A complementary partition law for all six inverse spines

Research support for C1433–C1482, September 29, 2026. Exact verification: 41 grouped checks passed. All coordinates are BC.

Main result

MT Regular and accepted SP Cumulative retain their inverse totals when a named 600-year segment is inserted explicitly or merged on either side. LXX Regular and MT Cumulative instead share a second pattern: both coarse partitions agree, while explicit refinement adds 990.

This is a relationship between complete selected path constructions, beyond the previously established reciprocal relationship between LXX and SP terminal heads. It also explains why matching terminal heads does not automatically imply additive refinement.

The 600-year interval has a different source role in the two chronological modes. In Regular chronology it runs from Noah's birth to the selected Flood. In Cumulative chronology it is Shem's serial lifespan from the Flood/Arphaxad seam back to cumulative Shem. Keeping those roles distinct reveals the shared arithmetic behavior.

Six named source paths

The ordinary source path is Conquest → Flood → head. The comparison replaces Flood with the named older node, 600 years earlier; the refinement retains both. The source heads remain ordinary Rounded Creation for MT/LXX and accepted Fall for SP.

All six comparisons retain the 1406 anchor and the declared original head. They change only the interior partition through an independently named node; they do not choose a new affine pivot or scan arbitrary partitions.

The six outcomes

Each entry below is a transformed duration measured from held Conquest. Adding 1406 gives its two-stage coordinate; adding 4106 gives its coordinate after the separately reversed original Nativity–Conquest leg is prepended.

The primary Flood-partition spines retain their accepted status. The alternate and refined constructions here are diagnostics that explain their behavior; they do not replace the accepted heads.

One local law explains the table

Let $a=F-1406$ and $b=C-F$. The original two legs are $(a,b)$. Moving the partition to Noah or Shem gives $(a+600,b-600)$. Retaining both gives $(a,600,b-600)$. Since $I(600)=600$, define the two original-duration merging defects:

Then

This gives three useful conclusions.

1. MT Regular ↔ SP Cumulative: both defects vanish. The accepted SP cumulative path therefore preserves its total through the same kind of named 600-year refinement that File52c establishes for MT Regular. SP's refinement uses the fixed span 8080 together with unchanged 600; its lower span 3310 becomes 1330, while 3910 becomes 1930.
2. LXX Regular ↔ MT Cumulative: both defects equal −990. Moving the partition preserves the terminal head because the two defects cancel. Retaining both nodes exposes the extra 990. The matching coarse heads do not require additivity at each merge.
3. LXX Cumulative and SP Regular: their unequal defects explain their changed alternate totals. LXX retains one additive merge. SP Regular crosses different trailing-zero registers and changes both.

The common 990 displacement follows from the place-value behavior of the three-digit significant cores. For $n=10(100A+10B+C)$, with $A,C\ne0$,

This formula explains why a carry between the original middle and leading columns can change a refined sum by 990. SP Regular includes two-digit cores with two trailing zeros; its corresponding displacement is $900(B-A)$ for $100(10A+B)$. No second reversal is applied.

Source control and novelty

• The original six source paths and MT/LXX cumulative Shem coordinates are supplied by the September 28 LXX/SP comparative study, §§2–7.
• MT Regular Noah and its existing three equivalent partitions are explicit in File51a §3.1 and File52c §§3.3–3.4.
• LXX Noah 3836 is already reconstructed inside the native Rounded model Moses_LXX_native_complete_path.json, with Rounded Creation 5486, Shem 3336 and Arphaxad 3236. No new 215-year translation or Actual Gear table is applied. SP Noah 3706 follows from its selected lower Rounded Flood 3106 and retained 600-year Noah–Flood leg; File47 §6.1 independently corroborates that local relation in its separately tagged SP state. This does not supply a new complete Rounded SP pre-Flood genealogy.
• Shem's lifespan 600 is explicit in all three File18 source tables: MT §2, SP §3, LXX §4. It is unchanged by five-year rounding. Cumulative placement therefore gives SP Shem 4716+600=5316; File53 §19 corroborates the paired 5316–5309 and 4716–4709 source envelopes. The Rounded member here remains explicitly selected; no regular SP person-station companion is imported into this cumulative path.
• Existing reader results include the primary head differences 0/+500/−190, cumulative contractions 1980/3960/1980, and the common stage shift 2700. These are not new discoveries in this note.
• MT Cumulative Flood+Shem refinement is also inherited: File52a §11.1 and File52b §2.6 already yield 13016 from 1406, cited as a secondary branch in File52c §3.4. Its +990 relation is used as a control. The Shem-only split, which returns 12026, remains distinct from that inherited refinement.
• The new contribution is the matched six-path comparison of one named 600-year refinement, together with the paired local laws that explain its outcomes. The companion partition_verification.json stores the original paths, reversed original legs, heads, two defects, exact outcomes and source hashes. verify_partitions.py reproduces it.

The most useful reader addition is the two complementary correspondences above. Detailed alternative heads can remain in the technical record so the explanation stays focused on the relation among the families.

Two SP upper branches translate intact after refinement

The refined SP Regular upper legs are 600 and 700; SP Cumulative upper legs are 600 and 8080. Each of these original durations is fixed by the one-pass reversal. Consequently the reconstructed upper three-node branches retain every internal interval and translate intact:

Prepending the reversed original Nativity–Conquest leg shifts each reconstructed node another 2700. The completed refined Regular branch is 11206 → 11806 → 12506, translated +8100 from its source. The completed Cumulative branch is 5436 → 6036 → 14116, translated +720 from its source.

The Regular refined head is 9806 from held Conquest, distinct from its primary Flood-only head 11606. This difference arises because the coarse original 1300 reverses to 3100, while its source-named 600+700 refinement stays 1300. The Cumulative upper total 8680 is fixed both whole and as 600+8080, so its primary head 11416 survives refinement.

These are translations of two explicitly named upper branches. The held Conquest coordinate remains 1406 and is not translated; neither result is a conversion map for an entire genealogy.