C69 — Reflection preserves the lifted algebra and reverses extension direction

22 September 2026 · Step 7 of the authorized ten-step Mirror continuation (C63–C72).

Canonical reflection is compatible with the phase-preserving affine maps when the anchor is reflected as well:

For an interval, reflection reverses endpoint order and swaps the two seasonal components. A leftward 300/299 extension holding the right endpoint therefore becomes a rightward extension holding the left endpoint:

The reflected 6877/6900/7475/7500 square closes under these rules. Keeping the extension leftward on both sides gives different endpoints even though it still multiplies a length by 300/299. The handedness and held endpoint are part of the operator identity.

These identities describe the geometry of declared paired states. Source admission remains a further condition. C66's scaled midpoint already shows that a valid paired image may lack the original event identity. A second boundary is the third 25/23 expansion of 8125:

This fractional width was already identified in C04. With the Creation anchors held here, its target centres have denominator 23; adding the exact ±¼ phase offsets gives component denominators 92. Each formal pair remains half a year wide and obeys the reflection identities, yet neither component lies on the current seasonal quarter-coordinate lattice for integer year labels. Ordinary integer member or two-year Gear shifts cannot remove that denominator.

Thus the algebra composes beyond the finite set of source-qualified calendar states. The two successful Noah landings do not make repeated dilation an unrestricted operation on the chronology. This report preserves the completed obstruction and adds its exact seasonal interpretation, without reopening the earlier proportional-continuation campaign.

Validation. 11 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. Test whether all width, midpoint and reflection identities together are sufficient to determine the required half-year phase.