Abstract
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, ψ_N(x,y)=e^-α r²/2P_N(x,y). Finite singularities, P_N=∇ 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_ dom, 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 b²=2ac, sharply detected by S_ dom 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_ dom 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.