# C60 — The 24-fold closure does not independently determine the six-year gap

22 September 2026 · Step 8 of the authorized ten-step Mirror continuation (C53–C62).

Hold the retained interval formulas and LXX source fields fixed, but formally release the MT-ON/LXX-OFF Moses-base displacement Δ. The MT ON membership condition becomes

\[
\Delta-7\le j\le\Delta.
\]

The other three cumulative fields still require 0≤j≤2 jointly. The regular Noah join selects j=2, which is admitted precisely when **2≤Δ≤9**.

| Formal Δ | Four-field cumulative j values | Joined j values | MT member offset at j=2 |
| --- | --- | --- | --- |
| 1 | [0, 1] | [] | 6 |
| 2 | [0, 1, 2] | [2] | 5 |
| 4 | [0, 1, 2] | [2] | 3 |
| 5 | [0, 1, 2] | [2] | 2 |
| 6 | [0, 1, 2] | [2] | 1 |
| 7 | [0, 1, 2] | [2] | 0 |
| 9 | [2] | [2] | -2 |
| 10 | [] | [] | -3 |

Therefore the existing 24-fold closure and endpoint bounds do not recover Δ=6 uniquely. Even requiring all four N01 offsets to lie in the 2+2+3 cut set permits Δ=2,4,6,9. The actual six-year value is supplied by the named source balance 460−454, not independently predicted by this family label or boundary classification.

At the source, the MT member and lower ON Shem member both have Moses-relative offset +1. Requiring that extra offset equality at j=2 would select Δ=6, but the equality would itself be an additional condition; it cannot be counted as a second derivation from the already selected endpoints.

These controls shift only a formal MT cumulative field while leaving the geometric construction fixed. They are not proposed chronology variants and do not alter any source lifespan. Their purpose is to distinguish a source explanation from an overstrong inverse claim.

**Validation.** 18 exact arithmetic/structural checks passed. The 13 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 §0.4 rules 21–22 and the Creation Cell / Aaron Phase guard (week anatomy), §3 (paired Mirror); File_18 §§4.2 and 6A–6D (lifespans and cumulative fields); frozen C15/C16/C22 and C43/C46/C48/C50/C52 certificates (retained graph, unit, and domains).

**Best next bounded step — High sufficient.** Audit the 6348 whole-year member and 6348.5 Aaron phase in the exact paired Mirror coordinates.