import json,math
for e in [.05,.1,.2,.5,1.]:
 assert abs(e*(1/e)-1)<1e-12
 for k in [0,1,3,7,15]:
  alpha=.1;den=e*e+k*alpha*alpha;v=[e/den]+[alpha/den]*k
  assert abs(e*v[0]+sum(alpha*x for x in v[1:])-1)<1e-12
  assert abs(sum(x*x for x in v)-1/den)<1e-10
for k in [1,2,5,20]:
 c=10.;w=[c/k]*k
 assert abs(sum(x*y for x in w for y in w)-c*c)<1e-10
 assert abs(sum(x*x for x in w)-c*c/k)<1e-10
es=[.05,.1,.2,.5,1.];ks=[0,1,3,7,15]
figures=[{"id":"small-direction-risk","kind":"analytical_curve","caption":"Exact author-constructed X=diag(1,epsilon), y=(0,1), zero target, identity test covariance: prediction risk epsilon^-2. Fixed design, no IID benchmark.","x":{"name":"Small observed scale epsilon","values":es},"series":[{"name":"Prediction risk","values":[1/e**2 for e in es]}]},{"id":"independent-vs-duplicate","kind":"analytical_curve","caption":"Author-constructed epsilon=alpha=.1. Independent coordinate model risk 1/(epsilon²+k alpha²); exact duplicate train/test feature model preserves risk 100. Different declared covariance models, no trained-network experiment.","x":{"name":"Additional coordinates k","values":ks},"series":[{"name":"Independent coordinates","values":[1/(.01+k*.01)for k in ks]},{"name":"Duplicate-feature risk","values":[100.]*len(ks)}]}]
print(json.dumps({"status":"PASS","checks":["Exact interpolation","Minimum-norm closed form","Duplicate covariance risk invariant while coordinate norm shrinks"],"empirical_benchmark":False,"source_proofs_verified":False,"figures":figures}))