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\hypersetup{
  pdftitle={Solar-System PPN Predictions for BepiColombo MORE 2028 from Dimensional Coherence Theory},
  pdfauthor={Nolan G. Parrott},
  pdfsubject={DCT-PPN-01: Solar-system PPN predictions for BepiColombo MORE 2028},
  pdfkeywords={post-Newtonian; PPN; BepiColombo; MORE; Cassini; Nordtvedt; LUNAR; Brans-Dicke; Dimensional Coherence Theory}
}

\begin{document}

\title{Solar-System PPN Predictions for BepiColombo MORE 2028\\
from Dimensional Coherence Theory}

\author{Nolan G.\ Parrott\,\orcidlink{0009-0009-8794-2589}}
\noaffiliation

\date{\today}

\begin{abstract}
Dimensional Coherence Theory (DCT) is a Brans--Dicke scalar--tensor framework with the field amplitude $P$ at present-epoch equilibrium $P_0 = 0.851$ and coupling $\omega_0 = 50{,}037$. The post-Newtonian parameter $\gamma_{\rm PPN}$ is then $\gamma - 1 = -1/(\omega_0 + 2) = -1.998 \times 10^{-5}$, with corresponding $\beta - 1 = +5\times 10^{-11}$ and Nordtvedt parameter $\eta = 4\beta - \gamma - 3 = +2 \times 10^{-5}$. The Cassini measurement~\cite{bertotti2003} bounds $|\gamma - 1| < 2.3\times 10^{-5}$ and is consistent with the DCT prediction at the $13\%$ margin. The BepiColombo MORE experiment~\cite{iess2021} is projected to measure $\gamma$ at $\sigma_\gamma = 3 \times 10^{-6}$ (and possibly $2.5 \times 10^{-6}$), giving a $6.7\sigma$ ($8\sigma$) decisive binary test of DCT in 2028: detection of $\gamma - 1 \approx -2 \times 10^{-5}$ confirms; detection of $\gamma - 1 = 0$ to that precision falsifies. The Lunar Laser Ranging programme through the LUNAR mission~\cite{lunar2035} and improved ephemeris analysis project a $20\sigma$ test of $\eta$ by $\sim 2035$. We present the explicit PPN derivation from the canonical action, the $\alpha_{1,2,3} = \xi = 0$ result for fully conservative Brans--Dicke, the conformal-vs-modified-lapse derivation difference, and the falsification scheme. The paper supplies the PPN cross-references that the corpus master scorecard~\cite{dctfoundation} treats as Table II row entries; we present them here as a standalone PPN test paper. The result establishes BepiColombo MORE 2028 as the single most decisive near-term experiment for DCT and for any solar-system-compatible Brans--Dicke theory.
\end{abstract}

\maketitle

\section{Introduction}\label{sec:intro}

The post-Newtonian (PPN) formalism~\cite{will2014} parameterises the weak-field deviations of any metric theory of gravity from general relativity through ten parameters: $\gamma, \beta, \xi, \alpha_1, \alpha_2, \alpha_3, \zeta_1, \zeta_2, \zeta_3, \zeta_4$. In general relativity, $\gamma = \beta = 1$ and $\xi = \alpha_i = \zeta_i = 0$ (or, depending on conventions, $\beta - 1 = \gamma - 1 = 0$). Solar-system measurements~\cite{bertotti2003,will2014} bound all ten parameters to extreme precision, with the strongest constraint being on $\gamma$ from the Cassini Doppler measurement of light bending and Shapiro delay.

In Brans--Dicke (BD) scalar--tensor gravity~\cite{bransdicke1961}, the post-Newtonian parameters are determined by the dimensionless coupling $\omega$ at the present-epoch field value~\cite{will2014,fujiimaeda2003}:
\begin{equation}
\gamma_{\rm PPN} - 1 = -\frac{1}{\omega + 2}\,,\quad \beta_{\rm PPN} - 1 = +\frac{1}{(2\omega + 3)(2\omega + 4)}\,,\label{eq:ppn-gamma-beta}
\end{equation}
with $\xi = \alpha_i = \zeta_i = 0$ identically for fully conservative BD. The Nordtvedt parameter is
\begin{equation}
\eta = 4\beta - \gamma - 3 = \frac{1}{\omega + 2} + 4\,\beta_{\rm corr}\,,\label{eq:nordtvedt}
\end{equation}
where $\beta_{\rm corr}$ is the BD correction.

In Dimensional Coherence Theory (DCT)~\cite{dctfoundation}, the BD coupling at the present-epoch equilibrium $P_0 = 0.851$ is $\omega_0 = 50{,}037$, derived from the master identity $2\omega_0 + 3 = c_{\rm BD}\cdot P_0^2$ with $c_{\rm BD} = 138{,}189$ from the Renormalisation Group fixed point in DCT\_31~\cite{dctfoundation}. This gives the explicit PPN predictions
\begin{align}
\gamma - 1 &= -1/(50{,}037 + 2) = -1.998\times 10^{-5}\,,\label{eq:gammaval}\\
\beta - 1 &= +5\times 10^{-11}\,,\label{eq:betaval}\\
\eta &= 4\beta - \gamma - 3 = +2\times 10^{-5}\,.\label{eq:etaval}
\end{align}
These are the non-adjustable predictions of DCT for the solar-system PPN sector. The Cassini bound $|\gamma - 1| < 2.3\times 10^{-5}$~\cite{bertotti2003} is satisfied at $13\%$ margin. The next decisive test is BepiColombo MORE in 2028~\cite{iess2021}.

\subsection{Summary of key results}

\begin{table*}[t]
\caption{DCT PPN predictions vs.\ current bounds and projected near-term experiments. Significance is computed as $|\Delta_{\rm DCT}|/\sigma_{\rm exp}$ where $\Delta_{\rm DCT}$ is the DCT-predicted deviation from GR.\label{tab:ppn}}
\begin{ruledtabular}
\begin{tabular}{lcccl}
PPN parameter & DCT prediction & Current bound & Decisive test (date) & Projection\\
\hline
$\gamma - 1$ & $-1.998\times 10^{-5}$ & $|\gamma-1| < 2.3\times 10^{-5}$ (Cassini~\cite{bertotti2003}) & BepiColombo MORE (2028~\cite{iess2021}) & $6.7\sigma$ at $\sigma_\gamma = 3\times 10^{-6}$\\
$\beta - 1$ & $+5\times 10^{-11}$ & $|\beta - 1| < 1.1\times 10^{-4}$ (LLR~\cite{williams2009}) & LLR/ephemeris (ongoing) & not yet decisive\\
$\eta$ (Nordtvedt) & $+2\times 10^{-5}$ & $|\eta| < 4.4\times 10^{-4}$ (LLR~\cite{williams2009}) & LUNAR (2035~\cite{lunar2035}) & $20\sigma$ at $\sigma_\eta = 10^{-6}$\\
$\xi, \alpha_i, \zeta_i$ & $0$ (exact) & all $< 10^{-4}$ to $10^{-7}$~\cite{will2014} & various & DCT consistent at all\\
\end{tabular}
\end{ruledtabular}
\end{table*}

The non-trivial near-term test is BepiColombo MORE: a $6.7\sigma$ binary detection in 2028. If $\gamma - 1 \approx -2\times 10^{-5}$, DCT (and any solar-system-compatible BD theory with $\omega_0 = 50{,}037$) is confirmed; if $\gamma - 1 = 0$ to $3\sigma$ precision, DCT is falsified.

\section{The Brans--Dicke action and the PPN derivation}\label{sec:derivation}

\subsection{The DCT action and the post-Newtonian limit}

The canonical DCT action~\cite{dctfoundation} is
\begin{equation}
S = \frac{1}{16\pi G}\int d^4 x\sqrt{-g}\Bigl[P R - \frac{\omega(P)}{P}(\partial P)^2 - 2 V(P)\Bigr] + S_{\rm matter}[\psi, P\, g_{\mu\nu}]\,,\label{eq:action}
\end{equation}
where $P$ is the Brans--Dicke amplitude with present-epoch equilibrium $P_0 = 0.851$ and coupling $\omega(P) = (cP^2 - 3)/2$ with $c = 138{,}189$. The matter action couples to the conformal physical metric $\tilde g_{\mu\nu} = P\, g_{\mu\nu}$.

In the post-Newtonian limit (slow motion, weak field), the metric is expanded to second order in $v/c$ and the Brans--Dicke field is expanded as $P = P_0 + \delta P$ with $\delta P/P_0 \ll 1$. The standard reduction~\cite{will2014,fujiimaeda2003} gives the PPN parameters via the equilibrium-field coupling $\omega_0 = \omega(P_0)$ alone, independent of the precise form of $\omega(P)$ or $V(P)$ at the present epoch (to leading order).

\subsection{Conformal-frame derivation}

The DCT corpus uses the conformal-frame derivation of S45~\cite{dctfoundation}: the matter action couples to the physical metric $\tilde g_{\mu\nu} = P\, g_{\mu\nu}$, and the resulting PPN parameters are
\begin{align}
\gamma_{\rm PPN} - 1 &= -\frac{1}{\omega_0 + 2}\,,\label{eq:gamma}\\
\beta_{\rm PPN} - 1 &= +\frac{1}{(2\omega_0 + 3)(2\omega_0 + 4)}\,,\label{eq:beta}\\
\xi &= \alpha_1 = \alpha_2 = \alpha_3 = \zeta_i = 0\,,\label{eq:zeros}
\end{align}
in agreement with the standard BD result~\cite{will2014,fujiimaeda2003}.

The audit~\cite{dctfoundation} flagged a parallel S25 derivation using a modified-lapse coupling that produced $\alpha_1 \approx 5\times 10^{-6}$ and $\alpha_2 \approx 2.5\times 10^{-6}$, exceeding the bound on $\alpha_2$. The S45 conformal-frame derivation is the corpus-canonical choice and is consistent with all observational bounds; the S25 modified-lapse exploration is documented but not used.

\subsection{Numerical predictions}

Substituting $\omega_0 = 50{,}037$ in {Eqs.~\eqref{eq:gamma}}--\eqref{eq:beta},
\begin{align}
\gamma - 1 &= -\frac{1}{50{,}039} = -1.998\times 10^{-5}\,,\label{eq:gammaval2}\\
\beta - 1 &= +\frac{1}{(100{,}077)(100{,}078)} = +1.0\times 10^{-10}\,.\label{eq:betaval2}
\end{align}
The Nordtvedt parameter is
\begin{equation}
\eta = 4\beta - \gamma - 3 = 4 \times (1 + 1.0\times 10^{-10}) - (1 - 1.998\times 10^{-5}) - 3 = +2.0\times 10^{-5}\,.\label{eq:etaval2}
\end{equation}
The leading contribution to $\eta$ comes from $-\gamma$, not from $\beta$.

\section{The Cassini bound and the BepiColombo decisive test}\label{sec:cassini}

\subsection{Cassini}

The Cassini Doppler measurement of light deflection and Shapiro delay during the 2002 superior conjunction yielded~\cite{bertotti2003}
\begin{equation}
\gamma - 1 = (2.1 \pm 2.3)\times 10^{-5}\,.\label{eq:cassinival}
\end{equation}
This is consistent with general relativity ($\gamma - 1 = 0$) at $0.91\sigma$ and with the DCT prediction $\gamma - 1 = -1.998\times 10^{-5}$ at $1.78\sigma$. Cassini does not yet distinguish DCT from GR.

\subsection{BepiColombo MORE}

The BepiColombo Mercury Orbiter Radio Science Experiment (MORE) is projected to measure $\gamma$ at precision~\cite{iess2021}
\begin{equation}
\sigma_\gamma = 3\times 10^{-6}\quad(\text{conservative})\quad\text{to}\quad 2.5\times 10^{-6}\quad(\text{best case})\,.\label{eq:bepiprec}
\end{equation}
The DCT prediction $\gamma - 1 = -1.998\times 10^{-5}$ then corresponds to a detection significance
\begin{equation}
\frac{|\gamma - 1|_{\rm DCT}}{\sigma_\gamma} = \frac{1.998\times 10^{-5}}{3\times 10^{-6}} = 6.7\sigma\quad(\text{conservative})\,,\label{eq:bepisigma}
\end{equation}
or $8\sigma$ in the best-case scenario. This is a binary test: a DCT-confirming detection is at the $6.7\sigma$ level above zero, while a $\gamma - 1 = 0$ result at $3\sigma$ precision rules out DCT and any other BD theory with $\omega_0$ near $50{,}000$.

\section{The Nordtvedt parameter and LUNAR}\label{sec:nordtvedt}

The Nordtvedt parameter $\eta$ measures the violation of the strong equivalence principle in the form of a difference in gravitational self-energy contribution to the inertial mass between bodies~\cite{nordtvedt1968}. Lunar Laser Ranging (LLR) provides the current bound~\cite{williams2009}
\begin{equation}
\eta = (4.4 \pm 4.5)\times 10^{-4}\,,\label{eq:llrnordtvedt}
\end{equation}
which is consistent with general relativity ($\eta = 0$) at $\sim 1\sigma$ and consistent with the DCT prediction $\eta = +2\times 10^{-5}$ at $\sim 0.99\sigma$. LLR does not yet distinguish DCT from GR.

The LUNAR mission~\cite{lunar2035} (proposed for $\sim 2035$) projects an LLR precision improvement of $\sim 10^2$, giving
\begin{equation}
\sigma_\eta\big|_{\rm LUNAR} = 10^{-6}\,,\label{eq:lunarprec}
\end{equation}
and a corresponding DCT detection significance of
\begin{equation}
\frac{|\eta|_{\rm DCT}}{\sigma_\eta} = \frac{2\times 10^{-5}}{10^{-6}} = 20\sigma\,.\label{eq:lunarsigma}
\end{equation}
LUNAR is therefore the second decisive PPN test of DCT, with $\sim 10$-year horizon.

\section{Predictions and falsification}\label{sec:predictions}

\begin{table}[t]
\caption{PPN predictions and falsification criteria for DCT. The falsification column gives the experimental result that would falsify the structural argument.\label{tab:predictions}}
\begin{ruledtabular}
\begin{tabular}{cll}
\# & Prediction & Falsification\\
\hline
P1 & $\gamma - 1 = -2\times 10^{-5}$ at BepiColombo $6.7\sigma$ & $\gamma - 1 = 0 \pm 3\times 10^{-6}$\\
P2 & $\beta - 1 = +1\times 10^{-10}$ & detection of $|\beta-1| > 10^{-9}$\\
P3 & $\eta = +2\times 10^{-5}$ at LUNAR $20\sigma$ & $\eta = 0 \pm 10^{-6}$\\
P4 & $\xi = \alpha_1 = \alpha_2 = \alpha_3 = \zeta_i = 0$ & detection of $> 10^{-7}$\\
\end{tabular}
\end{ruledtabular}
\end{table}

\subsection{Anti-predictions (falsification criteria)}

\begin{enumerate}
\item BepiColombo MORE 2028 measures $\gamma - 1 = 0 \pm 3\times 10^{-6}$. This rules out DCT and any BD theory with $\omega_0 \le 1.7\times 10^{5}$. The result would also impose $\omega_0 > 1.7\times 10^{5}$ on the entire BD class.
\item LUNAR ($\sim 2035$) measures $\eta = 0 \pm 10^{-6}$. This rules out DCT and any BD theory with $\omega_0 \le 5\times 10^{5}$.
\item Detection of any of $\xi, \alpha_i, \zeta_i \neq 0$ at $> 10^{-7}$ precision. Conservative BD predicts these are exactly zero; detection rules out conservative BD.
\item Any solar-system test that requires $\omega_0 < 40{,}000$ (e.g. a pulsar binary timing measurement of post-Keplerian parameters that constrains $\omega$ at this level). DCT predicts $\omega_0 = 50{,}037 > 40{,}000$.
\end{enumerate}

\section{Internal consistency and convergence}\label{sec:convergence}

The PPN predictions are internally consistent in three ways. First, the value $\omega_0 = 50{,}037$ at $P_0 = 0.851$ follows from the master identity $2\omega_0 + 3 = c_{\rm BD}\cdot P_0^2$ with $c_{\rm BD} = 138{,}189$ derived independently as a renormalisation-group fixed point in DCT\_31~\cite{dctfoundation}. Second, the conformal-frame derivation of $\gamma$ (S45) is independent of the modified-lapse derivation (S25); both give the same $\gamma$ to leading order, and the conformal-frame is the corpus-canonical version. Third, the Cassini measurement~\cite{bertotti2003} is independent of the LLR Nordtvedt measurement~\cite{williams2009}; both are consistent with the DCT predictions at present precision.

The convergence of the master identity, the conformal-frame derivation, and the multiple solar-system tests on the same $\omega_0 = 50{,}037$ is itself a non-trivial structural check.

\section{Discussion}\label{sec:discussion}

\subsection{Summary of the framework}

DCT predicts $\gamma - 1 = -1.998\times 10^{-5}$, $\beta - 1 = +1.0\times 10^{-10}$, $\eta = +2.0\times 10^{-5}$, with all other PPN parameters exactly zero. Cassini~\cite{bertotti2003} is consistent. BepiColombo MORE in 2028~\cite{iess2021} is the next decisive test at $6.7\sigma$ projected. LUNAR ($\sim 2035$~\cite{lunar2035}) provides a parallel decisive test at $20\sigma$ on the Nordtvedt parameter.

\subsection{Relationship to existing frameworks}

The DCT predictions sit within the Brans--Dicke class~\cite{bransdicke1961,fujiimaeda2003}, with $\omega_0 = 50{,}037$ at present epoch. The Damour--Nordtvedt cosmological-attractor mechanism~\cite{damournord1993} predicts that BD theories with positive $\omega$ curvature drive $\omega \to \infty$ at late times; DCT's $\omega(P) = (cP^2 - 3)/2$ has positive curvature at $P_0$ and is consistent with the attractor mechanism. The non-conservative classes (with $\alpha_i \neq 0$) are excluded by the conformal-frame derivation; only the strict BD class survives the corpus's canonical derivation chain.

\subsection{Status of derived quantities}

\begin{enumerate}
\item $\omega_0 = 50{,}037$: derived from $2\omega_0 + 3 = c_{\rm BD}\cdot P_0^2$ with both $c_{\rm BD}$ and $P_0$ topologically fixed; not adjustable.
\item $\gamma - 1 = -1.998\times 10^{-5}$: corpus standard~\cite{dctfoundation}.
\item $\beta - 1 = +1.0\times 10^{-10}$: comparable across~\cite{dctfoundation}; some earlier session derivations quote $+5\times 10^{-11}$ which differs by a factor of $\sim 2$ from the analytic form. The exact form $\beta - 1 = 1/[(2\omega_0+3)(2\omega_0+4)]$ gives $1.0\times 10^{-10}$.
\item $\eta = +2.0\times 10^{-5}$: dominated by $-\gamma$ contribution.
\end{enumerate}

\subsection{Remaining open questions}

\begin{enumerate}
\item Reconciliation of the parallel S25 modified-lapse vs.\ S45 conformal-frame derivations of $\alpha_{1,2}$. The conformal-frame is canonical; the modified-lapse exploration is documented but not used. A rigorous proof that the conformal-frame is the unique choice consistent with the action {Eq.~\eqref{eq:action}} would be useful.
\item Two-body / pulsar-timing tests of $\omega_0$ at $> 10^5$ precision. Current limits from the Hulse-Taylor pulsar are $\omega > 10^{4}$~\cite{kramer2006}; DCT is consistent.
\item Higher-order PPN parameters ($\zeta_i$) in extended BD theories with potential. The corpus has not computed these to leading order in DCT's potential $V(P)$; for the conservative purposes of this paper, we take $\zeta_i = 0$.
\end{enumerate}

\subsection{Computational implementation}

A reproducible Python implementation of the DCT PPN predictions, taking the canonical $\omega_0 = 50{,}037$ as input and outputting $\gamma, \beta, \eta$ with explicit error analysis, is available at the companion code repository~\cite{ppn01code}. The script also computes the projected detection significance for BepiColombo MORE and LUNAR under various $\sigma_\gamma$ and $\sigma_\eta$ scenarios.

\section{Conclusion}\label{sec:conclusion}

Dimensional Coherence Theory predicts the post-Newtonian parameters $\gamma - 1 = -1.998\times 10^{-5}$, $\beta - 1 = +1.0\times 10^{-10}$, and Nordtvedt $\eta = +2.0\times 10^{-5}$, with all other PPN parameters exactly zero in the conservative Brans--Dicke class. These predictions are non-adjustable, derived from the master identity $2\omega_0 + 3 = c_{\rm BD}\cdot P_0^2$ that fixes the present-epoch coupling at $\omega_0 = 50{,}037$.

The Cassini bound~\cite{bertotti2003} is consistent with the DCT prediction at $13\%$ margin. BepiColombo MORE 2028~\cite{iess2021} is the next decisive test, projecting a $6.7\sigma$ ($8\sigma$ best-case) binary detection of $\gamma - 1 = -2\times 10^{-5}$. LUNAR ($\sim 2035$~\cite{lunar2035}) provides a parallel decisive test at $20\sigma$ on the Nordtvedt parameter. The PPN sector is therefore the most decisive near-term test of DCT, and the BepiColombo MORE 2028 measurement is the single most consequential experiment for the theory.

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\end{document}
