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\newcommand{\paperid}{Paper 038}
\newcommand{\papertitle}{Finite-Difference and Retarded-Potential Solvers for Gravity-Cogravity Fields}
\newcommand{\papersubtitle}{A computational proposal for mass-current source ledgers, causal kernels, and benchmark geometries}
\newcommand{\paperdate}{June 14, 2026}
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{\Large \textbf{\paperid}}\\[0.5em]
{\LARGE \textbf{\papertitle}}\\[0.35em]
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JMP0X1B Research Group\\
Working paper draft / \paperdate
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\begin{abstract}
This proposal specifies a computational solver for Jefimenko-style gravity/cogravity. The solver evaluates retarded potentials and direct Jefimenko integrals from mass-current source histories, records continuity and gauge residuals, and exports parity-aware signatures for experiments and observational fits. The paper argues that solver metadata - retarded interpolation, mesh convergence, singular-cell rules, and residual gates - is part of the scientific claim. A first benchmark suite is proposed for static spheres, rotating rings, mass-current pulses, and binary toy models.
\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}
Analytic examples are not enough for a research program. A cogravity proposal needs an auditable solver that can compute fields from arbitrary mass-current histories, quantify residuals, and export the predicted signatures used by experiments and observational fits. The problem is to define a solver architecture that is causal, conservative, and difficult to fool.

The solver is not an engine for creating anomalies. It is an error-accounting tool. Its first job is to fail loudly when a source violates continuity or when a numerical shortcut has replaced retarded propagation with instantaneous action.

\section{Model}
Discretize the source into cells $a$ with volume $V_a$, position $\xp_a$, mass density $\rho_a(t)$, and mass current $\J_a(t)$. For an observation node $i$ at $\x_i$ and time $t_n$, define
\begin{equation}
  R_{ia}=|\x_i-\xp_a|,\qquad t_{{\rm r},n}^{ia}=t_n-R_{ia}/c.
\end{equation}
The retarded potential sums are
\begin{align}
\Phi_i^n&=-G\sum_a V_a\frac{\rho_a(t_{{\rm r},n}^{ia})}{R_{ia}},\\
\A_i^n&=-\frac{G}{c^2}\sum_a V_a\frac{\J_a(t_{{\rm r},n}^{ia})}{R_{ia}}.
\end{align}
Fields may be obtained either by differentiating $(\Phi,\A)$ on the observation mesh or by direct quadrature of the Jefimenko integrals. The first route tests gauge and derivative consistency; the second route is useful for sparse detector points.

A solver state should carry residuals:
\begin{align}
\epsilon_{\rm cont}&=\frac{\norm{\partial_t\rho+\nabla\cdot\J}_2}{\norm{\partial_t\rho}_2+\norm{\nabla\cdot\J}_2+\varepsilon},\\
\epsilon_{\rm gauge}&=\norm{\nabla\cdot\A+c^{-2}\partial_t\Phi}_2,\\
\epsilon_{\rm field}&=\sum_{j=1}^4 w_j\epsilon_j,
\end{align}
where $\epsilon_j$ are the four field-equation residuals and $\varepsilon$ is a small scale guard. No predicted anomaly should be exported if the residual gate fails.

\section{Claims}
\begin{claim}[Retarded interpolation is a scientific assumption]
The interpolation used to evaluate $\rho(\tr)$ and $\J(\tr)$ is part of the model. Linear, spline, spectral, or event-based interpolation can produce different high-frequency fields. The solver must publish interpolation metadata.
\end{claim}

\begin{claim}[Continuity-preserving sources are better than corrected sources]
Whenever possible, source histories should be generated from variables that satisfy continuity by construction, such as particles, incompressible flow maps, or finite-volume fluxes. Projecting a bad source onto a conserved source can hide the error that an experiment actually needs to know.
\end{claim}

\begin{claim}[Two field paths are required]
The solver should implement both potential differentiation and direct retarded-field quadrature. Agreement between the two paths is a strong internal check; disagreement is an error signal rather than a tunable parameter.
\end{claim}

\begin{claim}[The first benchmark suite should be boring]
The benchmark suite should start with a static sphere, a moving compact blob, a rotating ring, and a binary toy source. A solver that cannot reproduce simple cases should not be trusted on exotic apparatus geometries.
\end{claim}

\section{Evidence Plan}
The release should provide four benchmark families.

\textbf{Newtonian sphere.} A static uniform sphere tests Gauss behavior inside and outside the source. Outside the sphere, $\gfield=-GM\mathbf r/r^3$ and $\kfield=0$.

\textbf{Steady rotor.} A rigid rotating sphere or ring tests cogravity dipole scaling and parity under angular-velocity reversal.

\textbf{Mass-current pulse.} A compact source with time-dependent current tests retardation and cogravitational induction.

\textbf{Binary toy model.} Two point-like masses on prescribed circular orbits test phase lag, source interpolation, and far-zone behavior.

Each benchmark should be run under mesh refinement. The publication should include convergence plots for $\epsilon_{\rm cont}$, $\epsilon_{\rm gauge}$, and $\epsilon_{\rm field}$, plus a field-energy ledger integrated over a finite domain.

\section{JMP0X1B Implementation Surface}
The solver belongs naturally beside AtlasForge and digital-twin tooling. A minimal interface is
\begin{verbatim}
record SourceHistory { cells, rho(t), J(t), continuity_certificate }
record RetardedKernel { c, interpolation, singular_rule }
record FieldSolution  { Phi, A, g, K, residuals, provenance }
fn solve(source, detector_grid, kernel) -> FieldSolution
fn export_signature(solution, controls) -> SignatureLedger
\end{verbatim}
The solver should emit a signed digest of source files, mesh files, solver version, constants, and residual thresholds.

\section{Release and Review Plan}
Release the LaTeX proposal, a small reference implementation, benchmark source files, and a manifest that records constants and residual gates. The first implementation can be slow and transparent. Optimized kernels are secondary to auditability.

\section{Open Questions}
\begin{itemize}
\item What singular-cell rule is least misleading for compact high-density sources?
\item Can finite-volume continuity certificates be made simple enough for apparatus designers?
\item Which reduced-order models preserve cogravity parity without solving full retarded integrals?
\item How should gravitational self-interaction be excluded or approximated in a linear solver?
\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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