TFP Section 6 (v9)

SECTION 6 — EMERGENT ELECTROMAGNETISM AND FORCE DYNAMICS (v9.7)

Forces from Flow Gradients and Coherence Asymmetry
By John Gavel

6.0 Overview

In Temporal Flow Physics (TFP), forces emerge from spatial gradients in flow asymmetry and temporal decoherence between localized motifs (Section 4). No force law is postulated — all interactions derive from:

• Binary flow update rules (Section 2)
• Operational distance \( d_{\text{eff}} \) (Section 3)
• Charge and current as coarse-grained flow observables (Section 4)

The continuum limit yields classical electromagnetic fields and Maxwell-like equations, with all physical units anchored via Section 5.

6.1 Dimensionless Sources

All forces originate from two dimensionless observables:

6.1.1 Charge Proxy

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

\[ q_{\text{dim}} = \frac{1}{|M|} \sum_{i \in M} F_i \tag{6.1} \]

Volume-averaged charge density over region \( R \) (Section 2.3.1):

\[ \rho_{\text{dim}}(\mathbf{x}) = \frac{1}{N_R} \sum_{i \in R} q_i \tag{6.2} \]

6.1.2 Current Proxy

Motif velocity \( \mathbf{v}_C \) is centroid motion (Section 4.10). Current density is:

\[ \mathbf{J}_{\text{dim}}(\mathbf{x}) = \rho_{\text{dim}}(\mathbf{x}) \mathbf{v}_C(\mathbf{x}) \tag{6.3} \]

Charge conservation follows from motif stability (Section 4.5):

\[ \frac{\partial \rho_{\text{dim}}}{\partial t} + \nabla \cdot \mathbf{J}_{\text{dim}} = 0 \tag{6.4} \]

6.2 Physical Sources via Calibration

Using the calibration hierarchy (Section 5):

• Length: \( L_c = a_{\text{phys}} \) (Eq 5.7)
• Time: \( T_c = \tau_{\text{phys}} \) (Eq 5.9)
• Mass: \( M_c \) (Eq 5.3)
• Action: \( \hbar_c = M_c L_c^2 / T_c \)

Physical sources are:

\[ \rho_{\text{phys}} = \rho_{\text{dim}} \cdot \frac{q_0}{L_c^3} \quad \text{[C m}^{-3}\text{]} \tag{6.5} \]

\[ \mathbf{J}_{\text{phys}} = \mathbf{J}_{\text{dim}} \cdot \frac{q_0}{L_c^2 T_c} \quad \text{[C m}^{-2}\text{ s}^{-1}\text{]} \tag{6.6} \]

where \( q_0 \) is the elementary charge unit (determined in Section 12).

6.3 Emergent Scalar and Vector Potentials

Static fields arise from instantaneous correlation distance \( r_{\text{dim}} = |\mathbf{x} - \mathbf{x}'| / L_c \) (Section 3.2).

6.3.1 Scalar Potential

The dimensionless potential is the convolution:

\[ \Phi_{\text{dim}}(\mathbf{x}) = \int \frac{\rho_{\text{dim}}(\mathbf{x}')}{r_{\text{dim}}(\mathbf{x}, \mathbf{x}')} dV_{\text{dim}}' \tag{6.7} \]

where \( dV_{\text{dim}}' = d^3x' / L_c^3 \). The physical potential introduces the effective permittivity \( \varepsilon_{0,\text{eff}} \) (Section 5):

\[ \Phi_{\text{phys}} = \Phi_{\text{dim}} \cdot \frac{q_0}{\varepsilon_{0,\text{eff}} L_c} \quad \text{[V]} \tag{6.8} \]

6.3.2 Vector Potential

Similarly, the dimensionless vector potential is:

\[ \mathbf{A}_{\text{dim}}(\mathbf{x}) = \int \frac{\mathbf{J}_{\text{dim}}(\mathbf{x}')}{r_{\text{dim}}(\mathbf{x}, \mathbf{x}')} dV_{\text{dim}}' \tag{6.9} \]

Physical mapping uses the effective permeability \( \mu_{0,\text{eff}} \) (Section 5):

\[ \mathbf{A}_{\text{phys}} = \mathbf{A}_{\text{dim}} \cdot \frac{\mu_{0,\text{eff}} q_0}{T_c} \quad \text{[T·m]} \tag{6.10} \]

6.4 Emergent Electromagnetic Fields

Fields are derived from potentials as in classical EM:

\[ \mathbf{E}_{\text{phys}} = -\nabla_{\text{phys}} \Phi_{\text{phys}} - \frac{\partial \mathbf{A}_{\text{phys}}}{\partial t} \tag{6.11} \]

\[ \mathbf{B}_{\text{phys}} = \nabla_{\text{phys}} \times \mathbf{A}_{\text{phys}} \tag{6.12} \]

where \( \nabla_{\text{phys}} = \nabla / L_c \).

6.5 Maxwell-Like Equations

From the Green’s function identity for the Laplacian (Section 3.2):

\[ \nabla_{\text{phys}}^2 \left( \frac{1}{r_{\text{phys}}} \right) = -4\pi \delta_{\text{phys}}(\mathbf{r}) \tag{6.13} \]

with \( r_{\text{phys}} = L_c r_{\text{dim}} \). Applying \( \nabla_{\text{phys}} \cdot \) to (6.11) and using (6.7)–(6.8):

\[ \nabla_{\text{phys}} \cdot (\varepsilon_{0,\text{eff}} \mathbf{E}_{\text{phys}}) = \rho_{\text{phys}} \tag{6.14} \]

Similarly, taking \( \nabla_{\text{phys}} \times \) of (6.12) and using (6.9)–(6.10) with charge conservation (6.4):

\[ \nabla_{\text{phys}} \times \mathbf{B}_{\text{phys}} - \mu_{0,\text{eff}} \varepsilon_{0,\text{eff}} \frac{\partial \mathbf{E}_{\text{phys}}}{\partial t} = \mu_{0,\text{eff}} \mathbf{J}_{\text{phys}} \tag{6.15} \]

Consistency with the speed of light (Section 5.2.3) requires:

\[ c_{\text{pred}} = \frac{L_c}{T_c} = \frac{1}{\sqrt{\mu_{0,\text{eff}} \varepsilon_{0,\text{eff}}}} \tag{6.16} \]

which holds by construction.

6.6 Interaction Energy and Effective Force

Forces arise from the decoherence cost between motifs. Define motif coherence (Section 3.1.1):

\[ \chi_A = \frac{1}{|A|^2} \sum_{i,j \in A} T_{i \leftrightarrow j} \quad \in [0,1] \tag{6.17} \]

where \( T_{i \leftrightarrow j} = \max_\tau |C_{ij}(\tau)| \) (Section 3.1). Decoherence energy is:

\[ E_{\text{decoh}}(A) = W_A (1 - \chi_A) \tag{6.18} \]

For two motifs \( A, B \), interaction energy uses operational distance \( d_{\text{eff}} = -\ln T_{i \leftrightarrow j} \) (Section 3.2):

\[ E_{\text{int}}(A,B) = \kappa' \sum_{i \in \partial A} \sum_{j \in \partial B} \frac{H_{ij}^2}{d_{\text{eff}}(i,j)^2} \tag{6.19} \]

where \( H_{ij} \) is local holonomy (Section 3.4.2). The effective force on \( A \) is the gradient:

\[ \mathbf{F}_A^{\text{dim}} = -\nabla_{\mathbf{x}_A} E_{\text{int}}(A,B) \tag{6.20} \]

Physical force uses the calibration mass and acceleration (Section 5):

\[ \mathbf{F}_A^{\text{phys}} = \mathbf{F}_A^{\text{dim}} \cdot \frac{M_c L_c}{T_c^2} \quad \text{[N]} \tag{6.21} \]

6.7 Newtonian Limit

From the deterministic update rule (Section 2.1.3), momentum change at node \( i \) is:

\[ \Delta p_i = \sum_{j \sim i} \alpha_{ij} (F_j - F_i) \hat{\mathbf{e}}_{ij} \tag{6.22} \]

Summing over a motif \( C \), internal forces cancel by Newton’s third law, leaving only boundary contributions:

\[ \sum_{i \in C} \Delta p_i = \sum_{i \in \partial C} \cdots \tag{6.23} \]

Thus, in physical units (Section 5):

\[ \frac{\Delta P_C^{\text{phys}}}{\Delta t_{\text{phys}}} = \mathbf{F}_C^{\text{phys}} \quad \Rightarrow \quad M_C^{\text{phys}} \mathbf{a}_C = \mathbf{F}_C^{\text{phys}} \tag{6.24} \]

6.8 Axiomatic Closure

Force Concept	Substrate Origin	Axiom
Charge \( \rho \)	Net alignment \( \bar{F} \)	A2
Current \( \mathbf{J} \)	Motif velocity + charge	A2, A3, A9
Fields \( \mathbf{E}, \mathbf{B} \)	Gradients of potentials	Section 3
Maxwell Equations	Green’s function + calibration	Section 5
Force \( \mathbf{F} \)	Decoherence energy gradient	A6

6.9 Bridge to Section 12 — Gauge Structure

Section 6 provides:

• Classical EM fields from flow gradients
• Maxwell-like equations from correlation geometry
• Forces from decoherence cost

Section 12 derives the microscopic origin of the coupling constant \( \alpha_{\text{EM}} \) and shows how U(1) gauge symmetry emerges from phase coherence in multi-component flow multiplets.