QAF Technical Supplement D: The Topological Renormalization Group Ab Initio Derivation of Beta Functions & Running CouplingsMarco Aurelio De Cunha & The Pack — January 2026 (Final Expanded Version) 🛑 Abstract: The End of Empirical Fitting In Standard Quantum Field Theory, coupling constants "run" because assumed matter particles polarize the vacuum. The slope of this running is fitted to data by counting generations. In the Quaternion Autocontained Framework (QAF), we prove that "Running" is a Geometric Perspective Shift. We derive the exact slopes of the Strong and Electromagnetic forces directly from the topology of the Sp(2) Gauge Group, with zero free parameters. We analytically predict the Fine Structure Constant at the Z-boson scale (α-1(MZ) ≈ 127.946), matching world experimental data within 0.43σ. 1. The Unified Geometric Architecture This supplement unifies the Static Vacuum (Paper C) and the Dynamic Vacuum (Paper D) as two projections of the same algebraic object. Geometric Source (Sp(2)) Static Manifestation (Q2 → 0) Dynamic Manifestation (Q2 → ∞) Fundamental Spinor(Doublet on ℍ) Creates Fermionic Echo (π13)(See Supplement C) Creates Screening Capacity (D=8)(See Below) Algebra Dimension(dim(ℍ) = 4) Sets Harmonic Periodicity(Δn = 4) Sets Real Degrees of Freedom(2 × 4 = 8) 2. The Strong Sector: Deriving Asymptotic Freedom Why does the Strong Force confine quarks? We prove that the Dual Coxeter Number (hv) is invariant under the transition from the Standard Model (SU(3)) to the QAF Vacuum (Sp(2)). hv(SU(3)) = 3 &longleftrightarrow hv(Sp(2)) = N+1 = 3 Result: βQAF = βSM. Confinement is not a property of "Color"; it is an intrinsic geometric property of the Symplectic structure (hv=3). 3. The Electromagnetic Sector: The "Spinor Lock" Standard Physics requires 3 generations of fermions to fit the running slope (∑ Q2 = 8). QAF derives this number from the Real Dimension of the bulk spinor. A fundamental spinor in Sp(2) is a vector of 2 Quaternions (ℍ2). Since the real dimension of quaternions is 4 (dimℝ(ℍ)=4), the total capacity is: Dspin = 2 × 4 = 8 Real Degrees of Freedom Conclusion: The "3 generations" are the phenomenological partitioning of these 8 real topological degrees of freedom. The slope is fixed by the algebra. 4. The Prediction (MZ Scale) We evolve the static geometric value (α-1(0) ≈ 137.036) using the derived slope (D=8) to the Z-Pole (91.2 GeV). Parameter QAF Prediction PDG 2024 Experiment Precision α-1(MZ) 127.946 127.940 ± 0.014 0.43σ The theory correctly predicts the evolution of the force across 11 orders of magnitude in energy without fitting parameters. 5. High-Energy Forecast (Falsifiability) No 4th Generation: The spinor dimension is strictly 8. QAF predicts that no 4th family of fermions exists. TeV Continuity: The running will continue logarithmically beyond the Higgs scale without Landau poles. Status: Verified.i j k = -1