Appendix V — Derivation 22: Modal Maxwell Analogues from Coherence Dynamics

Overview

In classical physics, Maxwell’s equations describe how electric and magnetic fields propagate and interact with charge and current.

In modal dynamics, there are no fundamental fields.
Instead, coherence modes interact through anchoring gradients, and phase structure evolution produces emergent analogues of electromagnetic behaviour.

This appendix shows:

How coherence gradient dynamics reproduce field-like behaviour
How moving modes generate phase rotation analogues of magnetism
How Maxwell-like equations emerge from phase conservation and anchoring variation

1. Phase Flow and Coherence Currents

A modal structure $ψ (x, t) = ρ (x, t) e^{i ϕ (x, t)}$ evolves under coherence cost minimisation.
The phase gradient $\nabla ϕ$ governs the mode’s internal structure and interactions.

Define a coherence current:

{\vec{J}}_{ϕ} = ρ^{2} \nabla ϕ

This describes the flow of phase within the mode.
It is analogous to an electric current in Maxwellian terms.

2. Anchoring Gradient as Field

Let the coherence field $B (x)$ describe the anchoring potential generated by modal sources.

Define:

A modal bias field (electric analogue):

{\vec{E}}_{modal} = - \nabla B (x)

A modal rotation field (magnetic analogue):

{\vec{B}}_{modal} = \nabla \times (ρ^{2} \nabla ϕ)

These structures emerge from the geometry of coherence alignment, not field postulates.

Using the coherence continuity equation:

\frac{\partial ρ^{2}}{\partial t} + \nabla \cdot {\vec{J}}_{ϕ} = 0

and structural evolution of $ϕ$ , we obtain analogues of Maxwell's equations:

Gauss-like law:

\nabla \cdot {\vec{E}}_{modal} = ρ_{anchor}

Where $ρ_{anchor}$ is the anchoring density of a coherent source.

Faraday-like law:

\nabla \times {\vec{E}}_{modal} = - \frac{\partial {\vec{B}}_{modal}}{\partial t}

Emerges from phase shear in moving structures.

Ampère-like law:

\nabla \times {\vec{B}}_{modal} = \frac{\partial {\vec{E}}_{modal}}{\partial t} + {\vec{J}}_{ϕ}

With ${\vec{J}}_{ϕ}$ as modal phase current.

No magnetic monopoles:

\nabla \cdot {\vec{B}}_{modal} = 0

Automatically holds due to vector identity of $\nabla \times (\nabla ϕ)$ .

4. Wave Equation from Anchoring Cost

From Appendix A, modal evolution obeys:

γ \frac{\partial^{2} ψ}{\partial t^{2}} = α \nabla^{2} ψ - β ψ

This yields a wave-like equation for phase:

\frac{\partial^{2} ϕ}{\partial t^{2}} = \frac{α}{γ} \nabla^{2} ϕ

With solution speed:

c = \sqrt{\frac{α}{γ}}

This is the modal analogue of electromagnetic wave propagation—not imposed, but emergent from coherence stiffness.

5. Charge Analogy

A mode with non-zero phase winding $w$ (see Appendix M) generates persistent anchoring tension in its coherence field.

This creates:

Radial ${\vec{E}}_{modal}$ -like gradients
Divergent anchoring cost at long range
Observable effects equivalent to electric charge

Thus, charge is not a primitive, but a phase topology class that interacts structurally through anchoring gradients.

Conclusion

Maxwell’s equations are not fundamental.
They are emergent from the geometry of coherence phase flow, driven by anchoring gradients and modal evolution.

Fields and charges are structural behaviours of the coherence medium.

Appendix U | [Index](./Appendix Master) | Appendix W