First-Principles Derivation of the Isotropic Closure Factor: Aether-Unit Geometry, Closure Topology, and {Γᵢₛₒ = 8π/3} in QMU

This work completes the derivation of cosmological expansion scaling within the Aether Physics Model (APM) using Quantum Measurement Units (QMU) by providing a first-principles derivation of the isotropic closure factor. Previous papers established the expansion relation\[H = F_q \,α_a4/5√{Γᵢₛₒ},\]where the exponent $4/5$ arises from the closure-measure ratio $8/5$ associated with double-loxodromic transport geometry, and Γᵢₛₒ represents the isotropic normalization factor. While the exponent was derived from closure topology, the origin of Γᵢₛₒ had remained implicit. The present work derives\[Γᵢₛₒ = 8π/3\]directly from Aether-unit geometry and closure topology. The derivation begins with the total angular closure measure of a spherical orientation space,\[∫S^2 dΩ = 4π,\]which is normalized over the three independent volumetric closure axes to obtain the one-sided volumetric closure coefficient\[Γᵥₒₗ = 4π/3.\] Closure transport is shown to be intrinsically bidirectional: each formation channel is paired with a return channel required by chronovibrational closure. This doubles the volumetric coefficient, yielding the isotropic closure factor\[Γᵢₛₒ = 2Γᵥₒₗ = 8π/3.\] The result demonstrates that Γᵢₛₒ is not an empirical parameter and not imported from Friedmann-type cosmology, but arises from the topology of closure flux across the Aether-unit boundary. It represents the geometric density parameter governing isotropic projection of closure imbalance. Substitution into the closure-density expansion law gives the fully determined QMU cosmological scaling relation\[H = F_q \,α_a4/5√8π/3,\]with no external normalization factors. Together with prior derivations of the Aether fine-structure parameter αₐ and the closure-measure exponent $8/5$, this completes the internal QMU derivation of the Hubble expansion rate. The cosmological scaling is thus shown to emerge from closure geometry, closure topology, and QMU charge structure alone.