{"id": "c1064", "title": "C1064 — Common-pivot gap obstruction", "filename": "C1064.md", "role": "Research action", "step": 1064, "sha256": "7121285ddf22c79ff3ddb48ce3e6207eb1ebd3f7747cc6dd74ed75b082e0dbfe", "bytes": 1434, "origins": ["C480–C1634/Research_Cycles/C0932_C1131/journal.json#C1064"], "download": "/sources/7121285ddf22c79ff3ddb48ce3e6207eb1ebd3f7747cc6dd74ed75b082e0dbfe.md", "format": "md", "derived": true, "tags": [], "excerpt": "Can paired endpoints990 or1980 apart both have integral Key images about one integer pivot?", "content": "# C1064 — Common-pivot gap obstruction\n\n## Question\n\nCan paired endpoints990 or1980 apart both have integral Key images about one integer pivot?\n\n## Inputs\n\n```json\n{\n  \"gaps\": [\n    990,\n    1980\n  ],\n  \"denominators\": [\n    23,\n    69,\n    299\n  ]\n}\n```\n\n## Sources\n\n```json\n[\n  \"File52c paired branches\",\n  \"C850–C858\",\n  \"C998\"\n]\n```\n\n## Opened utc\n\n2026-09-28T04:10:48.994087+00:00\n\n## Results\n\n```json\n{\n  \"gap_tests\": [\n    {\n      \"gap\": 990,\n      \"residues\": {\n        \"23\": 1,\n        \"69\": 24,\n        \"299\": 93\n      }\n    },\n    {\n      \"gap\": 1980,\n      \"residues\": {\n        \"23\": 2,\n        \"69\": 48,\n        \"299\": 186\n      }\n    }\n  ],\n  \"proof\": \"Two integral images require both x−a and y−a divisible by the reduced denominator, hence require x−y divisible by it.\",\n  \"scope\": \"necessary-condition theorem; no pivot or target search\"\n}\n```\n\n## Finding\n\nNeither long-core gap allows both paired endpoints to expand integrally about any one common integer Key pivot. The impossibility follows from their difference alone.\n\n## Reassessment\n\nState the corresponding symbolic conditions for the remaining branch forms.\n\n## Checks\n\n```json\n{\n  \"all_obstructed\": true,\n  \"mod23\": true\n}\n```\n\n## Closed utc\n\n2026-09-28T04:10:48.994542+00:00\n\n## Record provenance\n\nExact record hash: `fdea2027b2a202ece22f7bece6c47567b85c0669a492e7372278169f1ab20285`\n\nSource journal: `Research_Cycles/C0932_C1131/journal.json`", "journalDownload": "/sources/f6bdc7aaaaa4fd40f1151b367204ddee7f0f9b77a9c43ae8764831b52d862670.json", "evidenceLinks": [{"id": "file52c", "title": "File 52c"}]}