# C1064 — Common-pivot gap obstruction

## Question

Can paired endpoints990 or1980 apart both have integral Key images about one integer pivot?

## Inputs

```json
{
  "gaps": [
    990,
    1980
  ],
  "denominators": [
    23,
    69,
    299
  ]
}
```

## Sources

```json
[
  "File52c paired branches",
  "C850–C858",
  "C998"
]
```

## Opened utc

2026-09-28T04:10:48.994087+00:00

## Results

```json
{
  "gap_tests": [
    {
      "gap": 990,
      "residues": {
        "23": 1,
        "69": 24,
        "299": 93
      }
    },
    {
      "gap": 1980,
      "residues": {
        "23": 2,
        "69": 48,
        "299": 186
      }
    }
  ],
  "proof": "Two integral images require both x−a and y−a divisible by the reduced denominator, hence require x−y divisible by it.",
  "scope": "necessary-condition theorem; no pivot or target search"
}
```

## Finding

Neither long-core gap allows both paired endpoints to expand integrally about any one common integer Key pivot. The impossibility follows from their difference alone.

## Reassessment

State the corresponding symbolic conditions for the remaining branch forms.

## Checks

```json
{
  "all_obstructed": true,
  "mod23": true
}
```

## Closed utc

2026-09-28T04:10:48.994542+00:00

## Record provenance

Exact record hash: `fdea2027b2a202ece22f7bece6c47567b85c0669a492e7372278169f1ab20285`

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