from research import *
s=begin(505,'The529 family is first a duration grammar','What does two-stage E expansion guarantee without any pivot choice?',{'E':E,'small_seed':1058,'macro_seed':10580},['C480 checkpoint','File52c §§3.8–3.9,6','C481'])
small=[F(1058)*E**i for i in range(3)];large=[F(10580)*E**i for i in range(3)]
finish(s,{'small':small,'macro':large,'general_integer_width_rule':'k*529 -> k*575 -> k*625','proof':'D_k,a(y)-D_k,a(x)=k*(y-x); second pivot cancels from the width too'},
 'Every supplied529-multiple width follows this ladder under two E expansions. Tenfold scaling commutes with it. Width agreement therefore connects C480 and the rounded macro family while leaving their source roles and pivots distinct.',
 'Derive the additional congruence condition needed for integral endpoints.',{'small':small==[1058,1150,1250],'large':large==[10580,11500,12500],'scale':large==[10*x for x in small]})