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

László János Németh

Published Aug 30, 2026• 0 citations
Publisher

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.

DOI
10.5281/zenodo.22179011

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This paper presents the first experimental validation of Causal-AI topological control on real superconducting quantum hardware. Addressing the fundamental bottleneck of ambient 1/f1/f1/f flicker noise and dephasing in quantum processors, we operationalize the Entropic-Resistance framework—modeling spacetime as a discrete network and time as a recursive relaxation process—to optimize transmon qubit performance. By extracting deterministic Taylor sensitivity operators (j1,j2j_1, j_2j1​,j2​) from real-time calibration data of IBM Quantum’s live ibm_fez processor, we compute and execute an analytically adjusted "Lasky-pulse" amplitude. Comparative hardware execution against standard factory baselines (validated across live QPU job execution logs: da294biein7c73bea68g and da29qpkdedkc73erutlg) demonstrates that topological network optimization successfully bridges discrete recursive spacetime theory with operational quantum control without requiring hardware modifications.
In this paper, we propose a mathematical formalism for describing the homogenization and renormalization group (RG) flow of a topologically entangled quantum vortex network (vortex tangle) in hierarchical Bose-Einstein condensates (BECs). Unlike traditional analogue gravity models, the spacetime microstructure is modeled as a discrete hierarchy of embedded, self-similar vacuums (Embedded Vacuums), where the coherent fluid of each subsequent level is situated within the vortex cores of the preceding level. Under non-equilibrium steady state (NESS) conditions, using rotationally invariant Parseval tight frames from Meyer's multiresolution analysis (MRA) of the Schwartz class, we define orthogonal projection operators in nested Hilbert spaces and derive a chain of homogenized variational equations of multi-layered effective action. Quasiparticle transport and vortex field evolution are reduced to a fully closed, self-consistent non-local integro-differential equation through the Biot-Savart relation. We mathematically demonstrate that the requirement of Lagrangian form-invariance and the prevention of causal short-circuits via a state-dependent Lieb-Robinson information bound are compatible, leading to the algebraic cancellation of the mass scaling parameter. In the extreme ultraviolet limit, we show a transition to Carrollian spacetime where, due to the vanishing of the coupling barrier, cascade energy transfer is sustained through a coherent quantum phase slip (CQPS) regime on a lattice of scales. Key Scientific Contributions The core framework of this paper establishes four novel, mathematically rigorous, and physically verifiable results: 1. Exact Scaling of Chiral Transport ( C₁ = K₀ m_n / ħ ): A direct mathematical derivation of the quasiparticle chiral transport coefficient, demonstrating the exact algebraic cancellation of the local interaction-density product (g_n ρ_n). This represents an independent, reproducible mathematical signature of the quantum transport equations. 2. Fractal Scaling Law of Instanton Action ( q(s_inst) = q_σ^D_f ): An analytical proof demonstrating that the scaling of the instanton action under renormalization group (RG) flow is determined uniquely by the fractal dimension (D_f) of the quantum vortex network. 3. Critical Step of Carrollian Phase Transition ( n_crit ): An experimentally verifiable analytical formula for the critical hierarchical step, n_crit = ln(√2 g₀ / (C_const λ₀)) / [(D_f* - 3) ln(q_σ)], beyond which classical acoustic transport collapses into Carrollian ultra-locality and transitions entirely to coherent quantum phase slips (CQPS). 4. Rigorous Homogenization Proofs (Lemmata 1, 2, and 3): Three complete mathematical proofs validating the multiresolution analysis (MRA) homogenization scheme, utilizing Jackson-type approximation theorems and bounded commutator estimates in Sobolev spaces H³(ℝ³) with strict L² error bounds.
We present a study of analog cosmological models in Bose-Einstein condensates (BEC) and in graphene, and superfluidity in a box-shaped traps. We start by examining the dynamics of a Bose-Einstein condensate (BEC) trapped inside an expanding toroid that can realize an analog inflationary universe. The expanding condensate forces phonons to undergo redshift and damping due to quantum pressure, owing to the thinness of the ring. We predict that such expanding BECs can exhibit spontaneous phonon creation from the vacuum state and show how it would manifest in the atom density and density correlations and discuss connections with the inflationary theory. We then extend this work to study entanglement of the spontaneously generated phonon pairs, for which we use the techniques of quantum continuous variables. We then develop a protocol to experimentally measure the correlations entering the covariance matrix, allowing an experimental quantification of the entanglement properties of the inflationary BEC. We then present a of how a spatially-varying quasiparticle velocity in honeycomb lattices, achievable using strained graphene or in engineered cold-atom optical lattices that have a spatial dependent tunneling amplitude, can yield the Rindler Hamiltonian embodying an observer accelerating in Minkowski spacetime. Within this setup, a sudden switch-on of the spatially-varying strain yields a spontaneous production of electron-hole pairs, an analog version of the Unruh effect characterized by the Unruh temperature. We discuss how this thermal behavior, along with Takagi's statistics inversion, can manifest themselves in photo-emission and scanning tunneling microscopy experiments. We show that the electronic conductivity grows linearly with frequency ω\omegaω, vanishing in the DC limit. Finally, we find that the total electronic energy at zero environment temperature looks like Planck's blackbody result for photons due to the aforementioned statistics inversion, whereas for an initial thermally excited state of fermions, the total internal energy undergoes stimulated particle reduction. Finally, we study ultracold Fermi gases trapped inside a three-dimensional box with vanishing boundary conditions, with interactions modeled using the Fermi-Huang pseudopotential. Taylor expanding the pairing to linear order in position, solves the gap equation yielding the transition temperature and local pairing amplitude that vanishes at the edges.
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