{"id": "c1209", "title": "C1209 — State the exact cap-round commutation condition", "filename": "C1209.md", "role": "Research action", "step": 1209, "sha256": "80572df91e09bab72615f4b1c499d99cb656c6ac38700adf268b17dd649b25a1", "bytes": 1094, "origins": ["C480–C1634/Research_Cycles/C1132_C1431_Recovered/journal.json#C1209"], "download": "/sources/80572df91e09bab72615f4b1c499d99cb656c6ac38700adf268b17dd649b25a1.md", "format": "md", "derived": true, "tags": ["Rounded"], "excerpt": "When does rounding commute with the cap?", "content": "# C1209 — State the exact cap-round commutation condition\n\n## Question\n\nWhen does rounding commute with the cap?\n\n## Inputs\n\n```json\n{}\n```\n\n## Sources\n\n```json\n[\n  \"C1208\",\n  \"C1131 inherited cap-round theorem\"\n]\n```\n\n## Opened utc\n\n2026-09-28T05:27:11.159718+00:00\n\n## Results\n\n```json\n{\n  \"commutation_scope\": {\n    \"path\": \"model/cap_round_commutation_scope.json\",\n    \"sha256\": \"3a28ca3976b2b3e0c3db3acc855fbab20e1484a32e4cf271b3787b19eb686a3f\",\n    \"bytes\": 450\n  }\n}\n```\n\n## Finding\n\nCap and rounding commute when both quantities are rounded by the same monotone rule. Keeping the capacity exact on only one route changes the operation; it is not a counterexample to that theorem.\n\n## Reassessment\n\nTest whether rounding preserves the strict fact that a life exceeded capacity.\n\n## Checks\n\n```json\n{\n  \"monotone_checks\": true,\n  \"witness_values\": true\n}\n```\n\n## Closed utc\n\n2026-09-28T05:27:11.160246+00:00\n\n## Record provenance\n\nExact record hash: `74a19343aab6121ea572fd8ed73239378aaa8634bdb5fe33ead4fdee495f8336`\n\nSource journal: `Research_Cycles/C1132_C1431_Recovered/journal.json`", "journalDownload": "/sources/6adb632ce171623c2bce60d7ed58ae319ccb379cf76b751c0a477c1742fd348c.json", "evidenceLinks": [{"id": "d-3a28ca3976b2b3e0", "title": "model/cap_round_commutation_scope.json"}]}