Inertial Bypass: Geodesic Symplectic Mechanics on Hyperspherical Manifolds

Modern neural architectures universally employ Root Mean Square Normalization (RMSNorm), constraining latent representations μ ∈ R^D to a hypersphere S^{D-1}. While standard manifold optimization relies on first-order retractions, these post-hoc projections fail to account for kinematic centripetal acceleration, injecting non-physical virtual work that destabilizes high-dimensional momentum dynamics. Furthermore, in high dimensions (D ≫ 1), optimization landscapes are dominated by strict saddle points rather than isolated local minima, rendering classical barrier-crossing intuitions obsolete. In this work, we formulate a first-principles geometric framework for hyperspherical active inference using Dirac second-class constraints on TS^{D-1}. We prove an exact dissipation cancellation identity, dH/dt = -Γ ||dμ/dt||^2 ≤ 0, ensuring zero spurious virtual work along admissible trajectories. By identifying an extrinsic curvature inversion mechanism in the Riemannian Hessian, we isolate the steepest negative curvature axis u_0 and prove the Inertial Bypass Theorem, establishing analytical bilateral momentum bounds [p_crit^lower, p_crit^upper) that guarantee deterministic saddle traversal and target basin capture. To eliminate non-linear iterations (such as SHAKE/RATTLE) and retractions, we introduce the Geodesic BAOAB integrator. Operating in O(D) complexity, it internalizes centripetal acceleration via closed-form great-circle rotations, preserving both holonomic and tangent constraints to machine precision (< 10^-14) alongside conformal symplectic decay. Numerical simulations at D = 2048 validate the predicted traversal window and demonstrate robust capture up to angular misalignments of θ_c ≈ 42°, while rigorously recovering flat canonical dynamics in the large-scale limit.