# C166 — The general square needs two diagonal residuals as well as δ

23 September 2026 · Step 11 of 50, continuing C155.

**The C155 square formula assumes both 17K diagonals are exact.** To test that condition explicitly, define α=D+X−17K and β=U+M−17K, retaining δ=X−M−2K.

\[
S=\begin{pmatrix}17K+\beta&19K+\beta+\delta\\15K+\alpha-\delta&17K+\alpha\end{pmatrix}.
\]

The regular side is X−M=2K+δ; the cumulative side is U−D=2K+δ+β−α. Therefore equal diagonals and equal sides require α=β. They need not both equal 17K unless α=β=0.

This generalization tests the selected source members without silently imposing the target diagonals. The certificate includes the prior thirty aligned comparisons and one-member source-admitted perturbations. It preserves the distinction between a coordinate being sourced and its chosen pairing satisfying an exact target relation.

## 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.