# C197 — The named 5+5 midpoint also needs the Noah first-leg condition

23 September 2026 · Step 42 of 50, continuing C155.

**An exact named Flood midpoint does not, by itself, force an exact3+2+5 Noah subdivision.** With ν=A−X−3K and ζ=A−Z−10K,

\[
A\to X\to M\to Z:
\quad(3K+\nu)+(2K+\delta)+(5K+\zeta-\nu-\delta).
\]

The midpoint condition is ν+δ=ζ/2. On an exact10K total it becomes ν+δ=0, reducing toδ=0 only when the Noah first leg also hasν=0.

For sourced A5294,X3708,M2649 and retained comparison landing Z4, the halves remain2645+2645, but the finer partition is1586+1059+2645. Thus the first two one-year deviations cancel inside the exact half.

Z remains a comparison landing without an independent historical birth claim. This diagnostic qualifies the equivalence among the10K,13K and17K constructions by all of their held alignment conditions.

## 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.