import numpy as np

def check_margin():
    # q(x)=x^2/2, xdot=u+w, |w|<=wbar: max_w grad(q)w=wbar|x|.
    x=.6; ws=np.linspace(0,.8,9)
    predicted=ws*abs(x)
    observed=np.array([max(x*w for w in (-w, w)) for w in ws])
    slope=np.polyfit(ws,observed,1)[0]
    return {'predicted_slope':float(abs(x)), 'observed_slope':float(slope),
            'max_abs_error':float(np.max(abs(predicted-observed))),
            'confirmed':bool(np.max(abs(predicted-observed))<1e-10)}

def check_projection():
    # Scalar QP min .5(u-u0)^2 subject to u<=b, u in [-1,1].
    u0=.8; b=.25; u=max(-1.,min(1.,min(u0,b)))
    return {'u0':u0,'bound':b,'projected':u,'constraint_residual':float(u-b),
            'confirmed':bool(u<=b+1e-12)}

if __name__=='__main__':
    import json
    out={'margin_scaling':check_margin(),'scalar_qp_projection':check_projection()}
    print(json.dumps(out,indent=2))