technical 05 Keys

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.