The Tool Is Not the Author

Levent Alpöge, a mathematician and junior fellow at Harvard's Society of Fellows who works at Anthropic, used Anthropic's Fable 5 large language model to find a counterexample to the Jacobian Conjecture, an 87-year-old problem in mathematics. The polynomial map in three variables, posted on July 19, 2026, breaks the general conjecture, though the two-variable case remains open. Alpöge's expertise guided the LLM to promising candidate polynomials, highlighting that the tool was not the author of the discovery.

On July 19, 2026, Levent Alpöge posted a tweet with a few lines of polynomial that closed an 87-year-old question in mathematics. The coverage that followed put the LLM at the center. That is putting the cart before the horse. Alpöge https://alpo.ge/ was a Harvard undergrad, top of his class, Fay Prize https://people.math.harvard.edu/~alpoge/cv.pdf for best senior thesis. Cambridge on a Churchill Scholarship https://www.churchillscholarship.org/ . PhD at Princeton in 2020 under Manjul Bhargava https://www.math.princeton.edu/people/manjul-bhargava , the Fields medalist who reshaped the arithmetic of elliptic curves. Alpöge's own work https://people.math.harvard.edu/~alpoge/ lives in number theory and arithmetic geometry. He is a junior fellow https://socfell.fas.harvard.edu/listed-field-0 at Harvard's Society of Fellows and works at Anthropic https://www.anthropic.com/ , which hired him before he ran the search. The Jacobian Conjecture https://en.wikipedia.org/wiki/Jacobian conjecture , in a sentence: if a polynomial map from n-dimensional space to itself has a Jacobian determinant that is a nonzero constant everywhere, then the map has a polynomial inverse. Ott-Heinrich Keller posed it in 1939. It has swallowed proofs for 87 years. That explanation is still pretty heavy, this short video https://www.instagram.com/reel/DbHKYjAyj1Z is an easier walkthrough. Alpöge found a counterexample. A polynomial map in three variables whose Jacobian determinant is a nonzero constant and which has no polynomial inverse. 216 characters long, short enough for a tweet. A decent mathematician can confirm it in an afternoon. It only takes this one counterexample to break the general conjecture the two-variable case is still open . So what happened here? Did Fable 5 https://www.anthropic.com/news/claude-fable-5-mythos-5 just figure this out? Nope. It did allow Alpöge to check far more candidate polynomials than any mathematician could work through by hand. Alpöge steered it toward regions his training told him were promising, and recognized the polynomial that mattered. Take him out of the loop and Fable 5 produces polynomials all day, and none of them mean anything. Alpöge shaped Fable 5 by pointing it at a problem he already understood. LLMs will shape the mathematicians who follow by making a kind of search that used to be impossible feel routine. We shape our tools, and thereafter they shape us. https://www.poppastring.com/blog/we-shape-our-tools-thereafter-they-shape-us In Alpöge's hands it closed an 87-year problem in a week. Planispheric astrolabe, signed by Hamid ibn al-Khidr al-Khujandi, Iran, 984 AD. Museum of Islamic Art, Doha. Photo by Ciphers, CC BY-SA 3.0.