C32 — Integer endpoints can have half-integer Mirror centres

22 September 2026 · Step 10 of the authorized 30-step Mirror continuation (C23–C52).

At the source, K=529 is odd. Consequently 13K and 17K are odd widths: integer endpoints give half-integer centres. The 20-fold interval has even width and the integer centre Z=−j.

In C22's scale parameter, W=299z and B=391z have the parity of z, while 2O=460z is always even. The source z=23 fixes the observed half-integer centres. This is a parity consequence, not evidence of a newly named half-year event or a change of Mirror axis.

The local Flood-to-MT-Noah interval is different: ℓ=598 is even, and its midpoint is the actual supported SP companion-death coordinate on N02/N04. Its two halves are 299 each. That source-supported local midpoint must not be confused with the geometric midpoint of the entire Creation-to-reflected-Flood span.

The first ten steps therefore leave the qualified endpoint inventory intact while supplying a compact model, an exact residual crosswalk, the branch-index action, the unequal 13/17 nesting, and the relevant integer distinctions. They make the next bounded extension possible: follow the already qualified native-ON cumulative Shem endpoint beyond the existing 17-fold interval.

Validation. 11 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. Compose the retained cumulative 2-fold prefix with the 17-fold Mirror interval and qualify the resulting 19-fold path.