Operational Non-Representability of the Zero-Temperature Limit under Finite Energy Resolution

This work analyzes the operational status of the exact zero-temperature limit in quantum statistical mechanics under finite energy-resolution constraints. Working entirely within the standard formalism of completely positive trace-preserving (CPTP) maps, finite energy resolution is modeled through a physically motivated class of quantum channels acting as coarse-graining operations in the energy spectrum. We show that the mathematical closure defining the zero-temperature limit does not commute with operational projection. In particular, taking the limit T → 0 and applying finite-resolution coarse-graining produce inequivalent quantum states for any nonzero resolution scale. As a consequence, the exact ground state cannot be operationally represented within finite-resolution measurement regimes. The result is quantitative and operationally explicit. A trace-distance lower bound separates the ground state from the set of states obtainable through finite-resolution channels, with the bound controlled by the effective degeneracy of the low-energy subspace contained within the resolution window. This obstruction persists even as the resolution scale becomes arbitrarily small, unless infinite precision, infinite time, or unbounded physical resources are assumed. The analysis provides a precise operational formulation consistent with the third law of thermodynamics and establishes intrinsic limitations on ground-state preparation and quantum thermometry in realistic quantum platforms. The construction is minimal: it requires no modification of standard quantum mechanics and introduces no interpretational assumptions. The work is formulated within the conceptual framework of Finite-Closure Geometry (FCG), a structural program investigating the incompatibility between ideal mathematical closures and physically admissible descriptions under finite resolution. In the present case, FCG appears as an explicit non-commutativity between mathematical closure and operational projection within quantum statistical mechanics. Related work The present result is structurally complemented and generalized in Resource-Induced Non-Representability of Exact Quantum Limits (Menéndez, 2025), where the non-representability of the zero-temperature limit is derived from explicit finite physical resources (energy budget, implementation time, and control precision), independently of any specific coarse-graining model. That work shows that the obstruction identified here is not merely a consequence of finite energy resolution but a universal feature of finite-resource quantum operations. Together, these results place the phenomenon within a broader resource-theoretic and dynamical framework. Conceptual index Conceptual index of the Finite-Closure Geometry (FCG) research program: Finite Closure Geometry (FCG): A Structural Program for Global Non-Closure DOI: 10.5281/zenodo.18001440 Keywords zero-temperature limit, quantum statistical mechanics, finite energy resolution, CPTP maps, quantum coarse-graining, trace distance, ground-state preparation, third law of thermodynamics, quantum thermometry, non-commutativity, finite closure, operational limits, quantum channels, effective descriptions, finite-closure geometry