C31 — The 23 and 529 patterns coexist without being identical
22 September 2026 · Step 9 of the authorized 30-step Mirror continuation (C23–C52).
Within one branch, the core direct nodes u,c,v,s have residue j modulo 529. Creation, Conquest and ordinary MT Flood share residue j−190. The opposite 17-fold pair cm,r shares residue j−598. The reflected partition points share −j. These classes explain equal-529 intervals among selected points; they are not an all-pairs endpoint search.
On either full construction the Creation expansion is:
| Endpoint reached | Width from Creation | 299 units | Remainder mod 529 |
|---|---|---|---|
| MT Flood | 6877 | 23 | 0 |
| SP companion Noah death | 7176 | 24 | 299 |
| MT Noah birth | 7475 | 25 | 69 |
| SP Noah birth | 8125 | 8125/299 | 190 |
The first three widths are also divisible by 23, with coefficients 299, 312 and 325. Only the first is an integer multiple of 529. The two 299-year steps retain the previously established 23→24→25 pattern; they are not three new 529 families.
The final 650-year extension changes residue by 6 modulo 23 and reaches 8125, which is not divisible by 23. This is consistent with p=190 having residue 6, while ℓ and κ are divisible by 23. Exact 529 closure therefore depends on which role supplies the endpoint. The SP Noah culmination belongs to the enclosing 17-fold interval from cumulative c, rather than becoming another 529 multiple from Creation b.
Validation. 8 exact arithmetic/structural checks passed. The ten frozen dependencies passed hash verification; all three generated outputs replayed byte for byte in isolation, and the original dependencies remained unchanged. This step reads completed certificates and does not execute their campaigns.
Scope and provenance. Controlling File_18 and File_09 remain unchanged. Regular Gears apply only to Noah/Shem/Flood; SP person companions only to Noah/Shem. Creation and Conquest fields remain fixed. Native LXX ON, explicitly derived OFF, independent Flood start/close, and paired suffix inversion at January AD 1 remain in force. No civil year zero, upstream Gear transport, primers, Rounded campaign, or localized −33 is introduced. These are new continuation reports, not recovered historical Reports 11–25. Evidence is conditional on the supplied project model, not a proof of literary intention or independent historical dating.
Source locators. File_09 §3.1 (paired Mirror); File_18 §§2.1.3, 3.1–3.2, 6B and 6D (qualified regular and cumulative fields); frozen C15 typed graph, C16 domain certificate, and C22 reduced scale/constraint certificates.
Best next bounded step — High sufficient. Separate integer endpoint coordinates from the parity of geometric midpoints.