C194 — Whole expanded widths do not guarantee whole anchored endpoints
23 September 2026 · Step 39 of 50, continuing C155.
The four exact squares have integer widths after25/23 scaling, yet dilation about4K gives fractional centers at all four vertices. Their signed vertices have common residues2,3,4,5 modulo23, whereas2116 has residue0.
For an integer anchor a, Pₐ(x) is integral exactly when x≡a(mod23). Each exact square has one common vertex residue because all its mutual spans are multiples of23. An anchor with that residue preserves the annual grid; the4K anchor does not.
Choosing an existing vertex as anchor gives integer images, but source-field membership still requires its own test. In contrast, choosing2116 ensures commutation with the8K complement but loses the annual vertex grid in these four cases.
This is a concrete compatibility limit between two attractive formal properties. Neither the exact529 widths nor the centered symmetry alone supplies a whole-source endpoint transformation.
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. 31 assertions passed; see RESULTS.json and CERTIFICATE.json. Assertions, selected field combinations and inherited identities are not independent historical witness counts.