[2603.22871] Dynamical Systems Theory Behind a Hierarchical Reasoning Model

Computer Science > Artificial Intelligence arXiv:2603.22871 (cs) [Submitted on 24 Mar 2026] Title:Dynamical Systems Theory Behind a Hierarchical Reasoning Model Authors:Vasiliy A. Es'kin, Mikhail E. Smorkalov View a PDF of the paper titled Dynamical Systems Theory Behind a Hierarchical Reasoning Model, by Vasiliy A. Es'kin and 1 other authors View PDF HTML (experimental) Abstract:Current large language models (LLMs) primarily rely on linear sequence generation and massive parameter counts, yet they severely struggle with complex algorithmic reasoning. While recent reasoning architectures, such as the Hierarchical Reasoning Model (HRM) and Tiny Recursive Model (TRM), demonstrate that compact recursive networks can tackle these tasks, their training dynamics often lack rigorous mathematical guarantees, leading to instability and representational collapse. We propose the Contraction Mapping Model (CMM), a novel architecture that reformulates discrete recursive reasoning into continuous Neural Ordinary and Stochastic Differential Equations (NODEs/NSDEs). By explicitly enforcing the convergence of the latent phase point to a stable equilibrium state and mitigating feature collapse with a hyperspherical repulsion loss, the CMM provides a mathematically grounded and highly stable reasoning engine. On the Sudoku-Extreme benchmark, a 5M-parameter CMM achieves a state-of-the-art accuracy of 93.7 %, outperforming the 27M-parameter HRM (55.0 %) and 5M-parameter TRM (87.4 %). Remarkabl...