Section 13 — Scale-dependent Gauge Dynamics and Emergent Fermionic Structure (TFP, v11.1)

Section 13 — Scale-dependent Gauge Dynamics and Emergent Fermionic Structure (TFP, v11.1)

By John Gavel

13.0 Overview

Non-Abelian gauge structure and fermionic matter are emergent, not postulated. They arise from multi-component causal motifs (multi-path flow bundles) under:

• K = 12 adjacency lattice
• H = 132 handshake budget
• Recursion/coherence rules (Sections 1–11)

This section presents a plain-symbol account of:

• Minimal four-path causal multiplets
• Discrete Dirac operator and Clifford algebra
• Scale-dependent running of gauge couplings
• Emergent SU(N) structure via loop holonomy
• Vacuum energy from latent substrate bandwidth

13.1 Fermionic Multiplets as Causal Recursion Paths

Minimal fermionic multiplet at node \(i\): \[ \Psi_i = \begin{bmatrix} F_i^{(0)} \\ F_i^{(1)} \\ F_i^{(2)} \\ F_i^{(3)} \end{bmatrix}, \quad F_i^{(a)} \in \{+1, -1\} \]

• Components are causal channels, not preexisting spacetime vectors
• Stability requires closed causal loops → handshake ledger H closes consistently
• Different species (Dirac/Weyl/flavors) correspond to phase-locking among channels and recursion depth \(d\)

13.2 Discrete Dirac Operator and Clifford Structure

Finite difference along path \(a\): \[ \Delta_a \Psi_i = \Psi_{i+a} - \Psi_i \] Directional (discrete gamma) operator: \[ \Gamma^a \Psi_i = \frac{1}{2} \left( \Psi_{i+a} - \Psi_{i-a} \right) \] Discrete Dirac evolution: \[ D \Psi = i \Gamma^a \Delta_a \Psi \] Local Clifford algebra: \[ \{ \Gamma^a, \Gamma^b \} \Psi_i = 2 \eta^{ab} \Psi_i, \quad \eta^{00} = +1, \eta^{ij} = -\delta^{ij} \]

Interpretation:

• Temporal channel(s) → irreversible coarsening → positive quadratic form
• Spatial channels → reversible → negative in emergent metric
• Anti-commuting differences + pigeonhole scheduling (Axiom 8) produce spinor-like transformations
• D Ψ = 0 → stable fermionic modes, discrete chirality encoded in phase ordering
• Half-integer spin arises from topological update constraints

13.3 Running Gauge Couplings from Scale-dependent Coherence

Effective coupling at recursion scale \(l\): \[ \alpha(l) \propto C_\theta(l)^2 \] \[ C_\theta(l) = \text{cluster phase coherence at depth } l \, (\text{Section 12.5}) \] Coherence-driven beta function: \[ \frac{d\alpha}{d\ln l} = 2 \alpha(l) \left[ \Gamma_\text{form}(l) - \Gamma_\text{decay}(l) \right] \]

Where: \[ \Gamma_\text{form}(l) = k_\text{form} \, C_\theta(l) \, \rho_\text{topo}(l), \quad \Gamma_\text{decay}(l) = k_\text{decay} \, \beta(l) \frac{D_n(l)}{H} \] \(\rho_\text{topo}(l) =\) topological reinforcement (persistent homology), \(\beta(l) = 1 - C_\theta(l)^2 =\) misalignment UV cutoff: \(l_\text{min} = a_\text{phys}\) (lattice pitch)

Interpretation: Running couplings reflect scale-dependent phase coherence, not fundamental scale-dependence.

13.4 Emergent SU(N) Gauge Structure from Loop Holonomy

Holonomy constraint (loop closure): \[ \sum_{(i,j) \in C_n} (\theta_i - \theta_j) = 2 \pi m, \quad m \in \mathbb{Z} \] 2-path loops → U(1), 3+ path loops → noncommuting phase adjustments

Infinitesimal multi-path rephasing: \[ \delta \Psi_i = i \epsilon_a T_a \Psi_i \] \([T_a, T_b] = i f_{abc} T_c, \quad f_{abc} = \sum_\text{loops p} w_p \sin(\text{phase mismatch}_p(a,b,c))\)

Intuition: Adjusting one path affects handshake bandwidth on overlapping paths → effective non-Abelian structure constants. Energy minimization of coherence tension selects stable algebras (U(1), SU(2), SU(3), …). 3-path quark triplets naturally → color SU(3).

13.5 Emergent Vacuum Energy (Latent Bandwidth as Λ)

Per-node latent quantities: \[ E_\text{idle}(i) = \text{energy of unresolved oscillations (idle bandwidth)}, \quad \xi(i) = \text{strain energy from partial alignment} \] Local vacuum energy density: \[ \Lambda_\text{phys}(i) = \frac{E_c}{L_c^3(i)} \left[ \langle E_\text{idle}(i) \rangle + \langle \xi(i) \rangle \right] \] E_c, L_c = substrate characteristic energy/length

Properties:

• Positive, extensive in nodes with unresolved bandwidth
• Global Λ = volume average over nodes
• Black-hole mediated emission cycles → local reductions in Λ

Interpretation: Vacuum energy = bookkeeping of unused recursive capacity. Small observed value → only tiny fraction of H = 132 budget unresolved.

13.6 Axiomatic Closure (Plain Summary)

Phenomenon	Substrate Origin	Axiom / Section
Fermions	4-path causal multiplets in emergent 3+1	A2, A3, A9
Dirac operator	Discrete differences + Clifford scheduling	Section 13.2
Running coupling	Scale-dependent phase coherence	Sections 13.3, 12.5
SU(N) gauge	Holonomy of multi-path loops	Sections 3.4.2, 13.4
Vacuum energy	Latent recursive degrees of freedom	A6, Section 13.5

13.7 Bridge to Section 14 — Unified Gauge Mediation

Section 14 will assemble:

• Fermionic matter as phase-locked four-path motifs
• Dynamical beta function expressed in substrate coherence variables (C_theta, beta, D_n)
• Holonomy → non-Abelian structure constants f_abc from geometry/timing
• Substrate account of vacuum energy as residual recursive bandwidth

These feed into gauge mediation, mass generation, and Standard Model structure as emergent from finite routing capacity and icosahedral phase geometry.