# C196 — An exact 20K total can conceal opposite 3K and 17K residuals

23 September 2026 · Step 41 of 50, continuing C155.

**The mixed20K total alone does not establish the3K Noah first leg and17K diagonal separately.** Put ν=A−X−3K and σ=A+D−20K. Then

\[
\boxed{\sigma=\nu+\alpha.}
\]

An admitted example is A5294,D5286,X3708. Its20K total is10580, while A−X=1586=3K−1 and D+X=8994=17K+1. The opposite residuals cancel without either component attaining its target.

The decomposition is a width identity; when read as a consecutive route it uses source Noah on the BC side and the reflected cumulative D endpoint on the AD side. Reflection relates that17K comparison to the opposite orientation used in the square.

This extends the residual audit to the twentyfold family and confirms why the earlier source hierarchy imposed the Noah first-leg condition separately.

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