{"id": "c507", "title": "C507 — Explain successful and fractional endpoint completions", "filename": "C507.md", "role": "Research action", "step": 507, "sha256": "931cf98e36110a94eadcd87c816660c1abe978cad1c3fa4d2f075e7396af138a", "bytes": 1815, "origins": ["C480–C1634/Research_Cycles/C0482_C0531/journal.json#C507"], "download": "/sources/931cf98e36110a94eadcd87c816660c1abe978cad1c3fa4d2f075e7396af138a.md", "format": "md", "derived": true, "tags": ["Mirror / 529"], "excerpt": "Does the pivot-residue theorem explain the two recorded529 realizations?", "content": "# C507 — Explain successful and fractional endpoint completions\n\n## Question\n\nDoes the pivot-residue theorem explain the two recorded529 realizations?\n\n## Inputs\n\n```json\n{\n  \"C480\": {\n    \"pair\": [\n      2148,\n      3206\n    ],\n    \"first_pivot\": 3620,\n    \"second_pivot\": 2756\n  },\n  \"Actual_fixed_A\": {\n    \"pair\": [\n      4114,\n      3056\n    ],\n    \"first_pivot\": 12026,\n    \"second_pivot\": 12026\n  }\n}\n```\n\n## Sources\n\n```json\n[\n  \"C480 checkpoint; C481 report\",\n  \"C506\"\n]\n```\n\n## Opened utc\n\n2026-09-27T22:45:59.468245+00:00\n\n## Results\n\n```json\n{\n  \"C480\": {\n    \"first\": [\n      2020,\n      3170\n    ],\n    \"first_mod23\": [\n      19,\n      19\n    ],\n    \"second_pivot_mod23\": 19,\n    \"second\": [\n      1956,\n      3206\n    ],\n    \"fractional_parts\": [\n      0,\n      0\n    ],\n    \"second_width\": 1250\n  },\n  \"Actual_fixed_A\": {\n    \"first\": [\n      3426,\n      2276\n    ],\n    \"first_mod23\": [\n      22,\n      22\n    ],\n    \"second_pivot_mod23\": 20,\n    \"second\": [\n      \"61598/23\",\n      \"32848/23\"\n    ],\n    \"fractional_parts\": [\n      \"4/23\",\n      \"4/23\"\n    ],\n    \"second_width\": 1250\n  }\n}\n```\n\n## Finding\n\nC480’s first endpoints and second pivot all have residue19, so the supplied completion is integral. The fixed12026 trial has first residue22 against pivot20, leaving a common4/23 fraction. Both widths still reach1250. This explains the recorded difference without adding a pivot.\n\n## Reassessment\n\nCompare another family where two different Keys produce the same gain: the SP rectangle.\n\n## Checks\n\n```json\n{\n  \"C480\": true,\n  \"fraction\": true,\n  \"widths\": true\n}\n```\n\n## Closed utc\n\n2026-09-27T22:45:59.468546+00:00\n\n## Record provenance\n\nExact record hash: `28e962c365ec018d66adeacf8a6fd99dae307eefa6fc9283c140097f2009abd0`\n\nSource journal: `Research_Cycles/C0482_C0531/journal.json`", "journalDownload": "/sources/6eaeccad4734eff2f6ac20ba847967877b944f9758c53c712ac8b5be2ddf61eb.json"}