{"id": "c1002", "title": "C1002 — Domain covariance under translation", "filename": "C1002.md", "role": "Research action", "step": 1002, "sha256": "d6edebf7093101fe761b2d227e458500ec3b74a7e8cabf3113812ea13f160c95", "bytes": 1082, "origins": ["C480–C1634/Research_Cycles/C0932_C1131/journal.json#C1002"], "download": "/sources/d6edebf7093101fe761b2d227e458500ec3b74a7e8cabf3113812ea13f160c95.md", "format": "md", "derived": true, "tags": ["Mirror / 529"], "excerpt": "Does translating both pivots and the whole field preserve the domain?", "content": "# C1002 — Domain covariance under translation\n\n## Question\n\nDoes translating both pivots and the whole field preserve the domain?\n\n## Inputs\n\n```json\n{\n  \"translation\": 215\n}\n```\n\n## Sources\n\n```json\n[\n  \"C998\",\n  \"C1001\"\n]\n```\n\n## Opened utc\n\n2026-09-28T03:55:48.267438+00:00\n\n## Results\n\n```json\n{\n  \"translated_residue\": 283,\n  \"unchanged_period\": 529,\n  \"first_output_shift\": 215,\n  \"second_output_shift\": 215,\n  \"general_rule\": \"Translate points and both pivots together; residue shifts by t.\"\n}\n```\n\n## Finding\n\nWhole-frame translation preserves the domain obstruction and period while shifting the residue. This is the same structural covariance seen in the SP cap capacities.\n\n## Reassessment\n\nRepresent a nonintegral source field exactly instead of changing its values.\n\n## Checks\n\n```json\n{\n  \"stage_shifts\": true,\n  \"same_pivot_difference\": true\n}\n```\n\n## Closed utc\n\n2026-09-28T03:55:48.267775+00:00\n\n## Record provenance\n\nExact record hash: `d7a873e98ab35ec6e9ae9f4c8fdbcf8e79677c5bad44640e49c90d50a5525ebc`\n\nSource journal: `Research_Cycles/C0932_C1131/journal.json`", "journalDownload": "/sources/f6bdc7aaaaa4fd40f1151b367204ddee7f0f9b77a9c43ae8764831b52d862670.json"}