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\newcommand{\paperid}{Paper 041}
\newcommand{\papertitle}{Gravity-Cogravity Operators for Projection-Field Research}
\newcommand{\papersubtitle}{A bridge between JMP0X1B projection electromagnetism and conserved mass-current analogies}
\newcommand{\paperdate}{June 14, 2026}
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\lhead{JMP0X1B Research}
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{\LARGE \textbf{\papertitle}}\\[0.35em]
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JMP0X1B Research Group\\
Working paper draft / \paperdate
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\begin{abstract}
This proposal connects the JMP0X1B projection-electromagnetism track to Jefimenko-style gravity/cogravity through conserved effective mass-current operators. It requires every speculative coupling to supply both delta rho and delta J, preferably through an antisymmetric superpotential whose divergence is automatically conserved. Candidate bulk, boundary, electromagnetic-energy, and rotation operators are framed as admissible mathematical classes rather than empirical claims. The deliverable is an operator dictionary with conservation proofs, dimensional analysis, retarded signatures, stress-energy ledgers, and null-test controls.
\end{abstract}

\textbf{Status.} Draft manuscript for review and revision. This paper is not an empirical claim of new physics or field-ready technology. It is a structured proposal, theory note, protocol, or application architecture intended to be auditable, falsifiable, and publishable with source.

\tableofcontents
\newpage


\section{Problem}
JMP0X1B already has a projection-electromagnetism track with scalar readouts, boundary fluxes, operator bases, null ledgers, and conservation-first audits. Gravity/cogravity should not become a disconnected fringe branch. The problem is to define a bridge: speculative couplings may be explored, but only through conserved mass-current operators and stress-energy ledgers.

This paper proposes an operator grammar for coupling projection-field ideas to a Jefimenko-compatible gravity/cogravity model. The grammar is designed to reject mathematically invalid proposals early.

\section{Model}
Let the ordinary mass-current four-vector be
\begin{equation}
  J_M^\mu=(c\rho,\J),\qquad \partial_\mu J_M^\mu=0.
\end{equation}
A speculative effective source $\delta J_M^\mu$ is admissible at the linear level if it is conserved:
\begin{equation}
  \partial_\mu\delta J_M^\mu=0.
\end{equation}
A sufficient construction is
\begin{equation}
  \delta J_M^\mu=\partial_\nu M^{\mu\nu},\qquad M^{\mu\nu}=-M^{\nu\mu},
\end{equation}
because $\partial_\mu\partial_\nu M^{\mu\nu}=0$ for smooth fields. Boundary-supported versions can be handled distributionally if the surface terms are published explicitly.

The effective source is then inserted into the same retarded equations:
\begin{align}
\nabla\cdot\gfield &= -4\pi G(\rho+\delta\rho),\\
\nabla\times\kfield &= -\frac{4\pi G}{c^2}(\J+\delta\J)+\frac{1}{c^2}\partial_t\gfield.
\end{align}
The conservation condition is not a philosophical preference; it is required by the divergence of the field equations.

Candidate operator families include:
\begin{align}
\delta J_{E}^{\mu}&=\partial_\nu\left[\chi_E u_{\rm EM} Q^{\mu\nu}\right],\\
\delta J_{B}^{\mu}&=\partial_\nu\left[\chi_B \delta_\Sigma B^{\mu\nu}_\Sigma\right],\\
\delta J_{\Omega}^{\mu}&=\partial_\nu\left[\chi_\Omega \Omega^{\mu\nu}(\rho,\vvec)\right],
\end{align}
where $u_{\rm EM}$ is electromagnetic energy density, $\delta_\Sigma$ denotes a boundary-supported term, and $Q^{\mu\nu}$, $B^{\mu\nu}_\Sigma$, and $\Omega^{\mu\nu}$ are antisymmetric structures fixed by the model. These are placeholders for operator classes, not claims that nature realizes them.

\section{Claims}
\begin{claim}[Conserved operators are the entry ticket]
Any projection-to-cogravity coupling must define $\delta\rho$ and $\delta\J$ together. A scalar-only effective mass term that changes in time without a matching current violates the field equations unless the missing flux is supplied.
\end{claim}

\begin{claim}[Boundary claims need distributional bookkeeping]
If a mechanism localizes at capacitor edges, seams, apertures, or material interfaces, the paper must publish the boundary measure, normal/tangent convention, jump conditions, and conservation proof.
\end{claim}

\begin{claim}[Electromagnetic energy couplings need stress-energy audits]
Using $u_{\rm EM}/c^2$ as an effective mass-density input is mathematically natural but physically dangerous. Any such coupling must be audited against ordinary stress-energy, support forces, thermal energy, and equivalence-principle constraints.
\end{claim}

\begin{claim}[Operator degeneracy should be expected]
Different conserved operators can produce the same detector channel under a limited control set. The correct output may be an equivalence class, not an identified mechanism.
\end{claim}

\section{Evidence Plan}
The first release should be an operator dictionary. Each operator entry should include:
\begin{enumerate}
\item mathematical definition of $M^{\mu\nu}$ or the equivalent conserved construction;
\item support: bulk, boundary, material interface, or time gate;
\item dimensional analysis and coupling constants;
\item induced $(\delta\rho,\delta\J)$ pair;
\item retarded field signature;
\item controls that separate it from conventional artifacts;
\item known observational or laboratory constraints;
\item kill criteria.
\end{enumerate}
The evidence is a set of conservation proofs and synthetic null experiments, not a claim of detected coupling.

\section{JMP0X1B Implementation Surface}
The operator grammar can become a shared library for speculative physics papers:
\begin{verbatim}
record ConservedOperator {
  name, M_antisymmetric, support, dimensions,
  delta_rho, delta_J, conservation_proof, controls
}
fn admit(op) -> AdmissionReport
fn project_to_gco(op, apparatus) -> SignatureLedger
\end{verbatim}
This directly connects to operator-basis papers, boundary-flux papers, null-ledger infrastructure, and conservation-first audits.

\section{Release and Review Plan}
Release a PDF, LaTeX source, operator dictionary, symbolic conservation checks, and example signatures for capacitor, rotor, cavity, and cryogenic apparatus. Review should reject any coupling that lacks units, a conservation proof, a stress-energy ledger, or a declared null-test control.

\section{Open Questions}
\begin{itemize}
\item Which projection-electromagnetism operators have a meaningful conserved mass-current analogue?
\item Can boundary-supported operators be made mesh-independent in finite-element simulations?
\item What equivalence-principle tests already rule out naive electromagnetic energy-density couplings?
\item How should operator equivalence classes be stored in NullLedger?
\end{itemize}


\section{Conclusion}
This paper turns one part of the gravity/cogravity idea into a named, reviewable work package. Its value does not depend on treating the framework as established physics. The value is the disciplined reduction of a speculative field analogy into equations, invariants, bounds, null tests, and release artifacts that can be audited and either extended or killed.

\section*{References}
\addcontentsline{toc}{section}{References}

\begin{thebibliography}{99}
\bibitem{jefimenko2000}
O. D. Jefimenko. \emph{Causality, Electromagnetic Induction, and Gravitation: A Different Approach to the Theory of Electromagnetic and Gravitational Fields}. Electret Scientific, 2nd ed., 2000.

\bibitem{jefimenko2006}
O. D. Jefimenko. \emph{Gravitation and Cogravitation: Developing Newton's Theory of Gravitation to its Physical and Mathematical Conclusion}. Electret Scientific, 2006.

\bibitem{heaviside1893}
O. Heaviside. A gravitational and electromagnetic analogy. \emph{The Electrician}, 31:281--282 and 359, 1893.

\bibitem{ruggiero2021}
M. L. Ruggiero. A note on the gravitoelectromagnetic analogy. arXiv:2111.09008, 2021.

\bibitem{kopeikin2005}
S. M. Kopeikin and E. B. Fomalont. Gravimagnetism, causality, and aberration of gravity in the gravitational light-ray deflection experiments. arXiv:gr-qc/0510077, 2005.

\bibitem{everitt2011}
C. W. F. Everitt et al. Gravity Probe B: Final results of a space experiment to test general relativity. \emph{Physical Review Letters}, 106:221101, 2011. arXiv:1105.3456.

\bibitem{jmp001}
JMP0X1B Research Group. Twelve Five-Dimensional Projection Models for Maxwell-Compatible Electromagnetism. Working paper, 2026. \url{https://research.jmp0x1b.com/papers/twelve-5d-projection-electromagnetism.html}

\bibitem{jmp005}
JMP0X1B Research Group. Weak-Field Radion Electrogravity from Maxwell-Compatible Five-Dimensional Projections. Working paper, 2026. \url{https://research.jmp0x1b.com/papers/weak-field-radion-electrogravity.html}

\bibitem{jmp017}
JMP0X1B Research Group. Conservation-First Stress-Energy Audits for Closed Electromagnetic Devices. Working paper, 2026. \url{https://research.jmp0x1b.com/papers/conservation-first-stress-energy-audits.html}

\bibitem{jmp023}
JMP0X1B Research Group. A Research Program for Maxwell-Compatible Projection Electromagnetism. Working paper, 2026. \url{https://research.jmp0x1b.com/papers/a-research-program-for-maxwell-compatible-projection-electromagnetism.html}
\end{thebibliography}


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