{"id": "c971", "title": "C971 — Bidirectional cap equation packet", "filename": "C971.md", "role": "Research action", "step": 971, "sha256": "9a535a7dd44632b6e2022e91e71a54a48f02c321257c139f695b1f4f21200c09", "bytes": 1387, "origins": ["C480–C1634/Research_Cycles/C0932_C1131/journal.json#C971"], "download": "/sources/9a535a7dd44632b6e2022e91e71a54a48f02c321257c139f695b1f4f21200c09.md", "format": "md", "derived": true, "tags": ["Regular", "Cumulative", "Inverse"], "excerpt": "How much independent information do forward and inverse reconstructions require?", "content": "# C971 — Bidirectional cap equation packet\n\n## Question\n\nHow much independent information do forward and inverse reconstructions require?\n\n## Inputs\n\n```json\n{\n  \"variables\": [\n    \"bJ\",\n    \"bE\",\n    \"bM\",\n    \"bL\",\n    \"lifeJ\",\n    \"lifeM\",\n    \"lifeL\",\n    \"k\"\n  ]\n}\n```\n\n## Sources\n\n```json\n[\n  \"C964\",\n  \"C965\",\n  \"C970\"\n]\n```\n\n## Opened utc\n\n2026-09-28T03:49:35.209792+00:00\n\n## Results\n\n```json\n{\n  \"equations\": [\n    [\n      1,\n      1,\n      1,\n      1,\n      -1,\n      0,\n      0,\n      1\n    ],\n    [\n      0,\n      0,\n      1,\n      1,\n      0,\n      -1,\n      0,\n      1\n    ],\n    [\n      0,\n      0,\n      0,\n      1,\n      0,\n      0,\n      -1,\n      1\n    ]\n  ],\n  \"rank\": 3,\n  \"unconstrained_dimension\": 5,\n  \"after_k_and_Enoch_held\": 3\n}\n```\n\n## Finding\n\nThe late-row relation has three independent equations on eight variables. Holding k and Enoch leaves equivalent three-number choices: the changed begettings or the capped lives. This is bidirectional constraint, not double evidence.\n\n## Reassessment\n\nPropagate the cap-derived reductions into the cumulative field.\n\n## Checks\n\n```json\n{\n  \"unique_life_columns\": true,\n  \"dimension\": true\n}\n```\n\n## Closed utc\n\n2026-09-28T03:49:35.210019+00:00\n\n## Record provenance\n\nExact record hash: `18f9402a8cb0d54db1c43591f00741da5a12074576156ff93e4888f59c086fb2`\n\nSource journal: `Research_Cycles/C0932_C1131/journal.json`", "journalDownload": "/sources/f6bdc7aaaaa4fd40f1151b367204ddee7f0f9b77a9c43ae8764831b52d862670.json"}