# 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:

| Interval | Left paired boundary | Right paired boundary | Both seasonal widths |
| --- | --- | --- | --- |
| A | 4422t/4421n BC | AD 2456t/2457n | 6877 |
| B | 4420t/4419n BC | AD 3056t/3057n | 7475 |
| C | 4445t/4444n BC | AD 2456t/2457n | 6900 |
| D | 4445t/4444n BC | AD 3056t/3057n | 7500 |

The two paths are

\[
6877\xrightarrow{25/23}7475\xrightarrow{300/299}7500,
\]
\[
6877\xrightarrow{300/299}6900\xrightarrow{25/23}7500.
\]

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,

\[
E_s(l_s,r_s)=\bigl(r_s-e(r_s-l_s),r_s\bigr).
\]

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.