C187 — The second expansion retains the square-of-23 obstruction

23 September 2026 · Step 32 of 50, continuing C155.

Two successive25/23 expansions carry K=23² to625=25², while the residual becomes625δ/529. Whole expanded widths require529 to divideδ, since625 and529 are coprime.

Onlyδ=0 satisfies that condition in the retained finite domain. The exact square scales to625×[[17,19],[15,17]]; P6 retains a fractional residual and does not become an exact source square by repeating the expansion.

This is the arithmetic reason the original529 unit matters across two expansion stages. It does not establish a source map for all four vertices, even where the widths are integers. No expanded endpoint is inserted into the historical graph.

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.