Section 6 — Emergent Electromagnetism and Force Dynamics (TFP, v11.1)

Section 6 — Emergent Electromagnetism and Force Dynamics (TFP, v11.1)

Forces from Handshake Gradients and Impedance
By John Gavel

6.0 Definition of Force in TFP

In Temporal Flow Physics, a force is not primitive. It emerges from the redistribution of a finite update budget when multiple motifs compete for local adjacency updates (A8–A10).

Let a site have:

• K = 12 neighbors (A3)
• H = 132 usable directed handshake addresses per tick (A10)
• \(\eta_\text{sub} = \frac{H}{K^2} = \frac{132}{144} = 11/12 \approx 0.916667\)

Redistribution of the handshake budget cannot propagate faster than one lattice pitch \(a_s\) per update tick \(\tau_s\) (A8–A9). When motifs approach, overlap, or move relative to each other, the spatial gradient of local efficiency produces operational forces.

6.1 Handshake-Efficiency Field

Define the global handshake efficiency: \(\eta_\text{sub} = H / K^2\) (A10)

Local handshake efficiency at site \(x\): \[ \eta_\text{local}(x) = \text{fraction of usable handshake addresses available at } x \]

Efficiency shadow felt by other motifs: \[ \delta \eta(x) = \eta_\text{sub} - \eta_\text{local}(x) \]

\(\delta \eta(x)\) is the primitive field mediating interaction. Gradients of \(\delta \eta(x)\) correspond to emergent forces: \[ \mathbf{F} \propto -\nabla (\delta \eta(x)) \]

6.2 Emergence of Charge

Motifs consist of nodes with binary relational states \(F_i \in \{+1,-1\}\) (A2). Define a dimensionless charge proxy for motif M: \[ q_\text{dim} = \frac{1}{|M|} \sum_{i \in M} F_i \]

Interpretation:

• \(q_\text{dim} = 0\) → symmetric motif, no long-range efficiency shadow
• \(q_\text{dim} \neq 0\) → persistent alignment bias, producing \(\delta \eta(x)\) over multiple hops

Charge arises as persistent asymmetry in local flows. Quantization occurs because a single handshake address is the minimal discrete packet of asymmetry (Section 5.3.4).

6.3 Inverse-Square Scaling of Interaction

Consider a shell of nodes at hop distance \(r\) from a motif (A3, A10): \[ N_\text{shell}(r) = 10 r^2 + 2 \] Handshake flux per far node: \[ I(r) = \frac{H}{N_\text{shell}(r)} \approx \frac{H}{10 r^2} \quad \text{for } r \gg 1 \]

This reproduces \(1/r^2\) scaling purely from combinatorial dilution of finite handshake flux.

6.4 Interaction Energy and Forces

Two motifs \(A\) and \(B\) with dimensionless charges \(q_A\) and \(q_B\) produce overlapping efficiency shadows: \[ E_\text{int}(A,B) \propto \frac{q_A q_B}{r_\text{dim}}, \quad r_\text{dim} = d_{ij} a_s \] Corresponding force: \[ F_{AB} = - \frac{d E_\text{int}}{dr_\text{dim}} \]

In the continuum limit, this reproduces Coulomb’s law scaled by substrate impedance \(\alpha\) (Section 5.3.4).

6.5 Fine-Structure Constant \(\alpha\)

Vacuum impedance arises from residual handshake mismatch: \[ \text{rough\_impedance} \approx H + \pi \delta \] Effective adjacency coupling (geometric factor): \(\zeta \approx \phi \approx 1.618\)

Folding formula: \[ \alpha_\text{inverse} = H + \delta (\pi + \zeta) \approx 137.036 \]

Interpretation: \(\alpha\) measures relational friction of the saturated substrate, affecting alignment currents and lepton mass scaling.

6.6 Emergent Electromagnetic Fields

At macroscopic scales, exact handshake bookkeeping is intractable. Define coarse-grained potentials: \[ \Phi(x) = \text{accumulated efficiency deficit from } q_\text{dim} \] \[ \mathbf{A}(x) = \text{transport of alignment imbalance due to motif motion} \] Derived fields: \[ \mathbf{E}(x,t) = -\nabla \Phi(x,t) - \frac{d \mathbf{A}(x,t)}{dt}, \quad \mathbf{B}(x,t) = \nabla \times \mathbf{A}(x,t) \]

6.7 Strong Interaction (Short-Range)

Hadrons saturate local handshake shells (\(\eta_\text{local} \ll \eta_\text{sub}\)). Small displacements produce steep efficiency gradients. From Section 5 hadron mass law: \[ M_\text{hadron} = \Psi \cdot \left( \frac{N_\text{total}}{N_\text{core}} \right)^{1.25} \cdot \text{Torque} \] Corresponding force: \[ F_\text{strong} = -\frac{d}{dr}[\text{volumetric tension}(M_\text{hadron}, r)] \] Exponent 1.25 encodes simplex/pentagonal impedance → short-range, strong coupling.

6.7.1 Law of Overlap (Nuclear Binding)

Overlap of handshake paths controls binding energy: \[ E_B = \frac{\Psi (L \cdot \Gamma)}{3 \alpha_\text{inverse}} \] Where:

• L = number of shared handshake paths (e.g., L = 6 for alpha–alpha binding)
• \(\Gamma = \Gamma_c = 4 / \pi \approx 1.2732\)
• 3 = valence axis anchor

Interpretation: "Gluons" are operationally shared handshake packets between overlapping shells.

6.8 Weak Interaction (Shell Transition)

Weak force arises from temporal handshake reconfiguration: motifs jump between recursion layers with different internal timing. RMS torque generated only at shell boundaries: \[ \text{Torque} = \frac{\pi}{2} \sqrt{\text{Gear\_Ratio}} (1 + 1/K) \] Transmission ceases beyond boundaries → short range, parity violation, chirality.

Interpretation: Magnetism = lateral redistribution of handshake pressure, Weak force = temporal redistribution across discrete layers.

6.9 Elementary Charge Scale

One elementary charge corresponds to a single effective handshake address normalized by substrate efficiency: \[ q_0 \propto \sqrt{\frac{\eta_\text{sub}}{H}} \] Explains charge quantization, universality, and electron scale.

6.10 Unified Force Table

Force	Substrate Mechanism	Regulator
Electromagnetism	Handshake alignment imbalance	\(\alpha_\text{inverse} \approx 137.036\)
Strong	Volumetric shell saturation, simplex projection	1.25 exponent, Law of Overlap
Weak	Shell-synchronization torque, RMS gear-shift	\((\pi/2) \sqrt{\text{Gear\_Ratio}} (1+1/K)\)
Gravity	Global handshake deficit	\(\eta_\text{sub} = 11/12\)

6.11 Astrophysical Implication

Neutron-star densities: extreme compression drives the Law of Overlap toward \(\Gamma_c = 4 / \pi\), setting a curvature limit. Collapse halts before substrate update starvation, providing an informational alternative to degenerate pressure.

Bridge to Section 7

Section 6 demonstrated that electromagnetic, strong, and weak interactions emerge as gradients and redistributions of the finite handshake budget. Section 7 will derive gravity from the geometric response of the substrate to local handshake exhaustion, using correlation-induced metrics and mass-density gradients.