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\hypersetup{
  pdftitle={Stiffness-Backreaction Duality: A General Bound on Cosmological Backreaction in Solar-System-Compatible Brans-Dicke Theories},
  pdfauthor={Nolan G. Parrott},
  pdfsubject={DCT-SBD-01: Stiffness-backreaction duality for solar-system-compatible Brans-Dicke theories},
  pdfkeywords={Brans-Dicke; scalar-tensor gravity; cosmological backreaction; Buchert averaging; Cassini bound; Jensen's inequality; dark energy; Dimensional Coherence Theory}
}

\begin{document}

\title{Stiffness--Backreaction Duality:\\
A General Bound on Cosmological Backreaction in Solar-System-Compatible Brans--Dicke Theories}

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

\date{\today}

\begin{abstract}
For any Brans--Dicke scalar--tensor theory satisfying the Cassini solar-system bound $\omega_0 > 40{,}000$ on the dimensionless coupling, we prove that the contribution of cosmological backreaction to the effective dark-energy density is bounded above by $6\times 10^{-10}\,\Lambda_{\rm obs}$. The bound follows from a Jensen-type inequality on the Brans--Dicke stiffness $(2\omega + 3)^{-1}$ averaged over Hubble-scale perturbations of the scalar field, combined with the standard Buchert averaging formalism extended to scalar--tensor cosmology. The result rules out backreaction-as-dark-energy explanations in any Brans--Dicke-class theory that respects solar-system constraints, independently of the specific form of the coupling function $\omega(\phi)$ or the potential $V(\phi)$. The duality is structural: the very stiffness that protects solar-system phenomenology mathematically forbids the scalar field from contributing meaningfully to large-scale acceleration through inhomogeneity-driven backreaction. We outline the application to Dimensional Coherence Theory and to general scalar--tensor extensions, including a comment on how the bound interacts with the disformal channel of theories that include both conformal and disformal couplings to matter. The result is a one-line theorem on a structurally simple class of theories; its proof requires only the Cassini bound, a convex Jensen inequality, and the linearised Buchert kinematic backreaction formula. We treat the result as a publishable side-result of the broader DCT cosmological program and present it standalone here so that it can be evaluated independently of any particular scalar--tensor model.
\end{abstract}

\maketitle

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

Cosmological backreaction~\cite{buchert2000,buchert2008} is the proposal that the apparent late-time acceleration of the universe arises not from a true cosmological constant but from the spatial averaging of an inhomogeneous gravitational field~\cite{rasanen2006,wiltshire2007}. In general relativity, the magnitude of the backreaction effect is sensitive to the variance of the local expansion rate $\theta$ and the local shear $\sigma$, captured by the kinematic backreaction $Q_{\mathcal{D}} = \frac{2}{3}\langle(\theta - \langle\theta\rangle_{\mathcal{D}})^2\rangle_{\mathcal{D}} - 2\langle\sigma^2\rangle_{\mathcal{D}}$ averaged over a spatial domain $\mathcal{D}$~\cite{buchert2000}. Whether this term can mimic a positive cosmological constant of magnitude $\Lambda_{\rm obs}$ at the level required by Type Ia supernova and BAO data~\cite{riess2022,desi2024} is an open question with extensive literature~\cite{ishibashi2006,green2014,buchert2015}.

In scalar--tensor extensions of general relativity, the cosmological backreaction acquires an additional contribution from the scalar field $\phi$ (or, in the parameterisation of Brans and Dicke~\cite{bransdicke1961}, the field $P = \phi/\phi_0$). The natural question is then: under what conditions does the scalar-field sector contribute enough backreaction to drive the observed acceleration?

The answer in this paper is: it cannot, in any solar-system-compatible Brans--Dicke theory. The Cassini bound $\omega_0 > 40{,}000$~\cite{bertotti2003} on the dimensionless coupling at the equilibrium field value places a hard ceiling on the Brans--Dicke ``stiffness'' factor $(2\omega + 3)^{-1}$, and a Jensen-type inequality~\cite{jensen1906} converts this local solar-system bound into a global cosmological-backreaction bound. The result is a one-line theorem that we prove in {Sec.~\ref{sec:duality}} and discuss in {Sec.~\ref{sec:implications}}.

We emphasise from the outset that this result is not specific to any particular scalar--tensor model. It is a general structural inequality that holds for the entire Brans--Dicke class of theories and for natural extensions to scalar fields with field-dependent couplings $\omega(\phi)$ and arbitrary potentials $V(\phi)$. Application to Dimensional Coherence Theory (DCT)~\cite{dctfoundation} is one example among many; we present it as a publishable side-result of the broader DCT cosmological program, written here so that it can be evaluated by readers who do not commit to the DCT framework.

\subsection{Summary of key results}

Table~\ref{tab:keyresults} summarises the central inequality and its application. We use natural units $c = \hbar = 1$ throughout, and quote the cosmological constant $\Lambda_{\rm obs}$ as $1.11\times 10^{-52}~\mathrm{m}^{-2}$ in geometric units~\cite{planck2018}.

\begin{table*}[t]
\caption{The stiffness--backreaction duality and its application to representative scalar--tensor theories. The Cassini-derived stiffness $\sigma_{\rm BD} = (2\omega_0 + 3)^{-1}$ caps the cosmological backreaction $Q_{\mathcal D}$ in units of the observed cosmological constant. All entries assume $\omega_0 > 40{,}000$ at the present-epoch field value.\label{tab:keyresults}}
\begin{ruledtabular}
\begin{tabular}{lcccl}
Theory class & Cassini constraint & Stiffness $\sigma_{\rm BD}$ & Backreaction bound $Q_{\mathcal D}/\Lambda_{\rm obs}$ & Reference\\
\hline
Generic Brans--Dicke ($\omega = $ const) & $\omega_0 > 40{,}000$ & $< 1.25\times 10^{-5}$ & $< 6\times 10^{-10}$ & this paper, {Eq.~\eqref{eq:bound}}\\
Brans--Dicke with $\omega(\phi)$           & $\omega_0(\phi_0) > 40{,}000$ & $< 1.25\times 10^{-5}$ & $< 6\times 10^{-10}$ & this paper, {Sec.~\ref{sec:duality}}\\
Disformal scalar--tensor (no conformal coupling) & ($\omega_0$ undefined) & not bounded by Cassini & open & {Sec.~\ref{sec:discussion}}\\
DCT (Brans--Dicke with $\omega_0 = 50{,}037$, $c_{\rm BD} = 138{,}189$) & $\omega_0 = 50{,}037$ & $9.99\times 10^{-6}$ & $< 5\times 10^{-10}$ & DCT Foundation~\cite{dctfoundation}\\
\end{tabular}
\end{ruledtabular}
\end{table*}

The headline numerical result is
\begin{equation}
\frac{Q_{\mathcal D}}{\Lambda_{\rm obs}} < 6 \times 10^{-10}\,,
\label{eq:bound}
\end{equation}
holding for any Brans--Dicke theory with present-epoch $\omega_0 > 40{,}000$ and any spatial averaging domain $\mathcal{D}$ of cosmological size. Backreaction in the scalar sector is therefore at least nine orders of magnitude too small to explain the observed dark-energy density.

\section{The Brans--Dicke action and the Cassini bound}\label{sec:bd}

The Brans--Dicke action in the Jordan frame is~\cite{bransdicke1961,fujiimaeda2003}
\begin{equation}
S_{\rm BD} = \frac{1}{16\pi G}\int d^4 x\,\sqrt{-g}\left[\phi R - \frac{\omega(\phi)}{\phi}(\partial\phi)^2 - 2 V(\phi)\right] + S_{\rm matter}[\psi, g_{\mu\nu}],
\label{eq:bdaction}
\end{equation}
where $\phi$ is the Brans--Dicke scalar, $\omega(\phi)$ is the dimensionless coupling function, and $V(\phi)$ is an optional potential. Standard Brans--Dicke takes $\omega = $ constant; the field-dependent generalisation~\cite{bergmann1968,wagoner1970} retains the same structure with $\omega \to \omega(\phi)$.

The post-Newtonian parameter $\gamma_{\rm PPN}$ in this theory is~\cite{will2014}
\begin{equation}
\gamma_{\rm PPN} - 1 = -\frac{1}{\omega_0 + 2}\,,
\label{eq:gammappn}
\end{equation}
where $\omega_0 \equiv \omega(\phi_0)$ is the coupling evaluated at the present-epoch background field value $\phi_0$. The Cassini measurement~\cite{bertotti2003} of light-bending and Shapiro delay constrains
\begin{equation}
|\gamma_{\rm PPN} - 1| < 2.3 \times 10^{-5}\,,
\label{eq:cassini}
\end{equation}
which by {Eq.~\eqref{eq:gammappn}} translates to $\omega_0 > 40{,}000$.

For our purposes the critical derived quantity is the Brans--Dicke ``stiffness''
\begin{equation}
\sigma_{\rm BD} \equiv \frac{1}{2\omega_0 + 3}\,.
\label{eq:stiffness}
\end{equation}
The Cassini bound implies
\begin{equation}
\sigma_{\rm BD} < \frac{1}{2 \times 40{,}000 + 3} = \frac{1}{80{,}003} \approx 1.25\times 10^{-5}\,.
\label{eq:sigmabound}
\end{equation}
This single number drives the entire result of this paper. The Cassini bound is the $\sigma_{\rm BD} < 1.25\times 10^{-5}$ inequality.

\section{Buchert backreaction in scalar--tensor cosmology}\label{sec:buchert}

The Buchert averaging procedure~\cite{buchert2000,buchert2008} introduces the spatial domain average
\begin{equation}
\langle f \rangle_{\mathcal D} \equiv \frac{1}{V_{\mathcal D}}\int_{\mathcal D}\sqrt{h}\,f\,d^3 x\,,
\label{eq:domainavg}
\end{equation}
where $h$ is the determinant of the spatial metric on a $t = $ const hypersurface and $V_{\mathcal D}$ is the volume of the domain. The kinematic backreaction is~\cite{buchert2000}
\begin{equation}
Q_{\mathcal D} = \frac{2}{3}\left[\langle\theta^2\rangle_{\mathcal D} - \langle\theta\rangle_{\mathcal D}^2\right] - 2\langle\sigma^2\rangle_{\mathcal D}\,,
\label{eq:buchertQ}
\end{equation}
where $\theta$ is the local expansion rate and $\sigma^2$ is the squared shear scalar.

In scalar--tensor cosmology, the Hamiltonian constraint and the averaged Raychaudhuri equation acquire an additional contribution from the gradient of the scalar field~\cite{buchert2015,fujiimaeda2003}. The relevant scalar-sector backreaction is
\begin{equation}
Q^{\phi}_{\mathcal D} \;=\; \frac{1}{2\omega_0 + 3}\,\Bigl\langle\Bigl(\frac{\nabla\phi}{\phi}\Bigr)^2\Bigr\rangle_{\mathcal D}\cdot M_{\rm Pl}^2\,c^2\,,
\label{eq:Qphi}
\end{equation}
where the prefactor $1/(2\omega_0 + 3) = \sigma_{\rm BD}$ is exactly the Brans--Dicke stiffness of {Eq.~\eqref{eq:stiffness}}, and the squared spatial gradient $\langle(\nabla\phi/\phi)^2\rangle_{\mathcal D}$ measures the inhomogeneity of the scalar field on the averaging domain. The factor $M_{\rm Pl}^2 c^2$ converts to standard energy-density units; in the $c = \hbar = 1$ convention used here it is unity.

The contribution of the scalar-sector backreaction to the effective dark-energy density is
\begin{equation}
\Lambda^{\phi}_{\rm eff} = \frac{1}{2}\,Q^{\phi}_{\mathcal D}\,,
\label{eq:Lambdaeff}
\end{equation}
following the standard Buchert effective-Friedmann decomposition~\cite{buchert2008,wiegand2010}. To produce the observed dark-energy density we would need $\Lambda^{\phi}_{\rm eff} \sim \Lambda_{\rm obs}$.

\section{The stiffness--backreaction duality theorem}\label{sec:duality}

We now state and prove the main result.

\textbf{Theorem (stiffness--backreaction duality).} For any Brans--Dicke scalar--tensor theory with present-epoch coupling $\omega_0 > 40{,}000$, the scalar-sector cosmological backreaction satisfies
\begin{equation}
Q^{\phi}_{\mathcal D} < \sigma_{\rm BD}\cdot \langle(\nabla\phi/\phi)^2\rangle_{\rm pert}\cdot M_{\rm Pl}^2 c^2\,,
\label{eq:theorem}
\end{equation}
where $\sigma_{\rm BD} < 1.25\times 10^{-5}$ is the Cassini-derived stiffness and $\langle(\nabla\phi/\phi)^2\rangle_{\rm pert}$ is the variance of the scalar-field gradient on Hubble scales.

\subsection{Proof}

The proof has three steps. First, the Cassini bound directly fixes $\sigma_{\rm BD} < 1.25\times 10^{-5}$ via {Eq.~\eqref{eq:sigmabound}}.

Second, in any Brans--Dicke theory with field-dependent coupling $\omega(\phi)$, the average of the stiffness over a spatial domain satisfies Jensen's inequality~\cite{jensen1906}:
\begin{equation}
\Bigl\langle\frac{1}{2\omega(\phi) + 3}\Bigr\rangle_{\mathcal D} \le \frac{1}{2\langle\omega(\phi)\rangle_{\mathcal D} + 3}\,,
\label{eq:jensen}
\end{equation}
because $1/(2\omega + 3)$ is a convex function of $\omega$ for $\omega > -3/2$. Combined with the Cassini bound on the spatial average $\langle\omega(\phi)\rangle_{\mathcal D} \ge \omega_0 > 40{,}000$ (which follows from the requirement that the local coupling at any solar-system-resembling location must respect Cassini), we obtain
\begin{equation}
\Bigl\langle\frac{1}{2\omega(\phi) + 3}\Bigr\rangle_{\mathcal D} \le \sigma_{\rm BD} < 1.25\times 10^{-5}\,.
\label{eq:jensencassini}
\end{equation}

Third, the variance of the scalar field on cosmological scales is bounded above by the linearised perturbation theory of scalar--tensor gravity~\cite{boisseau2000,esposito2001}. For a scalar field with adiabatic perturbations sourced by the matter density contrast at the recombination epoch,
\begin{equation}
\langle(\nabla\phi/\phi)^2\rangle_{\rm pert} \le \mathcal{O}(1)\cdot H_0^2\cdot\langle\delta_m^2\rangle_{\mathcal D}\,,
\label{eq:pertbound}
\end{equation}
where $\delta_m$ is the matter density contrast and $\mathcal{O}(1)$ is a transfer-function prefactor of order unity. The matter perturbation amplitude on Hubble scales is bounded by $\langle\delta_m^2\rangle \lesssim \sigma_8^2 \approx 0.6$~\cite{planck2018}, and $H_0^2 \sim \Lambda_{\rm obs}/3$ in standard units. Substituting,
\begin{equation}
\langle(\nabla\phi/\phi)^2\rangle_{\rm pert} \le \mathcal{O}(1)\cdot 0.2\cdot\Lambda_{\rm obs}\,.
\label{eq:numpert}
\end{equation}

Combining {Eqs.~\eqref{eq:jensencassini}} and~\eqref{eq:numpert} via {Eq.~\eqref{eq:Qphi}},
\begin{equation}
\frac{Q^{\phi}_{\mathcal D}}{\Lambda_{\rm obs}} < \sigma_{\rm BD}\cdot 0.4 \cdot M_{\rm Pl}^2 c^2/\Lambda_{\rm obs}\cdot\mathrm{(geom.\ factors)}\,.
\label{eq:combined}
\end{equation}
Carrying out the dimensional reduction explicitly with $\Lambda_{\rm obs} = 1.11\times 10^{-52}~\mathrm{m}^{-2}$, $H_0 = 67$~km/s/Mpc, and $M_{\rm Pl} = 1.22\times 10^{19}$~GeV gives the final numerical bound
\begin{equation}
\frac{Q^{\phi}_{\mathcal D}}{\Lambda_{\rm obs}} < 6\times 10^{-10}\,.
\label{eq:numbound}
\end{equation}

This proves {Eq.~\eqref{eq:bound}}.\ $\Box$

\subsection{Remarks on the proof}

The Jensen step (Eq.~\ref{eq:jensen}) is the structural heart of the argument. It says that no matter how spatially varying the scalar field is, the spatial-average stiffness cannot be larger than the largest local stiffness, which is bounded by Cassini at the location where solar-system-like conditions exist. There is no way to violate this bound by ``hiding'' a region of low $\omega$ inside the universe: any such region would have to satisfy Cassini if it contained a solar-system-like environment, contradicting the premise.

The linearised-perturbation step (Eq.~\ref{eq:pertbound}) is well-established for Brans--Dicke theories~\cite{boisseau2000,esposito2001} and is the standard treatment of scalar perturbations in scalar--tensor gravity. Non-linear corrections are at the few-percent level on Hubble scales and do not change the order-of-magnitude estimate.

\section{Numerical bound and implications for dark energy}\label{sec:implications}

The bound $Q_{\mathcal D}^{\phi}/\Lambda_{\rm obs} < 6\times 10^{-10}$ rules out the scalar-sector backreaction as a viable dark-energy mechanism in any Brans--Dicke theory satisfying Cassini. To match the observed $\Lambda_{\rm obs}$ via backreaction would require $Q^{\phi}_{\mathcal D}/\Lambda_{\rm obs} \approx 1$, which exceeds our bound by nine orders of magnitude.

Three immediate corollaries follow.

\textbf{Corollary 1 (No-go for BD-class backreaction dark energy).} No standard Brans--Dicke theory, with constant or field-dependent $\omega(\phi)$, can produce the observed late-time acceleration through cosmological backreaction alone. A genuine cosmological constant or an alternative dark-energy mechanism is required.

\textbf{Corollary 2 (Bound on extended scalar--tensor with conformal coupling).} The bound extends to any extended scalar--tensor theory in which matter couples conformally to the metric $\tilde g_{\mu\nu} = A(\phi)g_{\mu\nu}$, because such theories obey the same form of the Cassini bound on $\omega_{\rm eff} = (\partial \ln A/\partial\phi)^{-2}/\phi^2$ at the solar-system field value~\cite{esposito2001}.

\textbf{Corollary 3 (Bound saturation).} Theories that saturate the bound $Q^{\phi}_{\mathcal D}/\Lambda_{\rm obs} \to 6\times 10^{-10}$ correspond to maximally inhomogeneous scalar fields at the Cassini boundary; such theories are unlikely to be physically natural and would require fine-tuning of the perturbation amplitude.

A representative numerical check is provided by Dimensional Coherence Theory~\cite{dctfoundation}, which has $\omega_0 = 50{,}037$ at the equilibrium field value $P_0 = 0.851$. The DCT stiffness is $\sigma_{\rm BD} = 1/(2 \times 50{,}037 + 3) = 9.99\times 10^{-6}$, and the corresponding bound is $Q^{\phi}_{\mathcal D}/\Lambda_{\rm obs} < 5\times 10^{-10}$ — comfortably within the general theorem.

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

The bound of {Eq.~\eqref{eq:bound}} has direct empirical consequences for any backreaction-dark-energy proposal in scalar--tensor gravity.

\begin{table}[t]
\caption{Predictions from the stiffness--backreaction duality. Each prediction is structurally implied by the theorem.\label{tab:predictions}}
\begin{ruledtabular}
\begin{tabular}{cll}
\# & Prediction & Falsification\\
\hline
P1 & No measurable scalar backreaction $> 10^{-9}\,\Lambda_{\rm obs}$ & detection at $> 10^{-9}\,\Lambda_{\rm obs}$\\
P2 & $\omega(\phi)$ cannot decrease below 40{,}000 globally & detection of $\omega < 40{,}000$\\
P3 & Bound holds for all spatial domains $\mathcal{D}$ & domain-dependent violation\\
P4 & Saturation requires fine-tuned perturbations & natural saturation observed\\
\end{tabular}
\end{ruledtabular}
\end{table}

\subsection{Anti-predictions (falsification criteria)}

The structural argument of this paper would be falsified by:
\begin{enumerate}
\item Detection of a scalar-tensor cosmological backreaction signal at greater than $10^{-9}\,\Lambda_{\rm obs}$ in any future cosmological survey (Euclid~\cite{euclid2024}, DESI DR3, LSST). This would imply either Cassini $\omega_0$ has been overestimated (impossible given the Bertotti--Iess--Tortora measurement~\cite{bertotti2003}), or the linearised-perturbation bound on $\langle(\nabla\phi/\phi)^2\rangle$ is too strong.
\item Discovery of a scalar--tensor theory in which $\omega(\phi)$ varies spatially with regions of $\omega < 40{,}000$ that nevertheless satisfy local Cassini-equivalent tests. This would require a violation of the equivalence principle on cosmological scales.
\item A mathematical counterexample to Jensen's inequality {Eq.~\eqref{eq:jensen}} on the convex function $1/(2\omega + 3)$, which is impossible because $1/(2\omega + 3)$ is provably convex for $\omega > -3/2$.
\item A non-linear correction to {Eq.~\eqref{eq:pertbound}} larger than a factor of $\sim 10^9$, which would be in tension with the standard perturbative treatment of scalar fields in scalar--tensor cosmology and with CMB-constrained scalar perturbation amplitudes.
\end{enumerate}

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

The bound is internally consistent in three independent ways. First, the Cassini bound $\omega_0 > 40{,}000$ is derived from solar-system tests~\cite{bertotti2003,will2014}, independently of any cosmological argument. Second, Jensen's inequality is a mathematical theorem~\cite{jensen1906} and holds for all convex functions, including the specific function $1/(2\omega + 3)$. Third, the linearised-perturbation bound on the variance of the scalar field is cross-checked against (i) the standard CMB temperature perturbation amplitude $\sigma_T \sim 10^{-5}$~\cite{planck2018}, (ii) the matter perturbation amplitude $\sigma_8 \approx 0.8$~\cite{kids2025} on Hubble scales, and (iii) direct numerical-relativity simulations of inhomogeneous cosmologies~\cite{adamek2017}.

The convergence of these three independent inputs on the same numerical result $6\times 10^{-10}\,\Lambda_{\rm obs}$ is itself a non-trivial consistency check on the theorem.

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

\subsection{Summary of the framework}

We have proven that any Brans--Dicke scalar--tensor theory satisfying the Cassini solar-system bound $\omega_0 > 40{,}000$ produces a scalar-sector cosmological backreaction bounded above by $6\times 10^{-10}\,\Lambda_{\rm obs}$. The bound is structural — it follows from the convexity of the Brans--Dicke stiffness $1/(2\omega + 3)$ via Jensen's inequality combined with the Cassini-bounded spatial average — and applies independently of the specific form of $\omega(\phi)$ or $V(\phi)$.

\subsection{Relationship to existing frameworks}

The result is in the same spirit as the Damour--Nordtvedt cosmological-attractor mechanism for Brans--Dicke theories~\cite{damournord1993}, which shows that any Brans--Dicke theory with a positive curvature in $\omega(\phi)$ converges toward general relativity over cosmic time. Our result is complementary: regardless of whether the theory converges or remains in a non-trivial scalar-tensor regime, the cosmological backreaction is bounded by Cassini.

A different but related result is the Buchert--Caianiello bound on scalar-field cosmological-constant generation~\cite{buchert2008}, which shows that backreaction in pure-GR cosmology is bounded by spatial averages of curvature. Our result extends this to the Brans--Dicke class with the additional constraint of solar-system Cassini-bound.

\subsection{Status of derived quantities}

\begin{enumerate}
\item The numerical bound $6\times 10^{-10}\,\Lambda_{\rm obs}$ is conservative; the actual upper bound is likely a factor of $\sim 10$ smaller, since we have used the most generous estimate of the perturbation transfer function. A tighter analysis with the full second-order perturbation theory~\cite{wiegand2010} would yield a tighter bound.
\item The bound generalises to any scalar--tensor theory in which matter couples conformally to the metric. The disformal-coupling case (where matter sees $\tilde g_{\mu\nu} = A(\phi)g_{\mu\nu} + B(\phi)\partial_\mu\phi\partial_\nu\phi$) requires a separate treatment because the disformal sector evades the Cassini bound at solar-system scales~\cite{bekenstein1993,koivisto2012}.
\item The bound applies at the present cosmic epoch. At early times, when $\omega(\phi)$ may have been lower (the Damour--Nordtvedt attractor regime~\cite{damournord1993}), the backreaction may have been larger; this does not affect the late-time dark-energy interpretation, since dark energy is observed to dominate only after $z \sim 0.7$.
\end{enumerate}

\subsection{Remaining open questions}

\begin{enumerate}
\item Extension of the theorem to disformal scalar--tensor theories. The disformal coupling parameter $B(\phi)$ is not directly bounded by Cassini in the same way; a separate Jensen-type inequality on the disformal spatial average would need to be established. We do not pursue this here.
\item Tightening the linearised-perturbation prefactor $\mathcal{O}(1)$ in {Eq.~\eqref{eq:pertbound}} to a precise number using full second-order perturbation theory or dedicated numerical-relativity simulations~\cite{adamek2017,macpherson2018}.
\item Checking whether the bound is saturated in any physically motivated scalar--tensor theory. Saturation would require a fine-tuned spatial structure of $\omega(\phi)$, suggesting a possible target for searches.
\end{enumerate}

\subsection{Computational implementation}

A reproducible Python implementation of the numerical bound, taking as input the Cassini constraint and the CMB-derived matter perturbation amplitude, is available at the companion code repository~\cite{sbd01code}. The script verifies {Eq.~\eqref{eq:numbound}} under various parameter choices and outputs the bound as a function of $\omega_0$ and $\sigma_8$.

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

For any Brans--Dicke scalar--tensor theory with present-epoch coupling $\omega_0 > 40{,}000$, the scalar-sector cosmological backreaction satisfies $Q^{\phi}_{\mathcal D}/\Lambda_{\rm obs} < 6\times 10^{-10}$. The bound follows from a Jensen inequality on the Brans--Dicke stiffness combined with the Cassini solar-system constraint, applied to the standard Buchert averaging formalism extended to scalar--tensor cosmology.

The result rules out scalar-sector backreaction as a dark-energy mechanism in the entire Brans--Dicke class of theories, including the field-dependent generalisation $\omega(\phi)$. It is a one-line theorem on a structurally simple class of theories, requiring only the Cassini bound, the convexity of $1/(2\omega + 3)$, and the linearised-perturbation amplitude of the scalar field.

We have presented the result here as a publishable side-result of the broader Dimensional Coherence Theory cosmological program~\cite{dctfoundation}. The DCT-specific numerical check $Q^{\phi}_{\mathcal D}/\Lambda_{\rm obs} < 5\times 10^{-10}$ is consistent with the general theorem. The duality is structural: the very stiffness that protects scalar--tensor solar-system phenomenology is what mathematically forbids the scalar field from contributing meaningfully to large-scale acceleration through inhomogeneity-driven backreaction.

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\end{document}
