A Bessel-Geodesic Conjecture on Curvature-Induced Displacement of Oscillation Zeros

We present a novel mathematical construction relating an exponentially growing Bessel vector field Wⁱ on a closed 2-dimensional submanifold Ω² embedded within a compact 3-dimensional Riemannian manifold M to forced geodesic motion through M. The Bessel vector field Wⁱ is differentiated using the intrinsic Levi-Civita connection ∇ with respect to the induced metric h on Ω². The Christoffel correction term Γⁱⱼs Wʲ in ∇ₛ Wⁱ couples the Bessel oscillations to the intrinsic geometry of Ω², deforming the zero-structure of ∇ₛ Wⁱ away from that of the flat-space derivative ∂ₛ Wⁱ. We establish via rigorous proof that the zeros of the geodesic component of ∇ₛ Wⁱ are displaced from those of J'ₙ (s) by a correction controlled by the principal curvatures of Ω² and the sectional curvature of M along Ω² (Propositions 1–3). We further establish that exponential growth of this displacement at second order in curvature occurs when the force profile F^µ ~ e^ (kz) drives a z-dependent tilt in the projected tangent eⁱ along γ (λ) (Proposition 4). All structural gaps in this argument are resolved; the explicit computation of the geometric coefficient Aₖ is identified as the remaining open problem. The flat-space limit is cleanly recovered when all curvatures vanish. We refer to this structure as Smith's Bessel-Geodesic Conjecture and provide a worked example for the embedding S² (ρ) ↪ S³ (r).