Material-Dependent Gravitational Decoherence: Thermodynamic Evaluation of the Pikovski Time-Dilation Mechanism

We evaluate the macroscopic entanglement-loss implications of gravitational time dilation, first identified by Pikovski et al. (2015). By moving beyond the idealized zero-temperature single-mode approximation to a continuous Debye model in the canonical ensemble, we derive the explicit phononic heat-capacity limits for spontaneous gravitational decoherence. We find that the decoherence rate driven by internal time-dilation variance strongly scales with $\sigma \propto T^{5/2}$ at ultra-low experimental temperatures ($T \ll \Theta_D$), vanishing strictly at absolute zero. Furthermore, we demonstrate mathematically that translational invariance renders proper-time differences between spatially distinct branches of an identical matter distribution identically zero. Thus, contrary to the Diósi-Penrose approach, an isolated mass in a pure superposition cannot undergo solely self-gravitational state-collapse. Decoherence requires either an external gravitational potential gradient or internal structural asymmetry. Lastly, we prove that the topological spectral dimension $d_s$ of the macroscopic test mass determines the decoherence rate scaling, creating a new class of ultra-sensitive 'fracton-gravity sensors'.