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

(v1.1 • 2025-06-12)

0 Symbols and units (recap)

Symbol	Meaning	SI unit
$α$	spatial anchoring energy–density	J m⁻¹
$β$	envelope penalty (drops out for $k r ≪ 1$ )	J m⁻³
$γ$	time-kinetic weight	J s² m⁻³
$Φ$	scalar anchoring potential	J m⁻¹
$ρ$	modal envelope amplitude	dimensionless
$n = ρ^{2}$	modal density	dimensionless
$J_{ϕ} = n \nabla ϕ$	phase current	m⁻¹

All “field” quantities below have the same units of J m⁻².

Scalar potential obeys

\begin{matrix} (1.1) & α \nabla^{2} Φ - β Φ = ρ_{anchor}, k r ≪ 1 \Rightarrow β Φ ≪ ρ_{anchor} . \end{matrix}

Define

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

Phase current

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

with continuity

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

Define

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

3.1 Gauss-like law

\begin{matrix} (3.1) & \nabla \cdot E = ρ_{anchor} / α \end{matrix}

3.2 Faraday-like law

From (2.3) and (2.2):

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

3.3 Ampère-like law

Start with the wave equation for $Φ$ (source-free)
$γ \partial_{t}^{2} Φ = α \nabla^{2} Φ$ ,
take $\nabla \times$ and substitute (1.2); adding sources furnishes

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

3.4 No monopoles

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

4 Wave speed

Combining (3.2) and (3.3) in vacuum
( $ρ_{anchor} = 0, J_{ϕ} = 0$ ) gives

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

Matches the photon speed derived in Appendix A.

5 Worked toy example — translating single-winding mode

Rest frame:
$n (r) = n_{0} e^{- r^{2} / σ^{2}}$ , $ϕ = φ$ (azimuth).
Boost along $\hat{x}$ : replace $x \to x - v t$ .

Then
$J_{ϕ} = n \frac{1}{r} \hat{φ}$ ,
$B = \nabla \times J_{ϕ}$ gives circular lines around
$\hat{x}$ . The pair $(E, B)$ satisfies (3.1)–(3.4) with
$ρ_{anchor} = 0$ , reproducing the familiar field of a moving charge
but entirely from phase flow.

Key points

All four Maxwell equations arise from phase conservation plus
anchoring gradients.
The displacement term is $1 / γ$ , not $ε_{0}$ .
“Charge” is topological winding; “fields” are secondary constructs.

Appendix U - CMB ← Modal-Maxwell → Appendix W

Appendix V — Derivation 22: Modal–Maxwell Equations