## 5. Keys, anchors and integer stages

An anchored Key acts as \(D_{k,a}(x)=a+k(x-a)\). In a common linear coordinate chart, \(D_kT_t=T_{kt}D_k\); anchor offsets must be carried when pivots differ. Rational spans compose under the exact factors. Requiring integer intermediate coordinates is an additional condition.

For a reduced factor \(p/q\) and integer pivot \(a\), an integer input \(x\) gives an integer output precisely when \(q\mid(x-a)\). This yields the Priestly modulus 23 and the two-stage duration ladder \(529k\to575k\to625k\).

A fixed-pivot example shows why rational composition alone is insufficient. Apply \(J=300/299\) about 14006, then \(P=70/69\) about 4836. The first integer stage requires \(x=14006+299t\) and yields \(y=14006+300t\). Its distance from the second pivot is \(9170+300t\), always 2 modulo 3. It is therefore never divisible by 69. This emptiness belongs to those pivots and integer-stage requirements, not to all J/P applications.

Uniform conversion commutes with addition. A block grouping \(B\) commutes with a diagonal selection of factors \(K\), in the sense \(BK=\bar K B\) for every input, precisely when each grouped block uses one common factor. Different partial-conversion routes can have equal totals while retaining different interior positions.

For crossing the civil BC/AD epoch, elapsed years use \(BC+AD-1\). The source’s Rounded endpoint-width convention \(BC+AD-2\) is a separately declared measurement. A formula valid in one convention is not silently transported to the other.

The original Keys satisfy \(336E=360P=364J=8400/23\). The Sothic companion \(H=2923/2921\) does not satisfy \(365H=8400/23\); the exact-K extension is \(G=1680/1679\). Sharing an operation family is broader than sharing exact calibration.

Source/proof route: inherited Key and fixed-pivot models; C1299, C1307–C1308; Strategy §§3.3–3.4.