# C182 — The complemented square can lie wholly on the AD side

23 September 2026 · Step 27 of 50, continuing C155.

**The8K complement preserves all four square lengths while placing their images on the AD side in this finite family.** With s=d+1,

\[
S(-U)=20K-j,\quad S(-D)=18K-s,
\quad S(M)=3K-j,\quad S(X)=K-s.
\]

Each transformed interval is measured by same-side subtraction; its length equals the original cross-Mirror span. P6 maps its paired centers to10575,9516,1582,523, giving the unchanged matrix [[8993,10052],[7934,8993]].

These positive AD coordinates are generated affine images, not claims of historical or predicted events in those years. The example shows why preservation of a numerical square does not establish preservation of its source chronology. The ordinary global Mirror remains R(x)=−x; S is its composition with the declared8K translation.

## 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. **49 assertions passed**; see `RESULTS.json` and `CERTIFICATE.json`. Assertions, selected field combinations and inherited identities are not independent historical witness counts.