C71 — One anchored half-year pair fixes the phase of each connected chain
22 September 2026 · Step 9 of the authorized ten-step Mirror continuation (C63–C72).
For the five retained nodes, write their two component times as tᵢ=xᵢ+aᵢ and nᵢ=xᵢ+bᵢ, where xᵢ is the recorded centre. Requiring the four consecutive widths on each seasonal rail gives
These eight independent equations leave two common offsets, a and b. Requiring the recorded x to be the actual pair centre gives a+b=0. Supplying File_09's half-year width gives b−a=1/2. Hence uniquely
| Constraints on ten component offsets | Exact matrix rank | Remaining freedom |
|---|---|---|
| Consecutive widths on both rails | 8 | two common offsets |
| Add the declared pair centre | 9 | one symmetric half-width |
| Add one half-year phase condition | 10 | none |
Equivalently, one correctly anchored source pair plus phase-preserving transport along the connected four-edge chain determines all ten component times. The exact 299 midpoint adds no new rank: its phase equations already follow from the consecutive-edge equations. Reflection is consistent with the centred offsets; it must not be counted as another independent determination of their half-width.
This is a minimal geometric certificate for these declared five-node chains. The typed source certificate is still required separately: which Creation member, which tradition and profile, which Gear, which Flood boundary, and which SP primary/companion record. C66 shows why a correct phase image can lack an event identity. Conversely, annual source membership alone does not determine an unstated birthday.
The result permits a compact reproducible representation: retain the source-tagged centres, one source phase convention, and the exact connected interval relations. Store component labels as derived values and check them against that convention. No source field, canonical file or historical graph is changed.
Validation. 7 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. Synthesize the ten phase-and-operator results with the retained 529 families and publish the next restart checkpoint.