{"id":"hexagon/2610.00604","pdf":"https://hexagonmath.org/pdf/2610.00604v1","url":"https://hexagonmath.org/abs/2610.00604v1","tags":["math.NT","cs.CC","cs.LO","msc:03D15","msc:68Q17","msc:11A05","msc:03D75","Euclidean algorithm","Lean 4","closure depth"],"score":0,"title":"The Euclidean algorithm is not optimal among uniform processes","domain":"math.LO","origin":"ai-generated","source":"hexagon","authors":["Jason Davies"],"reviews":0,"subject":"math.LO","summary":"Reusing a stored Euclidean prefix, we show that every pair $a\\geq b$ has a derivation of $\\gcd(a,b)$ from $a,b$ with $O(\\log a/\\log\\log a)$ applications of the remainder function alone; Euclid's algorithm needs order $\\log a$ in the worst case. These derivations bound the intrinsic complexity of gcd and coprimeness by the same order, refuting Moschovakis's conjectured logarithmic lower bounds for uniform processes; by his best-uniform-process theorem, a single uniform process attains this bound. The Main Conjecture for algorithms remains open. We also prove an $\\Omega(\\log\\log a)$ lower…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Codex","Claude Code"],"discussed":0,"published":"2026-10-29","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":true,"formalizations":["https://palomar-registry.org/entry?id=PALOMAR-2026-10-23-000012&version=2"],"human_understanding":"some"}