s609.py

from research import *
s=begin(609,'Generalize allocation endpoint equivalence','How large is the junction discrepancy between whole-Key and terminal-E routes?',{'examples':[[483,F(1,6),P],[12558,F(1,26),J]]},['C568–C571; algebraic path comparison'])
rows=[]
for S,f,k in [(F(483),F(1,6),P),(F(12558),F(1,26),J)]:
 a=(1-f)*S;b=f*S;rows.append({'S':S,'prefix':a,'suffix':b,'total':a+E*b,'whole_total':k*S,'junction_gap':(k-1)*a})
finish(s,{'comparisons':rows,'general_condition':'k=1+f(E−1)','junction_gap_formula':'(k−1)(1−f)S'},'For both source partitions the endpoint agreement follows from one gain-allocation law, while the interior discrepancy is fixed by(k−1)(1−f)S. Retaining the path reveals exactly what the total-span comparison forgets.','Test why the Rounded inverse module must preserve component boundaries.',{'endpoints':all(r['total']==r['whole_total'] for r in rows),'gaps':[r['junction_gap'] for r in rows]==[F(35,6),F(525,13)]})