import json,itertools,math
for M in [2,3,4,6]:
 for rho in [0.,.2,.8,1.]:
  vals=[];vectors=[]
  for zs in itertools.product([-1.,1.],repeat=M+1):
   es=[math.sqrt(rho)*zs[0]+math.sqrt(1-rho)*zs[i+1]for i in range(M)]
   vals.append(sum(es)/M);vectors.append(es)
  mean=sum(vals)/len(vals);var=sum((v-mean)**2 for v in vals)/len(vals)
  assert abs(mean)<1e-12 and abs(var-(rho+(1-rho)/M))<1e-12
  cov=sum(es[0]*es[1]for es in vectors)/len(vectors)
  assert abs(cov-rho)<1e-12
# Duplicate errors have unchanged variance; shared wrong means have no disagreement.
assert sum(v*v for v in [-1.,1.])/2==1
means=[2.]*5;assert sum((v-sum(means)/5)**2 for v in means)==0 and (sum(means)/5)**2==4
mus=[0.,2.];vs=[1.,3.];mu=sum(mus)/2
total=sum(v+m*m for v,m in zip(vs,mus))/2-mu*mu
assert abs(total-(sum(vs)/2+sum((m-mu)**2 for m in mus)/2))<1e-12
H=lambda p:-sum(v*math.log(v)for v in p if v>0)
ps=[[.9,.1],[.1,.9]];avg=[sum(p[j]for p in ps)/2 for j in [0,1]]
assert H(avg)-sum(H(p)for p in ps)/2>=0
ns=[1,2,4,8,16,32]
figures=[{"id":"correlation-floor","kind":"analytical_curve","caption":"Exact author-constructed equicorrelated scalar error variance with individual variance one; nonnegative rho is valid for all displayed M. No trained-network measurements.","x":{"name":"Ensemble members M","values":ns},"series":[{"name":"rho="+str(r),"values":[r+(1-r)/n for n in ns]}for r in [0.,.2,.8]]},{"id":"shared-bias-risk","kind":"analytical_curve","caption":"Exact author-constructed expected squared error, shared bias .5 and individual zero-mean error variance one. No classification or OOD detector scores are inferred.","x":{"name":"Ensemble members M","values":ns},"series":[{"name":"Bias-squared contribution","values":[.25]*len(ns)}]+[{"name":"Total MSE, rho="+str(r),"values":[.25+r+(1-r)/n for n in ns]}for r in [0.,.2,.8]]}]
print(json.dumps({"status":"PASS","checks":["Exact finite Rademacher mean, covariance and average variance","Duplicate-member variance control","Shared-bias zero-disagreement control","Predictive-mixture total variance","Categorical mixture entropy difference"],"empirical_benchmark":False,"source_proofs_verified":False,"figures":figures}))