C70 — Exact family closure and reflection do not determine half-year phase
22 September 2026 · Step 8 of the authorized ten-step Mirror continuation (C63–C72).
A coherent countermodel makes the phase requirement explicit. Replace every pair P(x)=(x−¼,x+¼) by Pₕ(x)=(x−h,x+h), using the same h throughout a branch. For both h=¼ and h=¾:
- every 6877/299/299/650 same-component interval is unchanged;
- the source-qualified midpoint's coordinate remains the numerical midpoint;
- reflection still satisfies R(Pₕ(x))=Pₕ(−x);
- the correspondingly defined component-wise affine expansions reach the same pair centres.
| Branch | Half-width h | Internal pair width | Geometric identities | File_09 phase admitted |
|---|---|---|---|---|
| N02 | 1/4 | 1/2 | pass | yes |
| N02 | 3/4 | 3/2 | pass | no |
| N04 | 1/4 | 1/2 | pass | yes |
| N04 | 3/4 | 3/2 | pass | no |
The h=¾ model has 1.5-year pairs. It fails File_09's seasonal phase convention even though all listed geometric identities survive. Assigning the old civil suffix labels to these altered component times would be incorrect. This is a formal countermodel, not an alternative chronology or a proposed calendar.
The reason is structural: equal phase offsets cancel in same-component differences, while symmetric offsets survive reflection. Width equations and reflection determine neither the internal width of every pair nor the calendar meaning of its labels. The half-year phase comes from the controlling seasonal model. Its agreement with the 529 geometry is a compatibility result, not another independent numerical determination of 529.
C61's one-sided suffix error could be detected by unequal component spans. This stronger control changes every pair coherently, so those span checks alone no longer detect it. A complete certificate must include an explicit source phase condition.
Validation. 17 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. Derive the smallest sufficient phase certificate for the retained five-node chain, keeping source membership as a separate requirement.