# Exact check of Lemma 2.3 (identity = 18 N(V)) and of the position-0 column (2.5), d=3, r=23. No arguments.
import random
from field_encoding import Field, rows, coef, Xs, k
r=23; F=Field(r,3)
print("X* =",Xs," k =",k,flush=True); assert (Xs,k)==(1439,1440)
th=[0,1,0]; t4=F.tr(F.pw(th,4))
rnd=random.Random(2026); bad=0
for t in range(10):
    xis=[[rnd.randrange(r) for _ in range(3)] for _ in range(3)]
    for x in xis:
        if not any(x): x[0]=1
    if t%4==3: xis[1]=xis[0][:]
    Ps=[rows(F,x) for x in xis]
    G=[[Ps[j][i] for j in range(3)] for i in range(3)]
    lhs=coef(G,k-1,r)
    V=F.mulmod(F.mulmod(F.sub(xis[1],xis[0]),F.sub(xis[2],xis[0])),F.sub(xis[2],xis[1]))
    rhs=(18*F.norm(V))%r
    col_ok=all([Ps[j][0].get(0,0),Ps[j][1].get(0,0),Ps[j][2].get(0,0)]==
               [F.norm(x)**2%r,(x[0]*F.norm(x)**2)%r,(-t4*F.norm(x)**2)%r] for j,x in enumerate(xis))
    bad+= (lhs!=rhs) or not col_ok
    print(" trial",t,"distinct" if len({tuple(x) for x in xis})==3 else "repeated",
          " lhs",lhs," 18N(V)",rhs," column0 ok:",col_ok,flush=True)
print("failures:",bad,flush=True)