C176 — An 8K BC preimage and its AD complement are different projections
23 September 2026 · Step 21 of 50, continuing C155.
The generated BC preimage used in 3+8+2 is q−8K in signed coordinates. Its reflection is the positive AD complement 8K−q. The equal numeral does not identify the two projections.
For P6, q=AD1403t/1404n has BC preimage 2830t/2829n BC, with signed center −2829. The local complement S(q)=4232−1403 is instead AD2829t/2830n, with signed center +2829.
Thus preimage formation is T₋₈K(q), whereas S=T₈K∘R and R(S(q))=T₋₈K(q). Both generated pairs retain their declared projections; neither is an independently identified historical event.
The certificate checks all seven held Conquest members and both seasonal components. This type distinction controls the following complement comparisons and prevents the BC middle station from being silently placed on the AD side of the 3+8+2 route.
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. 33 assertions passed; see RESULTS.json and CERTIFICATE.json. Assertions, selected field combinations and inherited identities are not independent historical witness counts.