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

TL;DR

This paper investigates the geometric rarity of symmetric solutions in G-invariant optimization landscapes, revealing that symmetric critical points dominate due to the super-exponential decay of asymmetric regions, and explains the prevalence of symmetry in natural phenomena.

Contribution

It introduces a measure-theoretic framework linking group symmetries to the geometric rarity of asymmetric solutions and explains the emergence of symmetry in optimization landscapes.

Findings

The volume of the real image L decays super-exponentially with group size.

Symmetric critical points are statistically dominant due to the rarity of asymmetric regions.

The global gradient directs solutions toward symmetric boundary strata, explaining symmetry emergence.

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.