C45 — The ON endpoint is an essential binding of the new paths
22 September 2026 · Step 23 of the authorized 30-step Mirror continuation (C23–C52).
At fixed retained K, replacing u with its derived-OFF Shem counterpart cm moves the BC coordinate forward by κ=460. Replacing U with the reflection of cm moves the AD coordinate backward by 460. Each affected new path shortens by 460:
| Path | Retained ON width | One endpoint changed to OFF | Change |
|---|---|---|---|
| 19: u to s | 10051 | 9591 | -460 |
| 17: u to v | 8993 | 8533 | -460 |
| 12: Z to U | 6348 | 5888 | -460 |
| 22: A to U | 11638 | 11178 | -460 |
| 24: H to U | 12696 | 12236 | -460 |
If both LXX ON endpoints of N01's 24-fold interval are replaced with their OFF counterparts, the result is 24K−920=11776. The 15-fold c→v intersection is unchanged by these particular substitutions, because neither of its endpoints was u or U.
These are explicit endpoint substitutions with the other assignments and K held fixed. They are not a rerun of an entire chronology under a newly fitted unit, and they do not equate MT restored ON with native LXX ON. The changed endpoints may be valid derived-OFF members, but they support different lengths and roles.
Thus the new families depend on the named Cainan insertion and the selected Shem member. The numeric family label cannot be preserved merely by switching the label ON to OFF while leaving its coordinates in place.
Validation. 8 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. Expose the regular/cumulative selected-length match in the new family residuals.