Axiom Math’s AI Agents Formally Verify the Stacks Project in Milestone for Mathematical Proofs

As artificial intelligence systems continue to expand their reach across complex technical disciplines, the quest to make machine-generated reasoning completely reliable has taken center stage. While major technology companies routinely publish massive volumes of mathematical claims and supposed proofs, even automated computer verification systems occasionally struggle to catch hidden errors, leaving researchers to question the absolute dependability of AI in advanced mathematics.

Against this backdrop, the well-funded startup Axiom Math is positioning itself to tackle the issue from the ground up. Rather than slowing down the pace of development, the $1.6 billion startup is focusing on building a rigorously verified mathematical foundation—a reliable substrate that autonomous AI agents can safely rely on to write flawless software and, ultimately, accelerate scientific discovery.

The company announced a significant milestone in this endeavor, revealing that it has formally verified the Stacks Project, widely recognized as one of the largest and most intricate bodies of modern mathematics ever compiled by humans. Mathematicians hold the Stacks Project in rare regard, with Columbia University mathematician Peter Woit famously observing that there is virtually nothing else comparable in the modern mathematical landscape. Woit has likened the project in scale and ambition to the famous Bourbaki books, the influential series that systematically rebuilt large swathes of mathematics from its foundational axioms during the 20th century.

The Stacks Project was originally launched in 2005 by Woit’s Columbia colleague, Johan de Jong, who conceived it as an exhaustive, open-source reference guide. Over the past two decades, numerous mathematicians from around the globe have contributed to the repository, steadily building out results that span from introductory college-level algebra all the way to the bleeding edge of modern geometry, advanced algebra, and sophisticated number theory.

To bring this massive human undertaking into the digital age of automated verification, Axiom Math deployed a fleet of specialized AI agents. These systems systematically translated the library’s roughly 7,654 pages of dense mathematical prose into approximately 16 million lines of highly structured, computer-checked code.

The process of formalizing mathematics is designed to ensure that the logical structure underpinning every theorem is entirely sound. In this practice, mathematicians or automated agents translate traditional mathematical proofs into code using a specialized programming language known as Lean. Once translated, a software program called a proof assistant meticulously evaluates the code line by line. If the proof successfully passes this rigorous computer check, mathematicians can rest assured that the argument is logically airtight and free of hidden flaws.

Software and hardware engineers have long relied on a similar methodology, known as formal verification, to confirm that complex microchip designs function precisely as intended without hidden layout flaws. In other public sector applications, researchers in Europe have successfully utilized formal verification techniques to root out and eliminate latent bugs in critical software systems responsible for calculating family welfare benefits.

However, for a sprawling reference library like the Stacks Project, merely checking individual theorems in isolation represents only a fraction of the challenge. The theorems and definitions contained within Stacks are deeply interconnected, with each new concept building directly upon those that came before it. Much like a biological family tree, countless advanced results in the Stacks repository trace their logical lineage directly back to foundational propositions established much earlier in the text.

This profound degree of interconnectedness initially gave the leadership team at Axiom Math serious doubts about whether the ambitious undertaking could actually succeed. The engineering and research team had to ensure that delicate logical dependencies remained intact across roughly 16 million lines of intricate Lean code without introducing contradictions or circular reasoning.

Carina Hong, the 25-year-old co-founder and CEO of Axiom Math, noted that combining massive scale with delicate mathematical structures is traditionally viewed as a risky proposition in software and research. Speaking about the nature of the repository, Hong emphasized that Stacks represents a core collection of foundational mathematical results where consistency is paramount, noting that researchers want to ensure individual theorems and definitions do not conflict with one another or harbor subtle paradoxes.

Hong dropped out of a Stanford University mathematics PhD program to launch the startup in March 2025. The company quickly captured the attention of the venture capital community, securing a $200 million Series A funding round led by Menlo Ventures, which highlighted a growing industry bet on the convergence of automated coding and rigorous mathematical proof systems.

Reflecting on the early stages of the translation process, Hong admitted that the team harbored initial reservations about whether automated methods—sometimes colloquially referred to as vibe coding—could successfully preserve the internal beauty, rigor, and logical coherence of the Stacks Project.

To overcome these hurdles, the Axiom team adopted a distributed approach. They provided their AI agents with a loose overarching structure while allowing the models to execute tasks in parallel. Ashvin Swaminathan, a member of Axiom’s technical staff, explained that while the team imposed a broad hierarchical framework for the Stacks translation, they deliberately avoided a rigid command-and-control structure that dictated every action of every individual agent. Instead, the autonomous agents actively searched for pending tasks, identified modular components they could tackle independently, and coordinated their work across the massive codebase.

Axiom Math also distinguished itself from other verification startups by training its own proprietary models specifically for mathematical reasoning. According to Hong, while the team initially relied on an in-house Lean expert to instill a human sense of mathematical taste and style in the early models, the AI agents eventually learned to independently recognize elegant and correct mathematical structures over time.

This technical breakthrough effectively fills one of the most persistent and conspicuous gaps in the field of formalized mathematics. Swaminathan pointed out that prior to Axiom’s recent milestone, researchers attempting to utilize Lean to verify new work in algebraic geometry faced a massive black hole with very few existing formal building blocks to lean upon. With the Stacks Project now fully formalized, researchers working in algebraic geometry finally have a robust, machine-readable foundation to build upon.

Axiom argues that each newly verified mathematical result serves as a reliable building block, enabling future proofs to stand on verified math rather than forcing researchers to reinvent the wheel for every new theorem. This cumulative compounding effect aligns with observations made by Terence Tao, the renowned UCLA mathematician and Fields Medal laureate, who has frequently noted that the more formalized proofs become available within the mathematical community, the easier it is to generate subsequent formalized proofs.

Importantly, professional mathematicians are not the only intended beneficiaries of Axiom’s work. Software developers have long struggled to guarantee that production software code is completely free of dangerous bugs and security vulnerabilities. That pressing software reliability challenge is rapidly intensifying as generative AI tools and autonomous coding agents take on an increasingly dominant role in writing the world’s software infrastructure.

Axiom is wagering that the next generation of enterprise software must rest on verifiable mathematical foundations in much the same way that the Stacks Project now rests on its computer-checked Lean code.

Swaminathan suggested that mathematics serves as the ideal proving ground for developing these advanced capabilities, noting that the underlying reasoning skills are broadly transferable. In Axiom’s vision, future AI agents will be able to read the company’s verified code, understand its underlying logic, and construct entirely new software infrastructure on top of it, turning the Lean formalization into a permanent substrate for automated growth.

The parallelized workflow utilized by Axiom’s agents to complete the Stacks Project offers a glimpse into how Hong expects complex AI tasks to be executed in the future. She argues that autonomous agents require well-structured, reusable libraries of verified knowledge to prevent them from outputting unreliable or error-prone results. If developers introduce poor engineering choices and unverified code early in the development cycle, it becomes exceedingly difficult for autonomous systems to expand the boundaries of reliable engineering later on.

Confident in the robustness and scalability of this verification paradigm, Hong has issued a direct challenge to the broader software development and AI community, urging other major coding companies to embrace formalization at a comparable scale.