C23 — The eight retained families on the ratio-and-scale system

22 September 2026 · Step 1 of the authorized 30-step Mirror continuation (C23–C52).

Every previously registered family uses K=23z, with the source fixing z=23. The coefficient of z is therefore the family multiplier times 23. This conversion adds no endpoint and creates no new witness.

The edge counts include reflections, translated placements and composed paths. They are inventory counts, not independent evidence counts. In particular, the 11-fold registry also includes three native Creation/Conquest placements outside the joined N01–N04 rows. The 13-fold registry includes its opening placement. These must not inherit the full Noah expansion merely because their widths agree.

The construction hierarchy remains: five base 10/17/20 rows; four qualified midpoint and Creation/Conquest joins; two full Creation–Flood–Noah expansions (N02/N04); one retained left-head 12/22 attachment (N01). The latter description refers to that particular attachment, not a theorem that no other 12- or 22-fold endpoint pair can be constructed.

The local 2-fold length is ℓ+κ. The 11-fold path subtracts that length from the 13-fold interval. The 12-fold left path adds it to a 10-fold path; the retained 22-fold path adds it to the mixed 20-fold interval. Those identities require their shared nodes and source memberships. The reduced scale alone cannot select a Creation field member, an SP companion, or a Flood boundary.

The next step translates the older graph residuals into C22's source residuals, so that arithmetic closure and endpoint admission can be tracked separately.

Validation. 10 exact arithmetic/structural checks passed. The ten 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 §3.1 (paired Mirror); File_18 §§2.1.3, 3.1–3.2, 6B and 6D (qualified regular and cumulative fields); frozen C15 typed graph, C16 domain certificate, and C22 reduced scale/constraint certificates.

Best next bounded step — High sufficient. Translate the C15 graph residuals into the C22 source residuals.