C68 — The 12/13-family square commutes on both seasonal components

22 September 2026 · Step 6 of the authorized ten-step Mirror continuation (C63–C72).

C05's sourced comparison square remains exact after lifting both operators to the seasonal components:

The two paths are

The 25/23 map uses the already retained apparent Creation anchor −4444. The 300/299 map extends the interval leftward while holding its right endpoint. On each seasonal rail s, with e=300/299,

Both endpoints carry the same phase offset, so the transformed left endpoint retains that offset. The extensions are exactly 23 and 25 years on both components. Affine covariance gives E∘D=D∘E when the same affine map acts on both endpoints. The checked square therefore closes in its complete paired endpoints, not only in its four scalar lengths.

Using the right centre in place of the corresponding right seasonal component would make the new left pair 150/299 year wide. Its components miss the retained pair by ±1/1196 year. This is the same distinction identified for 25/23: the ratio acts on a comparison span with its declared seasonal anchoring. The error fractions describe a rejected proof-coordinate operation; they are not new calendar residues.

The 12-fold carriers 6348→6900→7500 and 13-fold carriers 6877→7475→8125 were established earlier. The new result supplies a common phase-preserving rule for the source-qualified square joining them. It does not claim a new endpoint for the 6348 member or for every scaled interval.

Validation. 12 exact arithmetic/structural checks passed. The 14 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 and Appendix A (paired Mirror and exact phase); File_18 §§2.1.3, 2.4, 3.1 and 3.2 (regular source bindings and seasonal limits); frozen C15/C16/C22/C43/C50/C61/C62 certificates (retained graph and phase); C01/C04/C05 reports (completed expansions and their source limits). The declared phase model does not infer an unstated birthday or event month from an annual source row.

Best next bounded step — High sufficient. Determine how canonical reflection transforms these phase-preserving maps, including which endpoint is held by the 300/299 extension.