# C58 — The two equal 1058 prefixes use different source decompositions

22 September 2026 · Step 6 of the authorized ten-step Mirror continuation (C53–C62).

Write positive BC numerals as F₋=5290−j (LXX OFF Flood), F₊=5290+j (MT ON Flood), S₋=6348−j and S₊=6348+j (two native-ON Shem members). These subscripts describe the index sign, not Cainan state or a regular Gear.

Their Moses-side gaps differ. ON Shem minus LXX OFF Flood is 1060; ON Shem minus MT ON Flood is 1054. The retained member selections compensate:

\[
\boxed{S_- -F_-=600+460-2=1058,}
\]
\[
\boxed{S_+ -F_+=600+454+4=1058.}
\]

The first prefix uses a two-year reduction from its Moses-side gap. The second uses a four-year increase. The difference between these adjustments is six, exactly the source displacement Δ=460−454.

| j | LXX Flood offset | Lower Shem offset | MT Flood offset | Upper Shem offset |
| --- | --- | --- | --- | --- |
| 0 | 5 | 3 | -1 | 3 |
| 1 | 4 | 2 | 0 | 4 |
| 2 | 3 | 1 | 1 | 5 |

At N01 (j=2), these offsets are +3,+1,+1,+5. The right prefix spans the +3/+1 selection; the left spans +5/+1 across its differently based source rows. Both use positions in the 2+2+3 numeric template, but that does not assign this cumulative template to regular Gear 2.

This explains the equal margins of the 20/24 construction without pretending their source mechanisms are identical. Since 454=460−6 and the endpoints were already linked, the two decompositions do not add independent unit determinations. Their value is to preserve the actual chronology and member roles behind the equality.

**Validation.** 9 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.** Intersect all four cumulative source fields and identify the shared week-boundary template of the joined N01 rectangle.