# C594 — Derive grid translation and carrier laws

## Question

Which operations commute, and why do companion rails converge?

## Inputs

```json
{
  "rows": 49,
  "carriers": [
    70,
    72
  ],
  "civil_translation": 110
}
```

## Sources

```json
[
  "File68 Appendix C; C593"
]
```

## Opened utc

2026-09-28T00:23:08.448006+00:00

## Results

```json
{
  "BC_carrier_gaps": [
    98,
    96,
    94,
    92,
    90,
    88,
    86,
    84,
    82,
    80,
    78,
    76,
    74,
    72,
    70,
    68,
    66,
    64,
    62,
    60,
    58,
    56,
    54,
    52,
    50,
    48,
    46,
    44,
    42,
    40,
    38,
    36,
    34,
    32,
    30,
    28,
    26,
    24,
    22,
    20,
    18,
    16,
    14,
    12,
    10,
    8,
    6,
    4,
    2,
    0
  ],
  "commutation_instances": 98,
  "gap_law": "2(49−j)"
}
```

## Finding

Row progression and fixed civil translation commute. Carrier substitution preserves row incidence and common terminal while its metric displacement shrinks from98 to0 according to remaining count. The terminal convergence is generated, not an independent target.

## Reassessment

Check how the fixed-coordinate Cainan attribution overlay changes names.

## Checks

```json
{
  "commutation": true,
  "convergence": true,
  "step": true
}
```

## Closed utc

2026-09-28T00:23:08.448318+00:00

## Record provenance

Exact record hash: `fac98fabcb945b6ef73b6143386560940312f213c87892a1c2911c6c459b7415`

Source journal: `Research_Cycles/C0532_C0631/journal.json`