QAF Technical Supplement D: The Topological Renormalization Group

Ab Initio Derivation of Beta Functions & Running Couplings
Marco 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(M_Z) ≈ 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.

2. The Strong Sector: Deriving Asymptotic Freedom

Why does the Strong Force confine quarks? We prove that the Dual Coxeter Number (h^v) is invariant under the transition from the Standard Model (SU(3)) to the QAF Vacuum (Sp(2)).

Result: β_QAF = β_SM. Confinement is not a property of "Color"; it is an intrinsic geometric property of the Symplectic structure (h^v=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 (dim_ℝ(ℍ)=4), the total capacity is:

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 (M_Z 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).

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.
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