C72 — Ten-step synthesis: the Mirror operators preserve phase and retain source limits

22 September 2026 · Step 10 of the authorized ten-step Mirror continuation (C63–C72).

C63–C72 complete the authorized next ten steps. The full 6877 chains now have exact seasonal certificates, and the existing 25/23 and 300/299 constructions share a precise phase-preserving interpretation. The strongest advance is a compact rule for composing the numerical maps while keeping their anchors and source permissions explicit.

The two complete chains remain:

Both seasonal components give 6877→+299→+299→+650, and both pairs retain half-year spacing at every node. The adjacent-Gear and SP companion assignments remain part of the source identity. These are exact statements about File_09's declared Mirror states; annual records do not establish otherwise unstated birthdays or event months.

The reduced phase representation is

An affine numerical comparison sends x to a+r(x−a) while retaining ε. On actual component times this means using the corresponding anchor a+ε. Reflection sends (x,ε) to (−x,−ε). The numerical image receives an event identity only if it matches a qualified source record.

This explains both the success and the limit of the 25/23 chain. It maps the retained MT Flood pair to MT Noah and then SP Noah. It also scales the named 299 midpoint into a 325+325 midpoint, but the resulting centres 3381/3383 are not SP Noah-death stations. Exact midpoint and phase preservation therefore do not imply preservation of an event name.

The connection to C05 is more specific than equal ratios. The N02 fixed-Creation map and the apparent-anchor map satisfy

They reach the same 7475 interval from two already admitted 6877 placements. The complete 6877/6900/7475/7500 square then closes on both components under 25/23 dilation and the right-endpoint-held 300/299 extension. Canonical reflection preserves this algebra when the anchor is reflected and left/right extension direction is exchanged.

The limits are now explicit. A common dilation of all component times changes the internal seasonal width. A coherently widened pair can nevertheless preserve all family widths and reflection identities. The source half-year convention is therefore an independent input to the geometric certificate. For a five-node chain the two seasonal edge systems have rank 8; the declared centre and one half-year condition raise it to 10 and determine every phase offset. This is a reduction in the necessary phase data, not an additional independent calibration of 529.

The third 25/23 width after 8125 remains the previously known 203125/23. Its lifted target components have denominator 92 in the present quarter-coordinate system and do not become new source dates. No completed expansion campaign was rerun. Existing C01/C04/C05 results were read selectively and extended only by these bounded phase and operator checks.

The selected family register remains 2/10/11/12/13/15/17/19/20/22/24. The joined graph retains 81 typed nodes, 161 duration records and 19 reflection relations. Full Creation chains remain N02/N04; the separate joined 24-fold head remains N01. The formal expanded midpoint is not added as a named event. All canonical files remain unchanged.

All ten steps have individually saved reports and reproducible evidence. C72 freezes C63–C71 certificates and checks their integration with the retained graph; it does not execute those earlier steps. This is the next delta packet for the compact restart kit. Interrupted historical Reports 11–25 remain unestablished as completed.

The phase question is now sufficiently resolved to return to a sharper 529-family question: 11×625=6875 is exactly two years below 13×529=6877. Whether that relation has a compatible endpoint realization in the existing Conquest/Flood fields is a pending source test, not an established new construction in this batch. Stop here before undertaking it.

Validation. 12 exact arithmetic/structural checks passed. The 23 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 the two-year relation 11×625=6875=13×529−2 against the retained Conquest and Flood Mirror endpoints, using the verified phase lift and existing Gear scope. Await user instruction.