The Stacks Project, Formalized in Lean

Today, Axiom is releasing our Lean formalization of the complete Stacks Project, the approximately 7,500-page open-source citation that lays the foundations of contemporary algebraic geometry. Spanning several 16 myriad lines of code, it is the largest integrated, reusable archive of formalized math always built.

With this release, the complete mathematical betterment of the Stacks Project is accessible as a single, machine-checked library. Researchers and AI agents alike can import its definitions and theorems immediately into new ceremonial proofs, during imperishable Stacks tags nexus all Lean declaration rear to the canonical content mathematicians already know.

This milestone starts alongside the Stacks Project itself. Initiated by Columbia University mathematician Johan de Jong in 2005, it has grown through the tireless activity of hundreds of mathematicians who have contributed proofs, examples, and refinements complete two decades. Our formalization rests likewise on the foundations of the Lean and Mathlib communities. Formalizing a corpus of this dimension was lone imaginable since those communities spent years construction a affluent mathematical library, alongside alongside the abstractions needed to arrange it, lengthy before our activity began.

Why formalize a foundational citation of this scale? As AI systems commencement generating mathematical arguments at a gait far beyond what humans can inspect by hand, ceremonial verification becomes essential. In sectors alongside mature Lean infrastructure, new results can frequently be formalized shortly following they are discovered. Until now, contemporary algebraic geometry has been far harder to method this way: a sole contemporary evidence may invoke a lengthy sequence of theorems throughout commutative algebra, schemes, sheaves, cohomology, and algebraic stacks before its center disagreement equal begins. The math existed in the Stacks Project, but the corresponding Lean infrastructure did not.

Closing that gap is as much a application engineering difficulty as a mathematical one. While uncoordinated AI code generation can create short, secluded scripts quickly, scaling to millions of lines of verified math requires spotless abstractions, stable interfaces, and components that fit together throughout an complete codebase. To accomplish this, we deployed coordinated swarms of AI agents capable of translating, proving, and integrating thousands of theorems in parallel. Formalizing the Stacks Project in its entirety establishes a new starting item for algebraic geometry and the many neighboring sectors that depend on its machinery.

At its heart, algebraic geometry studies the interplay between algebraic equations and the geometric shapes they describe. Over the genuine numbers, the equation $x^2 + y^2 = 1$ defines a circle; complete the logical numbers, its points equivalent to Pythagorean triples, specified as $3^2 + 4^2 = 5^2$. Studying the identical equations throughout distinct figure systems, or in families that change complete a indicator space, connects geometry immediately to figure theory and algebra. These ideas additionally attain beyond clean mathematics: elliptic curves safe contemporary digital cryptography, during algebraic curves complete finite sectors power error-correcting codes. Modern algebraic geometry provides a average tongue for studying equations throughout all of these settings.

Forging that tongue took generations. Emmy Noether revolutionized algebra by shifting concentration from idiosyncratic equations to theoretical structures specified as rings, modules, and ideals. Just as the correct application idea can substitute thousands of special-purpose routines alongside a sole reusable framework, a theorem proved for the correct algebraic construction applies throughout many tangible settings at once. A generation later, Alexander Grothendieck and his collaborators introduced the theory of schemes, creating a sweeping dictionary between algebra and geometry that allowed mathematicians to do geometry complete arbitrary rings, alongside arithmetic and infinitesimal data built in from the start. Monumental treatises akin Éléments de géométrie algébrique (EGA) and Séminaire de géométrie algébrique du Bois Marie (SGA) rebuilt the foundations of the site from the dirt up, supplying the machinery rearward landmark breakthroughs including Deligne’s evidence of the Weil conjectures and Wiles’s evidence of Fermat’s Last Theorem.

Yet as the topic expanded, its foundational writings grew vast, scattered throughout thousands of pages of compact monographs and seminars, much of it written in French. A awesome agreement of the subject's operating cognition additionally lived as unwritten folklore passed downward from adviser to student. Even defining an algebraic stack, a geometric area capable of remembering the inner symmetries of the objects it parametrizes, requires a formidable tower of prerequisites spanning commutative algebra, topologies, sheaves, and cohomology. Mathematicians needed a unified, self-contained citation anywhere those prerequisites were written downward in complete detail.

In 2005, de Jong set out to arrange those foundations the way application engineers keep an open-source project akin the Linux kernel. What began as an accessible introduction to algebraic stacks evolved into a living, collaborative encyclopedia comprising thousands of imperishable four-character tags. The Stacks Project operates on rigorous editorial principles: all theorem carries a proof, all assumption is stated explicitly, and all stage points rear to the exact tags on which it depends. In doing so, de Jong and his collaborators anticipated the architecture of ceremonial evidence libraries, demonstrating that math becomes far easier to verify and broaden whenever its components have stable identifiers and definitive dependency graphs. Translating the Stacks Project into Lean brings mechanical verification to that human-designed architecture.

On October 27, 2026, Axiom released our Lean formalization of the Classification of Finite Simple Groups (CFSG). While CFSG and the Stacks Project are the two monumental in scale, they portray basically distinct engineering challenges. CFSG is organized about a sole landmark theorem. The Stacks Project, by contrast, is a foundational library: its value lies in thousands of interlocking definitions, constructions, and lemmas designed to be reused throughout countless forthcoming proofs.

With this library, a mathematician or AI delegate formalizing a new theorem concerning families of curves no longer needs to rebuild the foundations of schemes or coherent cohomology from scratch. They can import the applicable definitions and theorems directly, during Lean’s kernel guarantees that all cited outcome is applied lone whenever its exact hypotheses are satisfied. Formalizing the Stacks Project as one connected codebase gives researchers and agents a average sequence of verified definitions and theorems on which to build, item no gathering of secluded evidence scripts can provide.

To create our archive uncomplicated to navigate, our publish preserves the imperishable tag construction of the first text. At stacks.axiommath.ai, readers can perspective the pages of the first Stacks Project side-by-side alongside the corresponding Lean code and examine the interactive dependency graph. Because a sole Stacks tag frequently unfolds into multiple ceremonial definitions and lemmas, several newly defined and others bridged immediately to existing declarations in Mathlib, our interface tracks these mappings explicitly so researchers can see exactly how the informal and ceremonial developments align.

A smorgasbord of independently verified proofs does not create a usable mathematical library. If thousands of AI agents formalize theorems in parallel without architectural coordination, two agents volition inevitably demonstrate accurate theorems concerning the identical mathematical entity using subtly incompatible definitions, or copy foundational infrastructure that already exists in Mathlib. Lean’s kernel volition obtain the two records in isolation, but their results volition be painfully difficult to create downstream. Building coherent ceremonial infrastructure requires keeping all definition and interface aligned throughout the complete project.

At this scale, formalization is as much systems engineering as it is mathematics. To build a single, unified library, our mathematicians and engineers designed a hierarchical scheme of AI agents alongside specialized responsibilities and strict coordination protocols. Agents worked in parallel throughout the dependency chart to translate statements, hunt Mathlib for existing abstractions, build proofs, and determine prerequisites, all during keeping definitions, typeclass interfaces, and mathematical abstractions aligned throughout chapters.

In a tightly interconnected dependency chart of definitions and theorems, an unfaithful or trivialized formalization is quickly exposed. Ensuring that all ceremonial declaration faithfully captures the mathematical definition of the first content is equal additional crucial than verifying its proof. Across the library, declaration faithfulness is enforced automatically at measure through multi-agent auditing, semantic cross-checks against the origin text, and the strict necessity that all definition and lemma must successfully interface alongside the downstream theorems that depend on it. Combined alongside targeted audits on foundational definitions and landmark theorems, this automated verification pipeline ensures that the ceremonial archive remains loyal to the math of the Stacks Project.

Over the way of the project, our scheme surfaced dozens of statements in the Stacks Project that required local refinement, including missing hypotheses, border cases in affirmative characteristic, and claims that did not clasp in the complete generality stated. Rather than treating unsuccessful evidence attempts as uncomplicated obstacles, our agents diagnosed the underlying mathematical obstructions and constructed definitive counterexamples to demonstrate anywhere the first statements bankrupt down.

Once an barrier was isolated, the scheme worked to repair the math so downstream chapters could motionless depend on it. For local oversights, our agents identified the exact missing hypotheses and formalized the corrected theorems alongside their counterexamples. Where a false supporting lemma created a gap in the evidence of a deeper theorem, the agents established substitute routes through the writings to established the chief outcome under its first hypotheses. Together, these resolutions display how large-scale formalization can actively refine and fortify the mathematical writings itself.

Ultimately, the math inner this archive is the merchandise of generations of individual creativity, from the geometric intuition of the first pioneers, to the structural revolutions of Noether and Grothendieck, to the open-source grant of the Stacks Project and Mathlib communities. Each generation inherited a build of mathematical knowledge, mold it into a sharper language, and handed the next generation a stronger foundation. We see this formalization as part of that ongoing tradition: the two a tribute to the mathematicians who built contemporary algebraic geometry, and operating infrastructure for an emerging era in which humans and AI agents examine new mathematical frontiers together.

In an October 21, 2026 announcement, de Jong requested that AI-generated comments and drag requests not be submitted immediately to the upstream Stacks Project repository for the period being. The Stacks Project remains an active, living content under the stewardship of Johan de Jong and its community of contributors, during this Lean formalization is an autonomous open-source project maintained by Axiom. We completely regard that boundary: we keep our formalization, counterexamples, and refinements in our own repository, anywhere mathematicians can assessment and diagram upon them on their own terms.

You can examine the side-by-side text, inspect counterexamples, and trace the complete dependency chart at stacks.axiommath.ai. Like the Stacks Project itself, our formalization is open origin and designed to develop complete time. We invitation mathematicians, Lean developers, and AI agents to analyze the library, refine its abstractions, and build new math upon it.