# C131 — Flood-close rows complete the four-member doubled-offset ladder

22 September 2026 · One bounded extension of C130 to **N01/N03**, with the previously documented fifth C10 row retained as a boundary control. Set \(K=529\). C127's frozen graph and File_18/File_09 control source roles and paired seasons.

**Finding.** The two Flood-close rows obey the same *dependent* 20K Mirror midpoint and 17K circuit as the N02/N04 start rows. Their sourced LXX cumulative OFF member offsets from the **5290 BC** opening are **2** and **4**, giving doubled MT-ON/LXX-OFF remainders **4** and **8**. Across all four retained Creation–Conquest–Flood prefix paths, the opening offsets form **2, 3, 4, 5**, and the corresponding remainders form **4, 6, 8, 10**. Neither the complete earlier expansions nor the absent N03 Shem join follows from that arithmetic.

## The four source-qualified comparisons

Entries below are Nisan **BC pair bases**: a base 5292 represents `5293t/5292n BC`. The middle member **O=5290** is the LXX cumulative **derived Cainan OFF** Flood opening pair, `5291t/5290n BC`. The two differences in each row hold separately for Tishri and Nisan components.

| Arc | MT cumulative ON \(A\) → LXX OFF opening \(O\) → LXX OFF member \(C\) | Equal BC legs | \(A-C\) | C129 two-leg offset | Ordinary MT Flood continuation |
|---|---|---:|---:|---:|---|
| **N01** | **5292 → 5290 → 5288** | **2 + 2** | **4** | **868 + 4 = 872** | **G2 close, AD2457** |
| N02 | 5293 → 5290 → 5287 | 3 + 3 | 6 | 874 | G2 start, AD2458 |
| **N03** | **5294 → 5290 → 5286** | **4 + 4** | **8** | **868 + 8 = 876** | **G3 close, AD2459** |
| N04 | 5295 → 5290 → 5285 | 5 + 5 | 10 | 878 | G3 start, AD2460 |

N01 and N03 reproduce C130's signed-center proof. Let \(A,c,O\) denote the first, third, and opening nodes, with \(O=-5290\). The retained mixed 20K path \(A\to D=R(c)\), the opening self-Mirror 20K path, and their reflection edges give

\[
\boxed{A+c=2O,\qquad c-A=2(c-O).}
\]

The separately sourced edges \(O\to c\) are width **2** and **4**. Thus the remainders **4** and **8** are exact *consequences of those member selections*. Each five-node 20K subsystem has rank **4** without its member edge and **5** with it. The one-year omission witness remains mathematical only. The selected C10 Christ-period landing bases **2** and **4 BC** are the midpoints of their respective mixed 20K intervals, not fresh historical birth identifications.

The close-row **17K** paths also close through the same existing edges:

\[
\boxed{868+5819+1058+190+1058=8993=17K.}
\]

For N01, the paired LXX-OFF→reflected SP Noah comparison is `5289t BC → AD3705t` and `5288n BC → AD3706n`; **5289+3705−1 = 5288+3706−1 = 8993**. For N03, **5287+3707−1 = 5286+3708−1 = 8993**. Their paired mixed 20K widths are **5293+5288−1 = 5292+5289−1 = 10580** at N01 and **5295+5286−1 = 5294+5287−1 = 10580** at N03. The 11K Creation→Conquest and 8K generated-preimage comparisons pass the same component test.

Both close-row C10 cumulative squares retain actual LXX ON and derived-OFF cumulative members and the selected alternate MT Flood *close*. The ordinary and alternate MT Flood closes are **G2 at N01** and **G3 at N03**, separated by the regular-profile 190; the SP Noah births are separately **G1 companion** and **G2 companion**. The SP person companion does not create an MT/LXX companion or redefine Flood close. C10 supplies no corresponding historical MT *primary* arc for these close rows, and N03's complete Shem join is still absent. C122 retains their Creation→Conquest→Flood **3+8+2 prefix paths**, while the wider prior expansions would demand unsupported MT Noah births **3055/3057 BC**.

## A fifth-row boundary with narrower support

The earlier C12 evidence already admits a *fifth mixed cumulative 20K* comparison:

\[
\boxed{5296\ \longrightarrow\ 5290\ \longrightarrow\ 5284\ \text{BC}
\quad=6+6,}
\]

with a selected **6 BC** landing and paired mixed width **5297+5284−1 = 5296+5285−1 = 10580**. Its source-qualified endpoint path through actual SP Noah **3709 BC** gives C12's 3+17 signed comparison. This does **not** extend C131's particular 17K *cumulative square* or 13K *Conquest path*: the cumulative ON member **6342 BC** needed for the square lies outside the retained field, the proposed regular **2651n BC** midpoint is merely geometric, and the matching reflected Conquest **AD1398** lies outside its admitted field. The separately supported **2651t** boundary belongs to a different phase and Noah selection.

Consequently the available evidence supports **five mixed 20K comparisons**, **four fully sourced cumulative 17K squares**, and **four retained 13K Creation–Conquest–Flood prefixes**. The domains differ for specific source reasons; one cannot extend them together by shifting an event or applying a Gear to Creation.

## Evidence and status

The C131 checker passes **47 bounded checks** on frozen File_18/File_09, C10/C12, C122, C127 and C129/C130 records. It verifies exact 20K and 17K dependency combinations for N01/N03, both paired components, close/companion profile tags, the 4/8 remainder omission witnesses, and the fifth-row distinctions. One isolated replay reproduces its result and certificate byte for byte. The historical graph remains **208 nodes, rank 208**; C131 adds no equation, source Gear, or completed old campaign. Interrupted original Reports 11–25 remain unestablished.

**Best next bounded step:** use the fifth row as a finite incidence control: remove one required source attachment at a time from the four-row pattern and record precisely which of the **20K, 17K-square, and 13K-prefix** claims remain, without widening the domain. **Sol Max is sufficient.**