C1064 — Common-pivot gap obstruction

Question

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

Inputs

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

Sources

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

Opened utc

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

Results

{
  "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

{
  "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