# 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

\[
\alpha'=\beta,\qquad\beta'=\alpha,\qquad\delta'=\delta+\beta-\alpha.
\]

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.