# C1218 — Lift source translations to Rounded values plus residuals

## Question

How can a non-grid row change act coherently on the Rounded representation?

## Inputs

```json
{}
```

## Sources

```json
[
  "C1213–C1217",
  "prep/round_joint_fibres.md"
]
```

## Opened utc

2026-09-28T05:30:55.328517+00:00

## Results

```json
{
  "translation": {
    "path": "model/rounded_residual_translation.json",
    "sha256": "f9e2b0fdc53f2c54c40d63f1de6d2aaa0aa0eef6eb8f7fe79eb7cde3bd98fdee",
    "bytes": 936
  },
  "examples": [
    {
      "x": 777,
      "d": -24,
      "Q_x": 775,
      "rho": 2,
      "rounded_shift": -20,
      "new_rounded": 755,
      "new_residual": -2
    },
    {
      "x": 595,
      "d": -24,
      "Q_x": 595,
      "rho": 0,
      "rounded_shift": -25,
      "new_rounded": 570,
      "new_residual": 1
    },
    {
      "x": 438,
      "d": 27,
      "Q_x": 440,
      "rho": -2,
      "rounded_shift": 25,
      "new_rounded": 465,
      "new_residual": 0
    },
    {
      "x": 433,
      "d": 27,
      "Q_x": 435,
      "rho": -2,
      "rounded_shift": 25,
      "new_rounded": 460,
      "new_residual": 0
    }
  ]
}
```

## Finding

The retained residual determines how an off-grid change crosses rounded cells. This single rule explains why the same−24 change produces−20 in Lamech’s rounded lifespan and−25 in his rounded remainder.

## Reassessment

Verify the lifted rule across every ordinary existing tradition comparison.

## Checks

```json
{
  "all_examples": true,
  "different_shifts": true
}
```

## Closed utc

2026-09-28T05:30:55.328995+00:00

## Record provenance

Exact record hash: `ab653031c1d0b310364ae75a98de6bfb2be19a9338c613c6a4e47707f1f75ba4`

Source journal: `Research_Cycles/C1132_C1431_Recovered/journal.json`