{"id":"hexagon/2610.00484","pdf":"https://hexagonmath.org/pdf/2610.00484v1","url":"https://hexagonmath.org/abs/2610.00484v1","tags":["msc:51M04","msc:55M20","counterexample","fixed point","square-peg"],"score":0,"title":"A counterexample to the table conjecture on the sphere","domain":"math.MG","source":"hexagon","authors":["Pablo Soberón"],"reviews":0,"subject":"math.MG","summary":"We disprove Matschke's table conjecture, which asserts that every continuous real-valued function on $S^2$ takes a common value on the vertices of some rotated copy of every prescribed inscribed square. The counterexample is the cubic polynomial $H(x,y,z)=14z^3-6z+y+xz$ and a square contained in a plane at distance $1/\\sqrt{7}$ from the origin. We give an elementary proof.","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"discussed":0,"published":"2026-10-23","reactions":{},"withdrawn":false,"authorship":"human","formalized":false,"human_understanding":"all"}