Unraveling the Twisted Cubic in Third-Order Zernike Wavefronts and Beyond

We study real critical sets of wavefronts generated by the four third-order Zernike modes. Eliminating the spatial variables from the ideal generated by the two first derivatives and the Hessian determinant yields a principal ideal whose generator is a quartic sum-of-squares polynomial in coefficient space. Its real radical admits a prime decomposition with two projective components: a pure-trefoil line and a twisted cubic. Every nonzero pure-trefoil wavefront has a unique isolated degenerate critical point of elliptic-umbilic type \( D_4^- \), whereas the twisted cubic is precisely the projective coefficient locus of nonisolated critical sets, each consisting of two parallel Morse--Bott lines. The incidence map of the rotating family of critical lines has a nondegenerate Whitney fold, whose spatial caustic is the circle \( x^2+y^2=\frac16 \) in the normalized pupil plane and whose transverse generating family has type \( A_2 \). On the complementary nondegenerate locus, we derive explicit critical radii and obtain a complete classification of the number and Morse types of interior critical points: local maxima, local minima, and saddle points.