Closure of the UQGAF: Gauge Emergence, Galactic Dynamics, and Diophantine Consistency from Torus Knot Topology on T³

The Unified Quantum Gravity Architectural Framework (UQGAF) models fundamental particles as torus knots T (P, Q) on a three-torus vacuum manifold T³ = S¹ x S¹ x S¹ of unit radii with volume form 2π². The Geometric Impedance Formula m/mₑ = Q²/ (P·2π²) reproduces the masses of 40 Standard Model particles with a coefficient of variation of 0. 0006%. This paper completes the framework in six structural directions. Two-loop renormalisation group evolution from the UQGAF boundary conditions α⁻¹ (MZ) = 137. 036, sin²θW = 3/13, and αₛ (MZ) = 484/4117 shows that the three Standard Model gauge couplings do not converge at any scale below the Planck mass, consistent with the categorical tripartite gauge structure SU (3) ₂ × SU (2) ₂ × Z₆ derived here from Categorical Entanglement Minimisation. The zero-parameter acceleration scale a₀ = (15/13) cH₀/ (2π) = 1. 203×10⁻¹⁰ m/s² reproduces the empirical MOND acceleration scale to better than 1% across all 175 galaxies of the SPARC rotation-curve survey. The dark energy equation of state is predicted exactly as w₀ = -1, wₐ = 0, consistent with current DESI DR2 BAO-only data and awaiting adjudication of the contested combined BAO+CMB+SNe signal by DESI DR3. The absolute neutrino mass spectrum is derived from ground state energy minimisation on the vacuum manifold, yielding m₁ = 5. 19, m₂ = 10. 11, m₃ = 50. 55 meV (Σm_ν = 65. 85 meV) with strict normal ordering; the implied direct-measurement observable m_β = 10. 25 meV sits 44× below the current KATRIN bound. Two theorems on the catalogue intersection-number identity entering the CKM CP-violating phase are proved: a factorisation theorem expressing Iᵤs + Icb - Iᵤb - Ics as a sum of pairwise integer-shear cross-products, and a transcendence-based inexactness theorem establishing that no finite integer catalogue satisfies |Icombo|/Iᵤd = π/2 exactly. The bare CP phase π/2 is derived by two independent routes that agree exactly: a manifold-level derivation from T³ product structure giving arg (τ) = π/2, and a categorical-level derivation computing the SU (2) ₂ R-matrix eigenvalue in the triplet fusion channel yielding 4·arg R = π/2 with k = 2 uniquely selected by both Categorical Entanglement Minimisation and by the constraint itself. The Weinberg angle sin²θW = PW/PZ = 3/13 follows from the additive partition of the Z-boson winding between W³ and hypercharge cycles. The 15 Weyl fermions per generation, the Standard Model hypercharges (1/6, 2/3, -1/3, -1/2, -1), and the three-generation structure emerge from simple-object counting in the minimal fusion category, anomaly cancellation with electron-charge normalisation, and the uniqueness of Z₅ as the cyclic fusion ring producing exactly three non-degenerate complex-conjugate-paired generations. With this closure the UQGAF forms a self-consistent five-paper programme whose structural questions are resolved at the theorem level, with empirical resolution deferred to DESI DR2/DR3, JUNO, DUNE, KATRIN, Project 8, and Hyper-Kamiokande. This is Paper 5 of the five-paper UQGAF programme.