01 / Pullback
The five-dimensional Bianchi identity pulls back to the homogeneous Maxwell equation on the observed hypersurface.
Working paper / June 6, 2026
Boundary-Flux Electromagnetism on Embedded Four-Manifolds and Null Observables for Capacitor Residuals
A linearized boundary-flux branch of the five-dimensional projection program, built to turn capacitor residual searches into publishable null observables.
01 / Abstract
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.
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 protects the homogeneous Maxwell equation on \(\Sigma\), and projection of the source equation yields normal-flux and embedding-curvature terms.
In electrostatics this becomes a modified Gauss law with edge-sensitive effective charge densities. The paper defines pressure, polarity, symmetry, inversion, and shielding ratios that distinguish boundary-flux residuals from electrohydrodynamic, leakage, thermal, and scalar-energy effects.
02 / Mechanism
Projected Source Law
The paper makes the hypersurface language precise: specify what is pulled back, what is projected, which terms survive, and how those terms differ from ordinary electromagnetic artifacts.
\[
D_\mu\bar F^{\mu\nu}
= \mu_0\bar J^\nu
+ \partial_\perp\Pi^\nu
+ K_\mu\bar F^{\mu\nu}
+ \mathcal Q^\nu[K,\Pi]
+ \cdots
\]
02 / Normal Flux
Mixed normal components \(\Pi_\mu\) can contribute effective source terms when the fifth-direction flux does not vanish.
03 / Curvature
Embedding curvature terms such as \(K_iE^i\) are edge-sensitive rather than volume-uniform.
04 / Conservation
Any projected residual must be divergence-free or paired with an explicit exchange current with the bulk.
03 / Nulls
Null Observables
A good null experiment should report ratios that survive even when the measured force is compatible with zero.
- Pressure Separates gas-density EHD behavior from field-and-geometry residuals.
- Polarity Distinguishes \(K_iE^i\) polarity-odd terms from scalar-energy \(E^2\) terms.
- Symmetry Checks whether matched dummy devices suppress edge-sensitive residuals.
- Inversion Tests device geometry, support coupling, local gravity, and environmental fields.
- Shielding Separates ordinary external-field interaction from internal boundary flux.
04 / Files
Paper Files
The PDF is generated from the LaTeX source. Both are published for auditability and future revision.