from research import *
s=begin(993,'Universal coarsening criterion','When can a componentwise conversion be replaced by one factor per merged block?',{'coarsening':'sum fine components into declared blocks','conversion':'diagonal factors k_i'},['C991','C992','File63 retained partitions'])
proof={'necessity':'Test each unit vector e_i: the fine factor k_i must equal its containing coarse-block factor.','sufficiency':'If factors are constant within each block, factoring that common value out of the block sum proves equality.','criterion':'C·diag(k_i)=diag(k_B)·C iff k_i=k_B within every block'}
finish(s,proof,'Coarsening commutes with conversion for every vector exactly when the factors are constant inside each merged block. Equality on one supplied total is weaker.','Exhibit the information hidden by one equal-total partial/whole comparison.',{'uniform_example':E*(299+161)==E*299+E*161,'mixed_not_universal':J!=E})