AxiomProver AI Verifies the 246 Prime-Gaps Theorem
Axiom Math's AxiomProver has generated a machine-checked Lean 4 proof of the 246 prime-gaps theorem, demonstrating how AI can verify highly complex, research-grade mathematics.
On August 17, 2026, Axiom Math announced that its multi-agent AI system, AxiomProver, successfully produced a machine-checked proof of the 246 prime-gaps theorem. This theorem, which represents the current limit of human knowledge regarding the twin prime conjecture, states that infinitely many pairs of prime numbers differ by no more than 246. The project, credited to 41 contributors, translates James Maynard’s 2013 paper establishing a bound of 600 and the subsequent Polymath8b collaboration that tightened it to 246 into the Lean 4 programming language.
The formalization process utilized a three-stage pipeline. First, researchers mapped out the proof's structure in a detailed blueprint, creating a dependency graph of definitions and lemmas. AxiomProver then generated the machine-checkable Lean 4 code, building upon the community mathematics library Mathlib and the existing PrimeNumberTheoremAnd project led by Alex Kontorovich and Terence Tao. The resulting code was organized into a public library called PrimeGapsLib, which also includes Maynard's original 600 bound and a self-contained verification challenge for outside researchers.
Unlike previous AI achievements focused on short competition problems or one-off proofs, such as Math, Inc.’s Gauss agent formalizing sphere-packing results in dimensions 8 and 24, Axiom Math designed PrimeGapsLib for long-term reuse. Sidharth Hariharan, a contributor to both projects, noted that this library provides reusable formal machinery that other researchers can build upon. For practitioners, this shift from solving isolated math puzzles to constructing reusable infrastructure represents a significant leap forward in automated reasoning.
Ultimately, Axiom Math’s founding mathematician, Ken Ono, views this mathematical milestone as a critical testbed for software verification. If software properties can be translated into formal mathematical statements, systems like AxiomProver could eventually verify the safety of AI-generated code running critical infrastructure, financial systems, and security networks.
This is our own summary of reporting by Unite.AI
