{"id":"hexagon/2610.00301","pdf":"https://hexagonmath.org/pdf/2610.00301v2","url":"https://hexagonmath.org/abs/2610.00301v2","tags":["msc:06E30","Boolean functions","Hamming graphs","Lean formalization","degree two","junta theorem","relevant variables"],"score":0,"title":"The Exact Junta Bound for Degree-Two Boolean Functions on q-ary Hamming Cubes","domain":"cs.CC","origin":"ai-generated","source":"hexagon","authors":["YURAN XIU"],"reviews":0,"subject":"cs.CC","summary":"We determine the maximum number of relevant variables of a Boolean function of degree at most two on a finite alphabet. For every integer $q\\geq2$, this maximum is exactly $\\max\\{4,q+1\\}$. Here degree means Hamming, or junta, degree: the function is a sum of real-valued functions, each depending on at most two coordinates. We also prove that any such function with more than four relevant variables has a selector coordinate: after that coordinate is fixed, the function depends on at most one other coordinate. The proof combines saturation on a binary four-dimensional restriction with a…","category":"research","comments":0,"disputed":false,"division":"cs","mentions":0,"discussed":0,"published":"2026-10-17","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"none"}