## How the three inverse families connect internally

The next comparison asks what happens inside the spines. The source supplies a named 600-year interval in each mode: Noah's age at the Flood in Regular chronology, and Shem's serial lifespan in Cumulative chronology. We can divide the same original trunk at the Flood, move that division to Noah or Shem, or retain both internal nodes. Each operation still reverses only original source intervals once. The source head and held anchor stay fixed.

This reveals two complementary relationships. **MT Regular and SP Cumulative preserve their inverse heads through all three partitions. LXX Regular and MT Cumulative preserve the head when the division moves, but increase it by 990 when both nodes are retained.** The distinction concerns the structure of the selected paths, beyond the earlier agreements between terminal dates. [C1443–C1452]

### A named 600-year comparison

The Regular Noah nodes are MT 3056, LXX 3836 and SP 3706 BC. The Cumulative Shem nodes are MT 5436, LXX 6346 and SP 5316 BC. Each is 600 years earlier than its own declared Flood seam. MT's nodes are explicit in File51a; the completed native Rounded LXX model supplies Noah 3836, and the comparative study supplies cumulative Shem 6346. SP's nodes follow from its admitted Flood coordinates and the unchanged 600-year intervals. File47's local Noah pairing and File53's cumulative Shem envelopes corroborate their source roles. The accepted SP Fall heads remain 4406 and 13396. [File51a §§3.1,16.2; comparative study §4.1; File47 §6.1; File53 §19; C1443]

The following table gives the change from each primary Flood-partition head. It applies at either held-anchor stage because the same Nativity extension adds 2700 to every reconstructed head.

| Construction | Move division to Noah/Shem | Retain Flood and Noah/Shem |
|---|---:|---:|
| MT Regular | 0 | 0 |
| LXX Regular | 0 | +990 |
| SP Regular, accepted Fall | −6300 | −1800 |
| MT Cumulative | 0 | +990 |
| LXX Cumulative | +990 | +990 |
| SP Cumulative, accepted Fall | 0 | 0 |

MT Regular's three agreeing paths were already established. Its Cumulative refinement with Flood and Shem was also known: the Conquest-held head is 13016 rather than 12026. The new contribution is the systematic comparison with LXX and SP, including the different **Shem-only** MT partition, which does return 12026. These are different partitions of the same source trunk. The alternatives explain the primary constructions; they do not replace the accepted Flood spines. [File52c §§3.3–3.4; File52a §11.1; File52b §2.6; C1444–C1450]

For SP Cumulative the agreement is especially transparent. Its original legs are 3310 and 8680. Retaining Shem divides the latter into 600 and 8080, both unchanged by reversal:

\[
I(3310)+I(8680)=1330+8680=10010;
\]

\[
I(3910)+I(8080)=1930+8080=10010;
\]

\[
I(3310)+I(600)+I(8080)=1330+600+8080=10010.
\]

All three therefore reach 11416 BC from Conquest, or 14116 BC after the original Nativity leg is reversed and included. This is the accepted SP Cumulative counterpart to the MT Regular agreement among its Flood, Noah and combined partitions.

LXX Regular shows why agreement between the first two partitions is a weaker condition. Its Flood and Noah divisions give \(3810+5220=3420+5610=9030\), while retaining both gives \(3810+600+5610=10020\). The extra 990 appears even though the original source head, endpoint and total duration never changed.

### Two local differences explain all six responses

Let the original Flood-partition legs be \(a=F-1406\) and \(b=C-F\), with \(C\) denoting the selected Creation or Fall head. The three original interval lists are

\[
(a,b),\qquad(a+600,b-600),\qquad(a,600,b-600).
\]

Because \(I(600)=600\), only two local differences are needed:

\[
d_L=I(a+600)-I(a)-600,
\qquad d_U=I(b)-600-I(b-600).
\]

Moving the division changes the reversed total by \(d_L-d_U\); retaining both nodes changes it by \(-d_U\). For MT Regular and SP Cumulative, both differences vanish. For LXX Regular and MT Cumulative, both equal −990: the changes cancel when the division moves, while explicit refinement exposes the extra 990. The unequal differences in the remaining two rows explain their distinct responses. [C1451–C1452]

This is a local explanation of partition sensitivity. Decimal reversal is not generally additive, but it can preserve a particular named refinement or balance two equal failures of additivity. The repeated 990 has a place-value explanation: for a span with significant core \(ABC\) and one trailing zero, \(I(n)-n=990(C-A)\). The SP Regular core intervals use a different trailing-zero register, producing the larger changes shown above. [C1441–C1442]

### SP supplies a translated upper branch and named Shem connections

The refined SP Cumulative path also establishes a complete translation of its selected upper branch:

| Source role | Original SP coordinate | Conquest-held inverse | With Nativity leg |
|---|---:|---:|---:|
| Flood seam | 4716 | 2736 | 5436 |
| Shem birth boundary | 5316 | 3336 | 6036 |
| Accepted Fall head | 13396 | 11416 | 14116 |
| Change at every listed node | — | −1980 | +720 |

Only the original lower leg changes, from 3310 to 1330; the upper 600 and 8080 stay intact. Consequently all three listed nodes move together. This is an exact translation of this selected branch. The held Conquest or Nativity endpoint has its own role outside that branch. [C1453]

The internal coordinates connect the SP construction to both other traditions. Its Conquest-held inverse Shem node **3336 BC** equals native Rounded **LXX Regular Shem**. After the Nativity extension, the SP inverse Flood node **5436 BC** equals ordinary Rounded **MT Cumulative Shem**. Their numerical agreement supplies an interface between constructions while preserving each node's source role and anchor. [Native Rounded LXX model; File51a §16.2; C1458]

At the primary heads, MT and SP Cumulative also preserve their original 610-year difference through both inverse stages:

\[
14006-13396=12026-11416=14726-14116=610.
\]

Both source trunks contract by 1980, although the contraction occurs in different legs. Their original component differences \(120+490\) become \(2100-1490\); each totals 610. The head agreement thus has a local explanation, while the changed Flood difference shows why it is not a translation of both complete primary paths. [C1454]

### The 11270 bridge closes through the Covenant Key

Dean's observation joins the original source chronology to the SP inverse construction:

\[
14006-2736=11270=23\times490.
\]

The MT 14006 coordinate is available both in the Rounded construction and in the Actual internal Year-6/Adam comparison. The observation sits between two existing equal spans:

| Earlier node, BC | Later node, BC | Span |
|---|---|---:|
| MT inverse head 14726 | MT inverse cumulative Shem 3456 | 11270 |
| MT cumulative Creation 14006 | SP Conquest-held inverse cumulative Flood 2736 | 11270 |
| LXX Regular inverse head 13136 | MT Minimum Covenant 1866 | 11270 |

The rows translate both endpoints by −720 and then −870. The last translation also carries the third node 1406 to 536, preserving the partition \(12600=11270+1330\). These are linked placements of the same interval. [C1432]

Holding the **MT Minimum Covenant at 1866 BC**, the Priestly Key joins the separately constructed LXX and SP heads:

\[
\boxed{1866+\frac{25}{23}(13136-1866)=14116.}
\]

The input and output radii are \(11270=70\times161\) and \(12250=70\times175\). Thus the relation is 70 times File60's 161→175 Covenant expansion. The reciprocal factor 23/25 returns the LXX head about the same anchor. It reverses this Key scaling, without performing a second decimal reversal. The minimum Covenant is \(2081-215=1866\); the LXX source trunk retains its declared full-430 frame. [File60 §2.6; C1432]

The coefficient 490 also comes from the source rows: the MT cumulative upper leg 9170 exceeds the accepted SP leg 8680 by 490, comprising the rounded lifespan reductions 485 and the accepted five-year Fall adjustment. Together with the original Flood difference 120, it gives the preserved MT–SP head gap 610. The SP inverse Flood 2736 itself is already determined before that Fall adjustment. [C1432; C1461]

The earlier LXX–SP 1700 comparison now closes into the same result by a second route:

\[
13136\xrightarrow{P_{1406}}13306
\xrightarrow{-1700}11606
\xrightarrow{+2700}14306
\xrightarrow{-190}14116.
\]

Here \(P_{1406}\) is the Prophetic Key held at Conquest. The other changes are the original SP Flood–Conquest interval, the common Nativity-stage shift and the accepted SP Regular-to-Cumulative inverse head difference. The net change is \(170-1700+2700-190=980\), exactly the direct Covenant Key gain. This circuit connects the supplied source relations; its routes share those inputs. [Comparative study §§6.2,7.7; C1460]

The scope is precise. The Covenant Key connects the specified heads; it does not send their corresponding internal nodes to one another. The SP upper-branch translation is the separate multi-node result. Together they explain how the three traditions can share a construction, exchange named numerical junctions and meet through an anchored Key while retaining different internal paths.