# C186 — One 25/23 expansion separates the exact grid from residual widths

23 September 2026 · Step 31 of 50, continuing C155.

**A25/23 expansion carries K to575 but carries δ to25δ/23.** Hence the square becomes

\[
\begin{pmatrix}17\cdot575&19\cdot575+25\delta/23\\15\cdot575-25\delta/23&17\cdot575\end{pmatrix}.
\]

In the finite residual range−3…6, all four widths remain whole years only atδ=0. P6'sδ=1 produces opposite fractional-year residuals±25/23, while both diagonal widths are9775.

These are calculations on paired-center interval lengths. They do not yet specify a fixed dilation anchor or admit the resulting endpoints in a source field. The inherited convention lifts transformed centers back to the ordinary seasonal pair; it does not enlarge the half-year pair separation itself. Whole widths are therefore a necessary, but not sufficient, condition for an annual source-endpoint realization.

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