# C89 — The complete 2-fold sequence 1058→1150→1250 has retained endpoint pairs

22 September 2026 · Step 7 of the authorized ten-step Mirror continuation (C83–C92).

**N02 and N04 now support all three stages of the 2-fold expansion with named endpoint pairs.** Hold the ordinary Creation member and apply the same phase-preserving 25/23 map to both ends of each interval:

| Branch | 1058=2×529 | 1150=2×575 | 1250=2×625 |
| --- | --- | --- | --- |
| N02 | 1400→2458 | 1906→3056 | 2456→3706 |
| N04 | 1402→2460 | 1908→3058 | 2458→3708 |

The source roles progress as follows:

| Stage | Left reflected endpoint | Right reflected endpoint |
| --- | --- | --- |
| 1058 | Fixed Conquest member | Upper-Gear MT Flood start |
| 1150 | Original backward cumulative Joseph head | Lower-Gear MT Noah birth |
| 1250 | Lower-Gear MT Flood start | Same-Gear SP primary Noah birth |

This source qualification is specific to the present placement. C09's scalar table listed 1150 and 1250 as generated in its original 2-fold construction. The present interval realization follows from the newly named 11-fold endpoints together with the retained 13-fold endpoints. It does not retroactively transfer these identities to every expansion of a1058 interval.

The first product grid, measured from fixed Creation, is

\[
\begin{pmatrix}5819&6325\\6877&7475\end{pmatrix}
=23\begin{pmatrix}11\\13\end{pmatrix}
\begin{pmatrix}23&25\end{pmatrix}.
\]

Its row differences are 506/598 and column differences 1058/1150. The determinant vanishes because the two rows share 25/23, so the 22–24–26 progression is a dependent consequence of this grid.

The second 1250 interval also has the already verified same-Gear dissection 600+650. Interval widths grow by 92 and 100 because their right endpoints move 598/650 while their left endpoints move 506/550. This preserves the distinction between endpoint motion and the width of the moving interval.

The exact bounded source domains differ by stage:

| Stage | Admitted j | Qualification |
| --- | --- | --- |
| 1058 | [2, 3, 4, 5] | Both Conquest and upper Flood present |
| 1150 | [3, 5, 7] | Joseph and MT primary Noah present |
| 1250 | [2, 3, 4, 5, 6, 7] | Lower MT Flood plus SP primary/companion Noah |
| All three | [3, 5] | N02/N04 only |

For the three close-ending 1250 rows, the SP Noah endpoint is the corresponding SP person companion. That is an authorized SP-only binding, not an MT companion or a Flood person rail. The MT Noah endpoint needed by 1150 is absent on those close rows. G3 admits the 1150/1250 tail, while its 1058 head lacks upper Flood 2462.

Both endpoints and both seasonal components pass the two exact affine maps. These are source-qualified interval images; they do not license transporting unrelated records by a whole-chronology dilation.

**Validation.** 18 exact arithmetic/structural checks passed. The 17 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 and Appendix A (paired Mirror and exact phase); File_18 §§2.1.3, 2.4, 3.1 and 6/6A (regular fields and cumulative collateral source); File_61 §§3.3–3.4, 11.1–11.4 and 14.3 (fixed Jacob events, original Joseph portion and its 500-year Conquest comparison); File_62 collateral-scope guards; frozen C15/C16/C22/C43/C65/C74–C77/C82 dependencies (retained source, phase and family controls). The declared phase model does not infer an unstated birthday or event month from an annual source row.

**Best next bounded step — High sufficient.** Compare this 506+552 dissection of 1058 with the retained cumulative 460+598 dissection, keeping endpoint frames distinct.