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
\[
H = F_q \, \alpha_a^{4/5} \sqrt{\Gamma_H},
\]
where $F_q$ is the quantum frequency, $\alpha_a$ is the Aether fine-structure parameter, and $\Gamma_H$ 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 $\Psi(x,t)$ 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
\[
\lambda_n \propto n^2,
\]
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
\[
D_{\mathrm{cl}} = \frac{8}{5}.
\]

This leads to a closure-density scaling
\[
\Omega_{\mathrm{cl}} = \alpha_a^{8/5},
\]
which determines the magnitude of cosmological expansion. The Hubble parameter follows from the root-mean-square closure imbalance,
\[
H = F_q \sqrt{\Gamma_H \, \Omega_{\mathrm{cl}}}.
\]

Numerical evaluation yields
\[
\Gamma_H \approx \frac{8\pi}{3},
\]
so that
\[
H^2 = {F_q}^2 \alpha_a^{8/5} \Gamma_H.
\]

This expression reproduces the observed expansion rate,
\[
H \approx 2.51 \times 10^{-18}\ \mathrm{s^{-1}}
\quad
(77.5\ \mathrm{km\,s^{-1}\,Mpc^{-1}}),
\]
and establishes a direct correspondence between closure geometry and cosmological dynamics.

The factor $\Gamma_H$ acts as a geometric density parameter of Aether expansion, mirroring the $8\pi/3$ coefficient in Friedmann-type cosmology, while the source term is supplied by the dimensionless closure-density $\Omega_{\mathrm{cl}}$.

The result provides a unified framework in which geometry, field dynamics, and cosmological expansion arise from closure imbalance within the Aether-unit structure.