import numpy as np
rng=np.random.default_rng(113)
cp=(0,1,1,1,1,1); p=6
def build(u,K):
    T=len(u)*p
    d=[cp[t%p] for t in range(T+K+2)]
    e=[]
    for t in range(T):
        ph=t%p; e.append(u[t//p] if ph==0 else (1 if ph==5 else 0))
    cols={0:d,1:e}
    for k in range(1,K):
        x,y=cols[k],cols[k-1]; cols[k+1]=[x[t+1]^(x[t]|y[t]) for t in range(len(x)-1)]
    return cols
K=60; N=40
reach={}
for trial in range(60):
    u=list(rng.integers(0,2,size=N)); base=build(u,K)
    for j in range(N):
        v=u.copy(); v[j]^=1; alt=build(v,K)
        for k in range(1,K+1):
            if any(base[k][t]!=alt[k][t] for t in range(0,p)):     # does flipping u_j change V_k(0)?
                reach[k]=max(reach.get(k,-1),j)
print("depth k : furthest free bit u_j that can change V_k(0)")
print("  "+"  ".join(f"{k}:{reach.get(k,'-')}" for k in range(1,K+1,4)))