University of Toronto Mathematician Daniel Litt Says AI Proof Generation Limited by Verification Capability
Daniel Litt, a mathematician at the University of Toronto, stated that the current key bottleneck for AI models in mathematics is not text generation, but the inability to reliably verify the correctness of complex derivations; models can write proofs but may not know where the proof fails.
Litt noted that when asked for shorter proofs and then questioned about their correctness, models often deny or correct their previous conclusions; however, when proofs become sufficiently long, containing numerous definitions, lemmas, classifications, and citation chains, errors may be obscured by subsequent text, and the models lack a stable global consistency checking capability.
He mentioned a paper claiming to solve the singularity resolution problem in positive characteristic fields, which is about 800 pages long and generated by AI. This problem is a fundamental challenge in algebraic geometry: the goal is to transform singularities into smooth structures through a series of operations without altering the object's birational structure; there are classical results in the case of characteristic zero, while the positive characteristic case still contains difficult open problems.
Litt stated that he has not yet located specific errors in this 800-page text but believes it cannot constitute a valid proof: if the conclusion were true, it would represent a significant breakthrough in mathematics, and neither human experts have thoroughly reviewed it nor do existing models possess the ability to audit the entire complex proof, track all dependencies, and independently confirm conclusions.
This judgment contrasts with Litt's previous stance on AI's mathematical capabilities. He had previously bet on whether AI could independently write a number theory paper reaching the level of the 2025 Annals of Mathematics by 2030, and after AI demonstrated substantial progress on some mathematical problems, he acknowledged his earlier misjudgment; however, he still views "verifiable long-chain reasoning" as the dividing line from local problem-solving to autonomous mathematical research.
In market mechanisms, funding will prioritize formal verification, theorem provers, proof search, code verification, scientific computing, and high-quality training data infrastructure, rather than simply chasing models that can output longer reasoning texts. Beneficiaries include developers who can integrate model-generated results into machine-checkable systems like Lean, Coq, Isabelle, and institutions with mathematical, engineering, and scientific verification data; those under pressure are products that directly package natural language AI outputs as research conclusions, financial models, or safety-critical code conclusions, as their liability costs will rise rapidly with errors becoming harder to audit.
Source: Public Information
ABAB AI Insight
In the history of mathematics, "proposing conjectures" and "providing auditable proofs" have always been two completely different abilities. After Andrew Wiles announced the proof of Fermat's Last Theorem in 1993, a critical gap was found, which he and Richard Taylor took nearly a year to fill; Grigori Perelman proposed a proof of the Poincaré conjecture, and the mathematical community also spent years organizing experts to verify and compile it. Litt's 800-page example falls under the same logic: long texts are not a signal of proof strength; rather, they increase the probability of local hypothesis mismatches, symbol drift, omitted implicit premises, and overlooked lemma circular references.
The capital path is shifting from "generation capability" to "verification closed loops." Large model companies can train models to propose candidate proofs using papers, textbooks, code, and synthetic reasoning trajectories, but final productization requires compiling candidate answers into checkable formal systems: models are responsible for searching, breaking down subproblems, and generating formal drafts, while theorem provers determine whether each derivation meets axioms and rules. This path aligns with the evolution from GitHub Copilot in software engineering to static analysis, unit testing, continuous integration, and formal verification—after the marginal cost of generating code decreases, testing, auditing, and accountability become scarce resources.
The industry analogy is closest to the difference between AlphaGo and AlphaFold. Go has clear rules and win/loss determinations, allowing models to receive high-quality feedback through self-play; protein structure prediction can be evaluated using experimental structural data, but the predicted results still require experimental validation. Open mathematics lies between the two: formalized propositions can undergo binary checks by proof kernels, but fully translating real mathematical language, definitions, and research directions into formal systems still requires expensive human expert work. Current models are in the expansion phase of "generating candidate routes and local steps" and have not yet entered the control phase of "end-to-end, automatically producing and certifying major theories."
Essentially, this is a transfer of pricing power. Once models turn writing, retrieval, and local derivation into low-cost capabilities, credibility will no longer be determined by who can write the longest proof, but by who controls the verification systems, formal languages, authoritative datasets, and expert review networks. The supply side of generation will explode, while the manpower, tools, and attention on the verification side will remain relatively scarce; thus, the more AI can quickly produce complex conclusions, the higher the cost of independent verification, and the more likely the value of verification infrastructure will concentrate among a few platforms that possess toolchains and standard-setting authority.
ABAB News · Cognitive Laws
Being able to generate answers does not equal being able to bear the answers.
The cheaper the information, the scarcer verification becomes.
Complexity is not a barrier; credible verification is the barrier.