{"id": "c667", "title": "C667 — Explain retained-part conversion algebraically", "filename": "C667.md", "role": "Research action", "step": 667, "sha256": "15e7b15f3f9f56f88c23290975f88c213771c3c688bd1573d844f250f5cc25d7", "bytes": 1359, "origins": ["C480–C1634/Research_Cycles/C0632_C0731/journal.json#C667"], "download": "/sources/15e7b15f3f9f56f88c23290975f88c213771c3c688bd1573d844f250f5cc25d7.md", "format": "md", "derived": true, "tags": [], "excerpt": "What term distinguishes converting a core from converting its entire bracket?", "content": "# C667 — Explain retained-part conversion algebraically\n\n## Question\n\nWhat term distinguishes converting a core from converting its entire bracket?\n\n## Inputs\n\n```json\n{\n  \"core\": 690,\n  \"flanks\": [\n    30,\n    30\n  ],\n  \"operators\": [\n    \"P\",\n    \"E\"\n  ]\n}\n```\n\n## Sources\n\n```json\n[\n  \"C666\",\n  \"C631 retained-part synthesis\"\n]\n```\n\n## Opened utc\n\n2026-09-28T01:08:54.665397+00:00\n\n## Results\n\n```json\n{\n  \"rows\": [\n    {\n      \"retained\": 760,\n      \"whole\": \"17500/23\",\n      \"excess\": \"20/23\",\n      \"flank_gain\": \"20/23\"\n    },\n    {\n      \"retained\": 810,\n      \"whole\": \"18750/23\",\n      \"excess\": \"120/23\",\n      \"flank_gain\": \"120/23\"\n    }\n  ],\n  \"general_difference\": \"2(k−1)f\"\n}\n```\n\n## Finding\n\nWhole conversion adds exactly2(k−1)f more than core conversion with retained flanks. This is the same retained-subdivision distinction that explains the earlier partial-E and whole-J routes: matching totals never establish identical internal paths.\n\n## Reassessment\n\nReconstruct the native alternating ladder from its two rail parameters.\n\n## Checks\n\n```json\n{\n  \"P_excess\": true,\n  \"E_excess\": true,\n  \"identity\": true\n}\n```\n\n## Closed utc\n\n2026-09-28T01:08:54.665748+00:00\n\n## Record provenance\n\nExact record hash: `65ceaa2a2d6692224d37483637d748381cb2e0a3bee2146ee8594dc7d403a603`\n\nSource journal: `Research_Cycles/C0632_C0731/journal.json`", "journalDownload": "/sources/f0643e934c6a7443706351fa40040684d915ed0ae25c0f9188a4c4dba12814ff.json"}