{"id": "c608", "title": "C608 — Derive covariance of anchored Keys", "filename": "C608.md", "role": "Research action", "step": 608, "sha256": "8253cd69438ac4291e178acdedb8caa4deeb11475cc48011f420364a11345305", "bytes": 1170, "origins": ["C480–C1634/Research_Cycles/C0532_C0631/journal.json#C608"], "download": "/sources/8253cd69438ac4291e178acdedb8caa4deeb11475cc48011f420364a11345305.md", "format": "md", "derived": true, "tags": [], "excerpt": "When does a translation commute with an anchored dilation?", "content": "# C608 — Derive covariance of anchored Keys\n\n## Question\n\nWhen does a translation commute with an anchored dilation?\n\n## Inputs\n\n```json\n{\n  \"anchor\": 14004,\n  \"point\": 1446,\n  \"translation\": 2,\n  \"Key\": \"300/299\"\n}\n```\n\n## Sources\n\n```json\n[\n  \"General anchored-Key grammar; C572\"\n]\n```\n\n## Opened utc\n\n2026-09-28T00:27:14.184026+00:00\n\n## Results\n\n```json\n{\n  \"translate_output\": 1406,\n  \"move_anchor_and_point\": 1406,\n  \"move_point_only\": \"420396/299\",\n  \"point_only_residual\": \"2/299\",\n  \"identity\": \"T_t D(a,k)=D(a+t,k) T_t\"\n}\n```\n\n## Finding\n\nTranslation commutes with dilation when the anchor moves with the frame. Holding the anchor while translating only the point leaves residual(k−1)t, here2/299. This formalizes why whole-pair and one-end changes behave differently.\n\n## Reassessment\n\nDetermine how allocation equivalence changes when interior junctions are retained.\n\n## Checks\n\n```json\n{\n  \"covariance\": true,\n  \"residual\": true\n}\n```\n\n## Closed utc\n\n2026-09-28T00:27:14.184353+00:00\n\n## Record provenance\n\nExact record hash: `94c0c4984af8effb5bd24e52a28db64fa404a490e7f33f7148c45ef3db5eee21`\n\nSource journal: `Research_Cycles/C0532_C0631/journal.json`", "journalDownload": "/sources/7a2e1117efc741218b68e2db8bb20eadd33cab34e00384e9b903708031da1d2d.json"}