{"id": "c498", "title": "C498 — Decimal gains explain the path increments", "filename": "C498.md", "role": "Research action", "step": 498, "sha256": "692b1ebf180d855ce644571e7586a58e86128cffc4417bb5e6e40f100a857a61", "bytes": 1456, "origins": ["C480–C1634/Research_Cycles/C0482_C0531/journal.json#C498"], "download": "/sources/692b1ebf180d855ce644571e7586a58e86128cffc4417bb5e6e40f100a857a61.md", "format": "md", "derived": true, "tags": ["Rounded", "Inverse"], "excerpt": "Which algebraic properties of decimal reversal generate the observed increments?", "content": "# C498 — Decimal gains explain the path increments\n\n## Question\n\nWhich algebraic properties of decimal reversal generate the observed increments?\n\n## Inputs\n\n```json\n{\n  \"three_digit_core\": \"10*(100a+10b+c)\",\n  \"two_digit_core\": \"100*(10a+b)\",\n  \"source_segments\": [\n    1650,\n    1050,\n    9170,\n    3430,\n    1400\n  ]\n}\n```\n\n## Sources\n\n```json\n[\n  \"File52c §1.2\",\n  \"C481\",\n  \"C495–C497\"\n]\n```\n\n## Opened utc\n\n2026-09-27T22:43:04.561288+00:00\n\n## Results\n\n```json\n{\n  \"gains\": {\n    \"1650\": 3960,\n    \"1050\": 3960,\n    \"9170\": -1980,\n    \"3430\": 0,\n    \"1400\": 2700\n  },\n  \"three_digit_gain\": \"990*(c-a)\",\n  \"two_digit_gain\": \"900*(b-a)\",\n  \"general_invariant\": \"digit sum mod9; with10^k placeholders, gain divisible by9*10^k\",\n  \"not_23_preserving_example\": {\n    \"input\": 230,\n    \"inverse\": 320,\n    \"inverse_mod23\": 21\n  }\n}\n```\n\n## Finding\n\nThe primary990 increments and the2700 tail increment are decimal-place effects. Reversal does not by itself preserve the23-lattice; its conjunction with the Key families still requires particular source inputs.\n\n## Reassessment\n\nDetermine the exact compatible scaling and translation rules for the rounded path module.\n\n## Checks\n\n```json\n{\n  \"gains\": true,\n  \"negative23\": true\n}\n```\n\n## Closed utc\n\n2026-09-27T22:43:04.561498+00:00\n\n## Record provenance\n\nExact record hash: `80b38f1933d3c37d67d4e2333ccff17c6e0df5268b132a6a2c803e23a6f197f4`\n\nSource journal: `Research_Cycles/C0482_C0531/journal.json`", "journalDownload": "/sources/6eaeccad4734eff2f6ac20ba847967877b944f9758c53c712ac8b5be2ddf61eb.json", "evidenceLinks": [{"id": "file52c", "title": "File 52c"}]}