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\title{\textbf{Twelve Five-Dimensional Projection Models for Maxwell-Compatible Electromagnetism}\\
\large A Teaching-Oriented Mathematical Theory Tree with Speculative Extensions}
\author{JMP0x1B Research Lab}
\date{June 3, 2026}

\begin{document}
\maketitle

\begin{abstract}
This paper develops twelve speculative five-dimensional projection models for electromagnetism.  The goal is deliberately narrow: show, branch by branch, how a five-dimensional mathematical structure can reduce to the standard four-dimensional Maxwell equations, and identify what new electromagnetic terms appear when the fifth-dimensional sector does not decouple.  The presentation is pedagogical: each model states the problem it solves, the approach it uses, the exact Maxwell-recovery condition, the main tradeoff, and one possible new-electromagnetism idea.  The paper does not claim that these models are empirically established.  It treats them as mathematical prototypes whose first test is internal clarity: do they contain ordinary Maxwell theory as a clean limit?
\end{abstract}

\tableofcontents

\section{Orientation: What This Paper Is Trying to Teach}

\subsection{Plain explanation}

The central idea is simple.  Suppose ordinary spacetime is not the whole mathematical arena.  Suppose there is a fifth coordinate, called \(\ch\).  A five-dimensional field may have extra components that ordinary four-dimensional observers do not directly see.  When those extra components vanish, average out, or become constant, the observer should recover the Maxwell equations we already know.  When they do not vanish, the observer sees a modified electromagnetism.

The paper asks:
\begin{quote}
For each possible meaning of \(\ch\), what assumptions are needed so that four-dimensional Maxwell theory appears exactly, and what new terms appear when those assumptions are relaxed?
\end{quote}

\subsection{Mental model}

Think of a five-dimensional curve or field as a three-dimensional object casting a two-dimensional shadow.  The shadow can look distorted or accelerated even when the higher-dimensional object is simple.  In this paper, ordinary electromagnetism is the ``shadow'' of a five-dimensional field theory.  The shadow becomes standard Maxwell theory only when the projection is clean.

\subsection{Status of the construction}

This is a mathematical proposal paper, not a claim of experimental confirmation.  The established reference points are Maxwell electrodynamics \cite{Maxwell1865}, Kaluza--Klein style five-dimensional unification \cite{Kaluza1921,Klein1926,OverduinWesson1997}, gauge symmetry \cite{YangMills1954}, variational conservation laws \cite{Noether1918}, thermodynamic formalism \cite{Caratheodory1909}, renormalization-group scale flow \cite{Wilson1971}, quantization ideas connected to electromagnetic topology \cite{Dirac1931}, and Williams' archived Dynamic Theory report \cite{Williams1980}.  The twelve branches below are new mathematical variants assembled from those ingredients.

\subsection{What counts as success?}

For a branch to be Maxwell-compatible, it must pass five checks:
\begin{enumerate}[leftmargin=2em]
\item Identify the observed four-dimensional field \(F_{\mu\nu}\).
\item Show the homogeneous Maxwell equation follows:
\[
\nab_{[\lambda}F_{\mu\nu]}=0.
\]
\item Show the source equation reduces to
\[
\nab_\mu F^{\mu\nu}=\mu_0J^\nu.
\]
\item State the assumptions that make all fifth-dimensional correction terms vanish.
\item State what new electromagnetic term appears when those assumptions are not imposed.
\end{enumerate}

\section{The Maxwell Equations as the Target}

\subsection{Plain explanation}

Maxwell's equations have two jobs.  First, they say how charge and current source electric and magnetic fields.  Second, they impose geometric constraints on the fields, such as the absence of ordinary magnetic monopoles in the standard theory.

In four-dimensional notation, the electromagnetic potential is
\[
A_\mu=(A_0,A_1,A_2,A_3),
\qquad
\mu,\nu=0,1,2,3.
\]
The electromagnetic field tensor is
\begin{equation}
F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu.
\label{eq:4dF}
\end{equation}
Because \(F\) is built from derivatives of \(A\), it automatically satisfies the homogeneous Maxwell equation
\begin{equation}
\nab_{[\lambda}F_{\mu\nu]}=0.
\label{eq:maxwell_homogeneous}
\end{equation}
The source equation is
\begin{equation}
\nab_\mu F^{\mu\nu}=\mu_0J^\nu.
\label{eq:maxwell_source}
\end{equation}

In flat spacetime, these are equivalent to
\begin{align}
\grad\cdot\Efield&=\frac{\rho_e}{\epsilon_0},
&
\grad\cdot\Bfield&=0,\\
\grad\times\Efield+\frac{\partial\Bfield}{\partial t}&=0,
&
\grad\times\Bfield-\frac{1}{c^2}\frac{\partial\Efield}{\partial t}&=\mu_0\Jfield.
\end{align}

These equations are the target.  Every branch in this paper must recover \eqref{eq:maxwell_homogeneous} and \eqref{eq:maxwell_source} under a clear limit.

\subsection{Why the tensor form is useful}

The tensor form is not just notation.  It makes the projection problem easier.  In four dimensions, electromagnetism is built from a two-index antisymmetric tensor \(F_{\mu\nu}\).  In five dimensions, the analogous object is \(\mcF_{AB}\), where
\[
A,B=0,1,2,3,4.
\]
The question becomes: what part of \(\mcF_{AB}\) is seen as \(F_{\mu\nu}\), and what part becomes a correction?

\section{The Five-Dimensional Master Framework}

\subsection{Coordinates and projection}

Let the five-dimensional coordinate system be
\begin{equation}
X^A=(x^\mu,\ch),
\qquad
x^4=\ch.
\label{eq:5dcoords}
\end{equation}
The projection map to ordinary spacetime is
\begin{equation}
\pi:\Mfive\to\Mfour,
\qquad
\pi(x^\mu,\ch)=x^\mu.
\label{eq:projection}
\end{equation}
A five-dimensional curve is
\[
\Gamma_5(\lambda)=\bigl(x^\mu(\lambda),\ch(\lambda)\bigr),
\]
and the observed four-dimensional curve is
\[
\Gamma_4(\lambda)=\pi(\Gamma_5)=x^\mu(\lambda).
\]

The five-dimensional velocity is
\begin{equation}
U^A=\frac{\dd X^A}{\dd\lambda}
=
\left(\frac{\dd x^\mu}{\dd\lambda},\frac{\dd\ch}{\dd\lambda}\right).
\label{eq:5dvelocity}
\end{equation}
The fifth component
\[
U^4=\frac{\dd\ch}{\dd\lambda}
\]
is the quantity that changes interpretation from branch to branch.

\subsection{Coordinate scaling: a common source of mistakes}

Many possible fifth coordinates are not lengths.  For example, mass, energy, entropy, and density have different units.  To avoid dimensional confusion, we usually define \(\ch\) as a dimensionless coordinate:
\[
\ch=\frac{m}{m_0},
\qquad
\ch=\frac{E}{E_0},
\qquad
\ch=\frac{S_{\rm ent}}{k_B},
\qquad
\ch=\frac{\rho}{\rho_0}.
\]
Then the fifth derivative is not the same as the derivative with respect to the original physical variable.  For example,
\begin{equation}
\ch=\frac{m}{m_0}
\quad\Rightarrow\quad
\Dchi = \frac{\partial}{\partial\ch}=m_0\frac{\partial}{\partial m}.
\label{eq:mass_derivative_scaling}
\end{equation}
Similarly,
\begin{equation}
\ch=\frac{E}{E_0}
\quad\Rightarrow\quad
\Dchi=E_0\frac{\partial}{\partial E},
\label{eq:energy_derivative_scaling}
\end{equation}
\begin{equation}
\ch=\frac{S_{\rm ent}}{k_B}
\quad\Rightarrow\quad
\Dchi=k_B\frac{\partial}{\partial S_{\rm ent}}.
\label{eq:entropy_derivative_scaling}
\end{equation}
This scaling matters because the fifth-gradient term is the source of the new electromagnetic correction.

\subsection{Metric ansatz}

A general split metric is
\begin{equation}
\dd\Sigma^2
=
G_{AB}\dd X^A\dd X^B
=
g_{\mu\nu}\dd x^\mu\dd x^\nu
+2G_{\mu4}\dd x^\mu\dd\ch
+G_{44}\dd\ch^2.
\label{eq:general_metric}
\end{equation}
A useful fiber form is
\begin{equation}
\dd\Sigma^2
=
g_{\mu\nu}\dd x^\mu\dd x^\nu
+
\eps\Phi^2\left(\dd\ch+B_\mu\dd x^\mu\right)^2.
\label{eq:kk_metric}
\end{equation}
Here \(\eps=\pm1\), \(\Phi\) is a scalar modulus, and \(B_\mu\) is a connection field that mixes spacetime motion with fifth-coordinate motion.

A length scale may be inserted when \(\ch\) is dimensionless:
\[
\dd\Sigma^2
=
g_{\mu\nu}\dd x^\mu\dd x^\nu
+
\eps L_\ch^2\Phi^2\left(\dd\ch+B_\mu\dd x^\mu\right)^2.
\]
This is not cosmetic.  It is what keeps the line element dimensionally meaningful.

\subsection{Five-dimensional potential and field strength}

Extend the electromagnetic potential to five components:
\begin{equation}
\mcA_A=(A_\mu,A_4).
\end{equation}
The five-dimensional field tensor is
\begin{equation}
\mcF_{AB}=\partial_A\mcA_B-\partial_B\mcA_A.
\label{eq:5dF}
\end{equation}
It has ten independent components.  They split into
\[
\mcF_{\mu\nu}=F_{\mu\nu}
\]
and
\begin{equation}
V_\mu\equiv \mcF_{\mu4}
=
\partial_\mu A_4-\Dchi A_\mu.
\label{eq:Vdef}
\end{equation}
The field \(V_\mu\) is the basic fifth-sector field.  Standard Maxwell theory is recovered when the effect of \(V_\mu\) on the four-dimensional source equation vanishes.

\subsection{Deriving the split field equations}

Start from the five-dimensional Maxwell-like equation
\begin{equation}
\nab_A^{(5)}\mcF^{AB}=\mu_5\mcJ^B.
\label{eq:5dmaxwell}
\end{equation}
The homogeneous equation is
\begin{equation}
\nab^{(5)}_{[A}\mcF_{BC]}=0.
\label{eq:5dbianchi}
\end{equation}

In an adapted coordinate chart, the \(B=\nu\) component has the form
\begin{equation}
\nab_\mu F^{\mu\nu}-\Dchi V^\nu
=
\mu_5J^\nu+\Delta^\nu_{\rm geom}.
\label{eq:split_source_general}
\end{equation}
Here \(\Delta^\nu_{\rm geom}\) collects connection, metric, and reduction terms that appear in a fully curved or mixed five-dimensional geometry.  In the simplest product case it is zero.  Rearranging,
\begin{equation}
\nab_\mu F^{\mu\nu}
=
\mu_5J^\nu+
\Dchi V^\nu+
\Delta^\nu_{\rm geom}.
\label{eq:source_with_correction}
\end{equation}
The \(B=4\) component gives
\begin{equation}
\nab_\mu V^\mu=\mu_5J^4+\Delta^4_{\rm geom}.
\label{eq:V_source}
\end{equation}

After normalizing constants so \(\mu_5\to\mu_0\) in the observed sector, define the new electromagnetic current
\begin{equation}
J_{\rm new}^\nu
\equiv
\frac{1}{\mu_0}\left(\Dchi V^\nu+\Delta^\nu_{\rm geom}\right).
\label{eq:Jnew}
\end{equation}
Then the observed source equation is
\begin{equation}
\boxed{
\nab_\mu F^{\mu\nu}=\mu_0\left(J^\nu+J_{\rm new}^\nu\right).
}
\label{eq:modified_maxwell}
\end{equation}

\subsection{Universal Maxwell-recovery criterion}

The standard Maxwell source equation is recovered when
\begin{equation}
J_{\rm new}^\nu=0.
\label{eq:Jnew_zero}
\end{equation}
Equivalently,
\begin{equation}
\Dchi V^\nu+\Delta^\nu_{\rm geom}=0.
\label{eq:decoupling_condition}
\end{equation}
This is the central rule of the paper.

The homogeneous Maxwell equation is recovered when the observed field is either the spacetime block of an exact five-dimensional field or the pullback of that field to the observed hypersurface:
\begin{equation}
F=\dd_4A
\qquad\text{or}\qquad
F=\iota^*\mcF.
\label{eq:homogeneous_recovery}
\end{equation}
Then
\begin{equation}
\dd_4F=0
\end{equation}
follows from
\begin{equation}
\dd_5\mcF=0.
\end{equation}

\subsection{The verification recipe used in every branch}

For each branch below, the verification follows the same pattern:
\begin{enumerate}[leftmargin=2em]
\item Choose what \(\ch\) means.
\item Identify \(F_{\mu\nu}\) or the pullback \(\bar F_{\mu\nu}\).
\item Show \(\dd F=0\), usually because the field is still a curl.
\item Compute or name the new current \(J_{\rm new}^\nu\).
\item Impose the branch-specific decoupling condition \(J_{\rm new}^\nu=0\).
\item The remaining equation is Maxwell:
\[
\nab_\mu F^{\mu\nu}=\mu_0J^\nu.
\]
\end{enumerate}

\section{A Guiding Example: Projection Turns Free 5D Motion into 4D Electromagnetic Force}

\subsection{Problem being solved}

Before listing twelve models, it helps to see what projection means in a concrete calculation.  The example below shows how a free five-dimensional geodesic can look like forced electromagnetic motion in four dimensions.

\subsection{Approach}

Use the metric
\begin{equation}
\dd\Sigma^2
=
\eta_{\mu\nu}\dd x^\mu\dd x^\nu
+
\left(\dd\ch+\kappa A_\mu\dd x^\mu\right)^2.
\label{eq:projection_metric_example}
\end{equation}
A free path has Lagrangian
\begin{equation}
L=\frac12\eta_{\mu\nu}\dot x^\mu\dot x^\nu+\frac12\left(\dot\ch+\kappa A_\mu\dot x^\mu\right)^2.
\label{eq:example_lagrangian}
\end{equation}
Define
\begin{equation}
Q=\dot\ch+\kappa A_\mu\dot x^\mu.
\label{eq:Qdef}
\end{equation}
Because \(L\) does not explicitly depend on \(\ch\), the conjugate momentum is conserved:
\begin{equation}
p_\ch=\frac{\partial L}{\partial\dot\ch}=Q=\text{constant}.
\label{eq:pchi_constant}
\end{equation}

\subsection{Derivation}

The Euler--Lagrange equation for \(x^\mu\) gives
\begin{equation}
\frac{\dd}{\dd\lambda}\left(\frac{\partial L}{\partial\dot x^\mu}\right)-\frac{\partial L}{\partial x^\mu}=0.
\end{equation}
One computes
\begin{equation}
\frac{\partial L}{\partial\dot x^\mu}
=
\eta_{\mu\nu}\dot x^\nu+
\kappa Q A_\mu,
\end{equation}
and
\begin{equation}
\frac{\partial L}{\partial x^\mu}
=
\kappa Q\,\partial_\mu A_\nu\dot x^\nu.
\end{equation}
Therefore
\begin{equation}
\eta_{\mu\nu}\ddot x^\nu
=
\kappa Q(\partial_\mu A_\nu-\partial_\nu A_\mu)\dot x^\nu.
\end{equation}
Using
\begin{equation}
F_{\mu\nu}=\partial_\mu A_\nu-\partial_\nu A_\mu,
\end{equation}
we get
\begin{equation}
\ddot x^\mu=\kappa QF^\mu{}_{\nu}\dot x^\nu.
\label{eq:projected_lorentz}
\end{equation}
If
\begin{equation}
\kappa Q=\frac{q}{m},
\end{equation}
then \eqref{eq:projected_lorentz} is the relativistic Lorentz-force equation.

\subsection{Lesson}

The five-dimensional path is free, but its four-dimensional projection is accelerated.  This is the core projection mechanism:
\begin{equation}
\boxed{
\text{free geometry in 5D can appear as force in 4D.}
}
\end{equation}
The twelve branches below reuse this idea in different ways.

\section{The Twelve Theories}

\subsection{Theory 1: Static Projection Theory \texorpdfstring{\(T_0\)}{T0}}

\paragraph{Problem solved.}
This branch asks for the simplest possible five-dimensional extension that still contains Maxwell theory.

\paragraph{Approach.}
The fifth coordinate exists, but physical fields do not change along it:
\begin{equation}
\frac{\dd\ch}{\dd\lambda}=0,
\qquad
\Dchi A_\mu=0.
\end{equation}
The five-dimensional space is effectively a stack of four-dimensional slices.

\paragraph{Why Maxwell is produced.}
Since \(\Dchi A_\mu=0\),
\begin{equation}
V_\mu=\partial_\mu A_4.
\end{equation}
If \(A_4\) is constant, or if its scalar sector decouples, then
\begin{equation}
\Dchi V^\nu=0,
\qquad
\Delta^\nu_{\rm geom}=0.
\end{equation}
Thus
\begin{equation}
J_{\rm new}^\nu=0,
\end{equation}
and \eqref{eq:modified_maxwell} becomes
\begin{equation}
\nab_\mu F^{\mu\nu}=\mu_0J^\nu.
\end{equation}
The homogeneous equation follows from \(F=\dd A\).

\paragraph{New electromagnetism idea.}
If \(A_4\) is not constant, the model gives Maxwell theory plus a scalar field:
\begin{equation}
\Boxfour A_4=\mu_0J^4.
\end{equation}
This suggests scalar-modulus electromagnetism: standard \(F_{\mu\nu}\) plus a hidden scalar sector.

\paragraph{Tradeoff.}
This branch is mathematically clean but conservative.  Since \(\ch\) is frozen, it explains little unless the scalar sector \(A_4\) has observable consequences.

\subsection{Theory 2: Mass-Coordinate Dynamic Theory \texorpdfstring{\(T_m\)}{Tm}}

\paragraph{Problem solved.}
This branch asks whether mass variation can be represented as motion along a fifth coordinate.

\paragraph{Approach.}
Let
\begin{equation}
\ch=\frac{m}{m_0}.
\end{equation}
Then
\begin{equation}
\Dchi=m_0\frac{\partial}{\partial m},
\qquad
U^4=\frac{1}{m_0}\frac{\dd m}{\dd\lambda}.
\end{equation}
Motion through the fifth coordinate means mass-flow or mass-conversion.

\paragraph{Why Maxwell is produced.}
The source equation is
\begin{equation}
\nab_\mu F^{\mu\nu}
=
\mu_0J^\nu+m_0\partial_mV^\nu+\Delta^\nu_{\rm geom}.
\label{eq:mass_branch_source}
\end{equation}
On a fixed-mass sheet, impose
\begin{equation}
\partial_mV^\nu=0,
\qquad
\Delta^\nu_{\rm geom}=0,
\qquad
J^4=0.
\end{equation}
Then \eqref{eq:mass_branch_source} reduces to Maxwell's source equation.

\paragraph{New electromagnetism idea.}
The correction current is
\begin{equation}
J_m^\nu=\frac{m_0}{\mu_0}\partial_mV^\nu.
\end{equation}
This suggests mass-dispersive electromagnetism: electromagnetic response could depend on mass-layer, becoming relevant only in regimes where mass or binding energy changes strongly.

\paragraph{Tradeoff.}
Mass is usually a property, not an independent coordinate.  A complete theory must explain when \(m\) is an independent coordinate and how the units of \(A_4\), \(V_\mu\), and \(J_m^\nu\) are fixed.

\subsection{Theory 3: Mass-Density Sheet Theory \texorpdfstring{\(T_\rho\)}{Trho}}

\paragraph{Problem solved.}
This branch is designed for continua: fluids, plasmas, shocks, phase transitions, and media whose density changes across spacetime.

\paragraph{Approach.}
Let
\begin{equation}
\ch=\frac{\rho}{\rho_0}.
\end{equation}
Instead of a particle moving through \(\ch\), the observed world is a density sheet embedded in five dimensions:
\begin{equation}
\iota:\Mfour\to\Mfive,
\qquad
\iota(x^\mu)=\bigl(x^\mu,f(x)\bigr),
\qquad
f(x)=\frac{\rho(x)}{\rho_0}.
\end{equation}

\paragraph{Observed field.}
The observed potential is the pullback
\begin{equation}
\bar A_\mu=(\iota^*\mcA)_\mu=A_\mu+A_4\partial_\mu f.
\end{equation}
The observed field is
\begin{align}
\bar F_{\mu\nu}
&=(\iota^*\mcF)_{\mu\nu}\\
&=F_{\mu\nu}+V_\mu\partial_\nu f-V_\nu\partial_\mu f.
\label{eq:density_pullback}
\end{align}

\paragraph{Why Maxwell is produced.}
Because pullbacks commute with exterior derivatives,
\begin{equation}
\dd_4\bar F=\iota^*(\dd_5\mcF)=0.
\end{equation}
The source equation reduces to Maxwell when the normal/fifth flux vanishes:
\begin{equation}
\Dchi V^\nu+\Delta^\nu_{\rm geom}=0,
\end{equation}
and when either density gradients are negligible or the fifth field is negligible:
\begin{equation}
\partial_\mu f\approx0
\qquad\text{or}\qquad
V_\mu\approx0.
\end{equation}
Then \(\bar F_{\mu\nu}\approx F_{\mu\nu}\), and the Maxwell equations are recovered on the sheet.

\paragraph{New electromagnetism idea.}
Density gradients create a new field contribution
\begin{equation}
\Delta F_{\mu\nu}=V_\mu\partial_\nu\left(\frac{\rho}{\rho_0}\right)-V_\nu\partial_\mu\left(\frac{\rho}{\rho_0}\right).
\end{equation}
This suggests geometry-induced polarization or magnetization from density gradients.

\paragraph{Tradeoff.}
The branch is useful for media, but the observed field is no longer simply the spacetime block \(F_{\mu\nu}\).  It is the pullback \(\bar F_{\mu\nu}\), so care is needed when comparing to ordinary material electrodynamics.

\subsection{Theory 4: Energy-Coordinate Theory \texorpdfstring{\(T_E\)}{TE}}

\paragraph{Problem solved.}
This branch asks whether electromagnetic fields can depend on an energy coordinate, not only on spacetime.

\paragraph{Approach.}
Let
\begin{equation}
\ch=\frac{E}{E_0},
\qquad
\Dchi=E_0\frac{\partial}{\partial E}.
\end{equation}
The fields are
\begin{equation}
A_\mu=A_\mu(x,E),
\qquad
A_4=A_4(x,E).
\end{equation}

\paragraph{Why Maxwell is produced.}
The source equation becomes
\begin{equation}
\nab_\mu F^{\mu\nu}
=
\mu_0J^\nu+E_0\partial_EV^\nu+\Delta^\nu_{\rm geom}.
\end{equation}
At a fixed energy layer, impose
\begin{equation}
\partial_EV^\nu=0,
\qquad
\Delta^\nu_{\rm geom}=0.
\end{equation}
Then ordinary Maxwell theory follows.

\paragraph{New electromagnetism idea.}
The energy correction current is
\begin{equation}
J_E^\nu=\frac{E_0}{\mu_0}\partial_EV^\nu.
\end{equation}
This suggests energy-layer electromagnetism: fields may have energy-dependent side structure, while standard Maxwell theory is the energy-independent limit.

\paragraph{Tradeoff.}
Energy is frame-dependent unless carefully defined.  A serious version of this branch must specify whether \(E\) means rest energy, local field energy, Hamiltonian energy, or another invariant quantity.

\subsection{Theory 5: Compact Fifth-Dimension Theory \texorpdfstring{\(T_5\)}{T5}}

\paragraph{Problem solved.}
This branch asks what happens if the fifth coordinate is real but compact, like a circle.

\paragraph{Approach.}
Let
\begin{equation}
\ch\sim\ch+2\pi R.
\end{equation}
Fields have Fourier expansions:
\begin{equation}
A_\mu(x,\ch)=\sum_{n=-\infty}^{\infty}A_\mu^{(n)}(x)e^{in\ch/R}.
\end{equation}
The zero mode is
\begin{equation}
A_\mu^{(0)}(x)=\frac{1}{2\pi R}\int_0^{2\pi R}A_\mu(x,\ch)\,\dd\ch.
\end{equation}

\paragraph{Why Maxwell is produced.}
For the zero mode,
\begin{equation}
\Dchi A_\mu^{(0)}=0.
\end{equation}
If the scalar zero mode \(A_4^{(0)}\) decouples, then
\begin{equation}
F_{\mu\nu}^{(0)}=\partial_\mu A_\nu^{(0)}-\partial_\nu A_\mu^{(0)}
\end{equation}
satisfies
\begin{equation}
\nab_\mu F^{(0)\mu\nu}=\mu_0J^{(0)\nu},
\qquad
\nab_{[\lambda}F_{\mu\nu]}^{(0)}=0.
\end{equation}

\paragraph{New electromagnetism idea.}
For \(n\neq0\), a five-dimensional wave equation gives
\begin{equation}
(\Boxfour+\partial_\ch^2)A_\mu=\mu_5J_\mu
\quad\Rightarrow\quad
\left(\Boxfour-\frac{n^2}{R^2}\right)A_\mu^{(n)}=\mu_5J_\mu^{(n)}.
\end{equation}
The higher modes behave like massive vector fields with characteristic mass scale
\begin{equation}
m_n\sim\frac{\hbar |n|}{cR}.
\end{equation}
Thus ordinary Maxwell theory is the zero mode, while new electromagnetism is a tower of hidden massive electromagnetic modes.

\paragraph{Tradeoff.}
If \(R\) is large, the massive modes should be observable.  If they are not observed, \(R\) must be small, the modes must be weakly coupled, or the branch needs another suppression mechanism.

\subsection{Theory 6: Phase-Action Theory \texorpdfstring{\(T_\theta\)}{Ttheta}}

\paragraph{Problem solved.}
This branch asks whether the fifth coordinate can be quantum-like phase rather than a spatial or material coordinate.

\paragraph{Approach.}
Let
\begin{equation}
\ch=\theta=\frac{S_{\rm act}}{\hbar},
\end{equation}
where \(S_{\rm act}\) is action, not entropy.  Phase is periodic:
\begin{equation}
\theta\sim\theta+2\pi.
\end{equation}
Expand
\begin{equation}
A_\mu(x,\theta)=\sum_{n=-\infty}^{\infty}A_\mu^{(n)}(x)e^{in\theta}.
\end{equation}

\paragraph{Why Maxwell is produced.}
The classical observed field is the phase average:
\begin{equation}
\langle A_\mu\rangle_\theta
=
\frac{1}{2\pi}\int_0^{2\pi}A_\mu(x,\theta)\,\dd\theta
=A_\mu^{(0)}(x).
\end{equation}
The zero mode has no \(\theta\)-gradient correction, so
\begin{equation}
J_{\rm new}^{(0)\nu}=0
\end{equation}
when \(A_4^{(0)}\) decouples.  Therefore \(F_{\mu\nu}^{(0)}\) satisfies Maxwell equations.

\paragraph{New electromagnetism idea.}
The nonzero terms
\begin{equation}
F_{\mu\nu}(x,\theta)=F_{\mu\nu}^{(0)}(x)+\sum_{n\neq0}F_{\mu\nu}^{(n)}(x)e^{in\theta}
\end{equation}
represent phase-sideband electromagnetism: hidden electromagnetic structure around the classical phase average.

\paragraph{Tradeoff.}
This branch does not automatically derive quantum mechanics.  It only gives a clean mathematical place where phase-dependent electromagnetic modes could live.

\subsection{Theory 7: Entropic Fifth-Dimension Theory \texorpdfstring{\(T_S\)}{TS}}

\paragraph{Problem solved.}
This branch asks whether irreversible thermodynamic behavior can enter electromagnetism through a fifth coordinate.

\paragraph{Approach.}
Let
\begin{equation}
\ch=\frac{S_{\rm ent}}{k_B},
\qquad
\Dchi=k_B\frac{\partial}{\partial S_{\rm ent}}.
\end{equation}
The path condition is monotonic:
\begin{equation}
\frac{\dd\ch}{\dd\lambda}\ge0.
\end{equation}
This is different from the compact phase branch.  Entropy has a direction; phase has a cycle.

\paragraph{Why Maxwell is produced.}
The source equation becomes
\begin{equation}
\nab_\mu F^{\mu\nu}
=
\mu_0J^\nu+k_B\partial_{S_{\rm ent}}V^\nu+\Delta^\nu_{\rm geom}.
\end{equation}
On an isentropic or reversible slice,
\begin{equation}
\partial_{S_{\rm ent}}V^\nu=0,
\qquad
\Delta^\nu_{\rm geom}=0.
\end{equation}
Then Maxwell's equations are recovered.

\paragraph{New electromagnetism idea.}
The entropy correction current is
\begin{equation}
J_S^\nu=\frac{k_B}{\mu_0}\partial_{S_{\rm ent}}V^\nu.
\end{equation}
This suggests dissipative electromagnetism: irreversible entropy production could appear as an effective current term.

\paragraph{Tradeoff.}
Entropy is not normally a coordinate of microscopic spacetime.  A complete branch must specify whether \(S_{\rm ent}\) is local entropy density, total system entropy, coarse-grained entropy, or another thermodynamic variable.

\subsection{Theory 8: Charge-Fiber Theory \texorpdfstring{\(T_q\)}{Tq}}

\paragraph{Problem solved.}
This branch asks whether electric charge can be interpreted as conserved fifth-dimensional momentum.

\paragraph{Approach.}
Use the fiber metric
\begin{equation}
\dd\Sigma^2
=
g_{\mu\nu}\dd x^\mu\dd x^\nu
+
\Phi^2\left(\dd\ch+kA_\mu\dd x^\mu\right)^2.
\label{eq:charge_fiber_metric}
\end{equation}
The cross-term contains the electromagnetic potential:
\begin{equation}
G_{\mu4}\propto A_\mu.
\end{equation}

\paragraph{Charge from fifth momentum.}
The geodesic Lagrangian is
\begin{equation}
L=\frac12g_{\mu\nu}\dot x^\mu\dot x^\nu+\frac12\Phi^2(\dot\ch+kA_\mu\dot x^\mu)^2.
\end{equation}
The conjugate momentum is
\begin{equation}
p_\ch=\frac{\partial L}{\partial\dot\ch}
=
\Phi^2(\dot\ch+kA_\mu\dot x^\mu).
\end{equation}
If the metric is independent of \(\ch\), then
\begin{equation}
\frac{\dd p_\ch}{\dd\lambda}=0.
\end{equation}
The identification is
\begin{equation}
q\propto p_\ch.
\end{equation}

\paragraph{Why Maxwell is produced.}
When \(\Phi\) is constant and the cylinder condition holds,
\begin{equation}
\Dchi G_{AB}=0,
\end{equation}
the five-dimensional curvature action reduces schematically as
\begin{equation}
\int\dd^5X\sqrt{|G|}\,R_5
\longrightarrow
\int\dd^4x\sqrt{|g|}\left(R_4-\frac{C}{4}F_{\mu\nu}F^{\mu\nu}\right),
\end{equation}
where \(C\) is absorbed into normalization.  Variation with respect to \(A_\mu\) yields
\begin{equation}
\nab_\mu F^{\mu\nu}=\mu_0J^\nu.
\end{equation}

\paragraph{New electromagnetism idea.}
If \(\Phi\) varies, the source equation becomes
\begin{equation}
\nab_\mu(\Phi^2F^{\mu\nu})=\mu_0J^\nu.
\end{equation}
This acts like electromagnetism in a variable medium whose constitutive coefficient is the size of the fifth fiber.

\paragraph{Tradeoff.}
This is one of the most elegant branches, but it inherits the usual Kaluza-style challenge: why is the fifth fiber hidden, and how are charge quantization and observed coupling strengths fixed?

\subsection{Theory 9: Scale-Flow Theory \texorpdfstring{\(T_R\)}{TR}}

\paragraph{Problem solved.}
This branch asks whether the fifth coordinate can represent scale or resolution rather than an extra spacetime direction.

\paragraph{Approach.}
Let
\begin{equation}
\ch=\ln\frac{L}{L_0}
\qquad\text{or}\qquad
\ch=\ln\frac{E}{E_0}.
\end{equation}
Fields and couplings may depend on scale:
\begin{equation}
A_\mu=A_\mu(x,\ch),
\qquad
e=e(\ch).
\end{equation}
A scale-flow equation has the form
\begin{equation}
\Dchi e=\beta(e).
\end{equation}

\paragraph{Why Maxwell is produced.}
At a fixed observational scale \(\ch=\ch_*\), define
\begin{equation}
A_\mu^{\rm obs}(x)=A_\mu(x,\ch_*),
\end{equation}
\begin{equation}
F_{\mu\nu}^{\rm obs}=\partial_\mu A_\nu^{\rm obs}-\partial_\nu A_\mu^{\rm obs}.
\end{equation}
If
\begin{equation}
\Dchi V^\nu\big|_{\ch_*}+\Delta^\nu_{\rm geom}\big|_{\ch_*}=0,
\end{equation}
then
\begin{equation}
\nab_\mu F^{{\rm obs}\,\mu\nu}=\mu_0(\ch_*)J^\nu.
\end{equation}
This is Maxwell theory with constants evaluated at the observational scale.

\paragraph{New electromagnetism idea.}
The model promotes coupling flow to fifth-coordinate motion:
\begin{equation}
\nab_\mu F^{\mu\nu}=\mu_0(\ch)J^\nu+\Dchi V^\nu.
\end{equation}
This suggests scale-layer electromagnetism: Maxwell theory is local in scale, while new effects come from gradients across scale.

\paragraph{Tradeoff.}
Scale is not a physical location in the usual sense.  This branch is best viewed as a geometric language for effective theories, not necessarily as a literal extra dimension.

\subsection{Theory 10: Full Metric Dynamic Theory \texorpdfstring{\(T_g\)}{Tg}}

\paragraph{Problem solved.}
This branch asks whether electromagnetism can emerge from the five-dimensional metric itself.

\paragraph{Approach.}
Take the metric as fundamental:
\begin{equation}
G_{AB}=
\begin{pmatrix}
g_{\mu\nu}+\eps\Phi^2B_\mu B_\nu & \eps\Phi^2B_\mu\\
\eps\Phi^2B_\nu & \eps\Phi^2
\end{pmatrix}.
\end{equation}
Identify
\begin{equation}
B_\mu\sim A_\mu.
\end{equation}

\paragraph{Why Maxwell is produced.}
Use the geometric action
\begin{equation}
I_5=\frac{1}{2\kappa_5}\int\dd^5X\sqrt{|G|}\,R_5.
\end{equation}
Under
\begin{equation}
\Dchi G_{AB}=0,
\qquad
\Phi=\text{constant},
\end{equation}
dimensional reduction gives
\begin{equation}
I_4=\int\dd^4x\sqrt{|g|}\left[\frac{1}{2\kappa_4}R_4-\frac14C F_{\mu\nu}F^{\mu\nu}+\cdots\right].
\end{equation}
Variation with respect to \(A_\mu\) gives Maxwell's source equation after normalization.

\paragraph{New electromagnetism idea.}
If \(\Phi\), \(B_\mu\), or \(G_{AB}\) varies along \(\ch\), then
\begin{equation}
\nab_\mu\left(C(x,\ch)F^{\mu\nu}\right)=\mu_0J^\nu+\text{fifth-gradient terms}.
\end{equation}
This suggests geometry-induced nonlinear electromagnetism or effective vacuum response.

\paragraph{Tradeoff.}
This branch is powerful but mathematically demanding.  It must control the extra scalar \(\Phi\), metric signature, stability, and unwanted fifth-gradient terms.

\subsection{Theory 11: Constraint-Hypersurface Theory \texorpdfstring{\(T_C\)}{TC}}

\paragraph{Problem solved.}
This branch asks whether four-dimensional electromagnetism can arise on a constrained hypersurface inside five dimensions.

\paragraph{Approach.}
The observed universe is defined by
\begin{equation}
C(X^A)=0.
\end{equation}
For a graph embedding,
\begin{equation}
\ch=f(x^\mu).
\end{equation}
The induced metric is
\begin{equation}
h_{\mu\nu}=G_{AB}\partial_\mu X^A\partial_\nu X^B,
\end{equation}
so
\begin{equation}
h_{\mu\nu}
=
G_{\mu\nu}+G_{\mu4}\partial_\nu f+G_{\nu4}\partial_\mu f+G_{44}\partial_\mu f\partial_\nu f.
\end{equation}

\paragraph{Why Maxwell is produced.}
The observed field is the pullback
\begin{equation}
\bar F=\iota^*\mcF.
\end{equation}
The homogeneous equation follows immediately:
\begin{equation}
\dd_4\bar F=\iota^*(\dd_5\mcF)=0.
\end{equation}
The source equation reduces to Maxwell on the hypersurface if normal flux vanishes:
\begin{equation}
n_A\mcF^{A\nu}=0
\end{equation}
or, in adapted coordinates,
\begin{equation}
\Dchi V^\nu+\Delta^\nu_{\rm geom}=0.
\end{equation}
Then
\begin{equation}
\nab_\mu^{(h)}\bar F^{\mu\nu}=\mu_0\bar J^\nu.
\end{equation}

\paragraph{New electromagnetism idea.}
If normal flux is nonzero, the hypersurface observer sees an induced current:
\begin{equation}
J_{\rm normal}^\nu=\frac{1}{\mu_0}\left(\Dchi V^\nu+\Delta^\nu_{\rm geom}\right).
\end{equation}
This suggests electromagnetic sources induced by bending, motion, or curvature of the physical hypersurface.

\paragraph{Tradeoff.}
This branch depends heavily on the embedding function \(f(x)\).  Without an equation determining \(f\), it is descriptive rather than predictive.

\subsection{Theory 12: Fifth-Force Conversion Theory \texorpdfstring{\(T_F\)}{TF}}

\paragraph{Problem solved.}
This branch asks whether apparent four-dimensional charge or current creation can be reinterpreted as conserved flux through a fifth direction.

\paragraph{Approach.}
Use the equations
\begin{equation}
\nab_\mu V^\mu=\mu_0J^4+\Delta^4_{\rm geom},
\end{equation}
and
\begin{equation}
\nab_\mu F^{\mu\nu}=\mu_0J^\nu+\Dchi V^\nu+\Delta^\nu_{\rm geom}.
\end{equation}
The fifth current \(J^4\) measures conversion flux.

\paragraph{Why Maxwell is produced.}
If conversion is absent,
\begin{equation}
J^4=0,
\qquad
\Dchi V^\nu=0,
\qquad
\Delta^\nu_{\rm geom}=0,
\end{equation}
then
\begin{equation}
\nab_\mu F^{\mu\nu}=\mu_0J^\nu.
\end{equation}

\paragraph{New electromagnetism idea.}
Define
\begin{equation}
J_{\rm conv}^\nu=\frac{1}{\mu_0}\left(\Dchi V^\nu+\Delta^\nu_{\rm geom}\right).
\end{equation}
Then
\begin{equation}
\nab_\mu F^{\mu\nu}=\mu_0(J^\nu+J_{\rm conv}^\nu).
\end{equation}
The five-dimensional conservation law
\begin{equation}
\nab_A\mcJ^A=0
\end{equation}
becomes, schematically,
\begin{equation}
\nab_\mu J^\mu=-\Dchi J^4.
\end{equation}
So four-dimensional non-conservation can be interpreted as five-dimensional conservation.

\paragraph{Tradeoff.}
This is the most explicitly non-Maxwellian branch.  Its Maxwell limit is easy, but its new predictions would require careful constraints because ordinary charge conservation is very well tested.

\section{Unified Comparison of the Twelve Branches}

\begin{longtable}{>{\raggedright\arraybackslash}p{0.10\textwidth} >{\raggedright\arraybackslash}p{0.17\textwidth} >{\raggedright\arraybackslash}p{0.30\textwidth} >{\raggedright\arraybackslash}p{0.31\textwidth}}
\toprule
\textbf{Branch} & \textbf{Meaning of \(\ch\)} & \textbf{How Maxwell is recovered} & \textbf{New electromagnetic idea}\\
\midrule
\endhead

\(T_0\) & Hidden projection coordinate & Fixed slice; \(A_4\) decouples; \(J_{\rm new}^\nu=0\) & Scalar-modulus electromagnetism \\

\(T_m\) & Mass & Fixed mass sheet; \(\partial_mV^\nu=0\) & Mass-dispersive correction current \\

\(T_\rho\) & Density & Pullback field; density gradients or \(V_\mu\) vanish & Density-gradient polarization/magnetization \\

\(T_E\) & Energy & Fixed energy layer; \(\partial_EV^\nu=0\) & Energy-layer electromagnetic response \\

\(T_5\) & Compact coordinate & Zero Fourier mode \(n=0\) & Massive vector-mode tower \\

\(T_\theta\) & Phase/action & Phase average / zero winding mode & Phase-sideband electromagnetism \\

\(T_S\) & Entropy & Isentropic or reversible slice & Entropic/dissipative current \\

\(T_q\) & Charge fiber & Constant \(\Phi\); cylinder condition & Variable-fiber medium equation \\

\(T_R\) & Scale & Fixed observation scale \(\ch_*\) & Scale-flow current and running coupling \\

\(T_g\) & Full metric fiber & Kaluza-like reduction; constant \(\Phi\) & Geometry-induced nonlinear EM \\

\(T_C\) & Constraint variable & Pullback with vanishing normal flux & Hypersurface-induced current \\

\(T_F\) & Conversion flux & No conversion; \(J^4=0\); \(\Dchi V^\nu=0\) & Fifth-conversion current \\

\bottomrule
\end{longtable}

\section{A Unified New Electromagnetism}

\subsection{The shared equation}

All twelve branches can be summarized by one equation:
\begin{equation}
\boxed{
\nab_\mu F^{\mu\nu}=\mu_0\left(J^\nu+J_{\rm new}^\nu\right),
\qquad
J_{\rm new}^\nu=\frac{1}{\mu_0}\left(\Dchi V^\nu+\Delta^\nu_{\rm geom}\right).
}
\label{eq:unified_new_em}
\end{equation}

The homogeneous equation remains
\begin{equation}
\boxed{
\nab_{[\lambda}F_{\mu\nu]}=0
}
\end{equation}
whenever the observed field is a curl or a pullback of a closed five-dimensional field.

\subsection{Plain interpretation}

Maxwell theory is the clean limit:
\begin{equation}
J_{\rm new}^\nu=0.
\end{equation}
New electromagnetism begins when
\begin{equation}
J_{\rm new}^\nu\neq0.
\end{equation}
The twelve models differ only in what creates \(J_{\rm new}^\nu\): mass gradients, density gradients, energy gradients, compact modes, phase modes, entropy production, fiber geometry, scale flow, hypersurface curvature, or conversion flux.

\subsection{Six families of new electromagnetic effects}

\begin{enumerate}[leftmargin=2em]
\item \textbf{Scalar-sector effects:}
\[
A_4\neq0.
\]
Ordinary electromagnetism is accompanied by a scalar fifth-potential.

\item \textbf{Material-coordinate effects:}
\[
\ch=m,\rho,E.
\]
Fields acquire mass-, density-, or energy-layer dependence.

\item \textbf{Compact-mode effects:}
\[
\ch\sim\ch+2\pi R.
\]
The Maxwell field is the zero mode; higher modes are massive vector excitations.

\item \textbf{Irreversible effects:}
\[
\ch=S_{\rm ent}/k_B,
\qquad
\dd S_{\rm ent}/\dd\lambda\ge0.
\]
Entropy gradients contribute effective current.

\item \textbf{Scale-flow effects:}
\[
\ch=\ln(L/L_0).
\]
Coupling constants and field strengths evolve along scale.

\item \textbf{Flux-conversion effects:}
\[
\nab_\mu J^\mu=-\Dchi J^4.
\]
Apparent four-dimensional source/sink behavior is reinterpreted as five-dimensional conservation.
\end{enumerate}

\section{Internal Consistency and Verification Checklist}

A branch should not be treated as mathematically mature until it answers the following questions.

\subsection{Dimensional consistency}

\begin{enumerate}[leftmargin=2em]
\item What are the units of \(\ch\)?
\item If \(\ch\) is dimensionless, what length scale \(L_\ch\) enters \(\dd\Sigma^2\)?
\item Are \(A_4\), \(V_\mu\), and \(J_{\rm new}^\nu\) assigned compatible units?
\end{enumerate}

\subsection{Geometric consistency}

\begin{enumerate}[leftmargin=2em]
\item Is the fifth direction compact, infinite, monotonic, or constrained?
\item What is the signature of \(G_{44}\)?
\item Are the projected equations stable under small perturbations?
\end{enumerate}

\subsection{Gauge consistency}

The five-dimensional gauge transformation is
\begin{equation}
\mcA_A\mapsto\mcA_A+\partial_A\Lambda.
\end{equation}
A branch must explain how this reduces to the four-dimensional transformation
\begin{equation}
A_\mu\mapsto A_\mu+\partial_\mu\Lambda.
\end{equation}

\subsection{Maxwell-limit verification}

For each branch, verify:
\begin{equation}
F=\dd A
\qquad\text{or}\qquad
F=\iota^*\mcF,
\end{equation}
then verify
\begin{equation}
J_{\rm new}^\nu=0.
\end{equation}
Only then does the branch produce the current Maxwell equations.

\subsection{Risk statement}

The main mathematical risk is overfitting.  It is easy to invent a fifth coordinate and then tune the correction term to vanish.  The stronger test is whether the same branch also makes clear, constrained, non-arbitrary predictions when \(J_{\rm new}^\nu\neq0\).

\section{Conclusion}

The answer to the guiding question is precise:
\begin{quote}
All twelve branches can be made compatible with Maxwell equations, but only under explicit decoupling, projection, zero-mode, fixed-slice, or pullback assumptions.
\end{quote}

The universal source equation is
\begin{equation}
\nab_\mu F^{\mu\nu}=\mu_0\left(J^\nu+J_{\rm new}^\nu\right).
\end{equation}
The standard Maxwell equations appear when
\begin{equation}
J_{\rm new}^\nu=0.
\end{equation}
The innovative theory space begins when
\begin{equation}
J_{\rm new}^\nu\neq0.
\end{equation}

The clearest Maxwell-compatible branches are
\[
T_0,\quad T_5,\quad T_q,\quad T_g,\quad T_C,
\]
because they use fixed-slice, zero-mode, fiber, metric-reduction, or pullback mechanisms.  The most Dynamic-Theory-like branches are
\[
T_m,\quad T_\rho,\quad T_E,\quad T_S,\quad T_F,
\]
because they interpret fifth-dimensional flow as material, thermodynamic, or conversion current.

The pedagogical rule for future development is:
\begin{equation}
\boxed{
\text{First recover Maxwell cleanly.  Only then study the fifth-dimensional correction.}
}
\end{equation}

\appendix

\section{Reference Verification Note}

The references below are included for established historical and mathematical context.  The twelve-branch theory tree is speculative modeling in this paper.  The bibliographic data were checked against public metadata pages, DOI records, publisher pages, arXiv, EuDML, OSTI/UNT, or other discoverable records on June 3, 2026.

\begin{thebibliography}{99}

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\emph{Sitzungsberichte der Preussischen Akademie der Wissenschaften}, pp. 966--972, 1921.
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C. N. Yang and R. L. Mills,
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E. Noether,
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C. Carath\'eodory,
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\bibitem{Wilson1971}
K. G. Wilson,
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\bibitem{Dirac1931}
P. A. M. Dirac,
``Quantised Singularities in the Electromagnetic Field,''
\emph{Proceedings of the Royal Society A}, vol. 133, no. 821, pp. 60--72, 1931.
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\end{thebibliography}

\end{document}
