01 / Pullback
The five-dimensional Bianchi identity pulls back to the homogeneous Maxwell equation on the observed hypersurface.
PAPER-003 / June 6, 2026 / corrected September 18, 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 the projected source equation is represented by an exact residual. Expanding that residual into normal-flux and embedding terms requires a declared metric, foliation, orientation, and action.
In electrostatics this becomes a modified Gauss law with an effective projected charge density. Metal-edge curvature is not identified with spacetime extrinsic curvature. The paper defines calibrated pressure, polarity, symmetry, inversion, and shielding contrasts 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.
\[
\mathcal R^\nu
=h^\nu{}_B\nabla_A\mathcal F^{AB}
-D_\mu\bar F^{\mu\nu},\qquad
D_\mu\bar F^{\mu\nu}
=\mu_0(\bar J^\nu+J_{\rm proj}^\nu)
\]
02 / Normal Flux
Mixed normal components \(\Pi_\mu\) can contribute effective source terms when the fifth-direction flux does not vanish.
03 / Curvature
Spacetime extrinsic curvature and ordinary electrode-edge curvature are distinct quantities. Any map between them requires a material/embedding model.
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 paired contrasts normalized by a traceable nonzero calibration force, so the statistic remains defined when the measured force is compatible with zero.
- Pressure Separates gas-density EHD behavior from field-and-geometry residuals.
- Polarity Separates voltage-even and voltage-odd channels without assigning an un-derived curvature mechanism.
- 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.