## 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

\[
\ell=(175,180,147,137,133,137),\qquad
b_i=M+\sum_{r=i}^{6}\ell_r,\quad i=1,\ldots,7,
\]

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

\[
d=[L+K+A+(M-e)]-(B-e)
 =M+K+A-D=487-473=14.
\]

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

\[
C_K-d=D,\qquad C_L-d=B.
\]

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:

| Cumulative boundary | Original | Half-clutch | Full clutch |
|---|---:|---:|---:|
| Abraham | 2435 | 2428 | 2421 |
| Isaac | 2260 | 2253 | 2246 |
| Jacob | 2080 | 2073 | 2066 |
| Levi | 1933 | 1926 | 1919 |
| Kohath | 1796 | 1789 | 1782 |
| Amram | 1663 | 1656 | 1649 |
| Moses-side interface | 1526 | 1519 | 1512 |

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

| Cumulative boundary | Original exact phase | Half-clutch | Full clutch |
|---|---:|---:|---:|
| Abraham | 2438.5 | 2431.5 | 2424.5 |
| Isaac | 2263.5 | 2256.5 | 2249.5 |
| Jacob | 2083.5 | 2076.5 | 2069.5 |
| Levi | 1936.5 | 1929.5 | 1922.5 |
| Kohath | 1799.5 | 1792.5 | 1785.5 |
| Amram | 1666.5 | 1659.5 | 1652.5 |
| Aaron-side interface | 1529.5 | 1522.5 | 1515.5 |

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

\[
X_{ijk}=b_i+jp-kh,\qquad
j\in\{0,1\},\quad k\in\{0,1,2\}.
\]

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.