Constructing a Neuro-Symbolic Mathematician from First Principles

TL;DR

Mathesis introduces a neuro-symbolic system combining hypergraph-based mathematical state encoding with a differentiable logic engine, enabling complex reasoning through energy minimization and search algorithms.

Contribution

It presents a novel architecture that integrates hypergraph representations with a differentiable logic kernel for improved mathematical reasoning.

Findings

Enables multi-step deduction via Monte Carlo Tree Search.

Uses energy minimization for logical consistency.

Incorporates learned value functions for proof search guidance.

Abstract

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.