# C640 — Solve the complete three-calendar congruence family

## Question

Do the fixed fourteen-year rail and thirteen-year flanks select the source count triple as the smallest positive solution?

## Inputs

```json
{
  "conditions": [
    "T=182m",
    "T−14=168n",
    "T−26=180p"
  ],
  "counts": "positive integers"
}
```

## Sources

```json
[
  "File70 §6 geometry; derived integer compatibility theorem"
]
```

## Opened utc

2026-09-28T00:58:46.567888+00:00

## Results

```json
{
  "m_residue_mod180": [
    13
  ],
  "general_T": "2366+32760q",
  "general_counts": "m=13+180q; n=14+195q; p=13+182q",
  "examples": [
    {
      "q": 0,
      "T": 2366,
      "m": 13,
      "n": 14,
      "p": 13
    },
    {
      "q": 1,
      "T": 35126,
      "m": 193,
      "n": 209,
      "p": 195
    },
    {
      "q": 2,
      "T": 67886,
      "m": 373,
      "n": 404,
      "p": 377
    }
  ],
  "least_positive_solution": [
    2366,
    13,
    14,
    13
  ]
}
```

## Finding

With the source offsets fixed, the full system has T=2366+32760q. Its least positive solution is exactly the source2366 and13/14/13 half-count triple. This is a conditional local minimality result, not a claim of a globally minimal chronology.

## Reassessment

Relate the congruence period to the calendar lattice, then test the source no-Cainan companion.

## Checks

```json
{
  "fundamental_residue": true,
  "all_three_equations": true,
  "least_positive": true
}
```

## Closed utc

2026-09-28T00:58:46.568078+00:00

## Record provenance

Exact record hash: `623fd5df259934d6c3ff7cec267048b2c6c0b361546a0a0f59ddddd9c1f4a956`

Source journal: `Research_Cycles/C0632_C0731/journal.json`