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\title{\Large Paper 021\\[0.6em]\textbf{Observational Equivalence Classes in Maxwell-Compatible Projection Theories}\\[0.3em]\large When different fifth-dimensional stories predict the same four-dimensional data}
\author{JMP0X1B Research Group}
\date{Working paper draft / June 13, 2026}
\begin{document}
\maketitle
\begin{abstract}
The twelve-branch projection tree deliberately permits multiple five-dimensional narratives. This paper classifies when those narratives are observationally equivalent at the four-dimensional level. Two models are equivalent relative to an experiment if their predicted distributions over observables match after nuisance marginalization. The result is a quotient map from speculative branches to measurable operator families.
\end{abstract}

\noindent\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}
Speculative theories can proliferate faster than experiments. A mass-coordinate branch, charge-fiber branch, scalar branch, and metric-mixing branch may all reduce to the same effective current under laboratory conditions. The problem is to avoid multiplying names for indistinguishable observations.

\section{Model}
Define a model M by fields, projection map, recovery limit, operator reduction, and nuisance prior. An experiment E induces a pushforward distribution over observables. Equivalence is equality or near-equality of these distributions under the experiment class.


\[
  M_a \sim_E M_b \quad \Longleftrightarrow \quad
  d\left(P_E(y\mid M_a),P_E(y\mid M_b)\right) < \epsilon_E .
\]
\[
  \mathcal{Q}_E = \mathcal{M}/\sim_E,
\]
where $\mathcal{Q}_E$ is the set of experimentally distinguishable classes.


\section{Claims}
\begin{enumerate}
\item \textbf{Claim 1.} Projection theory papers should publish both branch-level stories and quotient-level observable classes.
\item \textbf{Claim 2.} Equivalence classes shrink when experiments add controls; they expand when controls are absent.
\item \textbf{Claim 3.} A positive residual cannot identify a microscopic branch unless it breaks equivalence under predeclared controls.
\item \textbf{Claim 4.} The quotient map is a useful bridge between mathematical theory trees and operator-basis anomaly testing.
\end{enumerate}

\section{Evidence Plan}
Evidence is formal and computational. The paper maps the twelve projection branches into effective operator templates, then computes equivalence classes for three experiment families: electrostatic capacitors, cavity balances, and material strain tests. The expected result is a taxonomy showing where branch names add value and where operator family names are more honest.

\section{JMP0X1B Implementation Surface}
A JMP0X1B library should encode branch metadata, operator reductions, experiment classes, and equivalence calculations. It can emit a reviewer-facing table: branch, recovery condition, operator family, distinguishability controls, and known degeneracies.

\subsection*{Illustrative JMP0X1B-style contract}
\begin{verbatim}
record Branch { name: Text, recovery: Conditions, operator: OperatorFamily, nuisance: PriorSet }
fn quotient(branches: List<Branch>, experiment: ExperimentClass, epsilon: Float) -> EquivalenceMap effects{Deterministic, Audit}
\end{verbatim}


\section{Release And Review Plan}
Release the branch-to-operator map, equivalence criteria, experiment-class definitions, and generated quotient tables. Future theory papers should update the map rather than creating isolated branch vocabularies.

\section{Open Questions}
\begin{itemize}
\item What distance metric best captures experimental indistinguishability?
\item How should nonlocal or boundary-supported operators enter the quotient?
\item Can equivalence maps guide which new controls are worth building?
\item Where do material claims fit in a projection-theory quotient?
\end{itemize}

\section{Conclusion}
Equivalence classes discipline speculation by keeping theory names tied to observables. The paper reduces narrative clutter and improves experimental planning.

\section*{References}
\begin{thebibliography}{99}
\bibitem{jmp001} JMP0X1B Research Group. \emph{Twelve Five-Dimensional Projection Models for Maxwell-Compatible Electromagnetism}. Working paper, 2026. \url{https://research.jmp0x1b.com/papers/twelve-5d-projection-electromagnetism.html}
\bibitem{jmp004} JMP0X1B Research Group. \emph{Gauge-Consistent Effective Currents in Five-Dimensional Projection Electromagnetism}. Working paper, 2026. \url{https://research.jmp0x1b.com/papers/gauge-consistent-projection-currents.html}
\bibitem{jmp006} JMP0X1B Research Group. \emph{Gauge-Consistent Effective Operators for Five-Dimensional Projection Electromagnetism}. Working paper, 2026. \url{https://research.jmp0x1b.com/papers/gauge-operators-spacefaring.html}
\bibitem{jmp012} JMP0X1B Research Group. \emph{Identifiability of Fifth-Sector Electromagnetic Operators Under Null Experimental Controls}. Working paper draft, 2026. \url{https://research.jmp0x1b.com/papers/identifiability-of-fifth-sector-electromagnetic-operators-under-null-experimenta.html}
\bibitem{maxwell} J. C. Maxwell. A dynamical theory of the electromagnetic field. \emph{Philosophical Transactions of the Royal Society of London}, 155:459--512, 1865.
\bibitem{kaluza} T. Kaluza. Zum Unit\"atsproblem in der Physik. \emph{Sitzungsberichte der Preussischen Akademie der Wissenschaften}, 966--972, 1921.
\bibitem{klein} O. Klein. Quantentheorie und f\"unfdimensionale Relativit\"atstheorie. \emph{Zeitschrift f\"ur Physik}, 37:895--906, 1926.
\bibitem{overduin} J. M. Overduin and P. S. Wesson. Kaluza-Klein gravity. \emph{Physics Reports}, 283(5--6):303--380, 1997.
\end{thebibliography}

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