# C189 — Dilation commutes with the 8K complement only at its center

23 September 2026 · Step 34 of 50, continuing C155.

**For a nontrivial fixed-anchor dilation, commuting with S requires anchor4K=2116.** Let Pₐ(x)=a+(25/23)(x−a). Then

\[
P_a(S(x))-S(P_a(x))=(25/23-1)(8K-2a).
\]

The difference vanishes exactly at a=4K. At C05's retained apparent-Creation anchor−4444, it is26240/23 rather than zero.

This does not contradict C05's commuting diagram. That report combines a common affine map with a right-endpoint-held relative interval extension; its second operation is not the fixed8K complement S. Distinguishing the operators prevents an established local commutation from being generalized to a different symmetry.

The result is affine algebra on comparison coordinates. It does not authorize a new historical dilation about2116.

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