Thermodynamic Compactness and Information-Geometric Bounds in Excluded-Volume Systems

We show that thermodynamic consistency in systems with finite excluded volume implies compact support of the grand canonical particle-number distribution. Understanding whether fundamental bounds on information and matter content can arise purely from statistical-mechanical principles—independent of gravitational dynamics—is of central interest in thermodynamics, information theory, and cosmology. For any nonzero excluded volume parameter b, the partition function vanishes identically beyond $N_{max}=V/b$, enforcing a strict upper bound on admissible macrostates. We demonstrate that this compactness induces bounded particle-number fluctuations and finite Fisher information with respect to the chemical potential, thereby rendering the associated statistical manifold effectively finite-dimensional. This informational compactness provides a structural mechanism limiting distinguishability of macrostates independently of gravitational considerations. We argue that such thermodynamically enforced bounds are compatible with entropy bounds and holographic scaling principles, suggesting that informational finiteness may arise from statistical-mechanical consistency alone. Cosmological implications are discussed cautiously: infinite matter content at fixed volume is incompatible with compact support induced by finite excluded volume. Accordingly, the Fisher metric and associated thermodynamic lengths remain bounded when particle-number fluctuations are restricted by excluded-volume constraints. These results show that excluded-volume constraints induce a natural information-geometric compactness of the thermodynamic manifold, providing a general mechanism by which statistical distinguishability and curvature remain finite in finite-occupancy systems.