Symmetry sets the dimensional limits of Gaussian rigidity

A pure optical state with a nonnegative Wigner function is Gaussian. We determine how this rigidity survives small impurity as the number of modes grows. At any fixed positive total-energy budget, we measure non-Gaussianity by relative entropy to the trace-norm-closed convex hull of all Gaussian states. Writing L=log(1/s) for an impurity bound s, the largest uniformly rigid mode scale is f₃=(L/log L)1/2 in the general class and f₄=(L²/log L)1/3 for parity-invariant states: the corresponding class supremum tends to zero exactly when m=o(f₃) or m=o(f₄). Matching Wigner-positive constructions have vanishing total energy. The distinction is set by the first nonlinear term in the logarithm of the dominant wavefunction: a cubic term is available in general, whereas parity exposes a quartic obstruction. For an unrecorded random parity flip with coherence factor κ, we also determine the complete crossover under an actual output-impurity constraint. Every fixed κ<1 recovers the larger parity scale; when κ approaches one, recovery depends on the competition between dimension and dephasing-induced impurity. These results identify symmetry and residual coherence as quantitative controls on the dimensional stability of Gaussian-mixture descriptions.