The phenomenon of angular non-uniformity in L2-normalized token embeddings, recently exploited for sublinear compute reduction in transformer architectures, suggests the existence of a broader geometric law. As detailed in the previous preprint “Angular Manifold Routing: Sublinear Compute Reduction via Hopf-Base Sector Discretization” 1, fixed geometric routing via Hopf fibration provides significant optimization in expert path effi-ciency. However, we posit that this non-uniformity is a specific manifestation of a hidden ”4th axis” - a dimension of geometric reconciliation. While the primary axes x, y, z quantify spatial extent, this 4th axis, defined by the closure metric σ, represents the informational entropy of the form’s distribution. The strategic objective is to identify the hidden dimension of geometric reconciliation that allows high-dimensional state spaces to collapse into stable, low-dimensional attractors. By transitioning from the computational utility of Hopf-base discretization to the fundamental information-geometric metric of σ, we define a universal coordinate of isotropic regression.
L. Charles Allard (Fri,) studied this question.