Metric Rarity and the Emergence of Symmetry in G-Invariant Potential Surfaces

Abstract

Let X be an irreducible complex affine algebraic variety defined over $\mathbb{R}$, equipped with a faithful action of a finite group G, and let Y = X // G denote the categorical quotient with projection $\pi$. We study the geometry of the real image $L = \pi(X(\mathbb{R})) \subset Y(\mathbb{R})$ and its consequences for G-invariant optimization. Equipping $Y(\mathbb{R})$ with the measure induced by a G-invariant metric on X, we prove that the relative volume of L in $Y(\mathbb{R})$ equals $(\#\mathrm{Inv}(G))^{-1}$, where $\mathrm{Inv}(G)$ is the set of involutions of G. For the symmetric group $S_n$ acting on $\mathbb{R}^n$, this ratio decays super-exponentially in n. In particular, L is metrically rare within the ambient real quotient. We apply this result to two phenomena observed in G-invariant optimization problems: Regime I (Rarity of asymmetric critical points). The super-exponential decay of the volume of L renders the interior $L^\circ$ statistically negligible as a locus for critical points. This geometric rarity provides a rationale for the observed prevalence of symmetry: generic critical points are constrained to the boundary strata of L, corresponding to orbits with non-trivial stabilizers. Regime II (Energetic ordering by symmetry). We formulate the Active Constraint hypothesis: due to the metric rarity of the real image L, the landscape is dominated by a global gradient that drives the deepest descent trajectories toward the boundary of L. This global gradient directs the global minimum into the high-codimension strata of the boundary -- corresponding to large stabilizers -- thereby establishing a structural link between low energy and non-trivial stabilizers. This mechanism rationalizes the funnel topography of Lennard-Jones clusters, where the system is funneled into a crystallized ground state.