Non-Euclidean Toroidal Topography in Large Language Models: Compact Riemannian Manifold Embeddings, Geodesic Attention, and 3D Photonic Co-Simulation Verification

Abstract

Conventional Transformer neural networks embed token representations within flat Euclidean latent spaces RD . Despite their empirical utility, Euclidean geometries ex- hibit severe isotropic distance concentration in high dimen- sions, lack periodic boundary-free inductive bias, and scale self-attention quadratically O(N2) over sequence length N. Furthermore, classical inner-product softmax atten- tion inherently suffers from cluster dominance, wherein dif- fuse groupings of irrelevant tokens overpower singular ex- act factual keys, inducing catastrophic hallucination. In this work, we introduce Toroidal Topography Lan- guage Models (TT-LLMs), an architectural paradigm that projects high-dimensional token representations onto a compact 2-Torus Riemannian manifold T2 ∼ = S1 ×S1 embedded within R3, paired with localized tangent vector fields T(θ,ϕ)T2 and 4D quaternionic hyperspherical S3 Hopf fibrations. We prove and demonstrate: (i) Elimination of dynamic coordinate clamping via continuous cyclic periodic coordinates (θ,ϕ) ∈[0,2π)2 evaluated under reg- ularized origin manifolds; (ii) Fractal Golden Ratio (Φ) Rotary Position Embeddings (F-RoPE) that asymp- totically suppress rational harmonic aliasing across arbi- trarily extended sequence horizons (N ≫105) as guaran- teed by Kolmogorov-Arnold-Moser (KAM) dynamical sta- bility theory; (iii) Deterministic Geodesic Resonance Handshakes via high-quality-factor (Q) Lorentzian ker- nels suppressing diffuse cluster interference by >105×; and (iv) Direct physical mapping to integrated silicon photonics via Gaspard Monge’s collinear circle theorem with inward-normal centering and an impedance-matched triple-ring cascade (100 mm →14.14 mm →2 mm). Full 3D Finite-Difference Time-Domain (FDTD) Maxwell vec- tor curl simulations verify an empirical loaded resonance factor Qloaded = 2,791.5 at 120 nm coupling gap and an op- tical passive switching energy of 0.0460 fJ/MAC, demon- strating a 21,758.1×core optical energy reduction over electronic GPU multiply-accumulate baselines.