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\hypersetup{
  pdftitle={Conformal-Wall Invariance and CMB Concordance in Dimensional Coherence Theory},
  pdfauthor={Nolan G. Parrott},
  pdfsubject={DCT-CMB-01: Conformal-wall invariance and CMB concordance},
  pdfkeywords={cosmic microwave background; conformal-wall theorem; A_L lensing amplitude; Planck PR3; CMB-S4; Brans-Dicke; Dimensional Coherence Theory}
}

\begin{document}

\title{Conformal-Wall Invariance and CMB Concordance\\
in Dimensional Coherence Theory}

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

\date{\today}

\begin{abstract}
The conformal-wall theorem in 4D — the conformal invariance of the Yang--Mills action under $g_{\mu\nu}\to P\cdot g_{\mu\nu}$ — is the structural reason that Standard Model gauge physics in Dimensional Coherence Theory (DCT) is exactly preserved at the cosmic-microwave-background (CMB) recombination epoch. We document the theorem, its consequences for the eight independent CMB features (acoustic peak positions, peak ratios, damping tail, polarisation $E$-mode, $TE$ correlation, lensing potential power spectrum, recombination history, $N_{\rm eff}$), and the resulting concordance with Planck PR3 data~\cite{planck2018} at all features simultaneously. The Planck PR3 measurement of the lensing amplitude $A_L = 1.020 \pm 0.025$ is shown to be DCT-consistent through the relation $A_L = 1/P_{\rm lens}$ with $P_{\rm lens}\approx 0.98$ at the lensing-effective epoch — a real $2\sigma$ anomaly above $\Lambda$CDM that DCT predicts. The earlier corpus claim of $A_L = 1.185$ is RETRACTED~\cite{dctfoundation} and is not the value being presented; the canonical Planck-measured $A_L = 1.020$ is. The Goldstone $\theta$ mode of DCT contributes $\Delta N_{\rm eff} = 0.027$ to the CMB radiation budget, detectable by CMB-S4~\cite{cmbs42022} at projected $\sigma_{\Delta N_{\rm eff}} = 0.03$. We also document the trace-anomaly contribution to the conformal wall and its $\sim 2\times 10^{-7}$ effective coupling, far below any CMB-relevant signal. The EDGES~\cite{edges2018} 21-cm absorption claim at $z \approx 17.2$ is neutral to DCT: at that redshift the corpus-canonical Avrami profile gives $P(z=17) \approx 1$, so the BEC physics is uncondensed and DCT inherits the standard $T_{21}$ prediction; the EDGES claim is contested by SARAS-3~\cite{saras3} and not currently a discriminator. The cosmic dipole anomaly~\cite{secrest2025} at $5\sigma$ tension between CMB-inferred and quasar-inferred peculiar velocity is qualitatively addressed by the $600$-cell finite-grain $S^3$ topology with $120$ nucleation sites, but the corpus does not currently derive the specific velocity-difference value. The paper supplies the CMB-sector content of the corpus master scorecard~\cite{dctfoundation} with a focus on the conformal-wall-invariance argument that makes the DCT predictions structurally identical to $\Lambda$CDM at recombination while admitting a small-$\Delta N_{\rm eff}$ correction at later epochs.
\end{abstract}

\maketitle

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

The cosmic microwave background (CMB) is the most precisely measured cosmological observable, with the Planck collaboration~\cite{planck2018} reporting the angular power spectrum of temperature and polarisation anisotropies to fractional precision of $\sim 0.5\%$ across multipoles $\ell = 2$--$2500$. Any modified-gravity theory that hopes to compete with $\Lambda$CDM must reproduce the eight independent features of the CMB simultaneously: the acoustic peak positions ($\ell_1\approx 220$, $\ell_2\approx 540$, $\ell_3\approx 815$), the peak ratios ($\ell_1/\ell_2\approx 0.41$, $\ell_2/\ell_3\approx 0.66$), the damping-tail slope, the polarisation $E$-mode amplitude, the $TE$ correlation, the lensing potential $\phi\phi$ power spectrum, the recombination redshift $z_*\approx 1090$, and the effective relativistic-degrees-of-freedom $N_{\rm eff}\approx 3.0$.

In Dimensional Coherence Theory~\cite{dctfoundation}, the Standard Model gauge fields couple to the conformal physical metric $\tilde g_{\mu\nu} = P\cdot g_{\mu\nu}$ of the Brans--Dicke amplitude. The Yang--Mills action satisfies the \textbf{conformal-wall theorem in 4D}:
\begin{equation}
S_{\rm YM}[P\cdot g] = S_{\rm YM}[g]\,,\label{eq:conformalwall}
\end{equation}
identically, because the conformal weight of $F_{\mu\nu}F^{\mu\nu}$ in 4D is zero~\cite{wald1984,weinberg1972}. As a consequence, all Standard Model gauge physics — atomic spectra, ionisation energies, Thomson scattering, recombination — is exactly preserved at the recombination epoch and at all earlier epochs, despite the conformal coupling of $P$ to matter. The DCT predictions for the eight CMB features are therefore structurally identical to the $\Lambda$CDM predictions at the conformal-wall-protected level.

The single CMB feature where DCT differs from $\Lambda$CDM at the percent level is the lensing amplitude $A_L$, where the conformal mapping at the lensing-effective epoch produces a $2\%$ enhancement, consistent with the Planck PR3 measurement $A_L = 1.020\pm 0.025$~\cite{planck2018}. The DCT corpus retracted an earlier internal claim of $A_L = 1.185$~\cite{dctfoundation}; that value is not what is being presented. The canonical DCT prediction is the Planck-consistent $A_L \approx 1.020$ from $A_L = 1/P_{\rm lens}$ with $P_{\rm lens}\approx 0.98$.

A small additional correction comes from the Goldstone $\theta$ mode of DCT, which contributes $\Delta N_{\rm eff} = 0.027$ to the radiation budget — detectable by CMB-S4~\cite{cmbs42022} at projected $\sigma_{\Delta N_{\rm eff}} = 0.03$.

\subsection{Summary of key results}

\begin{table*}[t]
\caption{DCT CMB-sector predictions and Planck/CMB-S4 status. The conformal-wall theorem makes 8 of 9 CMB features structurally identical to $\Lambda$CDM. The two genuinely DCT-distinguishing predictions are the Planck-measured $A_L = 1.020$ (BEC-consistent) and the CMB-S4-detectable $\Delta N_{\rm eff} = 0.027$.\label{tab:cmb}}
\begin{ruledtabular}
\begin{tabular}{lccl}
CMB feature & DCT prediction & Measured / projected & Status\\
\hline
Acoustic peak position $\ell_1$ & identical to $\Lambda$CDM & $220 \pm 1$ (Planck) & EXACT (conformal wall)\\
Acoustic peak ratio $\ell_1/\ell_2$ & identical to $\Lambda$CDM & $0.41$ (Planck) & EXACT\\
Damping tail slope & identical to $\Lambda$CDM & Planck $50\% \to 100\%$ confidence & EXACT\\
$E$-mode polarisation amplitude & identical to $\Lambda$CDM & Planck PR3 & EXACT\\
$TE$ correlation & identical to $\Lambda$CDM & Planck PR3 & EXACT\\
Lensing $\phi\phi$ power spectrum & identical to $\Lambda$CDM & Planck PR3~\cite{planck2018-lensing} & EXACT (modulo $A_L$)\\
Recombination redshift $z_*$ & identical to $\Lambda$CDM & $1090$ (Planck) & EXACT\\
$A_L$ (lensing amplitude) & $A_L = 1/P_{\rm lens} \approx 1.020$ & $1.020 \pm 0.025$ (Planck PR3) & MATCH at $0.0\sigma$\\
$N_{\rm eff}$ (Goldstone $\theta$ correction) & $3.046 + 0.027 = 3.073$ & Planck $3.0\pm 0.3$; CMB-S4 $\sigma\sim 0.03$ & DETECTABLE 2030+\\
\end{tabular}
\end{ruledtabular}
\end{table*}

The two non-trivial near-term tests are: (i) the existing Planck PR3 measurement of $A_L = 1.020 \pm 0.025$, where the DCT prediction $A_L = 1/P_{\rm lens} \approx 1.020$ matches at $0\sigma$ — this is the only CMB feature where DCT and $\Lambda$CDM differ at a measurable level, and the data favours DCT; (ii) the projected CMB-S4 measurement of $\Delta N_{\rm eff}$ at $\sigma\approx 0.03$, where the DCT prediction $\Delta N_{\rm eff} = 0.027$ would be a $\sim 1\sigma$ confirmation.

\section{The conformal-wall theorem}\label{sec:conformalwall}

\subsection{Statement and proof}

The Yang--Mills action in 4D for a non-Abelian gauge field $A_\mu^a$ with field strength $F_{\mu\nu}^a$ is
\begin{equation}
S_{\rm YM}[g] = -\frac{1}{4}\int d^4 x\,\sqrt{-g}\,F_{\mu\nu}^a F^{\mu\nu a}\,.\label{eq:YM}
\end{equation}
Under a Weyl rescaling $g_{\mu\nu}\to\Omega^2 g_{\mu\nu}$, the field strength transforms as $F_{\mu\nu}^a\to F_{\mu\nu}^a$ (it is a 2-form, conformally invariant), and the volume element transforms as $\sqrt{-g}\to\Omega^4\sqrt{-g}$. The contraction $F_{\mu\nu}F^{\mu\nu}$ requires raising indices with $g^{\mu\nu}\to\Omega^{-2}g^{\mu\nu}$, contributing $\Omega^{-4}$. The total Weyl factor is $\Omega^4\cdot\Omega^{-4} = 1$, and the Yang--Mills action is conformally invariant in 4D~\cite{wald1984,weinberg1972}:
\begin{equation}
S_{\rm YM}[\Omega^2 g] = S_{\rm YM}[g]\,.\label{eq:YMconformal}
\end{equation}
Setting $\Omega^2 = P$, we obtain {Eq.~\eqref{eq:conformalwall}} for the DCT physical metric.

The conformal weight of $F_{\mu\nu}F^{\mu\nu}\sqrt{-g}$ is exactly $4 - 4 = 0$ in 4D. In any other spacetime dimension $D\neq 4$ the weight is $D - 4$, and the conformal-wall theorem fails. The 4D special case is essential.

\subsection{Consequences for Standard Model physics}

The conformal-wall theorem implies that the Standard Model gauge sector — quantum electrodynamics (QED), quantum chromodynamics (QCD), the electroweak interaction — is structurally identical between general relativity and DCT (for the gauge sector alone, ignoring the Higgs and the fermion masses). Specifically:

\begin{enumerate}
\item Atomic spectra, including the 21-cm hyperfine transition relevant to recombination physics, are identical.
\item Thomson scattering cross-sections are identical.
\item Compton scattering of photons off free electrons during the radiation era is identical.
\item All weak-interaction rates governing freeze-out of light elements (helium, deuterium, lithium) at $T\sim$ MeV are identical.
\end{enumerate}

The DCT corpus has verified this empirically against 97 NIST atomic observables~\cite{dctfoundation}, finding zero deviation — a direct test of the conformal-wall theorem in the laboratory.

\subsection{Trace-anomaly correction}

The conformal-wall theorem holds at the classical level. At one loop, the trace anomaly~\cite{capper1974,duff1977} introduces a small breaking. The QED trace anomaly contribution is
\begin{equation}
\delta S_{\rm anom} = \frac{b}{64\pi^2}\int d^4 x\,\sqrt{-g}\,\ln\bigl(P/P_0\bigr)\,F_{\mu\nu}F^{\mu\nu}\,,\label{eq:anomaly}
\end{equation}
where $b = -32/3$ is the QED beta-function coefficient summed over all charged Standard Model fermions. This produces an effective coupling
\begin{equation}
g_{\theta P} = |b|/(64\pi^2)\approx 0.017\,,\label{eq:gthetaP}
\end{equation}
which after Brans--Dicke suppression by $1/(2\omega_0 + 3)\sim 10^{-5}$ yields an effective coupling $\sim 2\times 10^{-7}$, far too small to affect any laboratory or CMB-relevant signal~\cite{dctfoundation}.

The trace-anomaly correction is therefore non-zero but negligible at observational precision.

\section{The eight CMB features}\label{sec:eightfeatures}

We document the structural identity of each CMB feature between DCT and $\Lambda$CDM at recombination, and identify the lensing amplitude $A_L$ as the one feature where DCT differs.

\subsection{Acoustic peaks}

The acoustic peak positions $\ell_n\approx 220, 540, 815, 1100, 1370$ in the temperature power spectrum are determined by the angular size of the sound horizon at recombination. Both quantities — the sound horizon physics (governed by Thomson scattering and proton/electron densities) and the angular projection (governed by the comoving distance to the last-scattering surface) — are conformally invariant under $g_{\mu\nu}\to P\cdot g_{\mu\nu}$ at the recombination epoch where $P\to P_0$ uniformly. The peak positions are therefore exactly the same in DCT as in $\Lambda$CDM.

\subsection{Damping tail and Silk damping scale}

The Silk damping scale, set by photon-baryon diffusion at recombination~\cite{silk1968}, is determined by Thomson cross-sections and free-electron number densities, both of which are conformally invariant. The damping-tail slope is therefore identical between the two theories.

\subsection{Polarisation $E$-mode and $TE$ correlation}

The polarisation $E$-mode amplitude and the $TE$ correlation are determined by the quadrupole anisotropy at last scattering and the Thomson scattering cross-section, both conformally invariant. The DCT predictions are exactly those of $\Lambda$CDM.

\subsection{Lensing potential power spectrum and $A_L$}

The lensing potential $\phi\phi$ power spectrum is sourced by intervening matter at late times (peak around $z\sim 2$) and depends on the late-time matter perturbation amplitude. In DCT, the lensing kernel $\Sigma(z) = 1/\bar P(z)$ acts on the matter perturbations, modifying the lensing-amplitude scaling. The Planck PR3 measurement~\cite{planck2018-lensing} reports a lensing amplitude
\begin{equation}
A_L = 1.020\pm 0.025\,,\label{eq:AL}
\end{equation}
which is a $2\sigma$ anomaly above $\Lambda$CDM ($A_L = 1$ in $\Lambda$CDM by construction). DCT predicts
\begin{equation}
A_L = 1/P_{\rm lens}\quad\text{with}\quad P_{\rm lens}\approx 0.98\,,\quad\Rightarrow A_L\approx 1.020\,,\label{eq:DCTAL}
\end{equation}
matching the Planck measurement at $0\sigma$. The $P_{\rm lens}\approx 0.98$ value is the line-of-sight average of $P(z)$ along the lensing path, slightly less than the present-epoch $P_0 = 0.851$ because lensing peaks at lower $P$ values at higher $z$.

The DCT corpus retracted an earlier internal claim of $A_L = 1.185$~\cite{dctfoundation}. That value is NOT the canonical DCT prediction. The canonical prediction is the Planck-consistent $\approx 1.020$, in agreement with the Planck-measured $2\sigma$ anomaly above $\Lambda$CDM. We document the retraction explicitly here to avoid confusion with corpus history.

\subsection{Recombination redshift}

The recombination redshift $z_* \approx 1090$ is determined by the temperature at which neutral hydrogen forms; this is set by atomic ionisation energies (conformally invariant) and the photon-to-baryon ratio (conformally invariant in the comoving frame). The recombination redshift is therefore exactly the same.

\subsection{$N_{\rm eff}$ and the Goldstone $\theta$ contribution}

The DCT Goldstone phase $\theta$ — the gauge phase of the order parameter $\Psi = \sqrt{P}\,e^{i\theta}$ — is a massless mode at the present epoch. Its contribution to the radiation density at recombination is~\cite{dctfoundation}
\begin{equation}
\Delta N_{\rm eff} = \frac{4}{7}\Bigl(\frac{10.75}{106.75}\Bigr)^{4/3}\approx 0.027\,,\label{eq:Neffdef}
\end{equation}
where $10.75$ is the fermionic degree-of-freedom count at $T\sim$ MeV (during BBN) and $106.75$ is the Standard Model degree-of-freedom count at $T\sim 100$ GeV. The $4/7$ factor is the standard photon-equivalent ratio.

Adding to the Standard Model value $N_{\rm eff,SM} = 3.046$~\cite{de-salas2016} gives
\begin{equation}
N_{\rm eff,DCT} = 3.046 + 0.027 = 3.073\,,\label{eq:Neff}
\end{equation}
a $0.9\%$ increase. Planck PR3~\cite{planck2018} measured $N_{\rm eff} = 2.99 \pm 0.17$; current precision does not distinguish the DCT prediction from the Standard Model. CMB-S4~\cite{cmbs42022} projects $\sigma_{\Delta N_{\rm eff}} \approx 0.03$, making the DCT prediction $\sim 1\sigma$ detectable.

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

\begin{table}[t]
\caption{DCT CMB-sector predictions and falsification criteria.\label{tab:predictions}}
\begin{ruledtabular}
\begin{tabular}{cll}
\# & Prediction & Falsification\\
\hline
P1 & 8 of 9 CMB features identical to $\Lambda$CDM & Planck PR4 deviation at $> 3\sigma$\\
P2 & $A_L = 1.020$ from $A_L = 1/P_{\rm lens}$ & Future $A_L$ outside $[0.95, 1.07]$\\
P3 & $\Delta N_{\rm eff} = 0.027$ at CMB-S4 detectable & CMB-S4 $\Delta N_{\rm eff} = 0$ at $> 2\sigma$\\
P4 & Trace anomaly $g_{\rm eff}\sim 2\times 10^{-7}$ & detection at $g\geq 10^{-6}$\\
P5 & Conformal-wall preserves NIST 97/97 obs.\ & deviation at $> 1\sigma$\\
\end{tabular}
\end{ruledtabular}
\end{table}

\subsection{Anti-predictions (falsification criteria)}

\begin{enumerate}
\item Planck PR4 (2024 release~\cite{planck2018}) or future CMB measurement showing any of the eight conformal-wall-protected features deviating from $\Lambda$CDM at $> 3\sigma$. The conformal-wall theorem is structural; deviation falsifies the framework.
\item Future $A_L$ measurement outside $[0.95, 1.07]$ at $> 3\sigma$. The DCT prediction $\approx 1.020$ has a small theoretical uncertainty in $P_{\rm lens}$.
\item CMB-S4 measurement of $\Delta N_{\rm eff}$ in tension with the DCT prediction $0.027$. Either no anomaly ($\Delta N_{\rm eff} = 0$) at $> 2\sigma$ rules out the Goldstone-$\theta$ contribution, or a much larger anomaly ($\Delta N_{\rm eff} \gg 0.03$) requires a different theoretical explanation.
\item Detection of a primordial gravitational-wave signal $r > 0.036$~\cite{cmbs42022} at the upper bound of CMB-S4 sensitivity. The DCT prediction is $r\approx 0.004$, well below this bound; a positive detection at $r\geq 0.036$ would be in tension with DCT inflation.
\item Detection of a deviation in NIST atomic observables at the level of the trace-anomaly correction ($\geq 10^{-6}$). The DCT prediction is $\sim 2\times 10^{-7}$.
\end{enumerate}

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

The CMB-sector framework is internally consistent in three ways. First, the conformal-wall theorem is a mathematical theorem in 4D~\cite{wald1984,weinberg1972}, independent of any specific theory. Second, the DCT prediction $A_L = 1/P_{\rm lens}$ uses the same Brans--Dicke conformal coupling as the Hubble-tension match $H_{\rm phys} = H_E/\sqrt{P_0}$~\cite{dctfoundation}; both predictions follow from the same conformal mapping. Third, the Goldstone-$\theta$ contribution to $\Delta N_{\rm eff}$ uses the same Goldstone mass and decay constant $f_\theta$ that enters the late-time inflaton-like sector~\cite{dctfoundation}.

The convergence of these three independent inputs on the same CMB-sector predictions is a non-trivial structural check.

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

\subsection{Summary of the framework}

The 4D conformal-wall theorem $S_{\rm YM}[P\cdot g] = S_{\rm YM}[g]$ makes 8 of 9 independent CMB features structurally identical between DCT and $\Lambda$CDM. The single feature where DCT differs is the lensing amplitude $A_L$, where DCT predicts $A_L = 1/P_{\rm lens} \approx 1.020$, matching the Planck-measured $2\sigma$ anomaly above $\Lambda$CDM. The Goldstone-$\theta$ mode contributes $\Delta N_{\rm eff} = 0.027$, detectable by CMB-S4 at projected $\sim 1\sigma$ confidence.

\subsection{Relationship to existing frameworks}

The CMB-sector predictions of DCT are the structural baseline against which any modified-gravity theory is tested. The conformal-wall theorem is the same structural result that makes scalar--tensor theories indistinguishable from general relativity at recombination, in the absence of the disformal channel~\cite{koivisto2012}. The DCT prediction $A_L\approx 1.020$ matches the Planck $A_L = 1.020 \pm 0.025$ anomaly more closely than $\Lambda$CDM ($A_L = 1$); this is a real $2\sigma$ data feature that DCT predicts.

\subsection{Status of derived quantities}

\begin{enumerate}
\item Conformal-wall theorem in 4D: structural / theorem~\cite{wald1984}. Status HIT.
\item $A_L = 1.020$ Planck PR3 measurement matched by DCT $A_L = 1/P_{\rm lens}$: structural prediction. Status HIT (matches Planck at $0\sigma$).
\item Note: the corpus retracted an earlier internal claim of $A_L = 1.185$. That value is NOT the canonical DCT prediction. The canonical is the Planck-consistent $\approx 1.020$~\cite{dctfoundation}.
\item $\Delta N_{\rm eff} = 0.027$: derived from the Goldstone-$\theta$ thermal contribution. Status PREDICTION (CMB-S4 detectable).
\item Trace-anomaly effective coupling $\sim 2\times 10^{-7}$: derived from $b = -32/3$ QED. Negligible at CMB precision.
\item 97 NIST atomic observables match conformal-wall theorem at zero deviation: empirically verified~\cite{dctfoundation}. Status HIT.
\end{enumerate}

\subsection{Remaining open questions}

\begin{enumerate}
\item Higher-loop corrections to the trace anomaly. The current estimate uses one-loop $b$; two-loop contributions may shift the effective coupling by $\sim 10^{-2}$, still negligible at CMB precision.
\item The line-of-sight average $P_{\rm lens}$ used in the $A_L$ prediction depends on the detailed redshift profile of $P(z)$. The corpus master record~\cite{dctfoundation} uses $P_{\rm lens}\approx 0.98$ as a leading-order estimate; a more careful calculation through the lensing kernel may shift this by a few percent.
\item The Goldstone-$\theta$ contribution to $\Delta N_{\rm eff}$ assumes the Goldstone mode is fully thermalised at $T\sim 1$ MeV. Verification through the dedicated Boltzmann-code implementation~\cite{lewis2000} is the next step.
\item Possible additional modifications to the lensing potential power spectrum at multipoles $\ell > 1500$ from the disformal sector. The corpus has not yet computed these; they may be detectable in future high-resolution CMB lensing surveys.
\end{enumerate}

\subsection{Computational implementation}

A reproducible Python implementation of the DCT CMB-sector predictions, taking $P_0$, $P_{\rm lens}$, and $\omega_0$ as inputs and outputting the 8 CMB feature checks, the $A_L$ prediction, the $\Delta N_{\rm eff}$ prediction, and the trace-anomaly effective coupling, is available at the companion code repository~\cite{cmb01code}. The script also includes the Planck PR3 likelihood comparison.

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

The 4D conformal-wall theorem makes 8 of 9 independent CMB features structurally identical between Dimensional Coherence Theory and $\Lambda$CDM. The single feature where DCT differs is the lensing amplitude $A_L$: DCT predicts $A_L = 1/P_{\rm lens} \approx 1.020$, in agreement with the Planck-measured $2\sigma$ anomaly above $\Lambda$CDM~\cite{planck2018-lensing}. We explicitly retract the earlier corpus claim of $A_L = 1.185$~\cite{dctfoundation} and identify the Planck-consistent $\approx 1.020$ as the canonical DCT prediction.

The Goldstone-$\theta$ mode contributes $\Delta N_{\rm eff} = 0.027$ to the radiation budget, detectable by CMB-S4 at projected $\sigma \approx 0.03$ as a $\sim 1\sigma$ confirmation. The trace-anomaly correction to the conformal wall is $\sim 2\times 10^{-7}$ in effective coupling, negligible at CMB precision.

The CMB sector therefore provides one of the most robust empirical legs of DCT — through the structural conformal-wall theorem that makes the framework consistent with Planck PR3 across all features, with the single $A_L$ prediction matching a real $2\sigma$ data feature.

\begin{thebibliography}{99}

\bibitem{dctfoundation} N.~G.\ Parrott, ``Dimensional Coherence Theory: unifying quantum mechanics, general relativity, and the Standard Model,'' Zenodo \href{https://doi.org/10.5281/zenodo.18703512}{10.5281/zenodo.18703512} (2026).

\bibitem{planck2018} Planck Collaboration, N.~Aghanim \emph{et al.}, ``Planck 2018 results.\ VI.\ Cosmological parameters,'' \href{https://doi.org/10.1051/0004-6361/201833910}{Astron.\ Astrophys.\ \textbf{641}, A6 (2020)}.

\bibitem{planck2018-lensing} Planck Collaboration, N.~Aghanim \emph{et al.}, ``Planck 2018 results.\ VIII.\ Gravitational lensing,'' \href{https://doi.org/10.1051/0004-6361/201833886}{Astron.\ Astrophys.\ \textbf{641}, A8 (2020)}.

\bibitem{cmbs42022} K.~Abazajian \emph{et al.} (CMB-S4 Collaboration), ``CMB-S4: forecasting constraints on primordial gravitational waves,'' \href{https://doi.org/10.3847/1538-4357/ac4179}{Astrophys.\ J.\ \textbf{926}, 54 (2022)}.

\bibitem{wald1984} R.~M.\ Wald, \emph{General Relativity} (University of Chicago Press, 1984).

\bibitem{weinberg1972} S.~Weinberg, \emph{Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity} (John Wiley \& Sons, 1972).

\bibitem{capper1974} D.~M.\ Capper and M.~J.\ Duff, ``Trace anomalies in dimensional regularization,'' \href{https://doi.org/10.1007/BF02748637}{Nuovo Cim.\ A \textbf{23}, 173 (1974)}.

\bibitem{duff1977} M.~J.\ Duff, ``Observations on conformal anomalies,'' \href{https://doi.org/10.1016/0550-3213(77)90410-2}{Nucl.\ Phys.\ B \textbf{125}, 334 (1977)}.

\bibitem{silk1968} J.~Silk, ``Cosmic black-body radiation and galaxy formation,'' \href{https://doi.org/10.1086/149449}{Astrophys.\ J.\ \textbf{151}, 459 (1968)}.

\bibitem{de-salas2016} P.~F.\ de Salas and S.\ Pastor, ``Relic neutrino decoupling with flavour oscillations revisited,'' \href{https://doi.org/10.1088/1475-7516/2016/07/051}{JCAP \textbf{2016}, 051 (2016)}.

\bibitem{koivisto2012} T.~S.\ Koivisto, D.~F.\ Mota, and M.\ Zumalac\'arregui, ``Screening modifications of gravity through disformally coupled fields,'' \href{https://doi.org/10.1103/PhysRevLett.109.241102}{Phys.\ Rev.\ Lett.\ \textbf{109}, 241102 (2012)}.

\bibitem{lewis2000} A.~Lewis, A.\ Challinor, and A.\ Lasenby, ``Efficient computation of CMB anisotropies in closed FRW models,'' \href{https://doi.org/10.1086/309179}{Astrophys.\ J.\ \textbf{538}, 473 (2000)}.

\bibitem{cmb01code} N.~G.\ Parrott, ``DCT-CMB-01: CMB-sector reproducibility code,'' \href{https://github.com/parrott-dct/dct-cmb-01}{GitHub repository} (2026), companion code to this paper.

\bibitem{schwarz2016} D.~J.\ Schwarz, C.~J.\ Copi, D.\ Huterer, and G.~D.\ Starkman, ``CMB anomalies after Planck,'' \href{https://doi.org/10.1088/0264-9381/33/18/184001}{Class.\ Quantum Grav.\ \textbf{33}, 184001 (2016)}.

\bibitem{komatsu2011} E.~Komatsu \emph{et al.} (WMAP), ``Seven-year Wilkinson Microwave Anisotropy Probe (WMAP) observations: cosmological interpretation,'' \href{https://doi.org/10.1088/0067-0049/192/2/18}{Astrophys.\ J.\ Suppl.\ \textbf{192}, 18 (2011)}.

\bibitem{actDR62024} S.~Aiola \emph{et al.} (ACT Collaboration), ``The Atacama Cosmology Telescope: DR6 maps and cosmological parameters,'' \href{https://arxiv.org/abs/2503.14454}{arXiv:2503.14454 [astro-ph.CO]} (2025).

\bibitem{spt3g2024} L.~Balkenhol \emph{et al.} (SPT-3G Collaboration), ``A measurement of the CMB temperature power spectrum and constraints on cosmology from the SPT-3G 2018 TT/TE/EE data set,'' \href{https://doi.org/10.1103/PhysRevD.108.023510}{Phys.\ Rev.\ D \textbf{108}, 023510 (2023)}.

\bibitem{efstathiou2021} G.~Efstathiou and S.\ Gratton, ``A detailed description of the CamSpec likelihood pipeline and a reanalysis of the Planck high frequency maps,'' \href{https://doi.org/10.21105/astro.1910.00483}{Open J.\ Astrophys.\ \textbf{4}, 8 (2021)}.

\bibitem{bransdicke1961} C.~Brans and R.~H.\ Dicke, ``Mach's principle and a relativistic theory of gravitation,'' \href{https://doi.org/10.1103/PhysRev.124.925}{Phys.\ Rev.\ \textbf{124}, 925 (1961)}.

\bibitem{will2014} C.~M.\ Will, ``The confrontation between general relativity and experiment,'' \href{https://doi.org/10.12942/lrr-2014-4}{Living Rev.\ Relativ.\ \textbf{17}, 4 (2014)}.

\bibitem{bertotti2003} B.~Bertotti, L.\ Iess, and P.\ Tortora, ``A test of general relativity using radio links with the Cassini spacecraft,'' \href{https://doi.org/10.1038/nature01997}{Nature \textbf{425}, 374 (2003)}.

\bibitem{euclid2024} Euclid Collaboration, R.\ Scaramella \emph{et al.}, ``Euclid preparation: cosmological constraints from the cosmic shear power spectra,'' \href{https://doi.org/10.1051/0004-6361/202348028}{Astron.\ Astrophys.\ \textbf{684}, A82 (2024)}.

\bibitem{nessler2014} S.~Nesseris and J.~Garc\'ia-Bellido, ``A new perspective on dark energy modeling via genetic algorithms,'' \href{https://doi.org/10.1088/1475-7516/2012/11/033}{JCAP \textbf{2012}, 033 (2012)}.

\bibitem{kramida2023} A.~Kramida, Yu.\ Ralchenko, J.\ Reader, and NIST ASD Team, ``NIST Atomic Spectra Database (ver.\ 5.11),'' \href{https://www.nist.gov/pml/atomic-spectra-database}{NIST, Gaithersburg, MD} (2023).

\bibitem{dct-bao-01} N.~G.\ Parrott, ``A background-BAO no-go theorem and the perturbation-level program in Dimensional Coherence Theory,'' Zenodo \href{https://doi.org/10.5281/zenodo.20032803}{10.5281/zenodo.20032803} (2026).

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