# C188 — The 300/299 comparison singles out the 13-fold width

23 September 2026 · Step 33 of 50, continuing C155.

**The inherited6877→6900 comparison requires an exact13K width.** For the residual family,

\[
(13K-\delta)\frac{300}{299}=6900-\frac{300\delta}{299}.
\]

Whole-year widths require299|δ; onlyδ=0 qualifies in−3…6. At a general nK width, integrality requires13|n because299=13×23 and K=23².

This explains why300/299 is not an integer-width operation on every529 multiple. The C05 source-qualified6900/7500 construction remains valid under its declared anchoring; this test does not extend its source realization to the whole square or repair P6's6876 interval.

The ratios25/23 and300/299 therefore have different divisibility conditions, even though their retained family connections are exact.

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