import json,math
def kl(q,p):
 if q==0:return -math.log1p(-p)
 if q==1:return -math.log(p)
 return q*math.log(q/p)+(1-q)*math.log((1-q)/(1-p))
def upper(q,c):
 if q==1:return 1.
 lo=q;hi=1.
 for _ in range(100):
  mid=(lo+hi)/2
  if mid==1:hi=mid;continue
  if kl(q,mid)<=c:lo=mid
  else:hi=mid
 return (lo+hi)/2
assert abs(math.log(1/.25)-math.log(4))<1e-14
for q in [0.,.01,.1,.5,.9]:
 for c in [.01,.05,.1,.5]:
  p=upper(q,c);assert p>=q and abs(kl(q,p)-c)<1e-10
for c in [.01,.1,.5]:assert abs(upper(0,c)-(1-math.exp(-c)))<1e-12
# Explicit invalid shortcut: the sample-selected prior equals the point posterior.
posterior=[0.,0.,1.,0.]
adaptive_prior=list(posterior)
point_kl=sum(q*math.log(q/p)for q,p in zip(posterior,adaptive_prior)if q>0)
assert point_kl==0.
def fixed_prior_eligible(prior_dependencies):
 return not prior_dependencies.get("uses_certification_sample",False)
dependencies={"uses_certification_sample":True,"chosen_after_posterior":True}
assert not fixed_prior_eligible(dependencies)
assert fixed_prior_eligible({"uses_certification_sample":False})
# This audits the declared dependency premise; it is not a theorem prover.
ns=[20,50,100,500,1000];ms=[1,2,4,16,64,256];delta=.05
bound=lambda n,k,m=1:1-math.exp(-(k+math.log(2*math.sqrt(n)/delta)+math.log(m))/n)
assert bound(50,math.log(4))>bound(100,math.log(4))
figures=[{"id":"sample-prior","kind":"analytical_curve","caption":"Author-constructed fixed-prior certificate arithmetic: zero empirical stochastic loss, delta=.05, KL log4 or log100. No actual trained network is certified.","x":{"name":"Certification sample size n","values":ns},"series":[{"name":"KL=log4 upper risk","values":[bound(n,math.log(4))for n in ns]},{"name":"KL=log100 upper risk","values":[bound(n,math.log(100))for n in ns]}]},{"id":"selection-cost","kind":"analytical_curve","caption":"Author-constructed n=100, zero empirical stochastic loss, delta=.05, base KL log4. Simultaneous confidence over M independently prespecified priors adds log M; plotted series are upper risks, not adaptive-prior certificates.","x":{"name":"Prespecified prior count M","values":ms},"series":[{"name":"Valid family-selection upper risk","values":[bound(100,math.log(4),m)for m in ms]},{"name":"One prespecified prior upper risk","values":[bound(100,math.log(4))for m in ms]}]}]
print(json.dumps({"status":"PASS","checks":["Discrete point-mass KL","Bernoulli-kl bisection residual","Zero-loss exact inversion","Adaptive-prior negative control ineligible","Split-sample denominator sensitivity"],"eligibility_check_is_theorem_proof":False,"empirical_benchmark":False,"source_proofs_verified":False,"figures":figures}))