{"id": "c1441", "title": "C1441 — Explain gains by decimal register", "filename": "C1441.md", "role": "Research action", "step": 1441, "sha256": "6683feafa7006ff6d6926c29d499de07e43acbb50b2ef06d4cffbf3e2848f0af", "bytes": 2623, "origins": ["C480–C1634/Research_Cycles/C1433_C1482/journal.json#C1441"], "download": "/sources/6683feafa7006ff6d6926c29d499de07e43acbb50b2ef06d4cffbf3e2848f0af.md", "format": "md", "derived": true, "tags": ["Regular", "Cumulative", "Inverse"], "excerpt": "Can the signed reversal gains be computed from the source digits without treating them as unrelated constants?", "content": "# C1441 — Explain gains by decimal register\n\n## Question\n\nCan the signed reversal gains be computed from the source digits without treating them as unrelated constants?\n\n## Sources\n\n```json\n[\n  \"File52c decimal operation\",\n  \"Six registered source spines\"\n]\n```\n\n## Opened utc\n\n2026-09-29T01:13:46.070184+00:00\n\n## Results\n\n```json\n[\n  {\n    \"spine\": \"MT_regular\",\n    \"n\": 1050,\n    \"I_n\": 5010,\n    \"core\": \"105\",\n    \"zeros\": 1,\n    \"gain\": 3960,\n    \"formula_gain\": 3960\n  },\n  {\n    \"spine\": \"MT_regular\",\n    \"n\": 1650,\n    \"I_n\": 5610,\n    \"core\": \"165\",\n    \"zeros\": 1,\n    \"gain\": 3960,\n    \"formula_gain\": 3960\n  },\n  {\n    \"spine\": \"MT_cumulative\",\n    \"n\": 3430,\n    \"I_n\": 3430,\n    \"core\": \"343\",\n    \"zeros\": 1,\n    \"gain\": 0,\n    \"formula_gain\": 0\n  },\n  {\n    \"spine\": \"MT_cumulative\",\n    \"n\": 9170,\n    \"I_n\": 7190,\n    \"core\": \"917\",\n    \"zeros\": 1,\n    \"gain\": -1980,\n    \"formula_gain\": -1980\n  },\n  {\n    \"spine\": \"LXX_regular\",\n    \"n\": 1830,\n    \"I_n\": 3810,\n    \"core\": \"183\",\n    \"zeros\": 1,\n    \"gain\": 1980,\n    \"formula_gain\": 1980\n  },\n  {\n    \"spine\": \"LXX_regular\",\n    \"n\": 2250,\n    \"I_n\": 5220,\n    \"core\": \"225\",\n    \"zeros\": 1,\n    \"gain\": 2970,\n    \"formula_gain\": 2970\n  },\n  {\n    \"spine\": \"LXX_cumulative\",\n    \"n\": 4340,\n    \"I_n\": 4340,\n    \"core\": \"434\",\n    \"zeros\": 1,\n    \"gain\": 0,\n    \"formula_gain\": 0\n  },\n  {\n    \"spine\": \"LXX_cumulative\",\n    \"n\": 9150,\n    \"I_n\": 5190,\n    \"core\": \"915\",\n    \"zeros\": 1,\n    \"gain\": -3960,\n    \"formula_gain\": -3960\n  },\n  {\n    \"spine\": \"SP_regular\",\n    \"n\": 1700,\n    \"I_n\": 7100,\n    \"core\": \"17\",\n    \"zeros\": 2,\n    \"gain\": 5400,\n    \"formula_gain\": 5400\n  },\n  {\n    \"spine\": \"SP_regular\",\n    \"n\": 1300,\n    \"I_n\": 3100,\n    \"core\": \"13\",\n    \"zeros\": 2,\n    \"gain\": 1800,\n    \"formula_gain\": 1800\n  },\n  {\n    \"spine\": \"SP_cumulative\",\n    \"n\": 3310,\n    \"I_n\": 1330,\n    \"core\": \"331\",\n    \"zeros\": 1,\n    \"gain\": -1980,\n    \"formula_gain\": -1980\n  },\n  {\n    \"spine\": \"SP_cumulative\",\n    \"n\": 8680,\n    \"I_n\": 8680,\n    \"core\": \"868\",\n    \"zeros\": 1,\n    \"gain\": 0,\n    \"formula_gain\": 0\n  }\n]\n```\n\n## Finding\n\nFor abc0 spans the gain is990(c−a); for SP regular ab00 spans it is900(b−a). Source digits account for the six signed gains.\n\n## Reassessment\n\nUse these register laws to distinguish forced residues from exact cancellations.\n\n## Checks\n\n```json\n{\n  \"all_twelve_legs_explained\": true\n}\n```\n\n## Closed utc\n\n2026-09-29T01:13:46.070815+00:00\n\n## Record provenance\n\nExact record hash: `b2685555226504831a03c145f8b1eb7b80b45857db9192bd4a5a7d32a704d513`\n\nSource journal: `Research_Cycles/C1433_C1482/journal.json`", "journalDownload": "/sources/ca542c4d870fa9b9292d3dd24718f08766a5353e1b27cbb1e7c0d69061b00ecf.json", "evidenceLinks": [{"id": "file52c", "title": "File 52c"}]}