from research import *
s=begin(1067,'Uniform Key measurements add no lost digit information','Do E/P/J images of the reversed total enlarge the aggregate measurement rank?',{},['C943–C945','C1066'])
from algebra import Matrix
raw=[100,10,1];rev=[1,10,100];M=Matrix([raw,rev]+[[k*x for x in rev] for k in [E,P,J]])
finish(s,{'measurement_rank':M.rank(),'kernel':[list(v) for v in M.nullspace()],'row_count':M.rows,'qualification':'bounded integer aggregate recovery still holds; component allocation remains lost'},'Adding uniform Key outputs supplies scalar multiples of an existing measurement. It restores neither aggregate information beyond the two totals nor the hidden allocation among source components.','Classify all completed-path interface claims by dependency type.',{'rank_unchanged':M.rank()==2,'row_count':M.rows==5,'kernel_direction':list(M*Matrix([10,-101,10]))==[0]*5})