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)

rho=.8;sigma2=1.
hs=list(range(1,13))
valid=[sum(rho**(2*j)for j in range(h)) for h in hs]
leaky=[v/(rho*rho*v+1)for v in valid]
for h,v,w in zip(hs,valid,leaky):
 close(v,(1-rho**(2*h))/(1-rho*rho))
 # Conditional covariance Schur complement.
 cov=rho*v;future_var=rho*rho*v+sigma2
 close(w,v-cov*cov/future_var);assert w<v
for h in hs:close(sum(0.**(2*j)for j in range(h)),1)
C=80;L=12;horizons=[1,2,4,8,12,16,24]
crossing=[sum(s+L+h-1>=C for s in range(C))for h in horizons]
for h,c in zip(horizons,crossing):
 close(c,min(C,L+h-1))
 safe=[s for s in range(C)if s+L+h-1<C]
 assert all(max(range(s,s+L+h))<C for s in safe)
figs=[
figure('future-proxy','A future observation lowers error without better forecasting','Forecast horizon h','Conditional MSE (innovation variance units)',hs,[('Valid information at origin',valid),('Forbidden future neighbor supplied',leaky)],'Constructed stationary Gaussian AR(1), rho=0.8 and innovation variance1. Future-assisted curve conditions on X_(t+h+1), unavailable at origin t. Analytic error, not trained forecasting performance.'),
figure('window-cutoff','Start-time splitting leaks window support across cutoff','Forecast horizon H','Cross-cutoff training-window count',horizons,[('Naive start<C split',crossing),('Require complete support<C',[0]*len(horizons))],'Constructed starts0..79, cutoff80, look-back12; support ends at s+L+H-1. Counts measure overlap only, not forecast-score inflation.')]
print(json.dumps(dict(status='PASS',checks=['geometric variance sum','conditional covariance identity','future information strictly lowers variance','zero-correlation control','enumerated window support','safe windows end before cutoff'],figures=figs)))