# C697 — Derive directional convergence from the Key gap

## Question

What source offset makes the paired anchored outputs equal?

## Inputs

```json
{
  "span": 690,
  "held_forward": [
    3626,
    3576
  ],
  "held_reverse": [
    2216,
    2166
  ]
}
```

## Sources

```json
[
  "C696; Supplement A §8"
]
```

## Opened utc

2026-09-28T01:16:27.059306+00:00

## Results

```json
{
  "Key_output_gap": 50,
  "forward_residual": 0,
  "reverse_residual": 0,
  "individual_gains": [
    10,
    60
  ]
}
```

## Finding

The50 source offset equals(E−P)×690. With the declared direction and assignment, that one equality forces both convergences. The four outputs are structured consequences of the offset and operators, not four independent confirmations.

## Reassessment

Recover the complete70-wide quartet and its supplied1470 translation.

## Checks

```json
{
  "gap": true,
  "residuals": true,
  "gains": true
}
```

## Closed utc

2026-09-28T01:16:27.059594+00:00

## Record provenance

Exact record hash: `7d177b1e037e0e89f11da8a55776b08a414cbe3f4868ce0427c15c6c02d11037`

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