{"id": "c733", "title": "C733 — Derive the full mode-bridge recurrence", "filename": "C733.md", "role": "Research action", "step": 733, "sha256": "d3d2d1bac761df190c3c53183917e00199957a04b3d5fccdddda46f290068540", "bytes": 1679, "origins": ["C480–C1634/Research_Cycles/C0732_C0831/journal.json#C733"], "download": "/sources/d3d2d1bac761df190c3c53183917e00199957a04b3d5fccdddda46f290068540.md", "format": "md", "derived": true, "tags": ["Regular", "Cumulative"], "excerpt": "What row quantity generates the change in cumulative-minus-regular displacement between adjacent named boundaries?", "content": "# C733 — Derive the full mode-bridge recurrence\n\n## Question\n\nWhat row quantity generates the change in cumulative-minus-regular displacement between adjacent named boundaries?\n\n## Inputs\n\n```json\n{\n  \"definition\": \"G_i=C_i−R_i\",\n  \"regular_edge\": \"R_i−R_(i+1)=b_eff_i\",\n  \"cumulative_edge\": \"C_i−C_(i+1)=L_i\",\n  \"local_MT_example\": {\n    \"regular\": [\n      2166,\n      2066,\n      2006\n    ],\n    \"cumulative\": [\n      2435,\n      2260,\n      2080\n    ],\n    \"lifespans\": [\n      175,\n      180\n    ],\n    \"effective_begetting\": [\n      100,\n      60\n    ]\n  }\n}\n```\n\n## Sources\n\n```json\n[\n  \"File18 MT regular/cumulative rows; File61 root vector; Research Strategy §5B\"\n]\n```\n\n## Opened utc\n\n2026-09-28T01:40:40.628020+00:00\n\n## Results\n\n```json\n{\n  \"mode_bridge\": [\n    269,\n    194,\n    74\n  ],\n  \"bridge_increments\": [\n    75,\n    120\n  ],\n  \"row_remainders\": [\n    75,\n    120\n  ],\n  \"general_rule\": \"G_i−G_(i+1)=L_i−b_eff_i\"\n}\n```\n\n## Finding\n\nThe mode bridge has a local generator: lifespan minus the effective regular edge. Telescoping these row contributions reconstructs the whole bridge once the terminal displacement is supplied. Declared binding corrections belong in b_eff rather than being hidden in a global conversion.\n\n## Reassessment\n\nApply this recurrence to every shared MT boundary, retaining source roles and terminal frame.\n\n## Checks\n\n```json\n{\n  \"local_increments\": true,\n  \"projection\": true,\n  \"telescoping\": true\n}\n```\n\n## Closed utc\n\n2026-09-28T01:40:40.628246+00:00\n\n## Record provenance\n\nExact record hash: `d3dc745a31bc5ec3060088752416c05e84342c9b46746ae3d26e6001d289e35d`\n\nSource journal: `Research_Cycles/C0732_C0831/journal.json`", "journalDownload": "/sources/8ea2b882289fbcea5cf11177923617f0c1d1ad7e6eaa3bffb174c6f89699d97d.json", "evidenceLinks": [{"id": "d-68301ab760bfbe2e", "title": "File 18"}, {"id": "d-d580afd48d6e12d6", "title": "File 61"}]}