# C853 — Derive the complete paired gain field

## Question

Where does the anchor dependence enter the forward construction?

## Inputs

```json
{}
```

## Sources

```json
[
  "C852; inherited one-pass gain definition"
]
```

## Opened utc

2026-09-28T02:36:19.444521+00:00

## Results

```json
{
  "gains": {
    "path": "model/inverse_gain_field.json",
    "sha256": "093b354383add7d87f104fb0d8a32411bab1bb2465eddb6ccb4e78273938c8ba",
    "bytes": 3362
  },
  "paired_delta": [
    7200,
    990,
    990,
    990,
    990,
    990,
    990,
    990,
    1980,
    1980,
    990,
    990,
    990,
    -6930,
    -3330,
    -630,
    -6840,
    -4140,
    -1440,
    -4050,
    -3960,
    0,
    -360,
    -1800,
    -900,
    -5940,
    990,
    -6660,
    -5670,
    -2970,
    -2970,
    1980
  ],
  "identity": "A+I(D−A)=D+g(D−A), with g(s)=I(s)−s"
}
```

## Finding

All anchor dependence resides in the gain of the original selected duration. The32 differences are one generated field, not32 adjustable translations.

## Reassessment

Derive the long-core no-borrow and borrow branches from digit positions.

## Checks

```json
{
  "gain_difference": true,
  "anchor_gap": true
}
```

## Closed utc

2026-09-28T02:36:19.445226+00:00

## Record provenance

Exact record hash: `0a3fb28d7893be93212abd083bf09ca914de9749a357127f1872025dc202a944`

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