Topological Mapping of Quantum Vacuum Fluctuations: Interfacing the Entropic-Resistance Paradigm with BEC Analog Models

Abstract

This paper establishes a rigorous theoretical bridge between the Entropic-Resistance paradigm and recent experimental visualizations of quantum vacuum fluctuations (Zhang et al., 2026). Traditional physics often treats analog laboratory models as mere mathematical simulations; however, under this framework, spacetime itself is defined as a discrete, algorithmically expanding topological network. Consequently, a Bose-Einstein Condensate (BEC) is treated as a functioning, discrete physical network ($G=(V,E)$) in its own right. We demonstrate that the fluctuations observed in a two-dimensional BEC analog model are not chaotic thermal noise, but macroscopic constraint reactions of this discrete network. By mapping the experimental parameters directly to a recursive topological graph, we show that radio-frequency state coupling corresponds to a localized relaxation contrast ($\Delta\lambda$). The resulting density oscillations can be deterministically analyzed using Causal-AI Taylor sensitivity operators ($j_1, j_2, j_3$). Finally, the paper proposes the emergent $Q$-hierarchy temporal drift as a definitive, falsifiable laboratory test. Because spatial transport and localized relaxation do not commute on the discrete grid ($[L, P_B] \neq 0$), the normalized sensitivity ratios exhibit a systematic temporal drift. Identifying this specific drift in the raw telemetry of BEC vacuum experiments would provide definitive "white-box" proof that quantum vacuum dynamics are governed by discrete, recursive topological geometry.