The Complete Two‑Weight \(\mathbb{Z}/5\mathbb{Z}\) Goemans–Linial Cone

Abstract

For the weighted Cayley graph on \(\mathbb{Z}/5\mathbb{Z}\) whose difference‑\(\pm1\) edges have weight a and difference‑\(\pm2\) edges have weight b, we determine the uniform‑demand sparsest cut and the Goemans–Linial relaxation for every \(b\ge 0\) with \(a+b>0\). Both values are\(\frac{5 \min\{a+2b, 2a+b\}}{6(a+b)}.\)We classify every optimal cut up to dihedral symmetry and every optimal translation‑invariant feasible semimetric, including both one‑weight faces and the tie \(a=b\). The key reduction is an objective‑preserving translation average realized in an orthogonal direct sum. Fourier diagonalization then shows that the full invariant Goemans–Linial cone is the elementary two‑distance triangle cone.