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)

N=8
for j in range(N):
 close(math.cos(2*math.pi*j/N),math.cos(18*math.pi*j/N),1e-12)
x=[j/128 for j in range(129)]
u1=[math.cos(2*math.pi*t)/(2*math.pi)**2 for t in x]
u9=[math.cos(18*math.pi*t)/(18*math.pi)**2 for t in x]
dist=math.sqrt(.5*((2*math.pi)**-4+(18*math.pi)**-4))
for n in [128,256,512]:
 numerical=math.sqrt(sum((math.cos(2*math.pi*j/n)/(2*math.pi)**2-math.cos(18*math.pi*j/n)/(18*math.pi)**2)**2 for j in range(n))/n)
 close(numerical,dist)
for t in x:close(math.cos(6*math.pi*t)**2,(1+math.cos(12*math.pi*t))/2)
ks=[1,2,3,4,6,8,12,16]
exact=[];biased=[]
for K in ks:
 tail=.5*sum(k**-6 for k in range(K+1,17))
 retained=.5*.1**2*sum(k**-6 for k in range(1,K+1))
 exact.append(math.sqrt(tail));biased.append(math.sqrt(tail+retained))
 # Independent direct Fourier quadrature of residual with exact retained modes.
 n=2048
 err=math.sqrt(sum(sum(math.cos(2*math.pi*k*j/n)/k**3 for k in range(K+1,17))**2 for j in range(n))/n)
 close(err,exact[-1],1e-8)
close(exact[-1],0);assert biased[-1]>.07
figs=[
figure('alias-output-separation','Identical coarse input, different continuous output','Periodic coordinate x','Inverse-Laplacian output (dimensionless)',x,[('Input frequency 1 output',u1),('Input frequency 9 output',u9)],'Constructed periodic inputs cos(2 pi x) and cos(18 pi x) agree at all eight coarse samples. Their mean-zero inverse-Laplacian outputs differ. Dense lines are analytic, not neural predictions.'),
figure('spectral-error-budget','Truncation error and a multiplier bias floor','Retained cosine modes K','L2 output error (dimensionless)',ks,[('Exact retained multipliers',exact),('10% relative multiplier bias',biased)],'Finite constructed modes k=1..16, input amplitudes1/k, target multipliers1/k^2. Retained bias is+10%. Orthogonal error sums checked by independent quadrature; no trained operator.')]
print(json.dumps(dict(status='PASS',checks=['coarse alias equality','output separation analytic vs quadrature','squaring frequency identity','truncation norm quadrature','full-retention endpoint','nonzero bias floor'],alias_lower_bound=dist/2,figures=figs)))