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.
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.
Geometric quantum hydrodynamics merges geometric hydrodynamics with quantum hydrodynamics to study the geometric properties of vortex structures in superfluid states of matter. Here the vortex line acts as the fundamental building block and is a topological defect of the fluid medium about which the otherwise irrotational fluid circulates. In this thesis, we show that except for the simplest fluid flows, a vortex line seeks to decompose localized regions of curvature into helical configurations. The simplest flow, known as the local induction approximation, is also an integrable one. Integrability makes the transfer of energy into helical modes impossible. In the following, we demonstrate that any arclength conserving correction to this approximation defines a non-Hamiltonian evolution of the vortex geometry, which is capable of supporting dissipative solitons and helical wavefronts. Quantum turbulence in ultracold vortex tangles relies on energy transfer between helical or Kelvin modes to decay. Thus, models of vortex lines beyond the lowest order integrable cases are vitally important to our mathematical description of free decay of turbulent tangles. To motivate the results of this thesis we connect our theory of vortex line dynamics to continuum fluid mechanics. The Navier-Stokes equations are a statement of momentum balance for a fluid whose response to shear stress is proportional to the fluid's velocity gradients. Building off of Onsager's rather obscure work in fluid turbulence, others have shown that solutions to Navier-Stokes limit to Euler evolutions of distributional velocity profiles for large Reynolds number. Using this as our context, we show that the Euler equation can be transformed, in an inverse Madelung sense, to the Gross-Pitaevskii equation associated with the mean-field quantum dynamics of a dilute Bose gas. This theory predicts that an irrotational fluid is capable of circulating around regions of density depletion known as vortex lines. Furthermore, if a vortex line is used as the ansatz for the Gross-Pitaevskii equation then it is possible to show that the Biot-Savart integral over the vortex source results. This connection between the vortex line geometry and the induced velocity field provides the basis for the application of geometric quantum hydrodynamics to Bose-Einstein condensates. If the Biot-Savart integral is the basis of geometric quantum hydrodynamics, then Hasimoto's transformation is the structure built on top. The fundamental theorem of space curves states that up to rotations and translations, a curve is defined by its curvature and torsion. Through the Hasimoto transformation, it is possible to map flows defined by the Biot-Savart integral to scalar partial differential equations evolving the curvature and torsion of a vortex line. In this thesis, we conduct an asymptotic expansion of the Biot-Savart integral and apply the Hasimoto transformation to show that vortex lines prefer to relax curved abnormalities through the excitation of helical waves along the vortex. Thus, our…
The Bose-Einstein condensation of noninteracting particles restricted to move on the sites of hierarchical diamond lattices is investigated. Using a tight-binding single-particle Hamiltonian with properly rescaled hopping amplitudes, we are able to employ an orthogonal basis transformation to exactly map it on a set of decoupled linear chains with sizes and degeneracies written in terms of the network branching parameter q and generation number n. The integrated density of states is shown to have a fractal structure of gaps and degeneracies with a power-law decay at the band bottom. The spectral dimension d(s) coincides with the network topological dimension d(f) = ln(2q)/ln(2). We perform a finite-size scaling analysis of the fraction of condensed particles and specific heat to characterize the critical behavior of the BEC transition that occurs for q > 2 (d(s) > 2). The critical exponents are shown to follow those for lattices with a pure power-law spectral density, with non-mean-field values for q < 8 (d(s) < 4). The transition temperature is shown to grow monotonically with the branching parameter, obeying the relation 1/T(c) = a + b/(q - 2).
An Theory is founded on two original postulates: (1) a nearest-neighbor attraction rule for discrete field quanta, structurally analogous to van der Waals forces, nuclear binding, the Casimir effect, and Bose–Einstein condensation; and (2) a 3/3 topological configuration consisting of three orthogonal vortex slots in three-dimensional space, the only complete knot structure consistent with 3D rotational symmetry. All physical content presented herein is derived logically from these two principles, with no additional independent assumptions. From the nearest-neighbor attraction alone, the theory yields stable self-bound topological solitons with energy scaling as 1/r2, naturally explaining wave-particle duality, quantum nonlocality, and the Born rule. It recovers Newtonian gravity via macroscopic summation of microscopic attractions under 3D flux conservation. Its lattice action reduces to the nonlinear Klein–Gordon Lagrangian in the continuum limit, rendering Lorentz invariance a mathematical theorem rather than a postulate, with c2=K/ρ identified as lattice stiffness over inertial density. The nonrelativistic limit exactly reproduces the Schrödinger equation, while Goldstone modes and topological winding enforce integer charge quantization. A threshold-based topological collapse mechanism provides a locally realist description of quantum measurement, double-slit interference, quantum erasure, delayed choice, and entanglement. The strong force arises as high-excitation contact binding, the electric force as topological knot stress, and the magnetic force as the kinematic effect of moving charges. All elementary particles—photons, electrons, quarks—represent the same self-bound structure distinguished only by topological winding number. Probabilistic weak decay with fixed half-lives is derived from the stochastic TDGL equation via Kramers escape rates over topological energy barriers. From the 3/3 topological configuration follow fractional quark charges as geometric necessities, SU(3) color symmetry, and numerically verified quark confinement. The theory classifies all 12 Standard Model fundamental fermions by geometric complexity and excitation level, explains the mass hierarchy through coherence-length relaxation and strong-force compression, and derives spin-1/2 from the double-connectedness of SO(3). Three falsifiable predictions emerge: no fourth-generation fermions, no magnetic monopoles, and no fractional charges outside multiples of 1/3. A proposed experimental test concerns the threshold behavior of interference visibility under tunable path-marker coupling; deviations from the smooth V2+D2=1 envelope would carry significant theoretical implications. This work demonstrates that two fundamental principles may suffice to unify the core phenomena of fundamental physics, rather than relying on a set of disconnected empirical assumptions.
An Theory is founded on two original postulates: (1) a nearest-neighbor attraction rule for discrete field quanta, structurally analogous to van der Waals forces, nuclear binding, the Casimir effect, and Bose–Einstein condensation; and (2) a 3/3 topological configuration consisting of three orthogonal vortex slots in three-dimensional space, the only complete knot structure consistent with 3D rotational symmetry. All physical content presented herein is derived logically from these two principles, with no additional independent assumptions. From the nearest-neighbor attraction alone, the theory yields stable self-bound topological solitons with energy scaling as 1/r2, naturally explaining wave-particle duality, quantum nonlocality, and the Born rule. It recovers Newtonian gravity via macroscopic summation of microscopic attractions under 3D flux conservation. Its lattice action reduces to the nonlinear Klein–Gordon Lagrangian in the continuum limit, rendering Lorentz invariance a mathematical theorem rather than a postulate, with c2=K/ρ identified as lattice stiffness over inertial density. The nonrelativistic limit exactly reproduces the Schrödinger equation, while Goldstone modes and topological winding enforce integer charge quantization. A threshold-based topological collapse mechanism provides a locally realist description of quantum measurement, double-slit interference, quantum erasure, delayed choice, and entanglement. The strong force arises as high-excitation contact binding, the electric force as topological knot stress, and the magnetic force as the kinematic effect of moving charges. All elementary particles—photons, electrons, quarks—represent the same self-bound structure distinguished only by topological winding number. Probabilistic weak decay with fixed half-lives is derived from the stochastic TDGL equation via Kramers escape rates over topological energy barriers. From the 3/3 topological configuration follow fractional quark charges as geometric necessities, SU(3) color symmetry, and numerically verified quark confinement. The theory classifies all 12 Standard Model fundamental fermions by geometric complexity and excitation level, explains the mass hierarchy through coherence-length relaxation and strong-force compression, and derives spin-1/2 from the double-connectedness of SO(3). Three falsifiable predictions emerge: no fourth-generation fermions, no magnetic monopoles, and no fractional charges outside multiples of 1/3. A proposed experimental test concerns the threshold behavior of interference visibility under tunable path-marker coupling; deviations from the smooth V2+D2=1 envelope would carry significant theoretical implications. This work demonstrates that two fundamental principles may suffice to unify the core phenomena of fundamental physics, rather than relying on a set of disconnected empirical assumptions.
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 (Δλ). The resulting density oscillations can be deterministically analyzed using Causal-AI Taylor sensitivity operators (j1,j2,j3). 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,PB]=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.
This dissertation focuses on theoretical studies of relativistic Bose-Einstein condensation (BEC).Based on the complex scalar field theory, which serves as a fundamental model for relativistic BEC, we provide a detailed explanation of the analyses using the functional renormalization group (FRG), including both its formalism and the underlying physics.We also present the numerical implementation of the FRG flow equation and the results obtained from it, as well as the derivations of various related equations.Although previous studies rely on approaches that are not well justified, we develop a novel and physically consistent FRG framework.In this framework, we impose a condition on the FRG flow equation, analogous to that in the non-interacting case.By solving the equation under this condition, we find that quantum fluctuations significantly affect the phase transition of relativistic BEC in the strong coupling regime.We further investigate the dependence of the condensate on spatial dimensions, and in particular, confirm that the FRG analysis is consistent with the Mermin-Wagner theorem from both numerical and analytical perspectives.These results contribute to the development of the theoretical foundation of the FRG, and uncover an aspect of relativistic BEC under non-perturbative interactions and for various spatial dimensions.