# 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.

| Family | Width | Coefficient of z | Registered typed edges |
| --- | --- | --- | --- |
| 2 | 1058 | 46 | 17 |
| 10 | 5290 | 230 | 10 |
| 11 | 5819 | 253 | 7 |
| 12 | 6348 | 276 | 1 |
| 13 | 6877 | 299 | 5 |
| 17 | 8993 | 391 | 14 |
| 20 | 10580 | 460 | 6 |
| 22 | 11638 | 506 | 1 |

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.