from fractions import Fraction as F
N=6
def av(*v):return[sum(t,F(0))for t in zip(*v)]
def sc(c,v):return[c*x for x in v]
def sy(a,b):return[[(a[i]*b[j]+a[j]*b[i])/2 for j in range(N)]for i in range(N)]
def ms(*ts):return[[sum(c*A[i][j]for c,A in ts)for j in range(N)]for i in range(N)]
x,y,z,w,bal,r=[[F(i==j)for j in range(N)]for i in range(N)]
ell=av(x,y,z,w);q=av(sc(F(1,2),x),sc(-1,bal))
base=[[F([F(1,4),1,F(1,2),1,1,0][i])if i==j else F(0)for j in range(N)]for i in range(N)]
fR=ms((1,sy(r,av(x,z,sc(-3,y)))),(-3,sy(r,r)))
b=F(135283,10**6);lam=F(69682,10**6);mu=F(21247,10**6);G=F(43321,100000)
f=av(r,sc(F(-7,8),q));g=av(q,sc(-1,r))
A=ms((1,base),((1-b)/8,sy(q,q)),(-b,fR),(-lam,sy(ell,f)),(-mu,sy(f,g)),(-(1-2*G),sy(ell,ell)))
for i in range(N):
 piv=A[i][i];assert piv>0,(i,piv)
 for j in range(i+1,N):
  for k in range(j,N):A[j][k]-=A[j][i]*A[k][i]/piv;A[k][j]=A[j][k]
assert 0<=b<=1 and lam>=0 and mu>=0
print('PASS c>=q²/8 branch: one exact6D certificate; gamma <=',G,float(G))