Large Language Models (LLMs) exhibit persistent logical failures in complex reasoning due to the lack of an internal axiomatic framework. We propose Mathesis, a neuro-symbolic architecture that encodes mathematical states as higher-order hypergraphs and uses a Symbolic Reasoning Kernel (SRK)--a differentiable logic engine that maps constraints to a continuous energy landscape. By defining a global energy function E(G), where zero energy implies logical consistency, the SRK yields gradient-based signals to train a Hypergraph Transformer Brain, turning proof search into energy minimization. Multi-step deduction is enabled via Monte Carlo Tree Search and Evolutionary Proof Search, guided by learned value functions and semantic unification.
@article{arxiv.2601.00125,
title = {Constructing a Neuro-Symbolic Mathematician from First Principles},
author = {Keqin Xie},
journal= {arXiv preprint arXiv:2601.00125},
year = {2026}
}