The calendar Keys belong to the same grammar because they preserve a declared measure while producing different counts. With E=25/23, P=70/69 and J=300/299,
The year counts differ because their associated calendar units differ. This existing calibration does not identify every chronological use of a Key as a literal calendar conversion. It gives one exact relation that explains many otherwise separate-looking outputs.
The source subdivisions supply a second connection. In the 483 carrier, expanding only the final 80.5 by E gives the same total as applying P to all 483. In the 12558 path, expanding only the final 483 by E gives the same total as applying J to the whole. The selected fractions are 1/6 and 1/26. These inherited identities and the calendar calibration form a small dependent equation system: three independent equations determine the three Keys; a fourth relation is a compatibility check. Solving this system backward is another description of the established relationships, not independent evidence for their historical origin.
The difference between a shared total and a shared path remains essential. Selective expansion retains one source boundary that uniform expansion moves. Their two component differences cancel in the total, so an endpoint-only comparison cannot detect the distinction. Removing a subdivision is valid for every input only when all its components receive the same factor. Otherwise the apparent whole-span factor depends on the source weights.
That observation makes the retained-part families particularly economical. If u is the converted part and v the retained part, their native, Prophetic and Priestly totals obey
and therefore
Two totals recover the numerical parts; the third is dependent. This explains both the complete 690/30 bracket table and the 12600/12740/13440 family through one construction. The sources still identify what those parts represent and where their boundaries belong.
Placement adds another constraint. Integral widths do not guarantee integral endpoints when a Key acts about a held pivot. For two successive anchored Keys, the permitted inputs form a residue class determined by both pivots—or the class can be empty. In particular, J followed by P requires the two integer pivots to agree modulo 3 if both intermediate outputs must remain integral. The scalar ratios still commute; their staged, anchored realizations need not have the same domain. Rational outputs remain exact comparison coordinates where permitted, without rounding or moving an anchor to force agreement.
The resulting explanation keeps a small set of rules alongside the source information those rules require: component weights, order, measurement unit, retained boundaries, pivots and authorized operations. It reconstructs complete families and identifies dependent consequences. Mathematical agreement can then be stated precisely while the separate questions of source selection, historical transmission and theological interpretation remain visible.