{"id": "c505", "title": "C505 — The529 family is first a duration grammar", "filename": "C505.md", "role": "Research action", "step": 505, "sha256": "7eb5e68ed369dc1d399986309ed40b5e8bf9b691c9a5366ba8e507027d5e319a", "bytes": 1259, "origins": ["C480–C1634/Research_Cycles/C0482_C0531/journal.json#C505"], "download": "/sources/7eb5e68ed369dc1d399986309ed40b5e8bf9b691c9a5366ba8e507027d5e319a.md", "format": "md", "derived": true, "tags": ["Rounded", "Mirror / 529"], "excerpt": "What does two-stage E expansion guarantee without any pivot choice?", "content": "# C505 — The529 family is first a duration grammar\n\n## Question\n\nWhat does two-stage E expansion guarantee without any pivot choice?\n\n## Inputs\n\n```json\n{\n  \"E\": \"25/23\",\n  \"small_seed\": 1058,\n  \"macro_seed\": 10580\n}\n```\n\n## Sources\n\n```json\n[\n  \"C480 checkpoint\",\n  \"File52c §§3.8–3.9,6\",\n  \"C481\"\n]\n```\n\n## Opened utc\n\n2026-09-27T22:45:10.631228+00:00\n\n## Results\n\n```json\n{\n  \"small\": [\n    1058,\n    1150,\n    1250\n  ],\n  \"macro\": [\n    10580,\n    11500,\n    12500\n  ],\n  \"general_integer_width_rule\": \"k*529 -> k*575 -> k*625\",\n  \"proof\": \"D_k,a(y)-D_k,a(x)=k*(y-x); second pivot cancels from the width too\"\n}\n```\n\n## Finding\n\nEvery supplied529-multiple width follows this ladder under two E expansions. Tenfold scaling commutes with it. Width agreement therefore connects C480 and the rounded macro family while leaving their source roles and pivots distinct.\n\n## Reassessment\n\nDerive the additional congruence condition needed for integral endpoints.\n\n## Checks\n\n```json\n{\n  \"small\": true,\n  \"large\": true,\n  \"scale\": true\n}\n```\n\n## Closed utc\n\n2026-09-27T22:45:10.631494+00:00\n\n## Record provenance\n\nExact record hash: `aaca498f1807d3744a9f95aa62a07a6b8de9dd53b097979c73df677fb1b93bc7`\n\nSource journal: `Research_Cycles/C0482_C0531/journal.json`", "journalDownload": "/sources/6eaeccad4734eff2f6ac20ba847967877b944f9758c53c712ac8b5be2ddf61eb.json", "evidenceLinks": [{"id": "file52c", "title": "File 52c"}]}