import json,math
worlds=[{"a0":1,"a1":-100},{"a0":1,"a1":100}]
logged=[[(0,"a0",w["a0"],"terminal")for _ in range(10)]for w in worlds]
assert logged[0]==logged[1]
assert max(worlds[0],key=worlds[0].get)!=max(worlds[1],key=worlds[1].get)
for gamma in [.0,.5,.9,.99]:
 for t in [0,1,2,5,10,20,50,100]:
  e=0.
  for _ in range(t):e=.1+gamma*e
  assert abs(e-.1*(1-gamma**t)/(1-gamma))<1e-10
  z=0.
  for _ in range(t):z=0+gamma*z
  assert z==0
es=[.01,.02,.05,.1,.2,.5,1.]
for e in es:
 mean=e*(1/e);variance=e*(1/e-mean)**2+(1-e)*(0-mean)**2
 assert abs(mean-1)<1e-12 and abs(variance-(1/e-1))<1e-10
def importance_moments(e):
 if not 0<e<=1:raise ValueError('Positive behavior support required')
 return 1.,1/e-1
try:importance_moments(0.)
except ValueError:pass
else:raise AssertionError('Zero support incorrectly accepted')
ts=[0,1,2,5,10,20,50,100]
figures=[{"id":"backup-error","kind":"analytical_curve","caption":"Exact author-constructed persistent perturbation recurrence e_(t+1)=.1+gamma e_t, e0=0. No agent training or measured return.","x":{"name":"Backup iterations t","values":ts},"series":[{"name":"gamma="+str(g),"values":[.1*(1-g**t)/(1-g)for t in ts]}for g in [.5,.9,.99]]},{"id":"importance-variance","kind":"analytical_curve","caption":"Exact author-constructed one-step importance-estimator variance, fixed reward one, target action a1, behavior probability epsilon>0. No trajectory benchmark.","x":{"name":"Behavior probability epsilon","values":es},"series":[{"name":"One-observation estimator variance","values":[1/e-1 for e in es]}]}]
print(json.dumps({"status":"PASS","checks":["Identical logs, different optimal actions","Backup recurrence closed form","Importance mean and variance","Zero perturbation and zero-support controls"],"empirical_benchmark":False,"source_proofs_verified":False,"figures":figures}))