{"id": "c760", "title": "C760 — Prove rounding equivariance for admitted shifts", "filename": "C760.md", "role": "Research action", "step": 760, "sha256": "5ca21154aeef2b93889a76e07f0e8b7d5c5e8f7bbf942278030ee58a70f95b87", "bytes": 1311, "origins": ["C480–C1634/Research_Cycles/C0732_C0831/journal.json#C760"], "download": "/sources/5ca21154aeef2b93889a76e07f0e8b7d5c5e8f7bbf942278030ee58a70f95b87.md", "format": "md", "derived": true, "tags": ["Rounded", "Counts & NT"], "excerpt": "Which state changes commute with componentwise rounding?", "content": "# C760 — Prove rounding equivariance for admitted shifts\n\n## Question\n\nWhich state changes commute with componentwise rounding?\n\n## Inputs\n\n```json\n{\n  \"admitted_amounts\": [\n    60,\n    130,\n    215,\n    460\n  ]\n}\n```\n\n## Sources\n\n```json\n[\n  \"File51a variants; C752\"\n]\n```\n\n## Opened utc\n\n2026-09-28T01:55:50.707706+00:00\n\n## Results\n\n```json\n{\n  \"identity\": \"For k=5m, Q(n+k)=5 floor((n+5m+2)/5)=Q(n)+5m. Thus row residuals and corresponding boundary residuals persist under compatible admitted changes.\",\n  \"checked_amounts\": {\n    \"60\": true,\n    \"130\": true,\n    \"215\": true,\n    \"460\": true\n  },\n  \"scope\": \"compatible coordinates or rows; insertion retains topology; finite source states only\"\n}\n```\n\n## Finding\n\nRounding compatibility is an equivariance under multiples of five. It explains why several variants preserve residual profiles without making all chronology operations interchangeable.\n\n## Reassessment\n\nTest counted-versus-completed input order where the shift is not a multiple of five.\n\n## Checks\n\n```json\n{\n  \"all_shifts\": true,\n  \"one_unit_counterexample\": true\n}\n```\n\n## Closed utc\n\n2026-09-28T01:55:50.708572+00:00\n\n## Record provenance\n\nExact record hash: `56187441824de6877048d05176f46bedcef35df34d274dc6aa93645d11bbc0b3`\n\nSource journal: `Research_Cycles/C0732_C0831/journal.json`", "journalDownload": "/sources/8ea2b882289fbcea5cf11177923617f0c1d1ad7e6eaa3bffb174c6f89699d97d.json", "evidenceLinks": [{"id": "d-d8aa38afb2a2df8f", "title": "File 51a"}]}