{"id": "c150", "title": "C150 — Paired reflection preserves the finite residual square", "filename": "490d_C150_529_Mirror_Bounded_Step_20260923.md", "role": "Research action", "step": 150, "sha256": "83e909348b7712b079d04fa7791f717163140f981516a9ede4e3618977474b12", "bytes": 2381, "origins": ["C01–C479/reference/490d_C01_C265_Complete_Window_Archive_20260923.zip!/workspace/mirror_C136_C155_529_Square_Batch_20260923/C150/490d_C150_529_Mirror_Bounded_Step_20260923.md"], "download": "/sources/83e909348b7712b079d04fa7791f717163140f981516a9ede4e3618977474b12.md", "format": "md", "derived": false, "tags": ["Regular", "Cumulative", "Inverse", "Mirror / 529"], "excerpt": "C150 — Paired reflection preserves the finite residual square\n\n23 September 2026 · Bounded step 15 of 20 (C136–C155).\n\nReflecting all four vertices preserves the same numerical square when event order is restored. It exchanges the cumulative-side and regular-side roles; it does not identify them.\n\nThe original arrangem", "content": "# C150 — Paired reflection preserves the finite residual square\n\n23 September 2026 · Bounded step 15 of 20 (C136–C155).\n\n**Reflecting all four vertices preserves the same numerical square when event order is restored.** It exchanges the cumulative-side and regular-side roles; it does not identify them.\n\nThe original arrangement has BC rows U,D and reflected AD columns M,X. Global reflection gives BC rows X,M and reflected AD columns D,U. Its matrix is\n\n\\[\n\\begin{pmatrix}X+D&X+U\\\\M+D&M+U\\end{pmatrix}\n=\\begin{pmatrix}17K&19K+\\delta\\\\15K-\\delta&17K\\end{pmatrix}.\n\\]\n\nThe diagonal positions exchange, but their equal values make the displayed matrix unchanged. The two side lengths exchange roles and remain 2K+δ. This invariance follows from reflection and equal diagonals; it is not a second independent family of source witnesses.\n\nThe check uses File_09/C127's paired convention: BC base b has coordinates −b−¼, −b+¼, and reflected AD base b has b−¼, b+¼. Negation plus reversal of component order maps one pair to the other. Both corresponding civil-season counts remain b+a.\n\nFor retained U members without materialized graph nodes, the reflected display is a permitted projection of the admitted field member, not a newly stored historical event. The historical graph, axis and source profiles remain fixed.\n\n## Source and scope control\n\nAll numerical bases in this packet stand for paired coordinates: BC base B is (B+1)t/Bn BC; reflected AD base A is AD At/(A+1)n. Thus a cross-Mirror same-season span is B+A, after the civil no-year-zero subtraction. AD stations are reflected source events, not historical AD occurrences. K=529 throughout.\n\nRegular Gears remain restricted to Noah/Shem/Flood. Creation and Conquest are fixed source-field selections. C133–C135 corrections remain controlling. The C127 graph and canonical controls are frozen; source-field admission is distinguished from an existing graph node and from a generated comparison coordinate. No wider N01/N03 expansion or missing N03 Shem join is repaired. Original Reports 01–10 remain previously verified references; interrupted original Reports 11–25 remain unestablished. Check counts are computational assertions, not independent historical witnesses.\n\nEvidence: `CERTIFICATE.json`; **168 passing assertions** in `RESULTS.json`. The assertion list identifies the actual tests.\n", "evidenceLinks": [{"id": "d-45851004c5864229", "title": "File 09"}]}