{"id": "c998", "title": "C998 — Fixed-pivot stagewise congruence", "filename": "C998.md", "role": "Research action", "step": 998, "sha256": "e865d92d197e0714676623e493996d1904dc0cb7bb8efb5083cbfd740730e2a4", "bytes": 1153, "origins": ["C480–C1634/Research_Cycles/C0932_C1131/journal.json#C998"], "download": "/sources/e865d92d197e0714676623e493996d1904dc0cb7bb8efb5083cbfd740730e2a4.md", "format": "md", "derived": true, "tags": [], "excerpt": "What condition makes both stages of a two-Key affine route integral?", "content": "# C998 — Fixed-pivot stagewise congruence\n\n## Question\n\nWhat condition makes both stages of a two-Key affine route integral?\n\n## Inputs\n\n```json\n{\n  \"first\": \"a+(p1/q1)(x−a)\",\n  \"second\": \"b+(p2/q2)(y−b)\",\n  \"assumptions\": \"reduced fractions; integer x,a,b\"\n}\n```\n\n## Sources\n\n```json\n[\n  \"C997\",\n  \"C872–C889\"\n]\n```\n\n## Opened utc\n\n2026-09-28T03:54:36.535025+00:00\n\n## Results\n\n```json\n{\n  \"first_stage\": \"x=a+q1·t; y=a+p1·t\",\n  \"second_stage\": \"p1·t≡b−a (mod q2)\",\n  \"solvability\": \"gcd(p1,q2) divides b−a\",\n  \"period\": \"q1·q2/gcd(p1,q2)\"\n}\n```\n\n## Finding\n\nThe 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.\n\n## Reassessment\n\nEvaluate the complete fixed nine-pair diagnostic with its two source-appointed pivots.\n\n## Checks\n\n```json\n{\n  \"first_stage_reduction\": true,\n  \"potential_obstruction\": true\n}\n```\n\n## Closed utc\n\n2026-09-28T03:54:36.535318+00:00\n\n## Record provenance\n\nExact record hash: `7b7c8675f32232625f98746a6f0eed5ab3d45116706fab42eec457dbbf305aaa`\n\nSource journal: `Research_Cycles/C0932_C1131/journal.json`", "journalDownload": "/sources/f6bdc7aaaaa4fd40f1151b367204ddee7f0f9b77a9c43ae8764831b52d862670.json"}