Dynamic Derivation of Cosmological Scaling from Closure Field Eigenmodes: From Loxodromic Geometry to Ψ-Field Transport in QMU

This work provides a dynamical derivation of cosmological expansion within the Aether Physics Model (APM) using Quantum Measurement Units (QMU). Previous results established that the Hubble parameter follows the scaling relation\[H = F_q \, α_a4/5 √Γ_H,\]where Fq is the quantum frequency, αₐ is the Aether fine-structure parameter, and ΓH is a dimensionless geometric factor. The exponent $4/5$ arises from the intrinsic geometry of the Aether unit. In the present work, this scaling is derived from first principles by constructing a closure-field operator governing the propagation of closure imbalance. The closure field Ψ(x,t) represents the local degree of Aether completion and is described by a variational field theory with dimensionless QMU coordinates. Solving the closure-field eigenvalue problem along loxodromic transport paths yields a standard closed-loop spectrum\[λ_n ∝ n^2,\]showing that the eigenvalue structure alone does not generate the fractional scaling exponent. The exponent instead emerges from a geometric closure measure defined over the Aether unit. The transport structure consists of eight orientation-resolved loxodromic channels distributed across five independent volumetric--chronovibrational constraints, producing a geometric exponent\[Dcl = 8/5.\] This leads to a closure-density scaling\[Ωcl = α_a8/5,\]which determines the magnitude of cosmological expansion. The Hubble parameter follows from the root-mean-square closure imbalance,\[H = F_q √{Γ_H \, Ωcl}.\] Numerical evaluation yields\[Γ_H ≈ 8π/3,\]so that\[H^2 = {F_q}^2 α_a8/5 Γ_H.\] This expression reproduces the observed expansion rate,\[H ≈ 2.51 × 10⁻¹⁸\ {s⁻¹}(77.5\ {km\,s⁻¹\,Mpc⁻¹}),\]and establishes a direct correspondence between closure geometry and cosmological dynamics. The factor ΓH acts as a geometric density parameter of Aether expansion, mirroring the 8π/3 coefficient in Friedmann-type cosmology, while the source term is supplied by the dimensionless closure-density Ωcl. The result provides a unified framework in which geometry, field dynamics, and cosmological expansion arise from closure imbalance within the Aether-unit structure.