Schrödinger estimates with arbitrarily small periodic regularity on ${\mathbb{H}}^2\times \mathbb{T}$

Let $M={\mathbb{H}}^2\times \mathbb{T}$ carry its product metric. We prove that the Schrödinger evolution maps $L^2({\mathbb{H}}^2;H^s(\mathbb{T}))$ into $L^2([0,1];L^6(M))$ for every $s>0$, without any regularity assumption in the hyperbolic variable. The proof separates hyperbolic and periodic frequencies. A localized hyperbolic kernel estimate restricts the effective interaction time at hyperbolic frequency $L$ to order $L^{-1}$. Quadratic exponential sums then give a logarithmic bound when the periodic bandwidth is comparable to $L$. A mixed-norm decoupling argument, including an $L^6$-valued extension and a periodic transference step, reduces wider periodic bands to this case. The infimum of the admissible periodic regularities is therefore zero. The estimate at zero regularity is not established here.