{"id": "c1000", "title": "C1000 — Verify fixed-pivot domains", "filename": "C1000.md", "role": "Research action", "step": 1000, "sha256": "6e0c3ba1b4d5a0782244422c72e89c5864e35cde2dbeb6650c183bf74003971b", "bytes": 969, "origins": ["C480–C1634/Research_Cycles/C0932_C1131/journal.json#C1000"], "download": "/sources/6e0c3ba1b4d5a0782244422c72e89c5864e35cde2dbeb6650c183bf74003971b.md", "format": "md", "derived": true, "tags": [], "excerpt": "Do the declared residue classes preserve integrality at both stages?", "content": "# C1000 — Verify fixed-pivot domains\n\n## Question\n\nDo the declared residue classes preserve integrality at both stages?\n\n## Inputs\n\n```json\n{}\n```\n\n## Sources\n\n```json\n[\n  \"C999\"\n]\n```\n\n## Opened utc\n\n2026-09-28T03:55:11.412548+00:00\n\n## Results\n\n```json\n{\n  \"nonempty_class_checks\": 24,\n  \"pivot_difference_mod3\": 1,\n  \"J_to_P_reason\": \"gcd(300,69)=3 does not divide−9170\"\n}\n```\n\n## Finding\n\nAll nonempty class formulas pass exact staged substitution. J→P fails because the two fixed pivots differ modulo3; no endpoint choice repairs that within the declared integer domain.\n\n## Reassessment\n\nExtend the congruence result from one point to an entire labelled field.\n\n## Checks\n\n```json\n{\n  \"all_substitutions\": true,\n  \"obstruction\": true\n}\n```\n\n## Closed utc\n\n2026-09-28T03:55:11.413103+00:00\n\n## Record provenance\n\nExact record hash: `b0ce4d9805172c518f2bdc73dfa1d9406777f1135d5baa0f8fc5008d4eb164e4`\n\nSource journal: `Research_Cycles/C0932_C1131/journal.json`", "journalDownload": "/sources/f6bdc7aaaaa4fd40f1151b367204ddee7f0f9b77a9c43ae8764831b52d862670.json"}