Forty-two coordinates form one structured field
The cumulative phase-and-clutch table has seven genealogical rows, two phase states and three clutch positions. Its 42 coordinates follow one expression: X(i,j,k)=b(i)+jp−kh. Here b(i) is a row’s original boundary, p the phase offset, and h the half-clutch. The indices select Nisan or Aaron phase and the original, half-clutch or full-clutch position.
Before additional source equalities are imposed, seven row positions plus p and h give nine parameters. The 42 displayed coordinates therefore satisfy 33 independent linear relations. Seven original Nisan boundaries, one Aaron boundary and one half-clutch boundary suffice to recover the unrestricted field.
Additional source relations can reduce that description. The exact link h=2p, represented by 7=2×3.5, removes one freedom. The equality of Levi’s and Amram’s 137 lifespans removes another. These are relational constraints; separately fixing all their numerical values is a stronger specification. The whole-year phase 3 does not satisfy the same phase relation and retains its distinct display policy.
Finally, the row boundaries themselves can be generated from the Moses anchor and lifespan stack, and the clutch from the regular/cumulative comparison. The table then becomes an output of the coupled source model. Its derived interfaces do not relocate the historical people whose source dates anchor the construction.
Sources: C1048–C1051; C1056–C1057; File61 §§7.3A–7.6; File62 §§1.2–1.4.