Topological Invariants of Proof Structure

This paper introduces a novel framework for analyzing the structure of mathematical proofs by applying methods from algebraic topology. Traditional proof theory primarily assesses the logical validity and derivability of a theorem, often treating structurally distinct proofs of the same result as equivalent. We propose a methodology to transform formal proofs, represented as directed acyclic graphs, into simplicial complexes. From these topological representations, we compute established invariants such as Betti numbers and the Euler characteristic. These invariants capture higher-order structural properties of proofs, quantifying aspects like modularity, redundancy, and the complexity of logical dependencies. We demonstrate that these topological measures can distinguish between proofs that are logically equivalent but structurally divergent, offering a new classification scheme. The potential applications of this framework include enhancing automated theorem proving by guiding search towards proofs with simpler topological signatures, developing new metrics for proof similarity in large formal libraries, and providing a more nuanced understanding of proof complexity beyond mere length or step-count. This work bridges the gap between structural proof theory and algebraic topology, opening new avenues for the formal analysis of mathematical reasoning.