from research import *
s=begin(506,'Integral endpoints require pivot compatibility','What extra condition distinguishes an integral width ladder from integral dates?',{'map':'D_E,a(x)=(25x−2a)/23','integer_source_and_pivot':True},['Strategy §5C','C505; exact algebra'])
finish(s,{'one_stage_iff':'x congruent a mod23','two_stage_condition':'first outputs integral; second pivot congruent to every first output mod23','pair_condition':'if original width belongs to529Z, integral first outputs share one residue mod23','same_pivot_two_stage_iff':'x−a divisible by529','proof':'25x−2a ≡2(x−a) mod23 and2 is invertible mod23; for a repeated pivot D_E,a²(x)=a+(625/529)(x−a), gcd(625,529)=1'},
 'Widths lose the pivot term; endpoints retain it. A529-multiple width guarantees a shared second-stage residue after an integral first stage, but it does not supply a licensed pivot of that residue.',
 'Apply this criterion to the already supplied C480 pivots and the fixed12026 trial, without choosing new pivots.',{'coprime':25%23==2 and 625%23==4,'common_residue_width':(F(25,23)*529)%23==0})