{"id": "c506", "title": "C506 — Integral endpoints require pivot compatibility", "filename": "C506.md", "role": "Research action", "step": 506, "sha256": "ec3d8cb650df293f155d0b51224cb3723ea1247a6747201439ab7c787e98164d", "bytes": 1474, "origins": ["C480–C1634/Research_Cycles/C0482_C0531/journal.json#C506"], "download": "/sources/ec3d8cb650df293f155d0b51224cb3723ea1247a6747201439ab7c787e98164d.md", "format": "md", "derived": true, "tags": ["Mirror / 529"], "excerpt": "What extra condition distinguishes an integral width ladder from integral dates?", "content": "# C506 — Integral endpoints require pivot compatibility\n\n## Question\n\nWhat extra condition distinguishes an integral width ladder from integral dates?\n\n## Inputs\n\n```json\n{\n  \"map\": \"D_E,a(x)=(25x−2a)/23\",\n  \"integer_source_and_pivot\": true\n}\n```\n\n## Sources\n\n```json\n[\n  \"Strategy §5C\",\n  \"C505; exact algebra\"\n]\n```\n\n## Opened utc\n\n2026-09-27T22:45:39.171525+00:00\n\n## Results\n\n```json\n{\n  \"one_stage_iff\": \"x congruent a mod23\",\n  \"two_stage_condition\": \"first outputs integral; second pivot congruent to every first output mod23\",\n  \"pair_condition\": \"if original width belongs to529Z, integral first outputs share one residue mod23\",\n  \"same_pivot_two_stage_iff\": \"x−a divisible by529\",\n  \"proof\": \"25x−2a ≡2(x−a) mod23 and2 is invertible mod23; for a repeated pivot D_E,a²(x)=a+(625/529)(x−a), gcd(625,529)=1\"\n}\n```\n\n## Finding\n\nWidths lose the pivot term; endpoints retain it. A529-multiple width guarantees a shared second-stage residue after an integral first stage, but it does not supply a licensed pivot of that residue.\n\n## Reassessment\n\nApply this criterion to the already supplied C480 pivots and the fixed12026 trial, without choosing new pivots.\n\n## Checks\n\n```json\n{\n  \"coprime\": true,\n  \"common_residue_width\": true\n}\n```\n\n## Closed utc\n\n2026-09-27T22:45:39.171772+00:00\n\n## Record provenance\n\nExact record hash: `260e6718bf7692c169bc4c7e10531a4f8684d9ad927a762ce3e5610c50872518`\n\nSource journal: `Research_Cycles/C0482_C0531/journal.json`", "journalDownload": "/sources/6eaeccad4734eff2f6ac20ba847967877b944f9758c53c712ac8b5be2ddf61eb.json"}