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  pdftitle={Electrostatic Projection Couplings in Five-Dimensional Maxwell-Compatible Theory},
  pdfauthor={JMP0x1B Research Lab},
  pdfsubject={Speculative five-dimensional electrostatic and electrogravity models},
  pdfkeywords={five-dimensional projection, electrostatics, electrogravity, Maxwell equations, Biefeld-Brown effect, Kaluza-Klein, speculative physics}
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\title{\vspace{-1.0em}\textbf{Electrostatic Projection Couplings in Five-Dimensional Maxwell-Compatible Theory}\\[0.35em]
\large A speculative continuation toward Brown-style electrogravity and falsifiable fifth-sector residuals}
\author{JMP0x1B Research Lab\\\small Working paper / June 6, 2026}
\date{}

\begin{document}
\maketitle
\thispagestyle{empty}

\begin{center}
\fbox{\begin{minipage}{0.92\linewidth}
\textbf{Status.} This manuscript is a speculative mathematical prototype. It does not claim that electrogravity, anomalous capacitor thrust, or laboratory gravity control has been experimentally established. Its purpose is narrower: to define projection-coupling mechanisms that recover ordinary electrostatics and Maxwell electrodynamics in a clean limit, while making any proposed fifth-sector correction explicit enough to test and falsify.
\end{minipage}}
\end{center}

\begin{abstract}
The companion working paper \emph{Twelve Five-Dimensional Projection Models for Maxwell-Compatible Electromagnetism} develops a theory tree in which different five-dimensional structures reduce to ordinary four-dimensional Maxwell equations when the fifth sector decouples. This continuation asks what happens in the electrostatic limit when the fifth sector does not fully decouple. We formulate three candidate paths: a fifth-current electrostatics path, a scalar or radion readout path, and a metric-mixing path. Each path is designed to preserve ordinary Gauss-Poisson electrostatics as the zeroth-order limit while allowing a controlled residual term sourced by electrostatic field energy, field gradients, dielectric polarization, or projection curvature. We then reinterpret Brown-style asymmetric capacitor observations as a decomposition into conventional electrohydrodynamic forces plus a possible fifth-sector residual. The result is not a claim of antigravity. It is a falsifiable speculative theory template: any surviving residual must be pressure-independent, shield-compatible, momentum-conserving, geometry-sensitive in a predicted way, and bounded by modern nanonewton-scale null searches.
\end{abstract}

\tableofcontents
\newpage

\section{Motivation}

The previous JMP0x1B working paper builds twelve five-dimensional projection models whose first requirement is Maxwell compatibility: each branch must identify an observed four-dimensional field $F_{\mu\nu}$, recover the homogeneous and sourced Maxwell equations under stated assumptions, and show which fifth-dimensional correction terms appear when those assumptions are relaxed \cite{jmp_twelve}. This continuation narrows the question to electrostatics and asks whether an internally consistent path can be drawn from those correction terms toward the family of ideas historically called electrogravity.

The motivating empirical object is the Brown or Biefeld-Brown style asymmetric capacitor: a high-voltage electrode pair with different electrode geometries that experiences a force in air. Brown interpreted such effects as electrogravitic. Modern explanations usually treat the measurable force as electrohydrodynamic (EHD): ions are created in high electric field regions, accelerated through the gas, and transfer momentum to neutral air. Bahder and Fazi modeled high-voltage asymmetric capacitors and emphasized the need to separate ionic wind, ionic drift, plasma formation, gas species, pressure, voltage, and vacuum behavior \cite{bahder_fazi}. NASA-sponsored work also studied asymmetric capacitors for propulsion within this conventional high-voltage context \cite{canning_nasa}. Tajmar's 2004 analysis argued that the Biefeld-Brown effect was a misinterpretation of corona wind phenomena \cite{tajmar_2004}. A later high-resolution search by Tajmar, Koessling, and Neunzig found no anomalous steady-field forces or torques down to nanonewton or nanonewton-meter scale in shielded high-vacuum tests \cite{tajmar_2024}.

Those facts constrain the speculative task. A usable continuation should not simply relabel ion wind as gravity control. Instead, it should define a residual term that:
\begin{enumerate}
  \item vanishes exactly in the Maxwell-compatible limit;
  \item does not duplicate ordinary Maxwell stress, electrostatic attraction, heating, leakage currents, acoustic vibration, or EHD thrust;
  \item remains compatible with conservation of momentum;
  \item predicts a sharp difference between gas-dependent EHD and any fifth-sector residual;
  \item gives experimentalists a way to kill the theory cleanly.
\end{enumerate}

The guiding idea is therefore modest but fertile: electrostatic fields do not directly ``bend gravity'' in this model. Rather, electrostatic field energy or projection leakage sources an intermediate fifth-sector object. Gravity reads that object weakly through a scalar, an effective current, an extrinsic curvature, or a metric-mixing term. Ordinary Maxwell theory is recovered when the object decouples.

\section{Projection setup}

Let the five-dimensional coordinates be
\begin{equation}
  X^A=(x^\mu,\ch), \qquad A,B=0,1,2,3,5, \qquad \mu,\nu=0,1,2,3,
\end{equation}
where $\ch$ denotes the fifth coordinate. The five-dimensional electromagnetic potential and field strength are
\begin{equation}
  \Afive_A=(A_\mu, A_5), \qquad
  \Ffive_{AB}=\partial_A\Afive_B-\partial_B\Afive_A.
\end{equation}
The observed four-dimensional electromagnetic field is the projected component
\begin{equation}
  F_{\mu\nu}(x)=\Ffive_{\mu\nu}(x,\ch_0),
\end{equation}
while the mixed components are collected as
\begin{equation}
  \Pi_\mu(x)=\Ffive_{\mu 5}(x,\ch_0)=\partial_\mu A_5-\partial_5 A_\mu.
\end{equation}
The previous paper's Maxwell-compatible limit can be represented abstractly as
\begin{equation}
  \Pi_\mu\rightarrow 0, \qquad \partial_5 F_{\mu\nu}\rightarrow 0, \qquad \Gfive_{AB}\rightarrow \Gfive_{AB}^{(0)},
\end{equation}
where $\Gfive_{AB}^{(0)}$ is a projection geometry that introduces no surviving fifth-sector current in four dimensions.

A generic five-dimensional sourced equation has the form
\begin{equation}
  \nabla_A \Ffive^{AB}=\mu_5\Jfive^B.
\end{equation}
Projecting the $B=\nu$ component onto the four-dimensional sheet gives
\begin{equation}
  \nabla_\mu F^{\mu\nu} = \mu_0 J^\nu + \mu_0 J_5^\nu,
  \label{eq:projectedmaxwell}
\end{equation}
where $J^\nu$ is the ordinary current and $J_5^\nu$ is a projection current. For a flat product geometry this can be sketched as
\begin{equation}
  J_5^\nu \sim -\frac{1}{\mu_0}\partial_5\Pi^\nu,
\end{equation}
while a general projection can also contain connection, extrinsic curvature, radion, or density-sheet terms:
\begin{equation}
  J_5^\nu = -\frac{1}{\mu_0}\left(\D_5\Pi^\nu + C^\nu[F,\Pi,\Gfive,K,\phi]\right).
\end{equation}
Here $K$ denotes projection curvature and $\phi$ denotes a scalar measuring the size, stiffness, or permeability of the fifth direction. Equation \eqref{eq:projectedmaxwell} is the basic bridge from the twelve-branch framework to electrostatics.

\section{Electrostatic reduction}

In electrostatics,
\begin{equation}
  \B=0, \qquad \E=-\grad\Phie, \qquad \partial_t\E=0.
\end{equation}
The standard Maxwell limit gives Gauss-Poisson theory,
\begin{equation}
  \divg\E=\frac{\rho}{\eps}, \qquad \nabla^2\Phie=-\frac{\rho}{\eps}.
\end{equation}
The projected equation \eqref{eq:projectedmaxwell} modifies Gauss law to
\begin{equation}
  \divg\E=\frac{\rho+\rho_5}{\eps},
  \label{eq:modifiedgauss}
\end{equation}
where $\rho_5=J_5^0/c$ in units where the time component is normalized consistently. Equivalently,
\begin{equation}
  \nabla^2\Phie=-\frac{\rho+\rho_5}{\eps}.
\end{equation}
This is the simplest electrostatic connection: a fifth-sector leakage appears as an effective charge density. The theory is Maxwell-compatible only if
\begin{equation}
  \rho_5=0
\end{equation}
under ordinary laboratory conditions, or is so small that it lies below present constraints.

A useful weak-field parameterization is
\begin{equation}
  \rho_5 = \eps\left[a_1\partial_5\Pi_0+a_2\grad s\cdot\E+a_3 K_i E^i+a_4\ell^2\nabla^2(\divg\E)\right],
  \label{eq:rho5}
\end{equation}
where $s$ is a scalar fifth-sector field, $K_i$ is an effective projection-curvature vector, $\ell$ is a characteristic fifth-sector length scale, and $a_i$ are phenomenological couplings. The terms in \eqref{eq:rho5} have different experimental meanings:
\begin{itemize}
  \item $\partial_5\Pi_0$ means direct leakage along the fifth coordinate.
  \item $\grad s\cdot\E$ means electrostatics is modified by gradients of a scalar projection medium.
  \item $K_iE^i$ means an extrinsic geometric tilt converts electric flux into projected charge.
  \item $\ell^2\nabla^2(\divg\E)$ means sharp electrode curvature can produce small higher-gradient corrections.
\end{itemize}
The Brown-style geometry becomes interesting because asymmetric capacitors maximize sharp gradients, dielectric boundary layers, and electrode curvature. In this model, those features are not automatically antigravity. They are potential amplifiers of $\rho_5$ or $s$.

\section{Three speculative paths}

\subsection{Path A: fifth-current electrostatics}

The first path treats the fifth sector as a hidden charge reservoir. Ordinary charge conservation becomes
\begin{equation}
  \nabla_\mu J^\mu = -\nabla_\mu J_5^\mu.
\end{equation}
If the total five-dimensional current is conserved, this is not a violation of conservation. It is a statement that four-dimensional charge can appear nonconserved if current flows through the projection boundary. In a stable electrostatic device this should normally average to zero. A residual appears only if the electrode-dielectric system has a persistent fifth-sector polarization:
\begin{equation}
  \rho_5 = -\divg \mathbf{P}_5, \qquad \mathbf{P}_5=\eps\,\eta_P\,s\E+\eps\,\eta_K\,\ell K \E.
\end{equation}
Then Gauss law becomes
\begin{equation}
  \divg\left[\eps(1+\eta_Ps)\E\right]=\rho+\eps\eta_K\divg(\ell K\E).
\end{equation}
This is experimentally close to a nonlinear dielectric law. Its clean signatures are not thrust, but anomalous capacitance, field-dependent dielectric relaxation, and geometry-sensitive electrostatic shielding residuals.

\subsection{Path B: scalar or radion readout of electrostatic energy}

The second path introduces a scalar $s$ representing the local size, tension, phase stiffness, or permeability of the fifth dimension. A minimal effective Lagrangian is
\begin{equation}
  \Lagr = -\frac{1}{4\mu_0}Z(s)F_{\mu\nu}F^{\mu\nu}
           -\frac{1}{2}\partial_\mu s\partial^\mu s
           - V(s)
           +\Lagr_{\text{matter}},
  \label{eq:lagrangian}
\end{equation}
with
\begin{equation}
  Z(s)=1+\eta s+O(s^2).
\end{equation}
Varying \eqref{eq:lagrangian} gives
\begin{equation}
  \nabla_\mu\left[Z(s)F^{\mu\nu}\right]=\mu_0J^\nu,
  \label{eq:Zmaxwell}
\end{equation}
\begin{equation}
  \Box s - V'(s)=\frac{1}{4\mu_0}Z'(s)F_{\mu\nu}F^{\mu\nu}.
  \label{eq:scalarfield}
\end{equation}
For static electric fields with $\B=0$, the field invariant is proportional to $-E^2$ up to sign convention. Thus electrostatic field energy sources $s$:
\begin{equation}
  (\nabla^2-m_s^2)s=-\alpha_E\ue, \qquad \ue=\frac{1}{2}\eps E^2,
  \label{eq:scalarpoisson}
\end{equation}
where $m_s$ is an inverse range and $\alpha_E$ is a phenomenological coupling with dimensions chosen so that $s$ is dimensionless.

To turn this into a speculative electrogravity model, assume the weak gravitational potential reads $s$ through
\begin{equation}
  \delta\Phig=\frac{1}{2}\gamma_s c^2s,
  \qquad
  \delta\gvec=-\grad\delta\Phig=-\frac{1}{2}\gamma_s c^2\grad s.
  \label{eq:gravreadout}
\end{equation}
Equations \eqref{eq:scalarpoisson} and \eqref{eq:gravreadout} provide a concrete version of ``electrostatic gravity modification'': electrostatic energy sources $s$, and $s$ perturbs the effective gravitational potential. The model reduces to ordinary electrostatics and ordinary gravity when $\alpha_E\gamma_s\to0$, $m_s\to\infty$, or $s\to0$.

The solution of \eqref{eq:scalarpoisson} is approximately
\begin{equation}
  s(\mathbf{x})=\alpha_E\int \frac{e^{-m_s|\mathbf{x}-\mathbf{x}'|}}{4\pi |\mathbf{x}-\mathbf{x}'|}\,\ue(\mathbf{x}')\,\dd^3x'.
  \label{eq:yukawa}
\end{equation}
A highly asymmetric capacitor can create a strongly asymmetric $\ue(\mathbf{x})$. That asymmetry creates an asymmetric $s(\mathbf{x})$. However, by itself, a closed isolated device still cannot produce reactionless self-acceleration if the Lagrangian is local and translationally invariant. A net force requires exchange with an environment: Earth, a boundary condition in $s$, a background fifth-sector gradient, radiation, leakage current, gas, or support structure.

This point is crucial. The path is closer to weight modulation than free propulsion. A laboratory force would be modeled as
\begin{equation}
  \mathbf{F}_{5D}=\int_V \rho_m(\mathbf{x})\,\delta\gvec(\mathbf{x})\,\dd^3x
  +\int_V \ue(\mathbf{x})\,\grad \Gamma_{\oplus}(\mathbf{x})\,\dd^3x
  +\mathbf{F}_{\text{boundary}},
  \label{eq:f5d}
\end{equation}
where $\rho_m$ is the device mass density, $\Gamma_{\oplus}$ represents possible Earth/background coupling, and $\mathbf{F}_{\text{boundary}}$ accounts for momentum exchanged through the fifth-sector boundary. If all three integrals reduce to internal self-stresses, the net external force must vanish.

\subsection{Path C: metric mixing and Kaluza-style charge geometry}

In Kaluza-Klein style theories, the electromagnetic vector potential can be encoded in off-diagonal metric components. A schematic metric ansatz is
\begin{equation}
  \dd S^2=g_{\mu\nu}\dd x^\mu\dd x^\nu+\phi^2(\dd\ch+kA_\mu\dd x^\mu)^2.
  \label{eq:kkmetric}
\end{equation}
This type of construction historically motivates the idea that electromagnetism and gravity can be different projections of a higher-dimensional geometry \cite{collion_vaugon,liko}. In the strict Maxwell-compatible limit, the four-dimensional equations reduce to ordinary Einstein-Maxwell behavior. When $\phi$, $\partial_5A_\mu$, or the projection hypersurface becomes dynamical, additional terms appear.

For electrostatics $A_\mu=(\Phie/c,0,0,0)$, the mixed metric component $g_{05}$ carries electrostatic potential. A geodesic projected into four dimensions can acquire terms of the schematic form
\begin{equation}
  \frac{\dd^2x^i}{\dd\tau^2}=\frac{q}{m}F^i{}_{\nu}u^\nu
  + b_1(\partial^i\phi)(u^5)^2
  + b_2\partial_5F^i{}_{\nu}u^\nu u^5
  + b_3K^i{}_{\mu\nu}u^\mu u^\nu.
  \label{eq:projectedgeodesic}
\end{equation}
The first term is the Lorentz force. The remaining terms are speculative projection forces. A Brown-like device could then be treated as a generator of large $\partial_i\Phie$, large field-gradient curvature, and strong dielectric polarization, all of which might alter $\phi$ or $K$.

This path is mathematically attractive because it links directly to branch $T_g$ (full metric dynamic theory), branch $T_5$ (compact fifth-dimension theory), branch $T_q$ (charge-fiber theory), and branch $T_F$ (fifth-force conversion theory) in the earlier theory tree. Its weakness is that Kaluza-style electromagnetic-gravitational couplings are normally far too small to produce dramatic laboratory effects unless new coupling scales are introduced. Therefore the model must be framed as a constrained phenomenology, not a claim of established high-power gravity control.

\section{Brown-style capacitor decomposition}

A disciplined continuation should write any measured force on a high-voltage asymmetric capacitor as
\begin{equation}
  \mathbf{F}_{\text{obs}}=\mathbf{F}_{\text{EHD}}+\mathbf{F}_{\text{Maxwell}}+\mathbf{F}_{\text{thermal}}+\mathbf{F}_{\text{leak}}+\mathbf{F}_{\text{vibration}}+\mathbf{F}_{5D}.
  \label{eq:forcebudget}
\end{equation}
The conventional terms are not nuisance details. They are the dominant expected physics. The speculative theory only lives in $\mathbf{F}_{5D}$ after the others are bounded or subtracted.

A minimal Brown-compatible fifth-sector residual can be written as
\begin{equation}
  \mathbf{F}_{5D}=\zeta_1\int_V \ue\,\grad s_0\,\dd^3x
  +\zeta_2\int_V s\,\grad \ue\,\dd^3x
  +\zeta_3\int_V \ue\,\mathbf{K}\,\dd^3x
  +\zeta_4\int_{\partial V}\ue\,\mathbf{n}_5\,\dd A.
  \label{eq:brownresidual}
\end{equation}
Here $s_0$ is an ambient fifth-sector scalar background, $\mathbf{K}$ is projection curvature, $\mathbf{n}_5$ is a boundary normal into the fifth sector, and the $\zeta_i$ are phenomenological couplings. This is intentionally broad. Its purpose is to expose where momentum goes:
\begin{itemize}
  \item the $\zeta_1$ term exchanges momentum with a background gradient;
  \item the $\zeta_2$ term exchanges momentum between field energy and the scalar sector;
  \item the $\zeta_3$ term exchanges momentum with projection curvature;
  \item the $\zeta_4$ term exchanges momentum through a fifth-boundary flux.
\end{itemize}
If all backgrounds and boundaries vanish, \eqref{eq:brownresidual} should integrate to zero for a closed system.

\section{Predicted signatures}

The purpose of this paper is to create testable theory paths. The following signatures separate the proposed fifth-sector residual from known high-voltage artifacts.

\begin{table}[h]
\centering
\small
\begin{tabularx}{\linewidth}{>{\bfseries}p{0.22\linewidth}X X}
\toprule
Signature & Conventional EHD expectation & Fifth-sector residual expectation \\
\midrule
Gas pressure & Strong dependence on pressure, gas species, corona onset, and ion mobility & Ideally pressure-independent after leakage, outgassing, and discharge are suppressed \\
Polarity & Often polarity-sensitive because corona and ion species differ by electrode sign & Scalar $E^2$ couplings are polarity-even; direct fifth-current terms may be polarity-odd \\
Symmetric capacitor & Maxwell stress is internal; no net thrust in a closed balanced setup & Should vanish unless external background or fifth-boundary term is present \\
Orientation to Earth & Ordinary EHD follows device geometry and gas flow & Weight-modulation path may change under 180-degree inversion relative to local $\gvec$ \\
Dielectric response & Tracks leakage, heating, permittivity, and breakdown behavior & May show delayed relaxation if $s$ couples to polarization or density-sheet variables \\
Shielding & Faraday shielding removes external electric fields but not internal ion wind unless sealed & A confined field could still source $s$; external readout would be a key anomaly \\
Vacuum & EHD should collapse when gas transport is removed & Any real electrogravity residual must survive clean high vacuum within measurement bounds \\
\bottomrule
\end{tabularx}
\caption{Qualitative separation between conventional electrohydrodynamic forces and the speculative fifth-sector residual.}
\label{tab:signatures}
\end{table}

The sharpest prediction comes from pressure scaling. If a force falls with gas density, tracks corona current, changes with gas species, or disappears when ion transport is eliminated, it belongs in $\mathbf{F}_{\text{EHD}}$. If a residual persists in shielded high vacuum, it must then satisfy orientation, polarity, and symmetry tests before it can be treated as a candidate $\mathbf{F}_{5D}$.

\section{Minimal experimental protocol}

A speculative paper should make itself easy to falsify. The following protocol is a theory-driven null test rather than a claim of expected success.

\begin{enumerate}
  \item \textbf{Force budget first.} Measure voltage, current, electrode temperature, acoustic vibration, leakage paths, corona luminosity, ozone or ion production, and support-wire electrostatic forces.
  \item \textbf{Pressure ladder.} Repeat from atmospheric pressure to high vacuum. Fit the force to gas-dependent EHD models before looking for residuals.
  \item \textbf{Geometry pair.} Test an asymmetric capacitor and a symmetric capacitor with matched capacitance, stored energy, mass, dielectric, and leakage.
  \item \textbf{Polarity pair.} Reverse polarity at fixed $E^2$. A scalar energy path predicts even behavior; a charge-fiber path may predict odd behavior.
  \item \textbf{Earth orientation.} Rotate the sealed device by 180 degrees relative to local gravity. A weight-modulation interpretation must define the sign change or null result in advance.
  \item \textbf{Internal-field shielding.} Confine the high electric field inside a sealed Faraday enclosure mechanically isolated from the balance. Any external force must not be explainable by cable forces or field leakage.
  \item \textbf{Null coupling bound.} If no residual is observed, report an upper bound on the combined parameter products, such as $|\alpha_E\gamma_s|$ or $|\zeta_i|$.
\end{enumerate}

A useful reporting equation is
\begin{equation}
  \left|\mathbf{F}_{5D}\right| < \left|\mathbf{F}_{\text{obs}}-\mathbf{F}_{\text{EHD}}-\mathbf{F}_{\text{Maxwell}}-\mathbf{F}_{\text{thermal}}-\mathbf{F}_{\text{leak}}-\mathbf{F}_{\text{vibration}}\right|+\sigma_F.
\end{equation}
This turns a failed experiment into a publishable constraint.

\section{Connection to the twelve branches}

The electrostatic continuation does not need all twelve branches equally. The most relevant are summarized below.

\begin{table}[h]
\centering
\small
\begin{tabularx}{\linewidth}{>{\bfseries}p{0.16\linewidth}X X}
\toprule
Branch & Electrostatic role & Possible electrogravity role \\
\midrule
$T_0$ static projection & Defines the exact Maxwell/Gauss recovery limit & Baseline null theory; all residuals vanish \\
$T_m$ mass-coordinate & Lets charge/mass ratios become projection variables & Couples electrostatic energy to inertial or gravitational response \\
$T_\rho$ density sheet & Treats matter density as a fifth-sheet coordinate & Dielectric density gradients become projection sources \\
$T_E$ energy-coordinate & Makes field energy a natural fifth coordinate & Electrostatic energy $\ue$ can source scalar readout \\
$T_5$ compact dimension & Supports Kaluza-style metric ansatz & Charge as fifth momentum; residuals from noncompact leakage \\
$T_\theta$ phase-action & Links phase stiffness to potential & High-voltage phase gradients as scalar sources \\
$T_q$ charge fiber & Gives charge a fiber coordinate & Effective fifth charge density $\rho_5$ \\
$T_R$ scale flow & Permits scale-dependent coupling & Field-dependent capacitance and local $G_{\text{eff}}$ shifts \\
$T_g$ full metric dynamic & Promotes projection geometry to dynamics & Metric mixing, radion coupling, gravitational readout \\
$T_C$ constraint surface & Enforces Maxwell as a hypersurface condition & Anomalies are constraint violations or boundary flux \\
$T_F$ fifth-force conversion & Allows fifth-sector force to project into 4D & Direct candidate for Brown-style residual \\
\bottomrule
\end{tabularx}
\caption{How the electrostatic/electrogravity continuation maps onto branches of the earlier five-dimensional theory tree.}
\label{tab:branches}
\end{table}

The key design rule is the same as in the earlier paper: every branch must state its Maxwell-recovery condition. For the present continuation, the recovery condition is stronger:
\begin{equation}
  \rho_5=0, \qquad s=0, \qquad \grad s=0, \qquad K_iE^i=0, \qquad \mathbf{F}_{5D}=0.
\end{equation}
Only when one of these assumptions is deliberately relaxed may a new electrostatic or electrogravity term appear.

\section{A compact toy model}

For future papers, the cleanest single model is the scalar readout path. Define
\begin{align}
  \divg\left[Z(s)\E\right]&=\frac{\rho}{\eps}, \label{eq:toy1}\\
  \curl\E&=0, \label{eq:toy2}\\
  (\nabla^2-m_s^2)s&=-\alpha_E\frac{\eps E^2}{2}, \label{eq:toy3}\\
  \delta\gvec&=-\frac{1}{2}\gamma_sc^2\grad s. \label{eq:toy4}
\end{align}
With $Z(s)=1+\eta s$, the modified Poisson equation is
\begin{equation}
  \nabla^2\Phie = -\frac{\rho}{\eps} - \eta\grad s\cdot\grad\Phie + O(s^2).
\end{equation}
The scalar solution is the Yukawa convolution \eqref{eq:yukawa}. The anomalous acceleration at a point is
\begin{equation}
  \delta\gvec(\mathbf{x})=-\frac{1}{2}\gamma_sc^2\alpha_E\grad\int \frac{e^{-m_s|\mathbf{x}-\mathbf{x}'|}}{4\pi |\mathbf{x}-\mathbf{x}'|}\,\ue(\mathbf{x}')\,\dd^3x'.
\end{equation}
For a capacitor with electrode gap $d$, characteristic field $E_0=V/d$, and active volume $V_c$, the magnitude scales roughly as
\begin{equation}
  |\delta g| \sim \frac{1}{2}|\gamma_s\alpha_E|c^2\,\frac{\eps E_0^2}{2}\,L_{\text{asym}}\,\mathcal{Y}(m_sL),
  \label{eq:scaling}
\end{equation}
where $L_{\text{asym}}$ encodes the geometry-dependent gradient of field energy and $\mathcal{Y}$ is a Yukawa range factor. Equation \eqref{eq:scaling} is deliberately phenomenological. It gives experimentalists a parameter product to bound without pretending to know the microscopic fifth-sector scale.

The toy model makes three immediate statements:
\begin{enumerate}
  \item A parallel-plate capacitor with uniform field has little external gradient in $\ue$ and should produce no net force except tiny field-energy weight in standard general relativity.
  \item A sharp asymmetric capacitor can maximize $\grad\ue$, but in air this also maximizes corona and EHD, so it is the worst geometry for a clean anomaly search unless operated below discharge in high vacuum.
  \item If a residual exists, its leading scalar component is voltage-even because it depends on $E^2$. Polarity-odd residuals point to direct charge-fiber leakage, leakage currents, ion chemistry, or instrumentation error.
\end{enumerate}

\section{Paper program}

This continuation naturally splits into three publishable papers.

\subsection*{Paper I: Electrostatic projection couplings}
This manuscript. Goal: define $\rho_5$, $s$, $K_i$, and the Maxwell-compatible electrostatic limit. Deliverable: modified Gauss law, scalar readout equation, and falsification protocol.

\subsection*{Paper II: Radion-weighted electrogravity}
Goal: promote the scalar model to a covariant action with gravity,
\begin{equation}
  S=\int \sqrt{-g}\left[\frac{1}{2\kappa}R-\frac{1}{4\mu_0}Z(s)F^2-\frac{1}{2}(\partial s)^2-V(s)+\Lagr_m(A,g,s)\right]\dd^4x.
\end{equation}
Deliverable: stress-energy tensor, weak-field limit, predicted bounds on $\alpha_E\gamma_s$, and relation to Einstein-Maxwell theory.

\subsection*{Paper III: Brown devices as residual tests}
Goal: write a complete experimental force budget for asymmetric capacitors. Deliverable: pressure-scaling models, polarity/orientation tables, null-result parameter bounds, and a standard reporting template for research.jmp0x1b.com.

\section{Conclusion}

The speculative bridge from five-dimensional Maxwell-compatible projections to electrogravity should be built in layers. The first layer is ordinary electrostatics. The second is a fifth-sector correction such as $\rho_5$, $s$, or $K_i$. The third is a gravitational readout of that correction. The fourth is a carefully controlled Brown-style force budget in which ion wind and all conventional forces are removed before any residual is named.

This framing keeps the theory imaginative but disciplined. It allows the phrase ``gravity modified by electromagnetism'' to mean something precise: electrostatic field energy or projection leakage sources a fifth-sector degree of freedom, and that degree of freedom weakly perturbs gravitational response. If the fifth sector decouples, Maxwell and ordinary gravity return exactly. If it does not decouple, the theory predicts specific residuals that can be searched for and bounded.

\appendix
\section{Notation summary}

\begin{table}[h]
\centering
\small
\begin{tabularx}{\linewidth}{>{\bfseries}p{0.18\linewidth}X}
\toprule
Symbol & Meaning \\
\midrule
$\ch$ & Fifth coordinate \\
$\Afive_A$ & Five-dimensional electromagnetic potential \\
$\Ffive_{AB}$ & Five-dimensional field strength \\
$F_{\mu\nu}$ & Observed four-dimensional electromagnetic field \\
$\Pi_\mu$ & Mixed fifth-field component $\Ffive_{\mu5}$ \\
$J_5^\mu$ & Effective projected fifth current \\
$\rho_5$ & Effective fifth charge density in modified Gauss law \\
$s$ & Scalar fifth-sector/radion/projection-medium field \\
$K_i$ & Effective projection-curvature vector \\
$\ue$ & Electrostatic field energy density $\eps E^2/2$ \\
$\delta\Phig$ & Fifth-sector perturbation to gravitational potential \\
$\delta\gvec$ & Fifth-sector perturbation to local gravitational acceleration \\
\bottomrule
\end{tabularx}
\end{table}

\section{Internal consistency checklist}

Before any future version claims a physical effect, the following must be satisfied.

\begin{enumerate}
  \item \textbf{Maxwell recovery:} $\rho_5=0$ and $Z(s)=1$ under ordinary conditions.
  \item \textbf{Gauge consistency:} the projected equations must remain invariant under $A_\mu\mapsto A_\mu+\partial_\mu\Lambda$ unless a controlled symmetry-breaking term is explicitly introduced.
  \item \textbf{Momentum accounting:} any net force must identify the external field, boundary, radiation, gas, or fifth-sector channel carrying opposite momentum.
  \item \textbf{Equivalence-principle accounting:} if $s$ changes gravitational response, the model must say whether it couples to all mass-energy universally or only to electromagnetic/dielectric sectors.
  \item \textbf{Null-result survival:} modern vacuum null tests must be translatable into upper bounds on the couplings.
  \item \textbf{Experimental separability:} a proposed residual must differ from EHD in pressure, polarity, orientation, shielding, and scaling.
\end{enumerate}

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\end{document}
