A Rank Threshold for Exact Sums of Squares in Virtually Abelian Group Algebras

Let G have a finite-index subgroup isomorphic to Z^d. We give an explicit finite-index compression formula and use it to show that positivity in all finite-dimensional unitary representations is equivalent to a sum of Hermitian squares at every matrix level exactly when d<=2. In the positive cases, the representations induced by characters of the abelian subgroup suffice. The essential input is the corrected bivariate matrix factorization theorem of Dritschel, Klep, McCullough and Volcic. For d>=3, a trigonometric version of the classical Motzkin form gives an explicit obstruction, and coefficient projection transfers it to every group containing Z^3. We spell out the resulting two-dimensional representation test and factor-transfer formula for the Klein bottle group. The result is a deduction from credited factorization and positivity methods; it does not settle the exact-squares conjecture for general closed surface groups. No factor-degree bound or absolute priority claim is made. This is an unrefereed, AI-assisted preprint.