TFP Section 12 (v9)

SECTION 12 — EMERGENT GAUGE STRUCTURE AND ELECTROMAGNETIC COUPLING (v9.1)

Gauge Symmetry from Causal Holonomy and Cluster Coherence
By John Gavel

12.0 Overview

Electromagnetism emerges from the geometric structure of causal correlations in stable motifs (Section 4). No U(1) symmetry is postulated — gauge structure arises from:

• Loop holonomy of correlation lags (Section 3.4.2)
• Conservation of net alignment (Section 4.5.1)
• Cluster coherence as a stiffness parameter (Section 10)

The fine-structure constant \( \alpha_{\text{EM}} \) is the dimensionless measure of phase rigidity in the substrate.

12.1 Charge as Topological Winding

From Section 4.5.1, charge is net alignment over a motif \( M \):

\[ q = \sum_{i \in M} F_i \tag{12.1} \]

For a closed loop \( p \) enclosing the motif, define the charge winding number:

\[ n = \frac{1}{2\pi} \sum_{(u \to v) \in p} \tau_{u \to v} \cdot \omega \tag{12.2} \]

where \( \tau_{u \to v} = \arg\max_\tau C_{uv}(\tau) \) (Section 3.1.2) and \( \omega \) is the dominant oscillation frequency (Section 10.1). Since \( \tau_{u \to v} \) is quantized in units of \( \tau_0 \) and \( \omega \tau_0 = 2\pi / d \) (Section 4.3.3), \( n \) is integer-valued. Thus:

\[ q = n q_0 \tag{12.3} \]

where \( q_0 \) is the elementary charge unit (Section 5). Charge quantization is topological, not postulated.

12.2 Discrete Gauge Connection

The phase difference between nodes is encoded in the correlation lag. Define the discrete gauge connection:

\[ A_{ij} = \frac{\theta_j - \theta_i}{L_c} = \frac{\omega (\tau_{j} - \tau_{i})}{L_c} \tag{12.4} \]

where \( \theta_i = \omega \tau_i \) is the emergent phase (Section 10.4). This governs how influence propagates along edge \( i \to j \).

12.3 Loop Holonomy and Field Strength

For a closed loop \( p = (i_1 \to i_2 \to \cdots \to i_1) \), the holonomy is:

\[ H_p = \sum_{(u \to v) \in p} A_{uv} = \frac{\omega}{L_c} \sum_{(u \to v) \in p} (\tau_v - \tau_u) = \frac{\omega}{L_c} \Delta \tau_p \tag{12.5} \]

where \( \Delta \tau_p \) is the total lag around the loop. In the continuum limit (Section 3.2), this becomes:

\[ H_p \to \oint A_\mu dx^\mu = \int F_{\mu\nu} d\sigma^{\mu\nu} \tag{12.6} \]

where \( F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu \) is the electromagnetic field tensor (Section 6.4). Thus, **field strength is loop holonomy per unit area**.

12.4 Conserved Currents

From Section 6.1.2, charge conservation is exact:

\[ \frac{\partial \rho}{\partial t} + \nabla \cdot \mathbf{J} = 0 \tag{12.7} \]

The current arises from motif velocity \( \mathbf{v}_C \) (Section 4.10):

\[ \mathbf{J} = \rho \mathbf{v}_C \tag{12.8} \]

Noether’s theorem is not required — conservation follows from binary flow preservation (Axiom 2).

12.5 Cluster Coherence and Gauge Stiffness

The order parameter for a cluster is the magnitude of the total correlation vector:

\[ Z_{\text{cluster}} = \sum_{i \in M} e^{i \theta_i} \tag{12.9} \]

Normalized coherence:

\[ C_\theta = \frac{|Z_{\text{cluster}}|}{|M|} \tag{12.10} \]

This measures phase alignment — \( C_\theta = 1 \) for perfect synchronization, \( C_\theta = 0 \) for complete decoherence.

12.6 Electromagnetic Coupling Constant

The dimensionless coupling strength is proportional to phase stiffness. From the discrete action (Section 2.2.1):

\[ S = -J \sum_{\langle i,j \rangle} \cos(\theta_i - \theta_j) \tag{12.11} \]

where \( J = \alpha(i,j) (1 - \delta_{\text{geom}}) \) (Section 2.3.4). In the small-phase limit:

\[ S \approx \frac{J}{2} \sum_{\langle i,j \rangle} (\theta_i - \theta_j)^2 = \frac{\rho_s}{2} \int |\nabla \theta|^2 d^3x \tag{12.12} \]

where the **superfluid density** is \( \rho_s = J |M|^2 C_\theta^2 \). The linear response to a gauge field \( A_\mu \) is:

\[ \mathbf{J} = \rho_s (\nabla \theta - \mathbf{A}) \tag{12.13} \]

Integrating out phase fluctuations yields an effective mass for the gauge field:

\[ S_{\text{eff}}[A] = \frac{\chi_A}{2} \int |A|^2 d^3x, \quad \chi_A \propto \rho_s \tag{12.14} \]

The dimensionless coupling is the inverse stiffness:

\[ \alpha_{\text{EM}} = \frac{1}{\rho_s} \propto \frac{1}{C_\theta^2} \tag{12.15} \]

However, physical measurements occur in the **coherent regime** where \( C_\theta \approx 1 \), so:

\[ \alpha_{\text{EM}} = C_{\text{EM}}^* \cdot C_\theta^2, \quad C_{\text{EM}}^* \approx 1 \tag{12.16} \]

Thus, \( \alpha_{\text{EM}} \) measures the **phase coherence of the substrate**.

12.7 Axiomatic Closure

Gauge Concept	Substrate Origin	Axiom
Charge Quantization	Loop winding number	A2, A3
Gauge Connection	Correlation lag difference	A6, A9
Field Strength	Loop holonomy	Section 3.4.2
Coupling Constant	Cluster coherence \( C_\theta \)	Section 10

12.8 Bridge to Section 13 — Non-Abelian Gauge Structure

Section 12 provides:

• U(1) gauge structure from loop holonomy
• Charge quantization from topological winding
• \( \alpha_{\text{EM}} \) from cluster coherence

Section 13 extends this to SU(N) by considering multi-component flow multiplets and higher-order holonomies.