{"id": "c557", "title": "C557 — Derive Joseph overlap geometry", "filename": "C557.md", "role": "Research action", "step": 557, "sha256": "93a90b7eb61ce2607d275a25746f8e17c9a90ca5e6b93e5d67894f83fde30755", "bytes": 1040, "origins": ["C480–C1634/Research_Cycles/C0532_C0631/journal.json#C557"], "download": "/sources/93a90b7eb61ce2607d275a25746f8e17c9a90ca5e6b93e5d67894f83fde30755.md", "format": "md", "derived": true, "tags": ["Regular"], "excerpt": "Why do all six paired states share27|83|27?", "content": "# C557 — Derive Joseph overlap geometry\n\n## Question\n\nWhy do all six paired states share27|83|27?\n\n## Inputs\n\n```json\n{\n  \"trunk\": 137,\n  \"collateral\": 110\n}\n```\n\n## Sources\n\n```json\n[\n  \"File61 §11; File62 §§5–6\"\n]\n```\n\n## Opened utc\n\n2026-09-28T00:13:33.225665+00:00\n\n## Results\n\n```json\n{\n  \"partition\": [\n    27,\n    83,\n    27\n  ],\n  \"overlap\": 83,\n  \"translation_commutes_with_collateral_construction\": [\n    true,\n    true,\n    true,\n    true,\n    true,\n    true\n  ]\n}\n```\n\n## Finding\n\nThe27|83|27 geometry follows from lengths137 and110. Translation commutes with fixed-offset collateral construction, so twelve intervals reduce to one shape in six placements.\n\n## Reassessment\n\nCompare the moving collateral lives with fixed regular Joseph.\n\n## Checks\n\n```json\n{\n  \"shape\": true,\n  \"all_states\": true\n}\n```\n\n## Closed utc\n\n2026-09-28T00:13:33.226047+00:00\n\n## Record provenance\n\nExact record hash: `2f5aeccbea999cf71652786151b395b5347b29412bab2e753ad347c31b974f08`\n\nSource journal: `Research_Cycles/C0532_C0631/journal.json`", "journalDownload": "/sources/7a2e1117efc741218b68e2db8bb20eadd33cab34e00384e9b903708031da1d2d.json", "evidenceLinks": [{"id": "d-be42daca1d82dac9", "title": "File 62"}, {"id": "d-d580afd48d6e12d6", "title": "File 61"}]}