s559.py

from research import *
s=begin(559,'Locate the Joseph cyclic symmetry','Does a three-cycle act consistently on the complete component family?',{'rotation':'(a,b,c)→(b,c,a)'},['File62 §§6.1–6.4; C558'])
t=json.loads((ROOT/'prep/file61_62_tables.json').read_text())['joseph_partition_family'];checks=[]
for r in t:
 v=[F(r[k]) for k in ['forward_flank','backward_flank','overlap']];rots=[v,v[1:]+v[:1],v[2:]+v[:2]]
 checks.append(rots==[[F(x) for x in row] for row in r['cyclic_110_components']])
finish(s,{'states':9,'component_displays':27,'example':[[4,23,83],[23,83,4],[83,4,23]],'action_domain':'ordered duration triples'},'A genuine C3 action rotates duration components and returns after three applications. It does not establish a permutation of historical people or complete chronology states.','Test how whole-year and exact phase choices change crossed containers.',{'all_rotations':all(checks),'third_power_identity':(4,23,83)==tuple([4,23,83][3:]+[4,23,83][:3])})