Nodal algebraic curves and entropy diagnostics in degenerate two-dimensional harmonic-oscillator shells

Degenerate quantum eigenspaces can support substantial changes in nodal geometry at fixed energy. We show that, for the two-dimensional isotropic harmonic oscillator, this restructuring is organized by the Hermite-constrained algebraic curve $P_N(x,y)=0$ associated with each real shell state, ${\psi }_N(x,y)=e^{-\alpha r^2\mathrm{/}2}{P}_N(x,y)$. Finite singularities, $P_N=\mathrm{\nabla }{P}_N=0$, together with projective degeneracies of the leading homogeneous part, identify the strata where topology-changing events can occur. We combine these algebraic criteria with three information diagnostics: the nodal-domain entropy $S_{\mathrm{d}\mathrm{o}\mathrm{m}}$, the Cartesian mutual information $I(x;y)$, and the entropic uncertainty sum $S_r+S_p$. The first three shells reveal a clear hierarchy. The $N=1$ shell only rotates a nodal line; the $N=2$ shell exhibits a conic transition at $b^2=2ac$, sharply detected by $S_{\mathrm{d}\mathrm{o}\mathrm{m}}$ but not by global entropies; and the $N=3$ shell supports cubic close-branch regimes organized by the projective discriminant, with enhanced responses in $S_{\mathrm{d}\mathrm{o}\mathrm{m}}$ and $I(x;y)$. Thus algebraic stratification, rather than spectral ordering, organizes nodal geometry inside a degenerate eigenspace, while entropy diagnostics quantify the associated probability redistribution and coordinate correlations. The same stratification defines experimentally testable signatures in real-phase Hermite–Gaussian structured light and approximately isotropic trapped motional systems, and suggests a geometry-sensitive verification primitive for fixed-shell bosonic-qudit gates.