# 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.

| Branch | 13-fold centre | 17-fold centre | 20-fold centre |
| --- | --- | --- | --- |
| N01 | -1963/2 | -1583/2 | -2 |
| N02 | -1961/2 | -1581/2 | -3 |
| N03 | -1959/2 | -1579/2 | -4 |
| N04 | -1957/2 | -1577/2 | -5 |

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.