# C1442 — Locate the obstruction to cross-mode coincidence

## Question

Are LXX/SP nonzero head differences avoidable without altering their declared source states?

## Sources

```json
[
  "Registered source mode gaps",
  "Decimal gain laws"
]
```

## Opened utc

2026-09-29T01:13:46.072488+00:00

## Results

```json
{
  "MT": {
    "source_gap": 9900,
    "head_gap": 0,
    "modulus": 990,
    "residue": 0,
    "same_residue": true
  },
  "LXX": {
    "source_gap": 9410,
    "head_gap": 500,
    "modulus": 990,
    "residue": 500,
    "same_residue": true
  },
  "SP": {
    "source_gap": 8990,
    "head_gap": -190,
    "modulus": 90,
    "residue": 80,
    "same_residue": true
  }
}
```

## Finding

MT and LXX primary spans preserve gaps modulo990; LXX residue500 prevents convergence. Across SP digit widths the safe common modulus90 leaves residue80, also preventing convergence.

## Reassessment

The next test should change only source-named partitions, not tune inputs toward convergence.

## Checks

```json
{
  "all_congruences": true,
  "LXX_residue": true,
  "SP_residue": true
}
```

## Closed utc

2026-09-29T01:13:46.072973+00:00

## Record provenance

Exact record hash: `27cc30a3a28136cc09e0f2bd013e3b270d528c6144fa0b0b1da65f669f63ec56`

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