What retained parts let us recover
Several apparently separate families follow the same retained-part construction. Let u denote the component that receives a Key and v the component held unchanged. Their native, Prophetic and Priestly totals are T₀=u+v, Tₚ=Pu+v and Tₑ=Eu+v. Because E−1=6(P−1), the gains obey Tₑ−T₀=6(Tₚ−T₀). The three totals therefore contain only two independent numerical measurements.
Two totals recover the components exactly: u=69(Tₚ−T₀), then v=T₀−u. The third total checks the same relationship rather than adding a third independent fact. This one construction covers all three 690-core brackets: with retained flanks of 0, 30 or 60, the native/P/E totals are respectively 690/700/750, 720/730/780 and 750/760/810.
Supplement A’s cumulative calendar-body family uses the same construction on a larger scale. Its converted upper segment is 9660 and its fixed lower segment is 2940. The resulting totals are 12600, 12740 and 13440. These source-defined components make the family intelligible without treating its three totals as unrelated coincidences.
Numerical recovery still leaves chronological information to the source. It does not identify which named interval occupies each role, their order, their absolute placement, or permission to apply the operation. Recovering two magnitudes is therefore one part of reconstructing a declared chronological object.
Sources: 14-file_70-supplement-a-key-of-23-fine-resolution-720-30-rail-2-.md §1.2: complete690/30 bracket table; 14-file_70-supplement-a-key-of-23-fine-resolution-720-30-rail-2-.md §10.3: cumulative calendar-body matrix, lines2130–2185.