{"id": "c609", "title": "C609 — Generalize allocation endpoint equivalence", "filename": "C609.md", "role": "Research action", "step": 609, "sha256": "66962fe78faa165517deb090ebdd75e5bdd6d57b5c93eff95e7c3f316962a3fe", "bytes": 1473, "origins": ["C480–C1634/Research_Cycles/C0532_C0631/journal.json#C609"], "download": "/sources/66962fe78faa165517deb090ebdd75e5bdd6d57b5c93eff95e7c3f316962a3fe.md", "format": "md", "derived": true, "tags": ["Rounded", "Inverse"], "excerpt": "How large is the junction discrepancy between whole-Key and terminal-E routes?", "content": "# C609 — Generalize allocation endpoint equivalence\n\n## Question\n\nHow large is the junction discrepancy between whole-Key and terminal-E routes?\n\n## Inputs\n\n```json\n{\n  \"examples\": [\n    [\n      483,\n      \"1/6\",\n      \"70/69\"\n    ],\n    [\n      12558,\n      \"1/26\",\n      \"300/299\"\n    ]\n  ]\n}\n```\n\n## Sources\n\n```json\n[\n  \"C568–C571; algebraic path comparison\"\n]\n```\n\n## Opened utc\n\n2026-09-28T00:27:38.215901+00:00\n\n## Results\n\n```json\n{\n  \"comparisons\": [\n    {\n      \"S\": 483,\n      \"prefix\": \"805/2\",\n      \"suffix\": \"161/2\",\n      \"total\": 490,\n      \"whole_total\": 490,\n      \"junction_gap\": \"35/6\"\n    },\n    {\n      \"S\": 12558,\n      \"prefix\": 12075,\n      \"suffix\": 483,\n      \"total\": 12600,\n      \"whole_total\": 12600,\n      \"junction_gap\": \"525/13\"\n    }\n  ],\n  \"general_condition\": \"k=1+f(E−1)\",\n  \"junction_gap_formula\": \"(k−1)(1−f)S\"\n}\n```\n\n## Finding\n\nFor both source partitions the endpoint agreement follows from one gain-allocation law, while the interior discrepancy is fixed by(k−1)(1−f)S. Retaining the path reveals exactly what the total-span comparison forgets.\n\n## Reassessment\n\nTest why the Rounded inverse module must preserve component boundaries.\n\n## Checks\n\n```json\n{\n  \"endpoints\": true,\n  \"gaps\": true\n}\n```\n\n## Closed utc\n\n2026-09-28T00:27:38.216246+00:00\n\n## Record provenance\n\nExact record hash: `7f35356e54ac66cba7786f922af6e959e6a4780608e91cee1d35582d3b880bd6`\n\nSource journal: `Research_Cycles/C0532_C0631/journal.json`", "journalDownload": "/sources/7a2e1117efc741218b68e2db8bb20eadd33cab34e00384e9b903708031da1d2d.json"}