head": 14006,
  "hinge": 4346,
  "tail": 1406,
  "upper": 9660,
  "retained_lower": 2940
}
```

## Sources

```json
[
  "Supplement A §10.2; inherited source family"
]
```

## Opened utc

2026-09-28T01:14:17.581267+00:00

## Results

```json
{
  "paths": [
    {
      "upper": 9660,
      "lower": 2940,
      "total": 12600,
      "head": 14006
    },
    {
      "upper": 9800,
      "lower": 2940,
      "total": 12740,
      "head": 14146
    },
    {
      "upper": 10500,
      "lower": 2940,
      "total": 13440,
      "head": 14846
    }
  ],
  "calendar_bodies": [
    12600,
    12740,
    13440
  ],
  "upper_gain": [
    0,
    140,
    840
  ]
}
```

## Finding

Holding4346→1406 fixed while converting only the9660 upper arm generates all three source totals12600/12740/13440. The generated heads14006/14146/14846 retain their existing status. This realizes the micro register family through retained macro components, not one whole-span conversion.

## Reassessment

Consolidate the micro-to-Rounded bridge and decide which anchored routes best explain its placement.

## Checks

```json
{
  "source_paths": true,
  "heads": true,
  "gains": true
}
```

## Closed utc

2026-09-28T01:14:17.581549+00:00

## Record provenance

Exact record hash: `095d04922354663bba3291939fc2ccf9d76a4e0bc263e88c8a8d95761eacc34c`

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