{"id": "c993", "title": "C993 — Universal coarsening criterion", "filename": "C993.md", "role": "Research action", "step": 993, "sha256": "562258d166822ce5842cc47984e165c1b51fc073f5db347d8af974455fe8b2ad", "bytes": 1304, "origins": ["C480–C1634/Research_Cycles/C0932_C1131/journal.json#C993"], "download": "/sources/562258d166822ce5842cc47984e165c1b51fc073f5db347d8af974455fe8b2ad.md", "format": "md", "derived": true, "tags": [], "excerpt": "When can a componentwise conversion be replaced by one factor per merged block?", "content": "# C993 — Universal coarsening criterion\n\n## Question\n\nWhen can a componentwise conversion be replaced by one factor per merged block?\n\n## Inputs\n\n```json\n{\n  \"coarsening\": \"sum fine components into declared blocks\",\n  \"conversion\": \"diagonal factors k_i\"\n}\n```\n\n## Sources\n\n```json\n[\n  \"C991\",\n  \"C992\",\n  \"File63 retained partitions\"\n]\n```\n\n## Opened utc\n\n2026-09-28T03:54:01.790588+00:00\n\n## Results\n\n```json\n{\n  \"necessity\": \"Test each unit vector e_i: the fine factor k_i must equal its containing coarse-block factor.\",\n  \"sufficiency\": \"If factors are constant within each block, factoring that common value out of the block sum proves equality.\",\n  \"criterion\": \"C·diag(k_i)=diag(k_B)·C iff k_i=k_B within every block\"\n}\n```\n\n## Finding\n\nCoarsening commutes with conversion for every vector exactly when the factors are constant inside each merged block. Equality on one supplied total is weaker.\n\n## Reassessment\n\nExhibit the information hidden by one equal-total partial/whole comparison.\n\n## Checks\n\n```json\n{\n  \"uniform_example\": true,\n  \"mixed_not_universal\": true\n}\n```\n\n## Closed utc\n\n2026-09-28T03:54:01.791014+00:00\n\n## Record provenance\n\nExact record hash: `e154b01f56e7b52fcf5a1193c7078496c77ac689a15b3616b00dbaced8f3cb8f`\n\nSource journal: `Research_Cycles/C0932_C1131/journal.json`", "journalDownload": "/sources/f6bdc7aaaaa4fd40f1151b367204ddee7f0f9b77a9c43ae8764831b52d862670.json", "evidenceLinks": [{"id": "d-180803601a2db301", "title": "File 63"}]}