{"id": "c501", "title": "C501 — The three-Key integral domain", "filename": "C501.md", "role": "Research action", "step": 501, "sha256": "6620e74282529457b5a2a29496ce1d5e4bedd10c5360e11d6652c8620b0a4f27", "bytes": 1562, "origins": ["C480–C1634/Research_Cycles/C0482_C0531/journal.json#C501"], "download": "/sources/6620e74282529457b5a2a29496ce1d5e4bedd10c5360e11d6652c8620b0a4f27.md", "format": "md", "derived": true, "tags": ["Calendar Keys"], "excerpt": "Why does the middle member admit all three integral calendar outputs?", "content": "# C501 — The three-Key integral domain\n\n## Question\n\nWhy does the middle member admit all three integral calendar outputs?\n\n## Inputs\n\n```json\n{\n  \"Jared\": 8372,\n  \"Noah_G1\": 8970,\n  \"Flood_G2\": 9568,\n  \"Creation\": 7912\n}\n```\n\n## Sources\n\n```json\n[\n  \"File52c §3.15\",\n  \"C500\"\n]\n```\n\n## Opened utc\n\n2026-09-27T22:43:53.341661+00:00\n\n## Results\n\n```json\n{\n  \"common_input_lattice\": 897,\n  \"rows\": {\n    \"Jared\": {\n      \"E\": 9100,\n      \"P\": \"25480/3\",\n      \"J\": 8400,\n      \"all_integral\": false\n    },\n    \"Noah_G1\": {\n      \"E\": 9750,\n      \"P\": 9100,\n      \"J\": 9000,\n      \"all_integral\": true\n    },\n    \"Flood_G2\": {\n      \"E\": 10400,\n      \"P\": \"29120/3\",\n      \"J\": 9600,\n      \"all_integral\": false\n    },\n    \"Creation\": {\n      \"E\": 8600,\n      \"P\": \"24080/3\",\n      \"J\": \"103200/13\",\n      \"all_integral\": false\n    }\n  },\n  \"JNF_mod3_coefficients\": [\n    2,\n    0,\n    1\n  ]\n}\n```\n\n## Finding\n\nThe common integral span domain is897Z, since897=lcm(23,69,299). For598m, E and J are integral for all integer m; P additionally needs3|m. Of14,15,16 only Noah’s15 qualifies. This explains the three-Key middle role from the existing progression.\n\n## Reassessment\n\nUse those outputs to derive the shared2926 endpoint and assess which placement assumptions remain.\n\n## Checks\n\n```json\n{\n  \"lattice\": true,\n  \"middle_only\": true\n}\n```\n\n## Closed utc\n\n2026-09-27T22:43:53.341906+00:00\n\n## Record provenance\n\nExact record hash: `3ad039dfc81a6b222c3776f8f9384647015ccaf55a0585aaa31e4e508fecbee3`\n\nSource journal: `Research_Cycles/C0482_C0531/journal.json`", "journalDownload": "/sources/6eaeccad4734eff2f6ac20ba847967877b944f9758c53c712ac8b5be2ddf61eb.json", "evidenceLinks": [{"id": "file52c", "title": "File 52c"}]}