C174 — Reflection exchanges diagonal defects and adjusts the side residual

23 September 2026 · Step 19 of 50, continuing C155.

The generalized square remains reflection-compatible, but its residual labels transform. Reflect all vertices, reverse event order, and use new BC rows X,M with new AD columns D,U. Then

The square matrix is numerically unchanged precisely when α=β; otherwise its unequal diagonal entries exchange places. C150's invariance is recovered at α=β=0.

This transforms the square's numerical roles, not regular source profiles into cumulative ones. The separate native Creation and ordinary-Flood residuals are not silently carried to new historical identities. All reflected pair components retain File_09's negation-and-order-reversal rule.

Evidence and scope

K=529. Positive BC base B denotes (B+1)t/Bn BC; reflected AD base A denotes AD At/(A+1)n. Their same-season civil span is B+A, after the no-year-zero correction. AD source projections are not historical AD occurrences. Generated coordinates and interval midpoints do not acquire event identity from numerical coincidence.

Regular Gears remain confined to Noah/Shem/Flood. Creation and Conquest use fixed source-field members; no fourth Gear is admitted. File_18/File_09 and the C127 graph are unchanged. C133–C155 source corrections remain controlling, with the C10 ancestry of the next-Flood-Gear rule now explicit. No historical campaign is rerun, no quarantined primer is reopened, and no N01/N03 expansion or missing Shem join is repaired. Original Reports 01–10 remain previously verified; interrupted original Reports 11–25 remain unestablished.

Sources: controlling File_18 §§2–3, 6A–6D; File_09 paired Mirror rules; selectively retained C05, C08, C10, C123, C128 and C133–C155 evidence. Specific dependencies are named in the body/certificate. 101 assertions passed; see RESULTS.json and CERTIFICATE.json. Assertions, selected field combinations and inherited identities are not independent historical witness counts.