The Hessian of Planar Central Configurations in Pair Space: Decomposition, Morse Index and Symmetry Reduction

What the math gives to ML

The paper provides a constructive Hessian decomposition in pair space: the constrained Hessian is a congruence-restricted ambient Hessian, and it splits into a positive-semidefinite gap Laplacian plus a signed transverse Laplacian containing all negative curvature. For non-collinear planar four-body configurations, the signed component has rank two, yielding a sharp low-dimensional description of instability and Morse index. Reflection symmetry further reduces the generalized eigenproblem into parity sectors, including a non-orthogonal reduction when the symmetry is not Euclidean-orthogonal. These mechanisms transfer well to neural-network optimization as low-rank negative-curvature tracking and symmetry-adapted Hessian/preconditioner blocks with measurable instability thresholds.