# C998 — Fixed-pivot stagewise congruence

## Question

What condition makes both stages of a two-Key affine route integral?

## Inputs

```json
{
  "first": "a+(p1/q1)(x−a)",
  "second": "b+(p2/q2)(y−b)",
  "assumptions": "reduced fractions; integer x,a,b"
}
```

## Sources

```json
[
  "C997",
  "C872–C889"
]
```

## Opened utc

2026-09-28T03:54:36.535025+00:00

## Results

```json
{
  "first_stage": "x=a+q1·t; y=a+p1·t",
  "second_stage": "p1·t≡b−a (mod q2)",
  "solvability": "gcd(p1,q2) divides b−a",
  "period": "q1·q2/gcd(p1,q2)"
}
```

## Finding

The source pivots enter an exact congruence: an integral route exists iff gcd(p1,q2) divides their difference. When it exists, the domain is one residue class.

## Reassessment

Evaluate the complete fixed nine-pair diagnostic with its two source-appointed pivots.

## Checks

```json
{
  "first_stage_reduction": true,
  "potential_obstruction": true
}
```

## Closed utc

2026-09-28T03:54:36.535318+00:00

## Record provenance

Exact record hash: `7b7c8675f32232625f98746a6f0eed5ab3d45116706fab42eec457dbbf305aaa`

Source journal: `Research_Cycles/C0932_C1131/journal.json`