import math, json
def figure(id,title,xlabel,ylabel,x,series,caption):
 return dict(id=id,title=title,kind='analytical_curve',x_label=xlabel,y_label=ylabel,x=dict(name=xlabel,values=x),series=[dict(name=n,values=v)for n,v in series],caption=caption)
def close(a,b,tol=1e-10):
 assert abs(a-b)<=tol*max(1,abs(a),abs(b)),(a,b)

alphas=[0,.25,.5,.75,1,1.5,2,3]
risk=[a*a for a in alphas]
# A finite symmetric moment control has same zero mean/unit variance needed for MSE.
for a,r in zip(alphas,risk):
 errors=[(a*p)**2 for p in [-1,1]]
 close(sum(errors)/2,r)
for v1 in [.1,1,10]:
 for v2 in [.1,1,10]:
  post=1/(1+1/v1+1/v2);single=1/(1+1/v1)
  assert post<single
  # Duplicate sensor posterior is the single likelihood, not squared likelihood.
  duplicate=single
  naive=1/(1+2/v1)
  assert naive<duplicate
  close(duplicate-naive,v1/((v1+1)*(v1+2)))
  close(duplicate/naive,(v1+2)/(v1+1))
# Absolute gap vanishes at both extremes; relative gap approaches factor two.
for v in [1e-8,1e8]:
 single=v/(v+1);naive=v/(v+2)
 assert abs(single-naive)<1e-7
assert abs((1e-8+2)/(1e-8+1)-2)<1e-7
ps=[.25+i*.025 for i in range(21)];missing=[]
for p in ps:
 a=.25/p;b=.25/(1-p)
 assert 0<a<=1+1e-12 and 0<b<=1+1e-12
 obs=p*a+(1-p)*b
 close(obs,.5);close(p*a/obs,.5)
 m=p*(1-a)/(1-obs);close(m,2*p-.5);missing.append(m)
close(missing[0],0);close(missing[-1],1)
figs=[
figure('private-factor-risk','Missing private information creates irreducible risk','Private-factor coefficient alpha','Bayes MSE (dimensionless)',alphas,[('Only shared S observed',risk),('Both S and P observed',[0]*len(alphas))],'Constructed independent standard-normal S,P and targetY=S+alpha P. Missing P gives conditional variance alpha^2. A finite symmetric moment check verifies the MSE identity; no neural or clinical data.'),
figure('selection-ambiguity','Same observed cases, different missing outcomes','Population positive prevalence p','Positive rate',ps,[('Observed-case positive rate',[.5]*len(ps)),('Missing-case positive rate',missing)],'Constructed binary selection family with pa=(1-p)b=1/4. Observation rate and observed positive rate remain1/2; missing positive rate2p-1/2 ranges0..1. Outcomes for missing cases are assumed unavailable.')]
print(json.dumps(dict(status='PASS',checks=['private-factor moment risk','independent precision reduction','duplicate-evidence negative control','valid selection probabilities','fixed observed distribution','varying missing distribution'],figures=figs)))