{"id": "c873", "title": "C873 — Derive the exact lattice-preservation criterion", "filename": "C873.md", "role": "Research action", "step": 873, "sha256": "4c7ea2e1dc1b648f9591eb4e63c4e1796d511379a0749f3c6c2cb1e6ba6863a5", "bytes": 1152, "origins": ["C480–C1634/Research_Cycles/C0832_C0931/journal.json#C873"], "download": "/sources/4c7ea2e1dc1b648f9591eb4e63c4e1796d511379a0749f3c6c2cb1e6ba6863a5.md", "format": "md", "derived": true, "tags": ["Rounded"], "excerpt": "When does a reduced Key p/q preserve hZ?", "content": "# C873 — Derive the exact lattice-preservation criterion\n\n## Question\n\nWhen does a reduced Key p/q preserve hZ?\n\n## Inputs\n\n```json\n{\n  \"keys\": {\n    \"E\": \"25/23\",\n    \"P\": \"70/69\",\n    \"J\": \"300/299\"\n  }\n}\n```\n\n## Sources\n\n```json\n[\n  \"C872 declared domains\"\n]\n```\n\n## Opened utc\n\n2026-09-28T02:46:40.097441+00:00\n\n## Results\n\n```json\n{\n  \"theorem\": {\n    \"path\": \"model/one_key_grid_theorem.json\",\n    \"sha256\": \"99ef73648f71a47a6a4c02ada19ea51bdd1db7db3bd5264862b0864719ee7a34\",\n    \"bytes\": 232\n  },\n  \"reduced_denominators\": {\n    \"E\": 23,\n    \"P\": 69,\n    \"J\": 299\n  }\n}\n```\n\n## Finding\n\nA Key preserves a declared grid exactly on hqZ. This is an input-domain condition, not a demand to round rational outputs or a license to apply a Key to every chronology.\n\n## Reassessment\n\nClassify integer and Rounded domains for all three Keys and their intersection.\n\n## Checks\n\n```json\n{\n  \"coprime\": true,\n  \"grid_cancels\": true\n}\n```\n\n## Closed utc\n\n2026-09-28T02:46:40.097734+00:00\n\n## Record provenance\n\nExact record hash: `7874513c4f183894460c3b9ad772dfe9222ba95d4ef5223cbbcc59d2b79f937e`\n\nSource journal: `Research_Cycles/C0832_C0931/journal.json`", "journalDownload": "/sources/6a37be4e218f9f467cbde5e9db3723ce9c79dc78cc97d1a289422b96cb9971e0.json", "evidenceLinks": [{"id": "d-99ef73648f71a47a", "title": "model/one_key_grid_theorem.json"}]}