{"id": "c1001", "title": "C1001 — Whole-field placement and difference conditions", "filename": "C1001.md", "role": "Research action", "step": 1001, "sha256": "e4283dfc29647a8b4fed387b7332a678769980a5d1bdb8ad70f2ec9748a16f87", "bytes": 1512, "origins": ["C480–C1634/Research_Cycles/C0932_C1131/journal.json#C1001"], "download": "/sources/e4283dfc29647a8b4fed387b7332a678769980a5d1bdb8ad70f2ec9748a16f87.md", "format": "md", "derived": true, "tags": ["Mirror / 529"], "excerpt": "What makes every point of a source field belong to one admissible domain?", "content": "# C1001 — Whole-field placement and difference conditions\n\n## Question\n\nWhat makes every point of a source field belong to one admissible domain?\n\n## Inputs\n\n```json\n{}\n```\n\n## Sources\n\n```json\n[\n  \"C998\",\n  \"C1000\"\n]\n```\n\n## Opened utc\n\n2026-09-28T03:55:13.386588+00:00\n\n## Results\n\n```json\n{\n  \"criterion\": {\n    \"criterion\": \"All x_i lie in r+mZ iff x_0≡r mod m and every x_i−x_0 is divisible by m.\",\n    \"difference_condition\": \"Controls field shape up to translation.\",\n    \"placement_condition\": \"One retained point fixes the residue class.\"\n  },\n  \"inherited_witnesses_reclassified\": {\n    \"C480\": {\n      \"points\": [\n        2148,\n        3206\n      ],\n      \"residue\": 32,\n      \"period\": 529,\n      \"width_ok\": true,\n      \"placement_ok\": true\n    },\n    \"fixed12026\": {\n      \"points\": [\n        4114,\n        3056\n      ],\n      \"residue\": 388,\n      \"period\": 529,\n      \"width_ok\": true,\n      \"placement_ok\": false\n    }\n  }\n}\n```\n\n## Finding\n\nA whole-field test separates the difference lattice from one placement residue. The inherited529-width examples share the width condition but differ in placement.\n\n## Reassessment\n\nCheck how this domain transforms under a whole-frame translation.\n\n## Checks\n\n```json\n{\n  \"both_widths\": true,\n  \"placement_distinction\": true\n}\n```\n\n## Closed utc\n\n2026-09-28T03:55:13.386862+00:00\n\n## Record provenance\n\nExact record hash: `3c01eb998f1b0f440c676831c9ab5eab4e50200b7c7ea2de588c0cdc3d642b5b`\n\nSource journal: `Research_Cycles/C0932_C1131/journal.json`", "journalDownload": "/sources/f6bdc7aaaaa4fd40f1151b367204ddee7f0f9b77a9c43ae8764831b52d862670.json"}