s1366.py

from research import *
s=begin(1366,"Write the anchored-Key domain supplement","When do exact affine comparisons preserve an integer-year stage?",{},["C1299","C1307–C1308","Strategy.md"])
t=json.loads("\"## 5. Keys, anchors and integer stages\\n\\nAn 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.\\n\\nFor 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\\\\).\\n\\nA 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.\\n\\nUniform 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.\\n\\nFor 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.\\n\\nThe 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.\\n\\nSource/proof route: inherited Key and fixed-pivot models; C1299, C1307–C1308; Strategy §§3.3–3.4.\\n\"")
a=artifact("draft/technical_05_Keys.md",t)
finish(s,{'artifact':a,'words':len(t.split())},"The Key supplement distinguishes exact affine composition, common calibration and integer-stage admissibility without extending any chronological route.","Adopt the complete Covenant technical supplement.",{'nonempty':len(t)>100,'obstruction':9170%3==2 and 300%3==0})