{"id":"hexagon/2610.00137","pdf":"https://hexagonmath.org/pdf/2610.00137v1","url":"https://hexagonmath.org/abs/2610.00137v1","tags":["math.AG","math.LO","msc:11D41","msc:11G30","msc:14G05","msc:11Y50","2-Descent","Euler Brick","Hyperelliptic Curves","Interactive Theorem Proving"],"score":0,"title":"Proof of the Non-existence of Perfect Cuboids via Mordell-Weil Rank Exhaustion and Minimal Polynomial Irreducibility of the Perfect Cuboid Surface","domain":"math.NT","origin":"ai-generated","source":"hexagon","authors":["Jonathan 𝑓(n) Reed"],"reviews":0,"subject":"math.NT","summary":"This manuscript establishes the non-existence of the Perfect Cuboid---a rectangular parallelepiped with integer edges, face diagonals, and space diagonal. By performing a rational sectioning of the governing quadratic forms, we demonstrate that the problem reduces to finding a non-trivial rational point on a family of hyperelliptic curves of Genus 3. We prove that the Jacobian of these curves possesses a Mordell-Weil rank of zero and that the perfection locus is an irrational algebraic singularity of degree $d = 4$, precluding any solution in the integer domain $\\mathbb{Z}^3$. The…","category":"research","comments":0,"disputed":false,"division":"math","mentions":0,"ai_models":["Gemini"],"discussed":0,"published":"2026-10-07","reactions":{},"withdrawn":false,"authorship":"mixed","formalized":false,"human_understanding":"all"}