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)

angles=list(range(0,181,15));mismatch=[];cov_error=[]
for deg in angles:
 t=math.radians(deg);c=math.cos(t);s=math.sin(t)
 close(c*c+s*s,1)
 cov_error.append(max(abs(c*c+s*s-1),abs(c*(-s)+s*c)))
 direct=(c-1)**2+s*s
 close(direct,2*(1-c));mismatch.append(direct)
 close((0*c-0)**2+(0*s)**2,0)
phis=[0,5,10,20,30,45,60,75,90]
weak=[]
for deg in phis:
 t=math.radians(deg);c=math.cos(t);s=math.sin(t)
 # Gram [[1+c^2,cs],[cs,s^2]], trace2 and determinant s^2.
 tr=2;det=s*s
 eig=(tr-math.sqrt(max(0,tr*tr-4*det)))/2
 close(eig,1-abs(c));weak.append(eig)
close(weak[0],0);close(weak[-1],1)
# Invertible mixing changes intervention meaning.
z1,z2=1.,2.;w1,w2=z1,z2+z1*z1
new_w1=3.;back_z2=w2-new_w1**2
close(back_z2,-6);assert back_z2!=z2
figs=[
figure('rotation-intervention','Intervention labels distinguish observationally equivalent rotations','Rotation angle (degrees)','Squared mismatch (dimensionless)',angles,[('Observational covariance discrepancy',cov_error),('Fixed labeled unit mean-shift discrepancy',mismatch)],'Constructed isotropic two-dimensional Gaussian; all rotations preserve observational covariance. A labeled shift delta=1 along the fixed first coordinate yields2(1-cos theta). No learned causal model.'),
figure('intervention-diversity','Nearly collinear interventions leave a weak direction','Angle between unit intervention directions (degrees)','Smallest Gram eigenvalue',phis,[('Directional information proxy',weak)],'Constructed sum uu^T+vv^T of two unit-direction outer products. Smallest eigenvalue1-|cos phi| is a linear conditioning diagnostic, not a universal identifiability or sample-complexity result.')]
print(json.dumps(dict(status='PASS',checks=['rotation covariance','direct mean-shift discrepancy','zero-strength control','Gram characteristic polynomial','collinear and orthogonal endpoints','nonlinear intervention counterexample'],figures=figs)))