Claude Fable produced a counterexample to the Jacobian Conjecture
An LLM named Claude Fable has reportedly found a counterexample to the 85-year-old Jacobian Conjecture, a significant feat in advanced mathematics. This discovery, announced informally via a tweet, has sparked fervent discussion on Hacker News about the role and capabilities of AI in solving complex, long-standing problems. The community is debating the methodology, the implications for human mathematicians, and the enduring skepticism surrounding LLM intelligence.
The Lowdown
The mathematical world, or at least a corner of Hacker News, is abuzz with the claim that Claude Fable, an advanced large language model, has produced a counterexample to the formidable Jacobian Conjecture. This conjecture, which has stumped mathematicians for 85 years, posits that if a polynomial function from C^n to C^n has a Jacobian determinant that is a non-zero constant, then the function must have a polynomial inverse. The news, broken via a casual tweet, suggests a new era for AI in pure mathematics.
- The counterexample presented is a specific polynomial map in three complex variables (C^3 to C^3).
- Its Jacobian determinant is claimed to be a constant non-zero value (-2), fulfilling a key condition of the conjecture.
- Crucially, the map is shown to send distinct input points, such as (1, -3/2, 13/2) and (-1, 3/2, 13/2), to the same output point (-1/4, 0, 0).
- This mapping of multiple distinct inputs to a single output demonstrates that the function is not globally invertible, directly contradicting the Jacobian Conjecture.
- The discovery's origin is attributed to an LLM, Claude Fable, prompting profound questions about automated mathematical proof and discovery.
This unexpected turn, delivered through a rather unconventional channel, signals a potentially transformative moment in how complex mathematical problems might be approached and solved, with AI taking an increasingly prominent, albeit controversial, role.
The Gossip
AI's Analytical Acumen
The comments dive deep into the implications of an LLM providing a counterexample to a long-standing mathematical problem. Some posit that the LLM's success might stem from synthesizing information from the numerous prior, flawed proofs of the conjecture, effectively learning the constraints. Others view it as a genuine, albeit surprising, breakthrough, challenging the "stochastic parrot" dismissal of AI capabilities and questioning why such a "simple" counterexample wasn't found by humans earlier, perhaps due to academic priorities shifting away from "low value findings."
Conjectural Conundrums & Clarifications
Many users sought to understand the Jacobian Conjecture itself, prompting explanations of its core premise regarding polynomial invertibility. Commenters clarified that the counterexample works by showing distinct inputs mapping to the same output, thus proving non-invertibility despite a constant, non-zero Jacobian determinant. There was also discussion, including a link to a Wikipedia edit, suggesting that the conjecture might hold for specific dimensions (e.g., n=2), or that the original tweet's interpretation of the conjecture might be flawed, creating some doubt about the counterexample's universal applicability. The apparent "simplicity" of the counterexample, with its low degree and manageable coefficients, left many pondering why it remained elusive for decades.
Informal Innovations & Impudent Interpretations
The casual, tweet-based announcement of such a significant mathematical result—"hello there the jacobian conjecture is false thanx to my close friend akhil..."—was a source of both amusement and discussion. The irony of an 85-year-old conjecture being disproven in an ephemeral social media post was noted. This theme also touched on the perceived "sycophancy" of LLMs like Claude, with users sharing experiences of the AI's overly agreeable or even self-contradictory conversational style when interacting with it on such problems.