import numpy as np
rng=np.random.default_rng(173)
def spacetime(right,T,K):
    c=[t%2 for t in range(T+K+4)]                         # centre dark at even t (phase 0)
    R=list(right)+[0]*(T+K+8); x1=[]
    for t in range(T+K+2):
        x1.append(R[0]); new=[0]*len(R)
        for i in range(len(R)):
            l=c[t] if i==0 else R[i-1]; r=R[i+1] if i+1<len(R) else 0
            new[i]=l^(R[i]|r)
        R=new
    cols=[c[:T+K+2]]
    cols.append([c[t+1]^(c[t]|x1[t]) for t in range(T+K+1)])
    for k in range(1,K):
        a,b=cols[k],cols[k-1]; cols.append([a[t+1]^(a[t]|b[t]) for t in range(len(a)-1)])
    return cols,x1
checks={"A4(n)=0":0,"A3(n)=A1(n+1)":0,"A5(n)=[A1(n+1)==A1(n+2)]":0,"A7(n)=[A1(n+2)==A1(n+3)]":0,
        "A6(n)=notA5 and (A3 or A2)":0,"u(n+1)>=u(n) (monotone u=A1)":0}
law={m:0 for m in range(2,8)}; tot=0
for _ in range(150):
    w=int(rng.integers(1,12)); right=list(rng.integers(0,2,size=w))
    T=40; K=18; cols,x1=spacetime(right,T,K)
    A=lambda k,n: cols[k][2*n]
    for n in range(0,10):
        tot+=1
        checks["A4(n)=0"]+= A(4,n)==0
        checks["A3(n)=A1(n+1)"]+= A(3,n)==A(1,n+1)
        checks["A5(n)=[A1(n+1)==A1(n+2)]"]+= A(5,n)==int(A(1,n+1)==A(1,n+2))
        checks["A7(n)=[A1(n+2)==A1(n+3)]"]+= A(7,n)==int(A(1,n+2)==A(1,n+3))
        checks["A6(n)=notA5 and (A3 or A2)"]+= A(6,n)==int((1-A(5,n)) & (A(3,n)|A(2,n)))
        checks["u(n+1)>=u(n) (monotone u=A1)"]+= A(1,n+1)>=A(1,n)
        for m in range(2,8):
            law[m]+= A(2*m+1,n)==int(A(1,n+m-1)==A(1,n+m))
for k,v in checks.items(): print(f"  {k:34s}: {v}/{tot}")
print("  proposed law A_(2m+1)(n) = [A1(n+m-1)==A1(n+m)]:", {f"m={m} (depth {2*m+1})":f"{v}/{tot}" for m,v in law.items()})