\documentclass[11pt]{article} \usepackage[a4paper,margin=1in]{geometry} \usepackage{amsmath,amssymb,amsfonts} \usepackage{booktabs} \usepackage{hyperref} \usepackage{array} \usepackage{longtable} \usepackage{enumitem} \setlist{nosep} \hypersetup{colorlinks=true,linkcolor=black,citecolor=black,urlcolor=blue} \newcommand{\dd}{\mathrm{d}} \newcommand{\JJ}{\mathcal{J}} \newcommand{\PP}{\mathcal{P}} \newcommand{\MM}{\mathcal{M}} \newcommand{\Sig}{\Sigma} \newcommand{\Lie}{\mathcal{L}} \newcommand{\order}{\mathcal{O}} \newcommand{\eff}{\mathrm{eff}} \newcommand{\new}{\mathrm{new}} \newcommand{\bdry}{\mathrm{bdry}} \newcommand{\diag}{\mathrm{diag}} \newcommand{\vac}{\mathrm{vac}} \newcommand{\perpdir}{\perp} \numberwithin{equation}{section} \newcommand{\obs}{\mathrm{obs}} \newcommand{\EHD}{\mathrm{EHD}} \newcommand{\Maxwell}{\mathrm{Maxwell}} \newcommand{\thermal}{\mathrm{thermal}} \newcommand{\leak}{\mathrm{leak}} \newcommand{\vib}{\mathrm{vib}} \newcommand{\support}{\mathrm{support}} \newcommand{\BF}{\mathrm{BF}} \newcommand{\low}{\mathrm{low}} \newcommand{\atm}{\mathrm{atm}} \newcommand{\sym}{\mathrm{sym}} \newcommand{\asym}{\mathrm{asym}} \newcommand{\matched}{\mathrm{matched}} \newcommand{\shield}{\mathrm{shield}} \newcommand{\shielded}{\mathrm{shielded}} \newcommand{\unshielded}{\mathrm{unshielded}} \title{Boundary-Flux Electromagnetism on Embedded Four-Manifolds and Null Observables for Capacitor Residuals} \author{JMP0X1B Research} \date{\today} \begin{document} \maketitle \begin{abstract} One branch of five-dimensional projection electromagnetism treats observed four-dimensional electrodynamics as the pullback of a higher-dimensional field to an embedded hypersurface. This paper develops the linearized boundary-flux version of that idea. Let the observed world be an embedded four-manifold $\Sigma\subset\mathcal M_5$ with unit normal $n^A$ and induced metric $h_{AB}=G_{AB}-\sigma n_An_B$. A five-dimensional field strength decomposes as $\mathcal F=\bar F+n\wedge\Pi$, where $\bar F$ is tangential and $\Pi$ contains mixed normal components. Pullback of the five-dimensional Bianchi identity yields the homogeneous Maxwell equation on $\Sigma$. Projection of the source equation yields an inhomogeneous law with normal-flux and embedding-curvature terms, \begin{equation} D_\mu\bar F^{\mu\nu}=\mu_0\bar J^\nu+\partial_\perp\Pi^\nu+K_\mu\bar F^{\mu\nu}+\mathcal Q^\nu[K,\Pi]+\cdots. \end{equation} In electrostatics this becomes a modified Gauss law with edge-sensitive effective charge densities. We derive dimensionless null observables - pressure, polarity, symmetry, inversion, and shielding ratios - that distinguish boundary-flux residuals from electrohydrodynamic, leakage, thermal, and scalar-energy effects. The paper does not claim an anomalous capacitor force. It gives the boundary-flux branch a mathematical form and a falsification protocol. \end{abstract} \section{Introduction} The constraint-hypersurface branch in Ref.~\cite{JMPA} is among the most geometrically distinctive parts of the five-dimensional projection program. Instead of treating the fifth coordinate as a compact physical circle, scalar scale, entropy coordinate, or charge fiber, it treats four-dimensional physics as the result of restricting a higher-dimensional field to a hypersurface. In such a picture, new source terms may appear when normal flux, extrinsic curvature, or boundary conditions fail to vanish. This branch deserves separate treatment because it is easy to state vaguely and difficult to test. A phrase such as ``flux through the fifth direction'' is not yet physics. It becomes physics only when one specifies what is pulled back, what is projected, which terms survive, and how their signatures differ from conventional electromagnetic artifacts. This paper provides a linearized version of that program. We formulate the embedded geometry, decompose the field strength, derive the projected source law, reduce it to electrostatics, and build null observables that an experiment can report even when no anomaly is found. \section{Embedded hypersurface geometry} Let $(\mathcal M_5,G_{AB})$ be a five-dimensional manifold and let $\Sigma$ be a four-dimensional embedded hypersurface with embedding map $\iota:\Sigma\to\mathcal M_5$. The tangent basis is \begin{equation} e^A_{\mu}=\frac{\partial X^A}{\partial x^\mu}, \end{equation} and the induced metric is \begin{equation} h_{\mu\nu}=G_{AB}e^A_{\mu}e^B_{\nu}. \end{equation} Let $n^A$ be a unit normal, $G_{AB}n^An^B=\sigma=\pm1$, and define the extrinsic curvature \begin{equation} K_{\mu\nu}=e^A_{\mu}e^B_{\nu}\nabla_A n_B. \end{equation} The intrinsic covariant derivative on $\Sigma$ is denoted $D_\mu$. For intuition, one may write a graph embedding $\chi=f(x^\mu)$ in a coordinate chart. Then derivatives of $f$ determine the local tilt of the hypersurface, while second derivatives determine extrinsic curvature. The formalism below does not require that special chart, but the chart helps explain why edges, boundaries, and sharp electrode geometry can matter. \section{Field decomposition} Let $\mathcal F_{AB}$ be a five-dimensional two-form. Its tangential pullback is \begin{equation} \bar F_{\mu\nu}=e^A_\mu e^B_\nu\mathcal F_{AB}. \end{equation} The mixed normal component is \begin{equation} \Pi_\mu=e^A_\mu n^B\mathcal F_{AB}. \end{equation} In form notation, \begin{equation} \mathcal F=\bar F+n^\flat\wedge \Pi \end{equation} near the hypersurface, where $n^\flat$ is the one-form associated with $n^A$. If $\mathcal F=\dd\mathcal A$, the five-dimensional Bianchi identity $\dd\mathcal F=0$ holds. Pulling it back to $\Sigma$ gives \begin{equation} \dd_\Sigma\bar F=0, \end{equation} or in components, \begin{equation} D_{[\alpha}\bar F_{\beta\gamma]}=0. \label{eq:homog} \end{equation} Thus the homogeneous Maxwell equation is automatically protected by the pullback. Any viable boundary-flux theory must preserve Eq.~\eqref{eq:homog}. \section{Projected source equation} Start from \begin{equation} \nabla_A\mathcal F^{AB}=\mu_5\mathcal J^B. \label{eq:5sourceC} \end{equation} Projecting tangentially with $e^\nu_B$ yields, schematically, \begin{equation} D_\mu\bar F^{\mu\nu}=\mu_0\bar J^\nu+\mathcal S^\nu_\perp+\mathcal S^\nu_K+\mathcal S^\nu_{K\Pi}+\cdots. \label{eq:projectedC} \end{equation} The leading terms are \begin{equation} \mathcal S^\nu_\perp\sim \partial_\perp\Pi^\nu, \qquad \mathcal S^\nu_K\sim K_\mu\bar F^{\mu\nu}, \qquad \mathcal S^\nu_{K\Pi}\sim K^{\nu}{}_{\mu}\Pi^\mu-K\Pi^\nu, \end{equation} where $K=K^\mu{}_{\mu}$ and $\partial_\perp=n^A\nabla_A$. The precise coefficients depend on normalization, signature, and the chosen foliation. What is invariant at this stage is the structure: tangential divergence receives contributions from normal variation and embedding curvature. Equation~\eqref{eq:projectedC} must still satisfy a conservation identity. If $\bar J^\nu$ is the ordinary conserved current, then the sum of the projected source terms must be divergence-free on $\Sigma$, or else it must be accompanied by an explicit exchange current with the bulk: \begin{equation} D_\nu\left(\mathcal S^\nu_\perp+\mathcal S^\nu_K+\mathcal S^\nu_{K\Pi}+\cdots\right)=0 \end{equation} for a closed effective system. \section{Electrostatic reduction} Set $\bar F_{i0}=E_i/c$ and assume static fields. The projected source equation becomes \begin{equation} \nabla\cdot\mathbf E=\frac{\rho}{\epsilon_0}+\frac{\rho_\perp}{\epsilon_0}+\frac{\rho_K}{\epsilon_0}+\frac{\rho_{K\Pi}}{\epsilon_0}+\cdots. \label{eq:gaussC} \end{equation} A minimal parameterization is \begin{align} \rho_\perp&=\epsilon_0 a_\perp \partial_\perp\Pi^0,\\ \rho_K&=\epsilon_0 a_K K_iE^i,\\ \rho_{K\Pi}&=\epsilon_0 a_{K\Pi}\left(K^0{}_{\mu}\Pi^\mu-K\Pi^0\right). \end{align} The coefficients $a_i$ carry the conversion factors needed to translate the fifth-direction normalization into ordinary SI units. The curvature term $K_iE^i$ is especially important. It vanishes in ideal flat symmetric regions and becomes large near geometry changes, edges, boundaries, or effective embedding distortions. Therefore a boundary-flux residual should be edge-localized rather than volume-uniform. This differs from a scalar-energy model, where the source is $E^2$ throughout the high-field region. \section{Force budget} A measured force in a capacitor-like experiment can be written \begin{equation} F_{\obs}=F_{\EHD}+F_{\Maxwell}+F_{\thermal}+F_{\leak}+F_{\vib}+F_{\support}+F_{\BF}, \end{equation} where $F_{\BF}$ denotes a possible boundary-flux residual. A boundary-flux theory is scientifically meaningful only if it predicts transformations that separate $F_{\BF}$ from the other terms. The basic scaling expectations are: \begin{center} \begin{tabular}{p{0.24\linewidth}p{0.28\linewidth}p{0.34\linewidth}} \toprule Contribution & Typical diagnostic & Expected behavior \\ \midrule Electrohydrodynamic & pressure ladder & collapses in high vacuum \\ Leakage/current & current correlation & tracks $I$ or discharge events \\ Thermal & delayed response & follows heating and time constants \\ Maxwell stress & geometry and grounding & reproduced by conventional field simulation \\ Scalar energy & polarity even, volume energy & scales with $E^2$ and energy density \\ Boundary flux & edge and embedding sensitivity & enhanced by edges and null in matched symmetry \\ \bottomrule \end{tabular} \end{center} \section{Null observables} We define five dimensionless ratios. A good null experiment should report them even when the result is compatible with zero. \paragraph{Pressure ratio.} \begin{equation} R_p=\frac{F(P_{\low})}{F(P_{\atm})}. \end{equation} Electrohydrodynamic effects generally decrease strongly with pressure, while a genuine boundary-flux term tied only to fields and geometry should not follow the same gas-density scaling. \paragraph{Polarity ratio.} \begin{equation} R_\pm=\frac{F(+V)-F(-V)}{F(+V)+F(-V)}. \end{equation} Boundary-flux terms linear in $K_iE^i$ may be odd under polarity reversal, while scalar energy terms sourced by $E^2$ are even at leading order. This separates boundary-flux behavior from radion-energy behavior. \paragraph{Symmetry ratio.} \begin{equation} R_{\sym}=\frac{F_{\asym}}{F_{\matched}}. \end{equation} A true edge-sensitive residual should be suppressed by symmetry-matched dummy devices with comparable mass, voltage, leakage, and thermal behavior. \paragraph{Inversion ratio.} \begin{equation} R_\pi=\frac{F(\theta+\pi)}{F(\theta)}. \end{equation} Rotating the apparatus tests whether the residual is tied to device geometry, support coupling, laboratory gravity, or environmental fields. \paragraph{Shielding ratio.} \begin{equation} R_{\shield}=\frac{F_{\shielded}}{F_{\unshielded}}. \end{equation} If the residual is ordinary external-field interaction, shielding should suppress it. If it is internal boundary flux, shielding should affect it only through changes in capacitance, grounding, or boundary condition. \section{Coefficient extraction} Let the candidate boundary-flux force be modeled as \begin{equation} F_{\BF}=a_\perp\mathcal G_\perp+a_K\mathcal G_K+a_{K\Pi}\mathcal G_{K\Pi}, \end{equation} where the $\mathcal G_i$ are geometry functionals computed from field simulations and embedding assumptions. A null bound $|F_{\BF}|