{"id": "c516", "title": "C516 — The Cainan-associated trio as one transformed object", "filename": "C516.md", "role": "Research action", "step": 516, "sha256": "770a5021ca253d1fcfa27849bb4036159202e3d2009156aabd99e40ba9e089b9", "bytes": 1457, "origins": ["C480–C1634/Research_Cycles/C0482_C0531/journal.json#C516"], "download": "/sources/770a5021ca253d1fcfa27849bb4036159202e3d2009156aabd99e40ba9e089b9.md", "format": "md", "derived": true, "tags": ["Inverse"], "excerpt": "What does one common dilation explain about the complete trio?", "content": "# C516 — The Cainan-associated trio as one transformed object\n\n## Question\n\nWhat does one common dilation explain about the complete trio?\n\n## Inputs\n\n```json\n{\n  \"inverse_Eber_Shelah_Mahalalel\": [\n    9321,\n    2421,\n    1731\n  ],\n  \"anchor\": 14726\n}\n```\n\n## Sources\n\n```json\n[\n  \"File52c §3.12.1–3.12.2\"\n]\n```\n\n## Opened utc\n\n2026-09-27T22:49:14.668348+00:00\n\n## Results\n\n```json\n{\n  \"input_triple\": [\n    9321,\n    2421,\n    1731\n  ],\n  \"output_triple\": [\n    8851,\n    1351,\n    601\n  ],\n  \"input_gaps\": [\n    6900,\n    690\n  ],\n  \"output_gaps\": [\n    7500,\n    750\n  ],\n  \"outer_widths\": [\n    7590,\n    8250\n  ],\n  \"radius_23_coefficients\": [\n    235,\n    535,\n    565\n  ],\n  \"P_integral\": [\n    false,\n    false,\n    false\n  ],\n  \"J_integral\": [\n    false,\n    false,\n    false\n  ]\n}\n```\n\n## Finding\n\nThe6900:690 partition becomes7500:750, preserving10:1. Its7590 outer width becomes8250. All three E results are integral, while P and J are fractional on these radii. The trio is one structured object with dependent displays.\n\n## Reassessment\n\nTest the mixed original/generated Cainan midpoint, which is not forced by transforming the trio alone.\n\n## Checks\n\n```json\n{\n  \"outputs\": true,\n  \"ratio\": true,\n  \"outer\": true\n}\n```\n\n## Closed utc\n\n2026-09-27T22:49:14.668625+00:00\n\n## Record provenance\n\nExact record hash: `c0abbc9184128fcaa94ed52617f36dfb207e5bad7cd78c89052b62807b5e5ab8`\n\nSource journal: `Research_Cycles/C0482_C0531/journal.json`", "journalDownload": "/sources/6eaeccad4734eff2f6ac20ba847967877b944f9758c53c712ac8b5be2ddf61eb.json", "evidenceLinks": [{"id": "file52c", "title": "File 52c"}]}