# C10 — The 3+7 and 5+5 partitions share a 3+2+5 refinement

21 September 2026 · Continuation of C09, incorporating the attached 3+7×529 Flood–Noah–Christ handoff.

**The new Noah division and the retained midpoint construction fit one exact refinement:**

\[
\boxed{10=3+7=5+5=3+2+5\quad\text{in units of }529.}
\]

Across all five admitted SP Nisan chains, Noah's birth precedes the 5+5 midpoint by **1058 = 2×529**. Four interior midpoints also have named MT Flood start/close support in the previously retained **430 / Cainan ON / Terah +60** comparison frame. The opening **2651n BC** midpoint remains geometric within this finite Nisan domain. Its numerical agreement with a supported **2651t BC** Flood boundary requires a different phase and Noah binding.

The four named intervals satisfy the same adjacent-Gear selection that underlies the earlier 529 comparisons. Their cumulative counterparts give **8993 = 17×529** equal-span Mirror squares. The two primary Noah rows also connect directly to C09's **5819+1058=6877**, using the same cumulative 1058 intervals already retained in C08–C09. This integrates the new partition with the 2-, 10-, 11-, 13- and 17-fold constructions while preserving their distinct endpoints.

**Source scope and the two regular profiles.** The principal handoff's source corrections remain controlling: **5296 is MT cumulative ON**, and **3709n is SP regular G3 companion Noah birth**. LXX cumulative OFF has the different field 5290–5283; regular LXX OFF G3 Noah is 3710n.

| Component | Held source state | Use in C10 |
|---|---|---|
| Initial Flood member A | MT cumulative Cainan ON, Nisan field 5296–5289 | Left endpoint of the 5290 interval |
| Noah birth N | SP regular, full-430, Cainan OFF, Terah 0 | Five primary/companion selections |
| Additional Noah support | LXX regular **derived Cainan OFF**, full-430, Terah 0 | Two primary selections; native LXX is ON |
| Midpoint M | MT regular full-430, **Cainan ON, Terah +60** | Retained alternate-frame Flood comparison only |
| Landing Z | Previously retained Christ-period field | Selected 6–2 BC Nisan members, not five independent historical birth claims |

The MT midpoint profile comes from the earlier handoff's explicit **5293→2648→3 BC** construction and its **2648, MT G2 Flood, 430/ON/+60** identification. It is used here as a labeled comparison control. No broader selector sweep is opened. Regular Gear movement remains restricted to **Noah/Shem/Flood**; SP's −1 person rail remains restricted to **Noah/Shem**. Flood close is independently the end of the one-year Flood event. Creation retains its fixed Normal/G2 state.

**The five-row comparison.** Every entry in the four date columns below is **Nisan BC**. Let

\[
N=A-1587,\qquad M=A-2645,\qquad Z=A-5290.
\]

Then

\[
A-N=1587,\quad N-M=1058,\quad M-Z=2645;
\]
\[
A-M=M-Z=2645,\qquad N-Z=3703.
\]

| A: cumulative MT ON Flood | N: regular Noah | M: 5+5 midpoint | Z: landing | SP Noah binding | LXX OFF Noah | MT midpoint binding, 430/ON/+60 |
|---:|---:|---:|---:|---|---|---|
| **5296** | **3709** | **2651** | **6** | G3 companion | — | Geometric Nisan midpoint; no admitted Flood match |
| 5295 | 3708 | 2650 | 5 | G2 primary | G2 | G3 Flood start |
| 5294 | 3707 | 2649 | 4 | G2 companion | — | G3 Flood close |
| **5293** | **3706** | **2648** | **3** | G1 primary | G1 | G2 Flood start; explicitly retained earlier |
| 5292 | 3705 | 2647 | 2 | G1 companion | — | G2 Flood close |

Thus the principal numerical refinement is

\[
5296\xrightarrow{3\times529}3709
\xrightarrow{2\times529}2651
\xrightarrow{5\times529}6.
\]

Its fully event-qualified interior example is

\[
\boxed{5293\xrightarrow{1587}3706
\xrightarrow{1058}2648
\xrightarrow{2645}3\quad\text{BC}.}
\]

The five selections span **four elapsed years**. The last row requires **5292**, since 3705+1587=5292. The handoff's apparent “5295/3705/2” typo remains an arithmetic correction awaiting explicit author confirmation.

The domain is complete for the held Noah stations: six SP birth coordinates were checked against the eight-member MT cumulative field. G3 primary **3710n** would require **5297n→3710n→7n BC**, one year beyond its opening. The same exclusion removes LXX OFF G3; LXX OFF retains only G2 and G1. No intermediate annual coordinate is added to the LXX Noah field.

**Why the Flood identification selects the existing 598 layer.** Write i for Noah Gear and j for Flood Gear. Let \(N_i^{MT}\) and \(F_j^{MT}\) be ordinary full-430/Terah-0/Cainan-OFF MT Noah and Flood-start coordinates. File_18 gives

\[
N_i^{MT}-F_j^{MT}=600-2(j-i).
\]

SP primary Noah is MT Noah **+650**. The retained alternate MT Flood profile is ordinary MT Flood **+130+60 = +190**. Put c=1 for SP's Noah companion, otherwise 0; put r=1 for Flood close, otherwise 0. The resulting comparison is

\[
N=N_i^{MT}+650-c,\qquad M=F_j^{MT}+190-r,
\]
\[
\boxed{N-M=1060-2(j-i)-c+r.}
\]

Within the three Gears and the two permitted endpoint choices,

\[
N-M=1058\iff 2(j-i)+c-r=2
\iff\boxed{j=i+1\ \text{and}\ r=c.}
\]

This gives exactly four matches among the **36** SP-Noah/alternate-MT-Flood endpoint pairs: primary Noah to next-Gear Flood start, or companion Noah to next-Gear Flood close. The two LXX OFF matches are the primary/start cases. The equality r=c records a matched selection of **two independent mechanisms**; it does not transfer SP's companion operator to MT or to the Flood.

For these matches,

\[
1058=598+460,
\qquad460=650-(130+60).
\]

The **regular residual 460** is a difference between two explicitly selected regular profiles. Its numerical equality to the **cumulative Cainan lifespan insertion 460** makes the comparison possible; their operations and node types remain distinct. This is the same next-Flood-Gear rule selected by C08, now expressed inside the 5290 partition.

**The phase distinction resolves the apparent 2651 problem without changing the Nisan result.** In the alternate MT frame, the admitted Nisan Flood years are G1 2646→2645, G2 2648→2647 and G3 2650→2649. Thus 2651n is outside that field. G3 Flood start nevertheless has the supported pair **2651t/2650n BC**.

The two primary rows have these complete paired refinements:

| Noah Gear, SP primary / LXX OFF | Cumulative MT ON — BC | Noah birth — BC | MT ON/+60 Flood start — BC | Landing — BC |
|---|---|---|---|---|
| G2 | 5296t/5295n | 3709t/3708n | 2651t/2650n, G3 | 6t/5n |
| G1 | 5294t/5293n | 3707t/3706n | 2649t/2648n, G2 | 4t/3n |

Each same-season component has the 3+2+5 lengths. In particular, **5296t→3709t→2651t→6t BC** is supported by the first paired row. Its Noah binding is **G2 primary SP / G2 derived-OFF LXX**. The all-Nisan chain with those same numerals uses **G3 companion SP** and has no supported 2651n Flood midpoint. These are different phase-resolved constructions.

This independent phase check does not settle Dean's intended “5296–5293 → 3709–3706 → 6–3” seasonal envelope. The earlier proposed 2½-year Tishri-to-Nisan reading remains **unconfirmed**. The five-row Nisan result does not depend on that reading.

**The 1058 interval also sits inside the retained 8993 comparisons.** For each named midpoint row, define

\[
U=8993-M,\qquad D=8993-N.
\]

U lies in the retained **LXX cumulative ON Shem-birth/Noah-death field 6350–6343**; D lies in the **LXX cumulative OFF Flood/Shem-death field 5290–5283**. These cumulative role labels remain distinct from regular Noah birth and regular Flood.

| Regular SP Noah N, BC | Alternate MT Flood M, BC | Cumulative ON U, BC | Cumulative OFF D, BC | Retained SP certificate label | Retained MT primary label |
|---:|---:|---:|---:|---|---|
| 3708 | 2650, start | 6343 | 5285 | N04 | N08 |
| 3707 | 2649, close | 6344 | 5286 | N03 | — |
| 3706 | 2648, start | 6345 | 5287 | N02 | N07 |
| 3705 | 2647, close | 6346 | 5288 | N01 | — |

Every row gives

\[
\boxed{U-D=N-M=1058,\qquad U+M=D+N=8993.}
\]

The cumulative coordinates are BC; M and N in the **sums** are their regular **reflected AD paired centers**. For example, the 5293 row gives

\[
6345+2648=5287+3706=8993.
\]

Its explicit paired comparisons are **6346t/6345n BC → AD2648t/2649n** and **5288t/5287n BC → AD3706t/3707n**. Both same-season civil counts are 8993 after the no-year-zero subtraction. The Mirror axis remains January AD1.

For all four rows the square is

\[
\begin{pmatrix}U+M&U+N\\D+M&D+N\end{pmatrix}
=\begin{pmatrix}8993&10051\\7935&8993\end{pmatrix}
=529\begin{pmatrix}17&19\\15&17\end{pmatrix}.
\]

The off-diagonal coefficients 15 and 19 follow from 17±2. They are dependent entries of this square, not newly searched independent families.

The primary rows reuse endpoint selections already present in N04/N08 and N02/N07. The close rows use the existing SP N03/N01 data and independently admitted MT Flood closes; they require no MT Noah companion. These are finite comparison records in C10. The historical joined graph has not been enlarged or rerun.

The principal Nisan row fails this stronger sourced square at the same boundary: M=2651 would require **U=6342**, one year beyond the closing 6343 of the held cumulative ON field. D=5284 is admitted, but that alone does not complete the square. The formal arithmetic remains correct; the missing endpoint support is recorded.

**Connection to the Creation–Conquest–Flood result.** C09 already supplies two source-qualified reflected Conquest→MT Flood intervals of 1058. C10's two primary rows use exactly the same cumulative intervals that supported those comparisons:

| Old SP label | C10 regular BC Noah→alternate MT Flood | Cumulative ON→OFF BC comparison | C09 reflected Conquest→ordinary MT Flood | C09 SP Creation, BC |
|---|---|---|---|---:|
| N02 | 3706→2648 | 6345→5287 | AD1400t/1401n → AD2458t/2459n | 4419n |
| N04 | 3708→2650 | 6343→5285 | AD1402t/1403n → AD2460t/2461n | 4417n |

Every interval in the three middle columns is 1058. The alternate MT Flood coordinates in C10 are the corresponding ordinary MT Flood coordinates **+190**, retaining their respective Gears. C09's complete source path still has **5819+1058=6877**, or **11+2=13** in units of 529.

The resulting compact dependency is:

| Construction | Partition or relation in units of 529 | What is established |
|---|---|---|
| C10 cumulative Flood→Christ-period comparison | 10=3+7=5+5=3+2+5 | Five Noah divisions; four named Flood midpoints in the retained alternate frame |
| C09 Creation→reflected Conquest→Flood | 13=11+2 | Two contiguous source paths at Creation 4419/4417 |
| C10 cumulative/regular comparison square | Equal diagonals 17, side width 2 | Four source-qualified 8993 squares |
| C08 cross-family width identity | 17=13+2×2 | Established width relation; the unequal flanks are not relabeled as two contiguous 1058 event segments |

The common 2×529 interval now has explicit roles in both partitions and in the previous Mirror/Creation comparisons. The equal lengths do not by themselves identify the different events or establish a global source-preserving transformation. No 25/23 expansion campaign was run in C10.

**Evidence and checkpoint.** `evidence/check_c10.py` reads nine frozen inputs, derives the source domains from the supplied tables, checks the complete 36-pair midpoint domain, verifies all five partitions and the phase distinction, and compares the four squares and two primary links with the retained certificates. `C10_RESULTS.json` records every candidate and rejection. The completed packet includes **112 passing checks** and a byte-identical isolated replay of the result file, with input and artifact SHA256 manifests.

Source locators: File_18 §§2.1.3, 3.1.4, 3.2.1, 4.1 and 6B–6D; File_09 paired-versus-single Mirror controls; original Mirror handoff §§3.1, 4.2 and Appendix A.1; new 3+7 handoff §§1–4; original selection certificate N01–N08; C08 and C09 published results. File_18 and File_09 retain their verified controlling hashes. No canonical source was edited and no completed historical campaign was rerun.

The latest published continuation is now **C10**, following C01–C09 and the recovered original Reports 01–10. Interrupted Reports 11–25 remain **unestablished as completed**. The unresolved seasonal intention and explicit confirmation of the 5292 typo correction remain visible; neither blocks this bounded result.

**Best next bounded step:** return to the deferred coordinate-chart extension, adding the C09 Conquest point and these source-qualified 1058 correspondences with their profile/phase labels. Test whether the existing chart can express them and isolate any additional boundary condition needed for the proposed full-week 300→299 Creation compression. **High is sufficient.**