## 4A. Comparative MT, LXX and accepted SP spines

This September 28 extension applies the same operation to six declared source paths. Let \(a=1406\), \(b=6\), and let \(F,C\) be the selected Flood and Creation/Fall BC coordinates. For a positive original interval \(n=10^km\), with \(10\nmid m\), write \(I(n)=10^k\operatorname{rev}(m)\). Define

\[
u=I(F-a),\qquad v=I(C-F),\qquad t=I(a-b)=4100.
\]

The complete two-stage path is \((a,a+u,a+u+v)\); the three-stage path is \((b,b+t,b+t+u,b+t+u+v)\). Their earlier heads satisfy

\[
H_2=a+u+v,\qquad H_3=b+t+u+v=H_2+2700.
\]

| Source state | \(C\), BC | \(F\), BC | \(u\) | \(v\) | \(H_2\), BC | \(H_3\), BC |
|------------------------------|------:|------:|------:|------:|------:|------:|
| MT Regular | 4106 | 2456 | 5010 | 5610 | 12026 | 14726 |
| MT Cumulative | 14006 | 4836 | 3430 | 7190 | 12026 | 14726 |
| LXX Regular | 5486 | 3236 | 3810 | 5220 | 10436 | 13136 |
| LXX Cumulative | 14896 | 5746 | 4340 | 5190 | 10936 | 13636 |
| SP Regular, accepted Fall | 4406 | 3106 | 7100 | 3100 | 11606 | 14306 |
| SP Cumulative, accepted Fall | 13396 | 4716 | 1330 | 8680 | 11416 | 14116 |

The LXX state retains native Cainan and the stated full-430 comparative frame. SP's head inputs are the accepted derived Fall nodes \(4411-5=4406\) and \(13401-5=13396\), using lower Flood members throughout. The SP Cumulative source has Rounded Joshua terminal 1296 and lifespan sum 12105, supplying Creation 13401; 1406 is the separate held comparison anchor. These source and interpretation choices precede reversal.

For modes \(r,c\) of one tradition, the signed head difference is

\[
\Delta=H_{2,c}-H_{2,r}=(u_c-u_r)+(v_c-v_r)
       =H_{3,c}-H_{3,r}.
\]

The source rows give MT \(-1580+1580=0\), LXX \(530-30=500\), and SP \(-5770+5580=-190\). A common tail preserves these differences. It does not force their values or select the source partition. The repeated 2700 is one identity shared by six executions.

Reversal preserves MT's cumulative Flood–Conquest 3430, LXX's corresponding 4340, and SP's cumulative Fall–Flood 8680. The latter equals twice 4340 but occupies a different source segment. The complete source paths, rather than the heads alone, retain that distinction.

Two cross-family identities summarize the resulting connections. The accepted SP tuple satisfies

\[
(14306,4406,4416)=(14006,4106,4116)+300(1,1,1),
\]

so its 9900 and 9890 interval measurements follow from the same translation. The MT tuple uses Cumulative Rounded, Regular Rounded and Actual Year-6/Adam coordinates respectively; the SP tuple uses a generated Regular inverse head, adopted Fall and upper Rounded Creation. No role identification is implied by the translation.

For the LXX/SP comparison let \(X=1406+(70/69)(13136-1406)=13306\). Then

\[
14306-X=1000,\quad X-11606=1700=3106-1406,
\]

and subtracting the common LXX Creation 5486 gives 8820, 7820 and 6120. These displayed partitions are consequences of the selected nodes and appointed expansion. The return of 1700 is exact; it does not by itself define a reciprocal transformation between full chronologies.

Finally, the LXX paired Priestly landing satisfies

\[
13636-\frac{25}{23}(13636-1446)
=13136-\frac{25}{23}(13136-1406)=386.
\]

The agreement uses \(500-40=460\) and \((25/23)460=500\). It supplies a source-calibrated equality between two held-head expansions. The generated 386 coordinate is not assigned a historical event.

Source/proof route: *Reversed Creation–Flood Spines: LXX Comparison and the Accepted SP Solution*, September 28, §§2–7; fresh comparative arithmetic review included with this edition. This extension follows C1431 and is outside that checkpoint's original 779-check count. The inherited single-pass MT proof and source choices remain unchanged; the inverse of the inverse stays deferred.