{"id": "c638", "title": "C638 — Explain the Priestly minimum by factor structure", "filename": "C638.md", "role": "Research action", "step": 638, "sha256": "7953d92de636b51c2358eb008970b9c2710486de83c40990ea06bd427b6375a2", "bytes": 1231, "origins": ["C480–C1634/Research_Cycles/C0632_C0731/journal.json#C638"], "download": "/sources/7953d92de636b51c2358eb008970b9c2710486de83c40990ea06bd427b6375a2.md", "format": "md", "derived": true, "tags": ["Calendar Keys"], "excerpt": "Why does subtracting the14-year rail width from13×182 produce14×168?", "content": "# C638 — Explain the Priestly minimum by factor structure\n\n## Question\n\nWhy does subtracting the14-year rail width from13×182 produce14×168?\n\n## Inputs\n\n```json\n{\n  \"corresponding_count\": 13,\n  \"Enochian_half\": 182,\n  \"rail_width\": 14\n}\n```\n\n## Sources\n\n```json\n[\n  \"File70 §§6.2–6.5\"\n]\n```\n\n## Opened utc\n\n2026-09-28T00:57:56.936051+00:00\n\n## Results\n\n```json\n{\n  \"corresponding_factorization\": [\n    14,\n    169\n  ],\n  \"minimum_factorization\": [\n    14,\n    168\n  ],\n  \"Priestly_half\": 168,\n  \"identity\": \"13×182−14=14(169−1)=14×168\"\n}\n```\n\n## Finding\n\nThe Priestly minimum follows from182=14×13 and168=13²−1. The common14-year width turns the corresponding vector into fourteen Priestly halves. This explains the change from13 to14 calendar units through the rail geometry.\n\n## Reassessment\n\nTest whether the combined half-calendar conditions select a unique count or a congruence family.\n\n## Checks\n\n```json\n{\n  \"factorization\": true,\n  \"minimum\": true,\n  \"difference_of_squares\": true\n}\n```\n\n## Closed utc\n\n2026-09-28T00:57:56.936249+00:00\n\n## Record provenance\n\nExact record hash: `48e03ab500abe657bfcd30dd81ee12ec3744e5ab24b2d6c021bf1052845dc452`\n\nSource journal: `Research_Cycles/C0632_C0731/journal.json`", "journalDownload": "/sources/f0643e934c6a7443706351fa40040684d915ed0ae25c0f9188a4c4dba12814ff.json", "evidenceLinks": [{"id": "d-0fa80217c94b3951", "title": "File 70"}]}