# C190 — Conjugation preserves the ratio while transporting its anchor

23 September 2026 · Step 35 of 50, continuing C155.

**The exact operator relation is S∘Pₐ∘S=Pₛ₍ₐ₎.** Complementing a dilation preserves its ratio and moves the anchor to8K−a.

C05's−4444 anchor therefore moves to+8676. The algebra remains valid, but the transported anchor is generated and is not a member of the held apparent SP Creation field. At a=2116 the anchor is fixed, recovering the commutation in C189.

This separates formal closure of affine operations from closure of a chronology's source domain. A conjugate map can exist mathematically even when the source record that would justify its anchor is absent. The canonical Mirror axis and source profiles remain unchanged.

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