# C1441 — Explain gains by decimal register

## Question

Can the signed reversal gains be computed from the source digits without treating them as unrelated constants?

## Sources

```json
[
  "File52c decimal operation",
  "Six registered source spines"
]
```

## Opened utc

2026-09-29T01:13:46.070184+00:00

## Results

```json
[
  {
    "spine": "MT_regular",
    "n": 1050,
    "I_n": 5010,
    "core": "105",
    "zeros": 1,
    "gain": 3960,
    "formula_gain": 3960
  },
  {
    "spine": "MT_regular",
    "n": 1650,
    "I_n": 5610,
    "core": "165",
    "zeros": 1,
    "gain": 3960,
    "formula_gain": 3960
  },
  {
    "spine": "MT_cumulative",
    "n": 3430,
    "I_n": 3430,
    "core": "343",
    "zeros": 1,
    "gain": 0,
    "formula_gain": 0
  },
  {
    "spine": "MT_cumulative",
    "n": 9170,
    "I_n": 7190,
    "core": "917",
    "zeros": 1,
    "gain": -1980,
    "formula_gain": -1980
  },
  {
    "spine": "LXX_regular",
    "n": 1830,
    "I_n": 3810,
    "core": "183",
    "zeros": 1,
    "gain": 1980,
    "formula_gain": 1980
  },
  {
    "spine": "LXX_regular",
    "n": 2250,
    "I_n": 5220,
    "core": "225",
    "zeros": 1,
    "gain": 2970,
    "formula_gain": 2970
  },
  {
    "spine": "LXX_cumulative",
    "n": 4340,
    "I_n": 4340,
    "core": "434",
    "zeros": 1,
    "gain": 0,
    "formula_gain": 0
  },
  {
    "spine": "LXX_cumulative",
    "n": 9150,
    "I_n": 5190,
    "core": "915",
    "zeros": 1,
    "gain": -3960,
    "formula_gain": -3960
  },
  {
    "spine": "SP_regular",
    "n": 1700,
    "I_n": 7100,
    "core": "17",
    "zeros": 2,
    "gain": 5400,
    "formula_gain": 5400
  },
  {
    "spine": "SP_regular",
    "n": 1300,
    "I_n": 3100,
    "core": "13",
    "zeros": 2,
    "gain": 1800,
    "formula_gain": 1800
  },
  {
    "spine": "SP_cumulative",
    "n": 3310,
    "I_n": 1330,
    "core": "331",
    "zeros": 1,
    "gain": -1980,
    "formula_gain": -1980
  },
  {
    "spine": "SP_cumulative",
    "n": 8680,
    "I_n": 8680,
    "core": "868",
    "zeros": 1,
    "gain": 0,
    "formula_gain": 0
  }
]
```

## Finding

For abc0 spans the gain is990(c−a); for SP regular ab00 spans it is900(b−a). Source digits account for the six signed gains.

## Reassessment

Use these register laws to distinguish forced residues from exact cancellations.

## Checks

```json
{
  "all_twelve_legs_explained": true
}
```

## Closed utc

2026-09-29T01:13:46.070815+00:00

## Record provenance

Exact record hash: `b2685555226504831a03c145f8b1eb7b80b45857db9192bd4a5a7d32a704d513`

Source journal: `Research_Cycles/C1433_C1482/journal.json`