# Covenant/clutch technical companion proposal

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.

For integration, the main reader can retain the source-path explanation, the 161/299/460 origins and the three joins’ different dependency roles. Move the symbolic envelope cancellation, both complete phase tables, the indexed array/rank paragraph and the general Key condition here. The main text can report their consequences in words and point to this section for reconstruction. 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 proposal adds no numerical family or source operation.