As artificial intelligence continues to tackle an increasing number of mathematical challenges, growing concerns surround the reliability of automated outputs. Last month, OpenAI released numerous manuscripts claiming solutions to some of the toughest problems in the field, yet even automated verification tools can occasionally miss errors. To combat the rising tide of unreliable AI-generated material, a startup valued at $1.6 billion is working to establish a much firmer foundation, ensuring that future AI agents can depend on verified mathematics to write reliable software and advance scientific research.
The company, Axiom Math, announced that it has formally verified the Stacks Project, which stands as one of the largest comprehensive bodies of mathematics ever compiled by human authors. Mathematicians hold this library in exceptionally high regard, with Columbia University mathematician Peter Woit comparing the monumental undertaking to the famous Bourbaki books that systematically rebuilt mathematical foundations throughout the 20th century.
Johan de Jong, a colleague of Woit at Columbia, originally launched the Stacks Project in 2005 as an open-source reference guide. Since its inception, contributors from across the mathematical community have continually expanded its scope, establishing results ranging from introductory college algebra up to the most complex concepts spanning geometry, advanced algebra, and number theory. A fleet of specialized AI agents developed by the company systematically translated the library's roughly 7,654 pages into approximately 16 million lines of rigorous computer-checked code.
Formal Verification and Code Verification
The process of formalizing mathematics ensures that the underlying logical structure remains entirely sound. Mathematicians translate traditional proofs into a specialized programming language known as Lean, after which an automated program called a proof assistant meticulously examines every single line of code. Once a proof successfully passes these rigorous checks, mathematicians can place full trust in its logical integrity.
Engineers apply this exact methodology, known as formal verification, to guarantee the proper functioning of complex computer chip designs, while research teams in France have successfully utilized similar techniques to detect hidden bugs within software systems calculating family welfare benefits. Within an extensive library like the Stacks Project, verifying individual theorems represents only part of the challenge because every definition and theorem is deeply interconnected, forming an intricate lineage where modern results depend entirely on the foundational proofs established before them.
Managing Complex Interconnections
This immense degree of interconnectedness initially caused hesitation within the Axiom team regarding whether the ambitious project could successfully succeed. The engineering staff needed to maintain every single logical connection intact across roughly 16 million lines of Lean code without introducing contradictions. Carina Hong, the 25-year-old CEO of Axiom who left a Stanford math PhD program to found the firm in March 2025, noted that delicate systems and massive scale rarely coexist easily, especially when dealing with the fundamental bedrock of mathematics where definitions must remain entirely consistent and free of paradoxes.
In March, the startup secured a $200 million Series A funding round led by Menlo Ventures. Reflecting on the technical hurdles, Hong noted initial skepticism over whether standard vibe coding methods could adequately preserve the intrinsic beauty and structural integrity of the Stacks library.
Parallel AI Agents and Custom Models
To accomplish the monumental task, the engineering team provided their AI agents with a flexible overarching hierarchy while allowing them to execute tasks concurrently in parallel. Ashvin Swaminathan, a technical staff member at Axiom, explained that rather than dictating every minute action for each individual agent, the system allowed autonomous agents to discover tasks independently and identify areas where they could collaborate efficiently.
Furthermore, Axiom trained proprietary models specifically optimized for advanced mathematical reasoning, an approach Hong notes distinguishes her company from most other verification startups. Initially, the team relied on an in-house Lean expert to guide the agents toward developing a human-like aesthetic sense for valid proofs, though the models eventually cultivated this refined taste autonomously over time.
Filling the Algebraic Geometry Gap
This achievement effectively bridges one of the most prominent gaps within the landscape of formalized mathematics. Swaminathan pointed out that algebraic geometry previously represented a massive black hole within the formalization community, leaving researchers with very few foundational resources to verify novel work using Lean. Now, mathematicians working in these complex subfields finally have a robust starting point.
Axiom emphasizes that every newly verified mathematical result functions as an essential building block, allowing subsequent proofs to rely upon established computational certainty rather than starting from absolute zero. Terence Tao, the renowned Fields Medal-winning mathematician at UCLA, has previously summarized this cumulative advantage by noting that an abundance of formalized proofs directly enables the creation of even more verified mathematics.
Broad Software Implications and Open Challenges
Mathematicians are not the sole intended beneficiaries of this technological breakthrough. Software developers have perpetually grappled with the difficult task of ensuring that their codebase remains entirely free from critical bugs and security vulnerabilities, a challenge that intensifies as AI agents generate a larger share of global software. Axiom is wagering that future software infrastructure will increasingly demand rigorous formal verification foundations akin to the newly formalized Stacks library.
Expressing absolute confidence in the company's long-term trajectory, Hong issued an open challenge to all major software coding enterprises, inviting them to engage in formalization efforts at this unprecedented scale.
