import json,math
cs=[.1,.2,.5,1.,2.,5.,10.];rho=.1;N=20000
base=[.5*((1+rho*math.cos(2*math.pi*i/N))*(1+rho*math.sin(2*math.pi*i/N))-1)**2 for i in range(N)]
circle=[];ellipse=[];worst=0.
for c in cs:
 a=c;b=1/c
 assert abs(a*b-1)<1e-12
 assert abs((a*a)*(b*b)-(a*b)**2)<1e-10
 vals=[]
 for i in range(N):
  u=rho*math.cos(2*math.pi*i/N);v=rho*math.sin(2*math.pi*i/N)
  worst=max(worst,abs(.5*((a+c*u)*(b+v/c)-1)**2-base[i]))
  vals.append(.5*((a+u)*(b+v)-1)**2)
 circle.append(max(vals));ellipse.append(max(base))
assert worst<1e-12 and max(circle)-min(circle)>.1
figures=[{"id":"curvature-orbit","kind":"analytical_curve","caption":"Exact author-constructed nonzero Hessian eigenvalue c²+c⁻² at identical predictors f(x)=x. No measured generalization or deep-network benchmark.","x":{"name":"Positive rescaling c","values":cs},"series":[{"name":"Nonzero Hessian eigenvalue","values":[c*c+c**-2 for c in cs]}]},{"id":"perturbation-comparison","kind":"analytical_grid","caption":"Author-constructed exact nonlinear losses on 20,000 equally spaced angles, radius .1. Sampled maxima are lower bounds on continuous maxima. Transformed ellipses preserve every sampled loss; no trained-network measurement.","x":{"name":"Positive rescaling c","values":cs},"series":[{"name":"Ordinary circle sampled maximum","values":circle},{"name":"Corresponding ellipse sampled maximum","values":ellipse}]}]
print(json.dumps({"status":"PASS","checks":["Same predictor","Hessian determinant and trace","Pointwise transformed losses","Untransformed circle negative control"],"max_transformed_discrepancy":worst,"continuous_maximum_certified":False,"empirical_benchmark":False,"source_proofs_verified":False,"figures":figures}))