Geometry, Effort Metrics, and Holonomy in AI Reasoning Systems (v1.0)

This preprint develops a formal mathematical framework for analyzing AI reasoning dynamics using tools from metric geometry, optimization, and differential topology. The work introduces three core structures: 1. Voronoi partitioning — modeling semantic region formation in embedding spaces. 2. Effort‑weighted geodesics — representing transition costs through Riemannian or weighted‑graph metrics. 3. Holonomy of loops — capturing path‑dependent orientation changes analogous to state evolution in recurrent and transformer‑style models. The preprint includes a theorem–lemma–proof structure establishing completeness of Voronoi partitions, geodesic characterization of effort minimizers, and curvature‑induced holonomy. A unified theorem demonstrates how region membership, effort, and holonomy jointly determine reasoning outcomes. A Lean verification sketch formalizes Voronoi cells, effort functions, and path‑dependent state evolution within Mathlib, showing compatibility with mechanized reasoning. This artifact is part of the NDH‑Research‑Pilot and is intended for researchers studying geometric and topological perspectives on AI behavior. The document is provided in Markdown with GitHub‑safe LaTeX for stable rendering under Zenodo’s MathJax pipeline.