What the fivefold regular kernel explains

The fivefold formula joins two source-grounded constructions. The named MT/LXX row changes generate the entire Regular displacement field, including the Creation gap1250 and Flood gap650. The Terah, Cainan and Sojourn variants generate the whole405 placement field. Their connection by a common coefficient5 is exact if and only if C=1.3u and T=0.6u, where u=100. The source supplies these scales; the support patterns alone do not force their proportions.

The row mechanism comes first

File18 §§2.1.3–2.2 and4.1–4.2 supply the complete selected birth rows. With Cainan leveled for this comparison, six pre-Flood differences are+100: Adam, Seth, Enosh, Kenan, Mahalalel and Enoch. Jared, Methuselah and main Lamech contribute zero begetting difference. After the Flood, Arphaxad, Shelah, Eber, Peleg, Reu and Serug each contribute+100, while Nahor contributes+50. Noah, Shem and Terah preserve their selected effective intervals between traditions.

Consequently, the Creation difference is(6+6+1/2)u=1250; the shared Flood/Noah-side plateau is(6+1/2)u=650. The600 difference between those measurements comes from the six located early changes. Ordered accumulation also reconstructs every intermediate boundary. From Adam through Abraham the complete difference field is:

1250,1150,1050,950,850,750,750,650,650,650,650,650,550,450,350,250,150,50,0,0.

The JSON companion attaches each value to its named source row. This is the positive explanatory content beyond the two sums. The zero contributions and the position of each century matter: preserving the same two totals would not preserve all internal boundaries.

The comparison is explicitly Cainan-OFF on both sides. Literal source checks are5279−130−3899=1250 and3023−130−2243=650. Native LXX retains Cainan and gives1380 and780. Common Noah/Shem binding corrections cancel; no new Gear choice is needed.

The variant scales have their own source meanings

Terah’s reconstructed130 and Cainan’s130 are distinct source quantities. The shared numeral does not establish that one was generated from the other. Likewise, a source-derivedT is not an independently attested manuscript switch, and twelve equal age differences are not twelve statistically independent witnesses.

The source scopes are nested: Adam through Arphaxad/Flood receivesS,T,C; inserted Cainan through Terah receivesS,T; Abraham through Joseph/Levi receivesS; Aaron/Moses remains fixed. Kohath and Amram have no direct Regular pivot dates in this scheme. Cainan’s inserted coordinate does not receive an additional self-shift. File18 §1.3.1, lines618–630, controls these scopes.

The coefficient condition is exact and bounded

The row results and Strategy’s formula require

The second equality givesC=13u/10. Subtracting the equations gives6u=10T, henceT=3u/5. Conversely, those two conditions imply both equalities. Nothing in these equations constrainsS.

If the coefficient is initially unspecified, a common nonzero coefficientκ exists precisely when6C=13T, withκ=13u/(2C). Settingκ=5 adds the scale conditionC=13u/10. This distinguishes the compatibility of the two ratios from the numerical choice of five.

The conditions hold for the received100,60,130. They are two independent equations in this declared variable model, not two independent historical witnesses. The support masks explain the coefficients12.5 and6.5. Their numerical calibration toC andT remains an observed source relation. RetainingT,C as source primitives or replacing them by rational multiples ofu are equivalent descriptions unless the new relations have a simpler independent source rationale.

The familiar950±300 form follows from the same pair: the midpoint is9.5u and half-width3u. Noah’s separately supplied950-year lifespan is a numerical comparison with that midpoint. It does not give an additional proof of the same pair or explain the placement of the twelve age changes.

The405 envelope is a complete source field

The binary offsets are

Their consecutive gaps are60|70|60|25|60|70|60. Here70=C−T and25=S−C−T. The ordering depends on0<T<C andS>T+C, both satisfied by the selected sources. The width isW=S+C+T=405 and does not depend on the fivefold coefficient.

File07 §2.11.1a, lines984–1016, explicitly prints all eight states. Its Year-6 labels3901,3961,4031,4091,4116,4176,4246,4306 and corresponding endpoint labels verify the whole field. File18 §1.1’s0,60,215,275 is the Cainan-OFF restriction: its gaps60|155|60 and width275 are equally valid. It varies Terah and Sojourn; Strategy’s four-state block0,60,130,190 varies Terah and Cainan at fixed Sojourn.

Paired traditions using the same variant labels and support retain their baseline difference, because their added shifts cancel. Different labels change the comparison by the declaredT,C,S contributions. This connects the row-generated baseline with a controlled placement geometry, without making the source masks interchangeable.

The SP side confirms the need for the local mechanism

The post-Flood SP begetting changes supply the same+650 baseline displacement. Before Noah, the current G2 primary chain loses100 at Jared,120 at Methuselah and130 completed units at Lamech. Thus its Creation offset is650−350=300 while the Flood offset remains650. The nominal Lamech age difference129 gains one completed-unit correction because the source53rd counted year contributes52 completed units. The counting distinction is used once.

These complete source facts satisfy Strategy§4.1’s demand to retain the SP100/120/129 and Flood-year qualification. The fivefold LXX summary does not generate this SP mechanism. Current File18 main LXX182/753 is retained; changing lifespan753 does not alter the regular age182 used here.

Source cost and the resulting explanation

The row mechanism retainsu, the named support and half-century exception, order, source leveling, bindings and terminal. The variant mechanism retainsT,C,S, their finite choices and nested scopes. The compact fivefold expression additionally retains the two cross-layer calibration identities. RewritingT andC in terms ofu removes two stored numbers only by storing their coefficients and justification; it does not establish a globally smallest grammar.

This audit extends the explanation rather than discovering1250,650 or405. C739–741 already reconstruct the whole displacement fields; C772–788 already establish the finite state family; C1178–1189 recover the repeated amplitudes and source supports. The distinct contribution here is the exact condition connecting the row mechanism to Strategy’s fivefold summary, with the source burden visible.

A reader-facing formulation is:

The long genealogy and the variant field measure different source structures. Repeated, located age changes generate the1250-to650 displacement pattern; three source-appointed choices generate the405 envelope of possible placements. At the received values, their scales satisfy exactly the relations that express both head gaps as fivefold combinations of Terah and Cainan. The formula therefore summarizes a compatibility between two complete mechanisms. It does not replace the local rows, their placement, or the independent Sojourn choice.

No reviewed source passage supplies a historical rule multiplying Terah/Cainan adjustments by five to create the ancestral ages. That remains beyond the demonstrated reconstruction. The current result positively explains the internal fields and their relation while preserving that distinction.

Verification

The independent script checks all nineteen common-row changes, the twenty-boundary field, literal matched/native heads, the eight printed variants, the coefficient conditions, and the SP counting side. All24 checks pass. The JSON companion records exact inputs, source locators and SHA256 hashes. No root journal, source file or adopted model was changed.