{"id": "c1307", "title": "C1307 — Register Key compatibility with complete path sums", "filename": "C1307.md", "role": "Research action", "step": 1307, "sha256": "f7cccd4226f526c92e260138e7fbc5b9b2ccf78fc7d2476176fd4a8cb0c01c94", "bytes": 1137, "origins": ["C480–C1634/Research_Cycles/C1132_C1431_Recovered/journal.json#C1307"], "download": "/sources/f7cccd4226f526c92e260138e7fbc5b9b2ccf78fc7d2476176fd4a8cb0c01c94.md", "format": "md", "derived": true, "tags": [], "excerpt": "When does applyingKeys before oraftergrouping give thesamefield?", "content": "# C1307 — Register Key compatibility with complete path sums\n\n## Question\n\nWhen does applyingKeys before oraftergrouping give thesamefield?\n\n## Inputs\n\n```json\n{}\n```\n\n## Sources\n\n```json\n[\n  \"C1306\",\n  \"C984–C1007 inherited\",\n  \"inherited/model/completed_path_interface_ledger.json\"\n]\n```\n\n## Opened utc\n\n2026-09-28T06:10:27.058589+00:00\n\n## Results\n\n```json\n{\n  \"Key_sum_relation\": {\n    \"path\": \"model/relation_Key_sums.json\",\n    \"sha256\": \"ba7b84e3f9838473c66fb0ecd9dfd3cb849acc3fa0a8048d0e8ad48449a34954\",\n    \"bytes\": 1035\n  }\n}\n```\n\n## Finding\n\nUniformKeys commute withsumming aselectedpath. DifferentcomponentKeys commute withblockgroupingonlyunder aconstant-factorcondition withinthatblock, so retainedcutsremain explanatorydata.\n\n## Reassessment\n\nRegister anchoredcomposition andintegerintermediate-domain conditions.\n\n## Checks\n\n```json\n{\n  \"bound_evidence\": true,\n  \"three_limits\": true\n}\n```\n\n## Closed utc\n\n2026-09-28T06:10:27.061608+00:00\n\n## Record provenance\n\nExact record hash: `938fb61b3523e5340588102c78b22635fe7b45b9b396bdb6ab0f5ae611f6ff9f`\n\nSource journal: `Research_Cycles/C1132_C1431_Recovered/journal.json`", "journalDownload": "/sources/6adb632ce171623c2bce60d7ed58ae319ccb379cf76b751c0a477c1742fd348c.json", "evidenceLinks": [{"id": "d-ba7b84e3f9838473", "title": "model/relation_Key_sums.json"}, {"id": "d-5fbd42388730d6d4", "title": "inherited/model/completed_path_interface_ledger.json"}]}