# C150 — Paired reflection preserves the finite residual square

23 September 2026 · Bounded step 15 of 20 (C136–C155).

**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.

The 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

\[
\begin{pmatrix}X+D&X+U\\M+D&M+U\end{pmatrix}
=\begin{pmatrix}17K&19K+\delta\\15K-\delta&17K\end{pmatrix}.
\]

The 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.

The 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.

For 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.

## Source and scope control

All 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.

Regular 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.

Evidence: `CERTIFICATE.json`; **168 passing assertions** in `RESULTS.json`. The assertion list identifies the actual tests.