Homogenization of a Topologically Entangled Vortex Tangle and Renormalization Group Flow in Hierarchical Bose-Einstein Condensates

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 mₙ / ħ): A direct mathematical derivation of the quasiparticle chiral transport coefficient, demonstrating the exact algebraic cancellation of the local interaction-density product (gₙ ρₙ). This represents an independent, reproducible mathematical signature of the quantum transport equations. 2. Fractal Scaling Law of Instanton Action (q (sᵢnst) = q_σDf): An analytical proof demonstrating that the scaling of the instanton action under renormalization group (RG) flow is determined uniquely by the fractal dimension (Df) of the quantum vortex network. 3. Critical Step of Carrollian Phase Transition (ncrit): An experimentally verifiable analytical formula for the critical hierarchical step, ncrit = ln (√2 g₀ / (Cconst λ₀) ) / (Df* - 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.