Resolving AdaBoost Cycling with LLMs: A Computer-Assisted Counterexample

We give a computer-assisted counterexample to the open question posed by Rudin, Schapire, and Daubechies in COLT 2012, of whether exhaustive AdaBoost always converges to a finite cycle. The construction is based on a block-product gadget whose two factors share an exact period-2 orbit for their 5-step branch maps, but whose linearized return maps have dominant eigenvalues with an irrational logarithmic ratio. This irrationality forces the burst-winner sequence to have an irrational asymptotic frequency, precluding eventual periodicity. All assertions are certified by exact rational arithmetic in two independent computer algebra systems. We also document the collaborative workflow that led to the construction, in which one model produced the key technical arguments; another model summarized and critiqued different versions of that work, proposing new directions to pursue; and the authors orchestrated the process by selecting directions to prioritize, resolving ambiguities, heavily refining each argument, and verifying the final solution. We present this as one of the first in-depth case studies of LLM-assisted mathematical research in which a long-standing open problem from theoretical machine learning is resolved, and detail the advantages and challenges of such research.