Trichotomy for the HRT Conjecture for mixed integer configuration

We consider the HRT conjecture in the mixed-integer setting, where a finite configuration in $\R^d\times\R^d$ consists of $N-1$ points in $\Z^d\times\Z^d$ and one point $(\alpha,\beta)$ outside the lattice. Assuming a linear dependence among the corresponding time-frequency shifts of a nonzero Schwartz function, we apply the Zak transform to obtain a cocycle over translation by $\gamma=(-\alpha,\beta)$ on $\T^{2d}$ and study the orbit closure $$H=\overline{\{n\gamma \bmod \Z^{2d}:n\in\Z\}}.$$ We show that this reduction yields a trichotomy. The dense-orbit case is impossible because a Zak zero propagates to a dense zero set, forcing the Zak transform to vanish identically. The finite-orbit case reduces to a rational configuration, and hence to the lattice case covered by Linnell's theorem. Thus any mixed-integer counterexample for a Schwartz window must occur in the infinite proper case. For that remaining case, we prove that the nonvanishing set of the Zak transform is $H$-saturated, that the averaged logarithmic growth of the modulus cocycle along $H$ exists and vanishes identically, and that the restriction to each nonvanishing $H_0$-coset satisfies a smooth cohomological equation. This yields small-divisor compatibility conditions for the induced translation on $H_0$. We further obtain an arithmetic rigidity condition. These results isolate a collection of necessary dynamical, cohomological, and arithmetic constraints that any mixed-integer counterexample must satisfy.