Riemann GeoResolver: A Non-Euclidean Attention Framework from Euclidean Resolver to Hyperbolic-Spherical Geometry

We present a theoretical foundation for inverse-distance attention, from its Euclidean prototype (Resolver) to its non-Euclidean realization (Riemann GeoResolver). The Euclidean part establishes three core theorems: (1) circuit separation---IDA achieves exact retrieval with O(1) resources while softmax requires Ω((\log n)^2) width; (2) a Polyak--Lojasiewicz inequality with Ω(e^{Δ^2/\sqrt{d}}/Δ^2) stronger constant than softmax, implying linear convergence, O(\log n) Lipschitz scaling under a low-rank/clustering assumption, Θ(1) Hessian spread, and absence of spurious local minima; (3) a width-independent effective rank bound that limits noise memorization---softmax memorizes arbitrary labels when d_h\ge n, while IDA limits test error to O(η^2). The non-Euclidean extension then builds upon this prototype, replacing Euclidean distance with hyperbolic geodesic distance for storage and spherical geodesic distance for routing. The Riemann GeoResolver framework comprises ten integrated modules: four HIDA operators spanning Θ(n^2) to Θ(1) per token; Hyperbolic Curvature Compression (HCC) with provable error bounds; HyperGate with gradient lower-bound theorem; Spherical Inverse Distance Attention (SIDA) with sphere-analog PL inequalities; Dynamic Memory Genesis (DMG) with O(\log T) regret bounds; and Geodesic Sparse Routing (GSR) with quality and communication bounds. The Euclidean theorems are proved in full; the non-Euclidean extension theorems are proved with analogous arguments. This work establishes a theoretical arc: from Euclidean attention as a special case, to hyperbolic memory, to spherical retrieval.