LLM Math case Study: A Proof of the Lonely Runner Conjecture via the Minimal Inertia Principle and Information Geometry

This paper presents a complete proof of the conjecture for all real speeds. We first establish the conjecture
for rationally dependent speeds by introducing the Minimal Inertia Principle (MIP), which posits that the configuration minimizing the moment of inertia of the speed set maximizes the system’s stability (the minimum magnitude of the associated LRC polynomial on the unit circle). We prove the MIP by translating the problem into the framework of information geometry. The core of the proof is the establishment of a novel Covariance Inequality, Covp(log p(d), cos(dθmin)) ≥ 0, which demonstrates that Shannon entropy acts as a Lyapunov functional for the system’s stability. We provide a rigorous proof of this inequality based on concentration properties and the second-order conditions of the extremum. We further support the MIP with alternative frameworks based on combinatorial Ricci flow, Perelman entropy, and analogies to Quantum Chaos/Random Matrix Theory. Finally, we extend the proof to all real speeds by demonstrating that the set of potential counterexamples is closed in the space of speed configurations, thereby completing the proof of the full conjecture.

This work was performed in significant part by AI. The use of AI deductive capacity to speed up hypothesis elimination and refinement inspired by these findings, which showed that in spite of AI limitations in inductive reasoning, their deductive speed and capacity might nevertheless allow them to find use in the advancement of mathematics.