## Calendar Keys preserve a declared measure

The Keys explain how another kind of agreement crosses families. Write

\[
E=25/23,\quad P=70/69,\quad J=300/299.
\]

Their corresponding schematic year lengths give

\[
336E=360P=364J=8400/23.
\]

Consequently different year counts can carry the same modeled day-volume. The completed 14720 duration gives 16000 under E, 44800/3 under P, and 192000/13 under J. Multiplying those counts by 336, 360 and 364 respectively gives 5376000 in every case. File52c appoints the E completion here; the P/J values illustrate the inherited calibration conditionally. Equal modeled volume does not itself appoint a new chronological route or a historical calendar. [Strategy §3.3; C1062]

Placement is specified by a held anchor: \(D_{k,a}(x)=a+k(x-a)\). This acts on a distance from the anchor, not a naked BC date. Source units, moving components and anchor together define the chronological comparison.

For the appointed E route, holding 6 BC converts 14720 to 16000 and locates the expanded endpoint at 16006 BC. The intermediate positions need not become integers. Every proper prefix of the four transformed paths has a nonzero residue modulo 23. For example, the shared first 4100-year prefix becomes 102500/23; the whole 14720 becomes 16000. Uniform multiplication remains additive over exact rational durations, so componentwise conversion and whole-span conversion agree at the endpoint. What fails is the further requirement that every retained internal position lie on an integer-year grid. The rational path is a conditional display of the same uniform conversion, not a universal map between all original chronological date labels. [File52c §3.6; C1059–C1062]

This distinction gives 23 and 529 their structural roles. The first controls integral Priestly conversion. Its square 529 permits the duration ladder

\[
529k\longrightarrow 575k\longrightarrow 625k.
\]

For the 12026 backbone and Exodus 1446, the supplied interval is 10580=20×529, yielding 10580→11500→12500 under two successive Priestly expansions. The coefficient 20 is obtained from that span; the later values follow from the same factor. Endpoint placement additionally requires the declared anchor and counting convention. [File52c §§6.1–6.2]