{"id": "c640", "title": "C640 — Solve the complete three-calendar congruence family", "filename": "C640.md", "role": "Research action", "step": 640, "sha256": "20a832662283628bc76cf0787d152dd10b392a0d35b84480cee2a974c2c7b4cf", "bytes": 1688, "origins": ["C480–C1634/Research_Cycles/C0632_C0731/journal.json#C640"], "download": "/sources/20a832662283628bc76cf0787d152dd10b392a0d35b84480cee2a974c2c7b4cf.md", "format": "md", "derived": true, "tags": ["Calendar Keys"], "excerpt": "Do the fixed fourteen-year rail and thirteen-year flanks select the source count triple as the smallest positive solution?", "content": "# C640 — Solve the complete three-calendar congruence family\n\n## Question\n\nDo the fixed fourteen-year rail and thirteen-year flanks select the source count triple as the smallest positive solution?\n\n## Inputs\n\n```json\n{\n  \"conditions\": [\n    \"T=182m\",\n    \"T−14=168n\",\n    \"T−26=180p\"\n  ],\n  \"counts\": \"positive integers\"\n}\n```\n\n## Sources\n\n```json\n[\n  \"File70 §6 geometry; derived integer compatibility theorem\"\n]\n```\n\n## Opened utc\n\n2026-09-28T00:58:46.567888+00:00\n\n## Results\n\n```json\n{\n  \"m_residue_mod180\": [\n    13\n  ],\n  \"general_T\": \"2366+32760q\",\n  \"general_counts\": \"m=13+180q; n=14+195q; p=13+182q\",\n  \"examples\": [\n    {\n      \"q\": 0,\n      \"T\": 2366,\n      \"m\": 13,\n      \"n\": 14,\n      \"p\": 13\n    },\n    {\n      \"q\": 1,\n      \"T\": 35126,\n      \"m\": 193,\n      \"n\": 209,\n      \"p\": 195\n    },\n    {\n      \"q\": 2,\n      \"T\": 67886,\n      \"m\": 373,\n      \"n\": 404,\n      \"p\": 377\n    }\n  ],\n  \"least_positive_solution\": [\n    2366,\n    13,\n    14,\n    13\n  ]\n}\n```\n\n## Finding\n\nWith the source offsets fixed, the full system has T=2366+32760q. Its least positive solution is exactly the source2366 and13/14/13 half-count triple. This is a conditional local minimality result, not a claim of a globally minimal chronology.\n\n## Reassessment\n\nRelate the congruence period to the calendar lattice, then test the source no-Cainan companion.\n\n## Checks\n\n```json\n{\n  \"fundamental_residue\": true,\n  \"all_three_equations\": true,\n  \"least_positive\": true\n}\n```\n\n## Closed utc\n\n2026-09-28T00:58:46.568078+00:00\n\n## Record provenance\n\nExact record hash: `623fd5df259934d6c3ff7cec267048b2c6c0b361546a0a0f59ddddd9c1f4a956`\n\nSource journal: `Research_Cycles/C0632_C0731/journal.json`", "journalDownload": "/sources/f0643e934c6a7443706351fa40040684d915ed0ae25c0f9188a4c4dba12814ff.json", "evidenceLinks": [{"id": "d-0fa80217c94b3951", "title": "File 70"}]}