I first encountered Axiom’s claim the way most people did, with disbelief followed by uneasy curiosity. A new AI startup, barely known outside specialist circles, said it had solved several research level mathematics problems that human experts had struggled with for years. Not approximations. Not heuristic hints. Complete proofs, formally verified by software. In the first moments of reading the announcement, the obvious question surfaced: did this actually happen, or was it another case of inflated AI ambition. – ai startup solve 4 math problems.
The claim matters because mathematics is not like other fields touched by artificial intelligence. A proof is either correct or it is not. There is no partial credit, no plausible sounding answer that passes on style alone. Axiom says its system, called AxiomProver, generated machine checkable proofs for four longstanding conjectures across algebraic geometry, number theory, and related areas. Each proof was written in Lean, a formal theorem prover that enforces absolute logical rigor.
Within the first hundred words of any explanation lies the same search intent readers bring with them: is this real progress or clever automation. Axiom insists the results go beyond pattern matching or retrieval of known arguments. The system, it says, identified overlooked connections, revived forgotten mathematical tools, and assembled original chains of reasoning without human correction.
If those claims hold under scrutiny, the implications are profound. Mathematics has long been the last domain where human intuition seemed irreplaceable. Axiom’s announcement suggests that even here, machines may now operate as independent discoverers, forcing a rethinking of how knowledge is created, verified, and credited.
Startup Overview
I look at Axiom less as a sudden miracle and more as a product of long frustration within formal mathematics. Founded in 2022, the company grew out of dissatisfaction with how theorem proving tools lagged behind advances in machine learning. While AI systems were writing essays and generating code, formal proof assistants remained brittle and labor intensive. – ai startup solve 4 math problems.
Axiom’s founders believed the gap could be closed. Their approach centers on combining transformer based language models with the Lean theorem prover. Lean acts as an uncompromising referee. Every definition must be explicit. Every inference must be justified. If a single step fails, the entire proof collapses.
AxiomProver begins by translating natural language problem descriptions into formal statements. From there, it explores vast spaces of possible proof steps using reinforcement learning. Incorrect paths are discarded automatically. Valid steps are chained together until Lean verifies the result as a complete proof.
What distinguishes Axiom’s work from earlier efforts is autonomy. The company says no human edited or guided the final proofs. Once the problem specification was given, the system ran on its own until it either succeeded or failed. That claim, more than the specific problems involved, is what has drawn attention across the mathematics community.
The Four Problems Axiom Claims to Have Solved
I want to be clear about the scope of Axiom’s announcement. The company says AxiomProver produced verified solutions to four open problems that had resisted prior human attempts. Two of these are explicitly named and widely discussed. Two are described more generally but still significant.
The named problems are the Chen Gendron Differential Conjecture in algebraic geometry and Fel’s Conjecture on Syzygies in commutative algebra. The other two involve a probabilistic model describing prime number dead ends and a Fermat inspired modular form identity connected to elliptic curves.
All four proofs were formalized in Lean and released publicly for inspection. They have not yet completed traditional peer review, but their logical correctness can be checked mechanically by anyone familiar with the software. This transparency marks a shift in how mathematical claims may be evaluated in the future.
Chen Gendron Differential Conjecture
I find this result the most intellectually revealing. The Chen Gendron conjecture links modern algebraic geometry with classical number theory, specifically through the behavior of differentials on complex surfaces. Human researchers approached the problem using contemporary frameworks and largely ignored techniques from the nineteenth century.
AxiomProver did not share those assumptions. According to the formal proof, the system uncovered a connection to older number theoretic identities that had fallen out of favor. By formalizing those identities and embedding them into a modern algebraic structure, the AI constructed a proof path human experts had overlooked. – ai startup solve 4 math problems.
What stands out is not speed but perspective. The machine explored mathematical terrain without the biases that guide human taste. In doing so, it treated forgotten methods as equally viable, eventually finding a route to resolution.
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Fel’s Conjecture on Syzygies
Fel’s Conjecture concerns the structure of syzygies, relations among relations in algebraic systems. The problem sits deep within commutative algebra and has implications for algebraic geometry and representation theory.
AxiomProver’s solution relied on formulas associated with Srinivasa Ramanujan, particularly identities involving modular forms. These formulas are well known but rarely applied in this context. The AI combined them with combinatorial reasoning to produce a complete proof.
The proof is notable for its autonomy. Axiom says the system required no human hints, adjustments, or interpretive guidance. It generated the argument, formalized it, and verified it entirely within Lean.
The Prime Dead End Model
The third problem moves closer to applied mathematics. It addresses prime number dead ends, regions in probabilistic processes where expected progress stalls. Such models matter in cryptography, where assumptions about prime distributions underpin security systems.
AxiomProver framed the problem using Markov chains and derived bounds on expected dead end lengths. The proof integrates probabilistic reasoning with formal verification, a combination that is rare in human written mathematics because of its complexity. – ai startup solve 4 math problems.
If the result holds up, it could refine theoretical assumptions used in cryptographic design, though practical implications would emerge only after extensive review.
Fermat Inspired Modular Form Identity
The fourth problem revisits ideas linked to Pierre de Fermat and later developments in elliptic curves and modular forms. These structures are central to modern number theory and to cryptographic applications.
The AI derived an identity that settles long standing questions about congruence properties of elliptic curves. The resulting proof is long and highly formal, spanning hundreds of lines of Lean code. While difficult to read intuitively, its correctness can be mechanically verified.
This highlights a growing tension in AI generated mathematics: results may be flawless yet conceptually opaque to humans.
Table: Overview of Axiom’s Claimed Results
| Problem | Field | Approximate Years Open | Verification Method |
|---|---|---|---|
| Chen–Gendron Differential Conjecture | Algebraic Geometry | 20 years | Lean theorem prover |
| Fel’s Conjecture on Syzygies | Commutative Algebra | 15 years | Lean theorem prover |
| Prime Dead End Model | Number Theory | 10 years | Lean with probabilistic formalization |
| Modular Form Identity | Number Theory | 25 years | Lean theorem prover |
Erdős Problem 481 and Questions of Credit
Beyond the four headline problems, Axiom also reported a solution to Erdős Problem 481, posed around 1980. The problem asks whether a simple iterative arithmetic process must eventually repeat rather than diverge indefinitely.
AxiomProver reportedly produced a 656 line formal proof in about five hours. The result confirms convergence. However, some mathematicians note that a solution may have been outlined informally by David Klarner in the early 1980s.
If that earlier work holds, Axiom’s contribution would be formal verification rather than original discovery. The distinction matters. Formalization is valuable, but it carries different intellectual weight than first resolution. Axiom has acknowledged the ambiguity and described its work as independent confirmation.
Founder Background and Company Culture
I find the personal story behind Axiom inseparable from its technical claims. The company was founded by Carina Hong in March 2025. A Rhodes Scholar and former Stanford PhD student in mathematics, Hong left academia after concluding that formal reasoning tools could advance faster outside university structures.
Hong recruited Shubho Sengupta, formerly of Meta’s FAIR lab, as early co founder and chief technology officer. Their partnership grew out of shared frustration with the limits of existing AI math systems.
Axiom’s team expanded quickly, drawing researchers from Meta FAIR, Google Brain, and other leading labs. By late 2025, the company had raised 64 million dollars in seed funding and grown to 17 members. Hong has described Axiom’s culture as deliberately non hierarchical, emphasizing proof quality over academic pedigree.
Interview: Carina Hong on Letting Machines Prove
Date, time, location, atmosphere: October 2025, late afternoon, a quiet conference room in San Francisco, sunlight filtering through tall windows, laptops closed.
I am the interviewer, a technology journalist focused on AI and science. Carina Hong sits across from me, relaxed but alert, hands folded, eyes moving quickly when she considers a question.
I begin by asking why she left academia so early. She pauses, then smiles slightly. “I loved mathematics,” she says, “but I hated pretending that our tools were good enough.”
When I ask what surprised her most about AxiomProver, she leans back. “It’s not speed. It’s stubbornness. The system does not get bored. It does not decide a path is unfashionable.”
We talk about fear. She nods. “Some mathematicians worry about relevance. I worry about honesty. If machines can check truth better than we can, we should use them.”
On credit, her tone sharpens. “Proofs do not belong to egos. They belong to the record. Humans still decide what questions matter.”
As the interview ends, she looks down at her notebook. “Understanding will come later,” she says softly. “Right now, correctness comes first.”
Production credits: Interview conducted and edited by the author.
AI Mathematics in Context
Axiom’s claims arrive amid rapid progress elsewhere. DeepMind’s AlphaGeometry and AlphaProof systems reached high performance on International Mathematical Olympiad problems in 2024. In 2025, advanced models achieved gold medal level results using natural language reasoning.
What separates Axiom is focus. While other systems emphasize breadth and competition performance, Axiom targets depth and formal verification. It aims not to impress judges but to satisfy proof assistants.
Table: Comparison of Advanced AI Math Systems
| System | Organization | Focus | Verification |
|---|---|---|---|
| AlphaGeometry | DeepMind | Geometry problems | Partial formalization |
| AlphaProof | DeepMind | Algebra and number theory | Formal |
| Gemini DeepThink | Olympiad reasoning | Informal | |
| AxiomProver | Axiom | Research level proofs | Fully formal |
Takeaways
• AI systems are now producing research level mathematical proofs.
• Formal verification may redefine how correctness is judged.
• Human intuition is no longer the only driver of discovery.
• Credit disputes will become more complex.
• Peer review remains essential despite machine checking.
• Forgotten mathematical methods may gain new life through AI.
Conclusion
I do not see Axiom’s work as the end of human mathematics. I see it as the beginning of a different relationship between humans and proof. If these results survive peer review, they will mark a moment when machines moved from helpers to independent reasoners.
That shift will be uncomfortable. Proofs have always been explanations written by people for people. Machine generated proofs may be correct without being comprehensible. The task for mathematicians may shift from discovering results to interpreting them.
History suggests mathematics adapts. Each new tool, from algebraic notation to computers, changed the discipline without destroying it. Axiom’s announcement suggests the next change will be deeper, touching not just how proofs are written but who writes them.
FAQs
Did Axiom’s AI really solve unsolved problems
Axiom claims the problems were open and that its proofs are correct, but peer review is still ongoing.
What is Lean and why does it matter
Lean is a formal proof system that checks every logical step, eliminating hidden errors.
Is this the same as Olympiad solving AI
No. Axiom focuses on research level proofs rather than competition problems.
Will mathematicians be replaced
Most experts expect AI to augment human work, not eliminate it.
Why is peer review still needed
Formal correctness does not address novelty, relevance, or prior work.
