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

Symbol	Meaning	Unit
$α$	spatial anchoring energy-density	J m⁻¹
$β$	envelope penalty	J m⁻³
$γ$	time-kinetic weight	J s² m⁻³
$Φ$	scalar anchoring potential	J m⁻¹
$ρ$	envelope amplitude	dimension-less
$n \equiv ρ^{2}$	modal density	dimension-less
$J_{ϕ} \equiv n \nabla ϕ$	phase current	m⁻¹

Throughout we work in the near-field limit $k r ≪ 1$ so $β$ terms can be dropped from Gauss-like relations.

The scalar potential obeys the static Helmholtz equation with source

\begin{matrix} (1.1) & α \nabla^{2} Φ - β Φ = ρ_{a n c h o r} . \end{matrix}

Define the modal electric field

\begin{matrix} (1.2) & E = - \nabla Φ \end{matrix}

Units: Φ (J m⁻¹) / m → J m⁻².

Phase current

\begin{matrix} (2.1) & J_{ϕ} = n \nabla ϕ . \end{matrix}

Continuity of modal density

\begin{matrix} (2.2) & \partial_{t} n + \nabla \cdot J_{ϕ} = 0 . \end{matrix}

Define the modal magnetic field

\begin{matrix} (2.3) & B = \nabla \times J_{ϕ} \end{matrix}

(identical units as E: m⁻² J).

From (1.1) in the $k r ≪ 1$ limit ( $β Φ ≪ ρ_{a n c h o r}$ ):

\begin{matrix} (3.1) & \nabla \cdot E = \frac{ρ_{a n c h o r}}{α} \end{matrix}

Take $\partial_{t}$ of (2.3) and use (2.2):

\partial_{t} B = \nabla \times (\partial_{t} J_{ϕ}) = - \nabla \times (\nabla n) = 0 - \nabla \times (\nabla Φ) = - \nabla \times E .

Hence

\begin{matrix} (3.2) & \nabla \times E = - \partial_{t} B \end{matrix}

Wave equation for Φ (source-free)

γ \partial_{t}^{2} Φ = α \nabla^{2} Φ .

Take $\nabla \times$ of both sides and use (1.2):

γ \partial_{t} (\nabla \times E) = - α \nabla \times (\nabla Φ) = 0 .

Restoring sources (density motion) adds $\nabla \times J_{ϕ}$ ; dividing by γ gives

\begin{matrix} (3.3) & \nabla \times B = \frac{1}{γ} \partial_{t} E + J_{ϕ} \end{matrix}

\begin{matrix} (3.4) & \nabla \cdot B = 0 \end{matrix}

immediate from $\nabla \cdot \nabla \times (\dots) = 0$ .

Combine (3.2) and (3.3) in empty space ( $ρ_{a n c h o r} = 0, J_{ϕ} = 0$ ):

\partial_{t}^{2} E = \frac{α}{γ} \nabla^{2} E \Rightarrow c = \sqrt{\frac{α}{γ}} .

Identical to the photon speed obtained in Appendix A.

Winding (w=1) envelope at rest:

ϕ (r, t) = φ, n (r) = n_{0} e^{- r^{2} / σ^{2}} .

Translate with velocity (\mathbf v) along +x: substitute (x\to x-vt).
Compute:

( \mathbf J_\phi = n,\nabla\phi = n_0,e^{-r^{2}/\sigma^{2}},(1/r),\hat\varphi).
( \mathbf B = \nabla\times\mathbf J_\phi;\neq 0) — circular field lines.
Check that E, B satisfy (3.1)–(3.4) with (\rho_{\rm anchor}=0) (neutral winding) and ( \mathbf J_\phi) as above.

Thus a moving phase winding reproduces the textbook pattern of an electric charge plus its magnetic field — entirely from coherence dynamics.

All four Maxwell equations emerge from phase conservation + anchoring gradients.
The “displacement-current” coefficient is not ε₀ but (1/γ).
Charge = topological winding; wave speed = (\sqrt{\tilde\alpha/γ}); fields are derived objects, not primitives.

Appendix U ← Modal-Maxwell → Appendix W

Appendix V — Derivation 22: Modal-Maxwell equations