MRI Manifolds: Algebraic Methods for Curvature Analysis of Transformer Embeddings

We present a computational framework for measuring discrete curvature in transformer embeddingspaces using uniform idempotent right quasigroups. Building on algebraic dilation operationsinstead of classical derivatives, we construct discrete analogues of Riemann, Ricci, and sectional curvaturetensors. This enables geometric analysis of high-dimensional non-smooth spaces, providingcurvature-based diagnostics for semantic structure and training dynamics.1 IntroductionNeural embeddings occupy highly structured yet non-smooth spaces. Traditional Riemannian geometryrequires differentiability, whereas transformer embeddings exhibit piecewise-linear structuresand high-dimensional discontinuities. To bridge this gap, we employ emergent algebra [?] to definecurvature through algebraic operations that emulate differentiability via uniform scaling limits.Our curvature analyses reveal geometric signatures in transformer embeddings consistent withhyperbolic space, including negative sectional curvature and hierarchical organization patterns.This suggests that embedding geometry naturally captures semantic hierarchies, and that ouremergent curvature framework effectively uncovers these intrinsic structures.Key contributions:• Construction of discrete curvature tensors using uniform idempotent right quasigroups• Derivation of Ricci and scalar curvature via algebraic dilation operations• Computational framework for curvature analysis in embedding spaces• Geometric monitoring revealing hyperbolic-like structure in trained models