s1202.py

from research import *
s=begin(1202,'Derive the composite cap-plus-round inverse set','What original lifespans can produce a fixed rounded capped value when capacity is known?',{},['C1201','C1143','C1131 cap analysis'])
def Q(x):return 5*((x+2)//5)
examples=[]
for c in [10,11,12,13,14]:
 q=Q(c);pred=[L for L in range(0,31) if Q(min(L,c))==q];lower=max(0,q-2);examples.append({'c':c,'Q_c':q,'first_input':min(pred),'finite_scan':pred})
a=artifact('model/cap_round_fibre_law.json',json.dumps({'law':'For y below Q(c), fibre is {L>=0:Q(L)=y}, provided nonempty below c. For y=Q(c), fibre is all L>=max(0,Q(c)-2). Values y>Q(c) are impossible.','proof':'Below the capacity cell clipping cannot affect the value. At the capacity cell, all uncapped values in that cell and every capped input merge.','examples':examples,'scope':'integer nonnegativeL,c; nearest5; no upper baseline bound supplied'},indent=2)+'\n')
finish(s,{'fibre_law':a},'At the rounded capacity, a finite rounding cell joins an unbounded cap tail. The rounded capped output can therefore forget even whether clipping occurred.','Apply the threshold formula to the three SP clipped capacities.',{'thresholds':all(x['first_input']==max(0,x['Q_c']-2) for x in examples),'tails':all(x['finite_scan']==list(range(x['first_input'],31)) for x in examples)})