SECTION 13 — SCALE-DEPENDENT GAUGE DYNAMICS AND EMERGENT FERMIONIC STRUCTURE (v9.2)
Fermions and Non-Abelian Gauge Symmetry from Causal Multiplets in Emergent 3+1 Geometry
By John Gavel
13.0 Overview
Non-Abelian gauge structure and fermionic matter emerge from multi-component flow motifs with recursive alignment. All phenomena derive from:
- Multi-component causal multiplets in the emergent 3+1 geometry (Section 13.1)
- Recursive cluster stability (Section 4)
- Holonomy group structure (Section 3.4.2)
No Lie algebras, spinors, or pre-existing spacetime are postulated. The 3+1 geometry itself emerges from substrate correlations (Section 3) and irreversible coarsening (Section 2.6).
13.1 Fermionic Multiplets from Causal Recursion Paths
The substrate is fundamentally 1D in time (Section 2.6.2), with 3D space emerging from adjacency correlations (Section 3.3.1). Within this emergent 3+1 geometry, a stable fermionic motif requires four independent causal recursion paths to satisfy tension minimization (Axiom 6) and holonomy closure (Section 3.4.2). Define the minimal multiplet:
\[ \Psi_i = \begin{bmatrix} F_i^{(0)} \\ F_i^{(1)} \\ F_i^{(2)} \\ F_i^{(3)} \end{bmatrix} \tag{13.1} \]
where \( F_i^{(a)} \) is the flow along the \( a \)-th recursion path. These are not spacetime components — they are four independent causal channels within the emergent geometry.
13.2 Discrete Dirac Operator
The causal evolution of fermionic bundles is governed by:
\[ D \Psi = i \Gamma^a \Delta_a \Psi \tag{13.2} \]
where \( \Delta_a \Psi = \Psi_{i+a} - \Psi_i \) is the forward difference along path \( a \), and \( \Gamma^a \) are directional operators:
\[ \Gamma^a \Psi_i = \frac{1}{2} \left( \Psi_{i+a} - \Psi_{i-a} \right) \tag{13.3} \]
These satisfy the discrete Clifford algebra:
\[ \{ \Gamma^a, \Gamma^b \} \Psi_i = 2 \eta^{ab} \Psi_i \tag{13.4} \]
The metric signature \( \eta^{00} = +1, \eta^{ij} = -\delta^{ij} \) emerges from the distinction between irreversible temporal correlations (Section 2.6.2) and reversible spatial correlations (Section 3.2).
13.3 Running Gauge Couplings
The effective coupling at scale \( l \) (recursion depth) is determined by cluster coherence. From Section 12.6, \( \alpha(l) \propto C_\theta(l)^2 \), where \( C_\theta(l) \) is the phase coherence of motifs at depth \( l \). The beta function is:
\[ \frac{d\alpha}{d\ln l} = 2\alpha(l) \left[ \Gamma_{\text{form}}(l) - \Gamma_{\text{decay}}(l) \right] \tag{13.5} \]
where formation and decay rates depend on motif density and geometric misalignment (Section 11.6). The ultraviolet cutoff is set by the minimal resolvable scale \( l_{\text{min}} = a_{\text{phys}} \) (Section 5.5).
13.4 Emergent SU(N) Gauge Structure
For multi-component multiplets, phase closure over recursive loops enforces:
\[ \sum_{(i,j) \in C_n} (\theta_i - \theta_j) = 2\pi m, \quad m \in \mathbb{Z} \tag{13.6} \]
This defines a holonomy group. For 2-path motifs, the group is U(1) (Section 12). For 3-path motifs (e.g., quark triplets), non-commuting transformations emerge. Infinitesimal transformations:
\[ \delta \Psi_i = i \varepsilon_a T_a \Psi_i \tag{13.7} \]
yield generators satisfying:
\[ [T_a, T_b] = i f_{abc} T_c \tag{13.8} \]
where structure constants arise from recursive phase misalignment (Section 13.4). Minimizing coherence tension selects stable SU(N) sectors.
13.5 Emergent Vacuum Energy
Vacuum energy arises from latent degrees of freedom in the substrate. For a node \( i \), define:
- \( E_{\text{idle}}(i) \): energy of unresolved flow oscillations (Section 10.3)
- \( \xi(i) \): strain energy from partial alignment (Section 2.3.4)
The physical vacuum energy density is:
\[ \Lambda_{\text{phys}}(i) = \frac{E_c}{L_c^3(i)} \left[ \langle E_{\text{idle}}(i) \rangle + \langle \xi(i) \rangle \right] \tag{13.9} \]
This is unused recursive bandwidth, scaling with substrate topology (Section 7.10).
13.6 Axiomatic Closure
| Phenomenon | Substrate Origin | Axiom |
|---|---|---|
| Fermions | 4-path causal multiplets in emergent 3+1 geometry | A2, A3, A9 |
| Dirac Operator | Discrete differences + Clifford algebra | Section 13.1 |
| Running Coupling | Scale-dependent coherence | Section 12.6 |
| SU(N) Gauge | Holonomy group of multi-path loops | Section 3.4.2 |
| Vacuum Energy | Latent recursive degrees of freedom | A6 |
13.7 Bridge to Section 14 — Unified Gauge Mediation
Section 13 provides:
- Fermionic matter as 4-path causal motifs in emergent 3+1 geometry
- Running couplings from recursive coherence
- SU(N) gauge symmetry from multi-path holonomy
- Vacuum energy from substrate topology
Section 14 unifies these into a single framework for all Standard Model forces and matter, with no reference to fundamental spacetime.
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