AI agents claim Collatz breakthrough: positive proportion of numbers reach 1, Lean-formalized
AlexKontorovich · x · 2026-09-29
Amid the buzz around Anthropic's FLT formalization and OpenAI's Navier-Stokes announcement, a quieter AI milestone may have gone unnoticed: Lech Mazur announced an AI-generated result on Collatz — for sufficiently large X, at least cX positive integers n < X reach 1 within 10.46 ln(n) ordinary Collatz steps for a fixed c > 0, ruling out prior doubts (bounds of X^0.84 and X^0.90 left open whether the proportion could vanish). The proof is Lean-formalized with 42k added lines, built on earlier formalizations of Tao's almost-boundedness and its natural-density extension, developed primarily by AI agents using the codex-openai-harness plus Codex with high-level guidance from Mazur. Mathematician Alex Kontorovich reviewed the Lean proof (via his own AI), believes it's correct, and is now working on a "digestion" of the result; Naoufal El Jaouhari has also formalized the same approach for the 3x-1 variant.
Related event: AI Agent Claims New Collatz Progress, Formalized in Lean(2 posts)→
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