130k Lines of Formal Topology in Two Weeks: Simple and Cheap Autoformalization for Everyone?

Josef Urban automates mathematical translation

Researcher Josef Urban published a paper on January 6, 2026, detailing a method to translate 130,000 lines of mathematical topology into formal code in just two weeks. The paper, titled "130k Lines of Formal Topology in Two Weeks: Simple and Cheap Autoformalization for Everyone?", suggests that the high cost of verifying complex mathematics is plummeting. Urban’s work focuses on autoformalization, the process of using artificial intelligence to turn human-readable math into machine-verifiable logic.

The project targeted the field of topology, a branch of mathematics concerned with the properties of geometric objects that remain preserved under continuous deformation. Historically, translating these proofs into a language that computers can check for errors required years of manual labor by specialized mathematicians. Urban’s approach uses large language models (LLMs) and symbolic reasoning to bypass this manual bottleneck.

By achieving this scale in 14 days, the research challenges the idea that formalizing the world’s mathematical knowledge requires decades of work. The system demonstrates that automated tools can now handle dense, abstract reasoning with minimal human intervention. This shift moves formal verification from a niche academic pursuit into a scalable utility for software developers and mathematicians alike.

High speed and low costs

Urban’s methodology emphasizes affordability and simplicity over massive computational power. While previous attempts at autoformalization relied on expensive, proprietary models and closed-source datasets, this new framework aims for accessibility. The paper argues that "cheap" autoformalization is now possible for individual researchers rather than just well-funded corporate labs.

The speed of the translation—averaging nearly 10,000 lines per day—marks a significant departure from traditional formalization projects. For comparison, the Liquid Tensor Experiment, a high-profile effort to formalize a single theorem by Peter Scholze, required a massive community effort and nearly two years to complete. Urban’s system handles 130,000 lines of general topology theorems in a fraction of that time.

The efficiency of the system stems from its ability to bridge the gap between natural language and formal languages like Lean, Coq, or Isabelle. These formal languages are notoriously difficult for humans to write because they require absolute logical precision. Urban’s tool acts as a high-speed interpreter, catching the nuances of mathematical prose and converting them into rigid, executable code.

• 130,000 lines of code generated in 14 days.
• Zero-cost or low-cost model architectures utilized.
• Topology used as the primary test case for complex reasoning.
• Automated verification ensures the resulting code is logically sound.

The mechanics of autoformalization

The autoformalization pipeline works by feeding mathematical textbooks and papers into a specialized AI model that understands both LaTeX and formal logic. The model identifies definitions, lemmas, and proofs within the text. It then attempts to reconstruct those elements in a formal language that a computer's kernel can verify for correctness.

Urban’s system does not just guess the next word in a sentence; it interacts with a formal prover to check its own work. If the computer rejects a line of code as logically inconsistent, the AI iterates on the translation until the proof is "green-lit" by the system. This feedback loop ensures that the 130,000 lines produced are not just plausible-looking text, but mathematically valid code.

The "simple" nature of this approach suggests that the barrier to entry for formal methods is disappearing. Researchers no longer need to spend months learning the idiosyncratic syntax of a theorem prover to contribute to a formal library. Instead, they can provide the informal math and allow the AI to handle the syntactic heavy lifting.

Building a verified math library

The 130,000 lines produced in this experiment represent a massive expansion of the existing formal math libraries. These libraries serve as the foundation for modern computer science, ensuring that critical algorithms and mathematical constants are free of errors. By rapidly expanding the library of formalized topology, Urban provides a larger sandbox for other AI agents to learn from.

Formalized mathematics is often used as a training ground for AI models that need to learn how to reason. Because the computer can objectively verify if a proof is correct, it provides a "ground truth" that is often missing in other AI training data. Urban’s contribution significantly increases the volume of high-quality, verified data available for the next generation of reasoning models.

This project also has implications for software security. Many of the techniques used to formalize math are also used to prove that software is free of bugs or vulnerabilities. If autoformalization becomes "cheap and simple," companies could use it to verify millions of lines of mission-critical code, such as those found in aerospace, medical devices, or cryptography.

Implications for the math community

The arrival of rapid autoformalization forces a rethink of how mathematical research is published and archived. If 130,000 lines of topology can be formalized in two weeks, it may soon become standard for all new mathematical papers to be accompanied by a machine-verified version. This would effectively eliminate the possibility of errors in published proofs.

The Verge has tracked the rise of formal methods in math since the Lean prover gained popularity among Fields Medalists. Urban’s work represents a transition from the "craftsman" era of formalization to the industrial era. The focus is no longer on the beauty of the code, but on the volume and velocity of verified knowledge.

Urban’s paper is currently hosted on arXiv, the open-access repository for scientific research. The data and code associated with the project are being integrated with tools like Hugging Face and GitHub, allowing other researchers to replicate the two-week formalization sprint. This openness ensures that the "autoformalization for everyone" promise is more than just a headline.

• Lean and Isabelle are the primary target languages for these translations.
• Neural-symbolic methods combine the intuition of AI with the rigor of logic.
• Error reduction in scientific publishing is a primary goal of the project.
• Scalability is the key metric, moving beyond single-theorem proofs.

The future of machine reasoning

Urban’s work suggests that the "reasoning gap" in AI is closing through structured data generation. While current LLMs often struggle with multi-step logic, training them on 130,000 lines of verified topology proofs could provide the logical scaffolding they currently lack. This creates a virtuous cycle where AI helps formalize math, and formalized math helps build better AI.

The "cheap" aspect of this breakthrough is particularly relevant as the industry debates the energy and financial costs of AI. If high-level reasoning can be achieved through clever autoformalization rather than just adding billions of parameters to a model, the path to Artificial General Intelligence (AGI) might be more efficient than previously thought. Urban’s results indicate that symbolic logic remains a vital component of the AI stack.

As of January 2026, the mathematics community is beginning to digest the scale of this output. The 130,000 lines represent a significant percentage of the total formalized math currently in existence. If this pace continues, the entirety of undergraduate and graduate mathematics could be machine-verified by the end of the decade.