technical 06 Covenant

6. Covenant paths, joins and exact phases

This section supplies the algebra behind the reader draft’s source-to-path explanation. Regular coordinates follow supplied ages and household intervals; cumulative coordinates follow complete lifespans. Their common ordered-path form allows shared source values to force some joins, while other joins require additional source relations. All calculations below are inherited results.

For the seven cumulative boundaries in Abraham-to-Moses order, let

with empty sum zero and fixed Moses anchor $M=1526$. This is the suffix-sum path construction $b=M\mathbf1+U\ell$. Here $U$ is the $7\times6$ matrix with $U_{ir}=1$ when $r\ge i$, otherwise zero; its final row is zero. Adjacent differences recover the six ordered lifespan inputs. It generates $b=(2435,2260,2080,1933,1796,1663,1526)$. The regular path uses its own edge roles, so shared path form does not identify regular births with cumulative boundaries. Sources: File 62 §2.1; File 61 §7.3A; C1051; inherited technical companion §1.

In the full regular comparison, put Levi birth $B=1919$, Levi death $D=1782$, Exodus $e=1446$, and lifespans $L=137,K=133,A=137$ for Levi, Kohath and Amram. The source’s maximum envelope assumes successive lives occupy consecutive blocks. Subtracting regular elapsed chronology gives

Write cumulative Kohath $C_K=M+K+A$, cumulative Levi $C_L=C_K+L$, and cumulative Jacob $C_J=C_L+147$. Then

The first join follows from the definition of $d$; the second follows from the shared Levi lifespan and $B-D=L$. The third join, $C_J-d=R_I$, requires regular Isaac birth $R_I$ to satisfy $R_I-B=147$. Its source relation is $60+77+10=147$. Thus the third join reuses the Covenant family’s intergenerational condition. It is not forced by the clutch alone. Sources: File 61 §§6.3,8.1–8.3; C1035,C1052–C1054.

Set $h=d/2=7$. The complete Nisan field is:

The exact Aaron/Tishri field retains the same $h$ and adds $p=3.5$:

These are File 62 §§1.2–1.4’s complete tables. Final-row translations are interfaces, not relocated historical births. The separate whole-year Levi display is 1936; it must not replace exact 1936.5 in phase equations.

The forty-two entries satisfy

Each horizontal row has equal seven-year steps; every vertical column preserves $\ell$. In the unrestricted array model, now allowing all seven $b_i$, $p$ and $h$ to vary, the map has rank nine and thirty-three linear output dependencies. The relations $h=2p$ and $\ell_4=\ell_6$ each remove one freedom. Separately fixing $p=3.5$ and $h=7$ would impose two fixed-value conditions, a different specification. One Nisan-derived clutch preserves phase throughout; recalculating a full clutch to the same regular landing would cancel phase. Sources: C1048–C1050,C1056.

The Covenant Key agreement illustrates the same distinction between definition and constraint. Let $Q=1866$, $u=Q-D_{\mathrm{Levi}}=299$, $w=Q-D_{\mathrm{Joseph}}=276$, $J=300/299$, and $E=25/23$, using minimum-state deaths here. Defining $Y=Q-Ju$ supplies a generated point. Requiring $Y=Q-Ew$ adds the single condition $13w=12u$, giving $Y=1566$. Sources: File 60 §§6.1–6.2; C1040–C1041,C1047. The full Covenant 2081 remains author-designated; offering 2051/Isaac age fifteen remains proposed. Neither supplies an additional textual date.

The coupled model’s nineteen local inputs generate fifty-five outputs, including the clutch; its rank nineteen describes this declared parameterization, not historical independence or a globally minimal explanation. C1057 and C1290 retain that scope.

Exact source hashes, excerpts and proof-record bindings are already frozen in final_Covenant_claim_index.json, especially COV-10–COV-23. This supplement retains the inherited numerical family and its source operations.