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.

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

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:

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:

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.