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\hypersetup{
  pdftitle={Dimensional Coherence Theory: Unifying Quantum Mechanics, General Relativity, and the Standard Model - Version 2 (Revised)},
  pdfauthor={Nolan G. Parrott},
  pdfsubject={Dimensional Coherence Theory v2; revision of zenodo.18703512},
  pdfkeywords={Brans-Dicke; 600-cell; binary icosahedral group; McKay correspondence; Hubble tension; BAO; KiDS-Legacy; perturbation-level program; Dimensional Coherence Theory}
}

\begin{document}

\title{Dimensional Coherence Theory:\\
Unifying Quantum Mechanics, General Relativity, and the Standard Model\\
{\it Version 2 (Revised)}}

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

\date{\today}

\begin{abstract}
This is a revised version of the February 2026 deposit~\cite{dctfoundation}. The mathematical framework is unchanged. Three classes of revision are made: (i) the {\it Sec.~XIII.A} ``no tension exceeding 2$\sigma$'' headline is reconciled with the explicit acknowledgment of the background-frame BAO falsification at 5.8$\sigma$ versus DESI Year-1, the KiDS-Legacy 2025 closure of the $S_8$ tension, and the cosmic-chronometer per-$z$ ratio test favouring $\Lambda$CDM at $0.63\sigma$; (ii) the conformal-frame mapping that resolves the Hubble tension at $H_{\rm phys} = H_E/\sqrt{P_0} = 73.06$ km/s/Mpc is presented honestly as mutually exclusive with the BAO concordance, and the live perturbation-level programme on a $\Lambda$CDM background ($\mu_b = 1/P$, $\mu_{\rm DM} = 1/[P(1+\beta)]$, $\Sigma = 1/\bar P$) is the framework on which the surviving cosmological predictions rest; (iii) the cosmological-data references are updated through KiDS-Legacy 2025, DESI DR2, and ACT DR6. The empirical scorecard is updated from the v1 ``629 observables'' headline to the DCT\_27 master scorecard ($55$ HIT / $10$ SOFT / $5$ FLAG / $1$ MISS / $6$ OPEN across $80$ graded). The joint Bayesian preference is recomputed under post-2025 data: $14.12\sigma\,/\,7.01\sigma\,/\,5.40\sigma$ at the three nested-test levels (vs.\ pre-2025 $14.0\,/\,10.5\,/\,8.0$). Calibrated probability of correctness is bounded at $35$--$50\%$. All other content (the canonical action, the McKay derivation of the Standard Model gauge group, the Avrami radial-acceleration relation, the proton-to-electron mass ratio at $9.2\times 10^{-8}$, the 12 polytope-derived structural identities, the PPN $\gamma - 1 = -2\times 10^{-5}$ BepiColombo MORE 2028 prediction, and the smoking-gun $M_{\rm lens}/M_{\rm dyn}$ turnover at $z\sim 1.5$ peak $1.30$ for Euclid 2027--2029) is preserved from v1 with the per-paper detail moved to standalone companion deposits in the same DCT cluster.
\end{abstract}

\maketitle

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

Dimensional Coherence Theory (DCT)~\cite{dctfoundation} is a unified Brans--Dicke scalar--tensor framework~\cite{bransdicke1961} in which the universe is described by a single complex order parameter $\Psi = \sqrt{P}\,e^{i\theta}$ of a cosmic Bose--Einstein condensate whose lattice topology is fixed by the 600-cell, the unique convex regular 4-polytope with 120 vertices, 720 edges, and binary-icosahedral symmetry $2I$~\cite{coxeter1973,duval1964}. Through the McKay correspondence~\cite{mckay1980,slodowy1980}, the binary icosahedral group is associated with the affine Dynkin diagram of $E_8$, providing a discrete topological bridge from the polytope to the largest exceptional Lie algebra and, via the standard descent $E_8\to E_6\times\mathrm{SU}(3)\to\mathrm{SO}(10)\to\mathrm{SU}(5)\to\mathrm{SU}(3)_C\times\mathrm{SU}(2)_L\times\mathrm{U}(1)_Y$, to the Standard Model gauge structure~\cite{georgi1974,slansky1981}.

The framework was deposited in February 2026 as zenodo.18703512~\cite{dctfoundation}. The present paper is a revision (v2) that addresses three classes of issue identified in the corpus internal audit of May 2026~\cite{audit2026}. The revision does not modify the mathematical framework — the canonical action, the McKay derivation, the Avrami profile, the proton-to-electron mass ratio, the structural identities, and the PPN predictions are all preserved from v1. The revisions are: (i) reconciling the v1 ``no tension exceeding 2$\sigma$'' framing with the explicit acknowledgment of the background BAO falsification, the KiDS-Legacy $S_8$ closure, and the cosmic-chronometer per-$z$ negative; (ii) replacing the conformal-frame Hubble-tension framing at the background level with the perturbation-level programme on a $\Lambda$CDM background, which is the live framework after the audit; (iii) updating cosmological-data references and the joint-Bayesian recomputation through 2025 data.

Detailed treatment of each domain — the spectral identities, the Brans--Dicke stiffness--backreaction duality, the BAO no-go theorem, the cosmological tensions re-read, the solar-system PPN predictions, the dark-matter Avrami crystallization, the Standard Model gauge derivation, and the CMB conformal-wall theorem — has been moved to standalone companion deposits in the DCT cluster~\cite{dct-spi-01,dct-sbd-01,dct-bao-01,dct-cos-01,dct-ppn-01,dct-dm-01,dct-sm-01,dct-cmb-01}, each of which deposits the full per-domain technical detail. This v2 paper is the umbrella that ties the cluster together.

\subsection{Summary of revisions from v1 to v2}

\begin{table*}[t]
\caption{Revisions from DCT Foundation v1 (Feb 2026) to v2 (May 2026). Each revision corresponds to an audit finding documented in~\cite{audit2026}.\label{tab:revisions}}
\begin{ruledtabular}
\begin{tabular}{lll}
v1 statement & v2 status & Reference\\
\hline
``No tension exceeding 2$\sigma$ has been found'' & RECONCILED with explicit BAO/$S_8$/CC tensions & {Sec.~\ref{sec:tensions}}\\
$H_{\rm phys} = H_E/\sqrt{P_0} = 73.06$ km/s/Mpc resolves $H_0$ tension & MUTUALLY EXCLUSIVE with BAO; perturbation-level programme is live & {Sec.~\ref{sec:bao}}, \cite{dct-bao-01}\\
KiDS-1000 $S_8 = 0.766$ cited & UPDATED to KiDS-Legacy 2025 $S_8 = 0.815 \pm 0.018$ & {Sec.~\ref{sec:tensions}}, \cite{kids2025}\\
``629+ matched observables'' headline & RETRACTED; replaced by DCT\_27 master scorecard 55/10/5/1/6 across 80 graded & {Sec.~\ref{sec:scorecard}}\\
$\omega(P) = (cP^{5/2} - 3)/2$ with $c = 149{,}800$ pre-S35 form & SUPERSEDED by $n=2$, $c_{\rm BD} = 138{,}189$ canonical & {Sec.~\ref{sec:action}}\\
$\delta_{\rm CP} = 2\pi/3 = 120^\circ$ & RETRACTED; canonical $\pi/3 = 60^\circ$ from spinorial branch & \cite{dct-sm-01}\\
$m_p/m_e$ at $5.4\sigma$ chance probability & RETRACTED to $4.6\sigma$, $P_{\rm chance} = 2.6\times 10^{-5}$ & \cite{dct-spi-01}\\
$A_L = 1.185$ DCT-internal value & RETRACTED; canonical Planck-measured $A_L = 1.020 \pm 0.025$ is BEC-consistent & \cite{dct-cmb-01}\\
SPARC ``mean $\chi^2/\mathrm{N} = 0.97$'' headline & DOCUMENTED honestly: real-data per-galaxy $\sim 2.4$, synthetic SPARC-LIKE $1.141$ & \cite{dct-dm-01}\\
Joint Bayesian pre-2025 $14.0\,/\,10.5\,/\,8.0\sigma$ & RECOMPUTED post-2025 to $14.12\,/\,7.01\,/\,5.40\sigma$ & {Sec.~\ref{sec:bayesian}}, \cite{dct-cos-01}\\
\end{tabular}
\end{ruledtabular}
\end{table*}

\section{The canonical action and topological inputs}\label{sec:action}

The canonical DCT action 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}
with the conformal physical metric $\tilde g_{\mu\nu} = P\, g_{\mu\nu}$ for the matter coupling, the canonical post-S35 coupling function
\begin{equation}
\omega(P) = \frac{c_{\rm BD}\,P^2 - 3}{2}\,,\quad c_{\rm BD} = 138{,}189\,,\label{eq:omega}
\end{equation}
and the Gross--Pitaevskii quantum-droplet potential $V(P)$~\cite{dctfoundation}. The post-S35 $n = 2$ form replaces the pre-S35 $n = 5/2$ Lee--Huang--Yang form $\omega(P) = (149{,}800\,P^{5/2} - 3)/2$~\cite{audit2026}; the new value is uniquely selected by 5 internal consistency checks~\cite{dctfoundation} and $c_{\rm BD}$ is derived as a renormalisation-group fixed point in the corpus DCT\_31 derivation~\cite{audit2026}.

The equilibrium values are $P_0 = 0.851$ and $\omega_0 = 50{,}037$, related by the master identity
\begin{equation}
2\omega_0 + 3 = c_{\rm BD}\,P_0^2 = 100{,}077\,.\label{eq:master}
\end{equation}
The topological derivation gives $P_0 = (9/10)\cdot(19/20) = 171/200 = 0.855$ from the 600-cell vertex-figure structure, with the $0.47\%$ discrepancy from the algebraic $P_0 = 0.851$ remaining as an open question requiring a rigorous Bogoliubov-depletion proof~\cite{dctfoundation}.

The 600-cell topological inputs are: vertex count $V = 120$, edge count $E = 720$, triangular face count $F = 1200$, tetrahedral cell count $C = 600$, coordination number $z = 12$, vertex-figure face count $f_v = 20$. Through the McKay correspondence~\cite{mckay1980}, the binary icosahedral group $2I$ of order $120$ corresponds to the affine Dynkin diagram of $E_8$.

\section{The eight surviving DCT predictions}\label{sec:predictions}

The eight non-trivial surviving DCT predictions, presented one per companion deposit in the cluster, are:

\begin{enumerate}
\item \textbf{Twelve polytope-derived structural identities} — DCT-SPI-01~\cite{dct-spi-01}. The proton-electron mass ratio $m_p/m_e = z\cdot 153 + 1/\varphi^4 + 1/z^2 = 1836.152842$ at $9.2\times 10^{-8}$, the spectral identity $G_{\rm LHY} = 3701/6300$ with $3701$ prime, the Casimir integers $31$ and $154$, the CKM mixing angles, the neutrino mass-squared splitting ratio. Joint chance probability $3.34\times 10^{-28}$ under independence; $\log_{10}(\text{BF}) = +27.5$; conservative $3.9\sigma$; LEE-corrected $> 9\sigma$. Post-diction with explicit pre-registration protocol.

\item \textbf{Stiffness--backreaction duality theorem} — DCT-SBD-01~\cite{dct-sbd-01}. For any Brans--Dicke theory with $\omega_0 > 40{,}000$ (Cassini), the cosmological backreaction is bounded above by $6\times 10^{-10}\,\Lambda_{\rm obs}$ via Jensen's inequality on the convex stiffness $1/(2\omega + 3)$. General theorem on solar-system-compatible BD class; not DCT-specific.

\item \textbf{BAO no-go and the perturbation-level programme} — DCT-BAO-01~\cite{dct-bao-01}. Background-frame DCT is falsified at $5.8\sigma$ versus DESI Y1; the conformal-frame mapping that resolves the Hubble tension is mutually exclusive with the BAO concordance. The live framework is the perturbation-level programme on a $\Lambda$CDM background, with $\mu_b = 1/P$, $\mu_{\rm DM} = 1/[P(1+\beta)]$, $\Sigma = 1/\bar P$, $\beta = f_v/z = 5/3$, score $5.5/10$. Smoking gun: $M_{\rm lens}/M_{\rm dyn}(z\sim 1.5) = 1.30$, falsifiable Euclid 2027--2029.

\item \textbf{Cosmological tensions re-read} — DCT-COS-01~\cite{dct-cos-01}. Post-2025 joint Bayesian preference $14.12\sigma\,/\,7.01\sigma\,/\,5.40\sigma$ at the three nested-test levels, down from pre-2025 $14.0\,/\,10.5\,/\,8.0$. Drop dominated by KiDS-Legacy $S_8$ closure ($-3.49\sigma$) and cosmic-chronometer per-$z$ test ($-2.60\sigma$).

\item \textbf{Solar-system PPN predictions} — DCT-PPN-01~\cite{dct-ppn-01}. $\gamma_{\rm PPN} - 1 = -1.998\times 10^{-5}$, $\beta - 1 = +1.0\times 10^{-10}$, $\eta_{\rm Nordtvedt} = +2.0\times 10^{-5}$. BepiColombo MORE 2028 projects $6.7\sigma$ ($8\sigma$ best-case) binary detection of $\gamma$; LUNAR $\sim 2035$ projects $20\sigma$ on $\eta$. Cassini bound consistent at $13\%$ margin.

\item \textbf{Dark matter / radial-acceleration relation} — DCT-DM-01~\cite{dct-dm-01}. Avrami profile $P(g) = 1 - \exp(-\sqrt{g/g_\dagger})$ from Allen--Cahn condensation, MOND scale $g_\dagger = 1.130\times 10^{-10}$ m/s$^2$ (Einstein frame, $2.4\%$ match) or $1.224\times 10^{-10}$ (Jordan frame, $0.81\sigma$). SPARC fits comparable to MOND on real data~\cite{lelli2016}. Direct detection $\sigma_{\rm SI} = 0$ exactly.

\item \textbf{Standard Model from McKay $2I\to E_8$} — DCT-SM-01~\cite{dct-sm-01}. Gauge group $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ via the McKay chain, three fermion generations forced from $(\mathbf{27},\mathbf{3})$ representation. CP phase $\delta_{\rm CP} = \pi/3$ at $3.6\sigma$ (the $2\pi/3$ branch retracted). Post-hoc closures of $\sin\theta_{13}$ ($1.69\%$) and $1/\alpha$ ($0.16\%$) explicitly labelled. $\mathbb{Z}_3$ cosmic-string tension at $\sim 25\times$ Planck bound — open structural problem.

\item \textbf{Conformal-wall invariance and CMB concordance} — DCT-CMB-01~\cite{dct-cmb-01}. 8 of 9 CMB features identical to $\Lambda$CDM via the 4D conformal-wall theorem $S_{\rm YM}[P\cdot g] = S_{\rm YM}[g]$. The single feature where DCT differs is $A_L$: DCT predicts $A_L = 1/P_{\rm lens} \approx 1.020$, matching the Planck PR3 measurement $1.020 \pm 0.025$~\cite{planck2018}. Note the canonical DCT prediction is $\approx 1.020$, NOT the retracted $1.185$~\cite{audit2026}. Goldstone $\theta$-mode contributes $\Delta N_{\rm eff} = 0.027$, CMB-S4 detectable.

\end{enumerate}

\section{The Hubble--BAO coupling and the perturbation-level programme}\label{sec:bao}

The most consequential revision from v1 to v2 concerns the Hubble tension and its relationship to the BAO. In v1~\cite{dctfoundation}, the conformal-frame mapping
\begin{equation}
H_{\rm phys} = H_E/\sqrt{P_0} = 67.4/\sqrt{0.851} = 73.06\ \mathrm{km/s/Mpc}\label{eq:Hphys}
\end{equation}
was presented as the resolution of the $H_0$ tension at $0.04\%$, with zero free parameters. The v1 paper did not address the BAO consequences of the same mapping.

The DCT-BAO-01 companion paper~\cite{dct-bao-01} establishes that the conformal-frame mapping {Eq.~\eqref{eq:Hphys}} that resolves $H_0$ is the same mapping that produces a $5.8\sigma$ falsification of the background-frame BAO prediction against DESI Y1~\cite{desi2024}. The two effects are not independent — they are mutually exclusive within DCT's architecture. By Jensen's inequality on the convex function $1/\sqrt{P}$, no spatial profile $P(x,t)$ with $\langle P\rangle = P_0$ can rescue the background falsification.

The audit~\cite{audit2026} therefore reclassifies the Hubble tension status: it is not solved as a physical problem at the background level. The live framework is the perturbation-level programme, in which the background cosmology is identical to $\Lambda$CDM and the DCT modifications appear only in the linearised growth and lensing kernels:
\begin{align}
\mu_b(a) &= 1/P(a)\,,\label{eq:mub}\\
\mu_{\rm DM}(a) &= 1/[P(a)(1 + \beta)]\,,\label{eq:muDM}\\
\Sigma(a) &= 1/\bar P(a)\,,\label{eq:Sigma}
\end{align}
with $\beta = f_v/z = 5/3$ from the 600-cell vertex figure. The perturbation-level programme has score $5.5/10$ on the corpus master scorecard~\cite{dctfoundation,dct-bao-01}, with two free parameters above $\Lambda$CDM ($\alpha = 0.405$, $B_0' = 1.437\times 10^5$ Gyr$^2$).

The smoking-gun observable for the perturbation-level programme is the lens-to-dynamical mass ratio of galaxy clusters: a turnover at $z\sim 1.5$ peaking at $1.30$~\cite{dct-bao-01,dct-dm-01}, falsifiable by Euclid 2027--2029~\cite{euclid2024}. A null result (constant ratio at all $z$) falsifies the perturbation-level programme; a confirmed turnover at $z\sim 1.5$ peak $1.30 \pm 0.05$ confirms the framework.

\section{The honest tension list}\label{sec:tensions}

The v1 statement ``no tension exceeding 2$\sigma$ has been found''~\cite{dctfoundation} is reconciled with the explicit acknowledgement of the following live tensions identified in the corpus audit~\cite{audit2026}:

\begin{enumerate}
\item \textbf{Background BAO falsification at $5.8\sigma$ versus DESI Y1}~\cite{dct-bao-01,desi2024}. The mutually exclusive Hubble-tension/BAO coupling. Live framework is the perturbation-level programme.

\item \textbf{KiDS-Legacy 2025 $S_8 = 0.815 \pm 0.018$ at $\sim 2\sigma$ above DCT}~\cite{kids2025}. The pre-2025 KiDS-1000 $S_8 = 0.766$ was favourable; the 2025 release is in tension. The DCT-canonical $S_8 \approx 0.831$ (audit 2026-05-05; E24-binding under $c_{GW}=c$) is unchanged.

\item \textbf{Cosmic-chronometer per-$z$ test at $3.08\sigma$ favouring $\Lambda$CDM}~\cite{dctfoundation,moresco2022}. The cluster-mean ratio $H/H_{\Lambda{\rm CDM}}\approx 1.084$ is consistent with simple BEC, but the per-$z$ ratio test favours $\Lambda$CDM at $0.63\sigma$ and rules out the simple-BEC mapping at $3.08\sigma$.

\item \textbf{$\sin\theta_{13}$ post-hoc closure at $1.69\%$, not first-principles}~\cite{dct-sm-01}.

\item \textbf{$1/\alpha$ post-hoc closure at $0.16\%$, not first-principles}~\cite{dct-sm-01}.

\item \textbf{$\mathbb{Z}_3$ cosmic-string tension at $\sim 25\times$ Planck bound}~\cite{dct-sm-01}.

\item \textbf{Vacuum-catastrophe ($\sim 14\%$ closure) without runnable verification script}~\cite{dctfoundation}. The $2\pi^2$ geometric prefactor in $\Lambda_P = 2\pi^2\,\rho_{\rm QFT}\,(\ell_P/L)^2 = 5.15\times 10^{-10}$ J/m$^3$ is asserted but not derived from a holographic 3-sphere argument anywhere.

\item \textbf{$E_G$ structural ceiling at $5.5/10$}~\cite{dctfoundation}. The growth-of-structure $E_G$ test has a structural inconsistency at $3.9\sigma$: $S_8\to\beta > 0\to \Sigma > 1\to E_G > $ GR conflicts with data.

\item \textbf{DEBA Mardia kurtosis $K_2 = 6.09$ at $\sim 90\sigma$ from any DCT-predicted value}~\cite{dctfoundation}.

\item \textbf{Universal-Cause biology framework anchored on one retrodictive calibration point}~\cite{dctfoundation}. Cancer/aging/CVD/autoimmune protocols (Sessions 77--104) are theoretical proposals with one retrodictive calibration point (TheraBionic 10\% PR in HCC); no DCT-driven animal or human study exists.

\end{enumerate}

The reconciled v1 statement is: ``No tension exceeding 2$\sigma$ has been found in the matched observables, with the explicit exceptions listed in {Sec.~\ref{sec:tensions}}.''

\section{Empirical scorecard — the DCT\_27 master table}\label{sec:scorecard}

The v1 ``629+ matched observables'' headline~\cite{dctfoundation} is RETRACTED in favour of the corpus DCT\_27 master scorecard~\cite{audit2026}, which uses a categorisation distinguishing genuine HITs from supporting evidence.

\begin{table}[t]
\caption{The DCT\_27 master scorecard (post-S57 audit~\cite{audit2026}). 80 graded observables across cosmology, particle physics, dark matter, and atomic physics.\label{tab:scorecard}}
\begin{ruledtabular}
\begin{tabular}{lc}
Category & Count\\
\hline
HIT (confirmed match within stated precision) & $55$\\
SOFT (directional agreement, low precision) & $10$\\
FLAG (known tension within $2$--$3\sigma$) & $5$\\
MISS (falsified or wrong) & $1$\\
OPEN (untested) & $6$\\
\hline
TOTAL graded & $80$\\
\end{tabular}
\end{ruledtabular}
\end{table}

The single MISS is the background-frame BAO~\cite{dct-bao-01}. The five FLAGs are the post-2025 cosmological tensions identified in {Sec.~\ref{sec:tensions}}. The 55 HITs include the structural identities (12 from~\cite{dct-spi-01}), the McKay-derived gauge group, the three-generation count, the conformal-wall theorem 8/9 CMB features, the 97 NIST atomic observables, the SPARC RAR (real-data, comparable to MOND), and the DM direct-detection nulls.

The previous ``629'' framing collapsed multiple structural identities into a single weighted observable count. The 80-graded master table is the corpus-audited honest version.

\section{The post-2025 joint Bayesian preference}\label{sec:bayesian}

The full recomputation of the joint Bayesian preference under post-2025 cosmological data is presented in DCT-COS-01~\cite{dct-cos-01}. The headline result is:

\begin{table}[t]
\caption{Joint Bayesian preference of DCT over $\Lambda$CDM, pre- vs.\ post-2025.\label{tab:bayesian}}
\begin{ruledtabular}
\begin{tabular}{lcc}
Test level & Pre-2025 & Post-2025\\
\hline
Cosmology only & $14.0\sigma$ & $14.12\sigma$\\
Cosmology $+$ particle physics & $10.5\sigma$ & $7.01\sigma$\\
Cosmology $+$ particle $+$ atomic & $8.0\sigma$ & $5.40\sigma$\\
\end{tabular}
\end{ruledtabular}
\end{table}

The drop from pre-2025 to post-2025 is dominated by the KiDS-Legacy $S_8$ closure ($-3.49\sigma$ from the joint preference) and the cosmic-chronometer per-$z$ test contradicting simple BEC ($-2.60\sigma$). The pre-2025 $14.0\,/\,10.5\,/\,8.0$ figure is FROZEN HISTORICAL and should not be quoted as currently authoritative; the post-2025 figure is the audited band as of May 2026.

The probability of correctness is bounded by the calibrated band $35\%$--$50\%$~\cite{audit2026}, set by (i) the BAO falsification penalty, (ii) the KiDS-Legacy tension, (iii) the post-hoc nature of the structural identities (despite the $\log_{10}(\text{BF}) = +27.5$ figure), and (iv) the lack of a clinically validated applied programme.

\section{The decisive forward tests}\label{sec:forward}

\begin{table*}[t]
\caption{Decisive forward tests of DCT, with projected experiment and expected sigma. Each test has a clear binary or quantitative falsification criterion.\label{tab:forward}}
\begin{ruledtabular}
\begin{tabular}{lcccl}
Test & Year & Experiment & DCT prediction & Decisive criterion\\
\hline
$\gamma_{\rm PPN}$ measurement~\cite{dct-ppn-01} & 2028 & BepiColombo MORE~\cite{iess2021} & $\gamma - 1 = -2.0\times 10^{-5}$ & $6.7\sigma$ ($8\sigma$ best); $\gamma - 1 = 0$ at $3\sigma$ falsifies\\
$M_{\rm lens}/M_{\rm dyn}$ turnover~\cite{dct-bao-01,dct-dm-01} & 2027--2029 & Euclid DR1/DR2~\cite{euclid2024} & peak $1.30$ at $z\sim 1.5$ & no turnover or peak $\ne 1.30 \pm 0.05$ falsifies\\
$\Delta N_{\rm eff}$ Goldstone-$\theta$~\cite{dct-cmb-01} & 2030+ & CMB-S4~\cite{cmbs42022} & $\Delta N_{\rm eff} = 0.027$ & $\sim 1\sigma$ at $\sigma_{\rm S4} = 0.03$\\
Nordtvedt $\eta$~\cite{dct-ppn-01} & $\sim 2035$ & LUNAR~\cite{lunar2035} & $\eta = +2.0\times 10^{-5}$ & $20\sigma$ at $\sigma = 10^{-6}$\\
WIMP/axion direct detection~\cite{dct-dm-01} & continuous & LZ Phase 2, DARWIN~\cite{lz2023} & $\sigma_{\rm SI} = 0$ exactly & any positive detection\\
JUNO neutrino hierarchy~\cite{dct-sm-01} & $\sim 2027$ & JUNO~\cite{juno2026} & Normal & Inverted falsifies\\
\end{tabular}
\end{ruledtabular}
\end{table*}

The single most decisive near-term test is BepiColombo MORE 2028~\cite{iess2021}, a $6.7\sigma$ ($8\sigma$ best-case) binary detection of the post-Newtonian parameter $\gamma$. The smoking-gun cluster-lensing test from Euclid 2027--2029~\cite{euclid2024} is the second decisive test. The Goldstone-$\theta$ contribution to $\Delta N_{\rm eff}$ provides a $\sim 1\sigma$ test at CMB-S4. The longer-horizon Nordtvedt test from LUNAR ($\sim 2035$~\cite{lunar2035}) is the third decisive test on the PPN sector.

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

Each companion paper in the DCT cluster~\cite{dct-spi-01,dct-sbd-01,dct-bao-01,dct-cos-01,dct-ppn-01,dct-dm-01,dct-sm-01,dct-cmb-01} carries its own predictions and falsification criteria. The umbrella anti-predictions are:

\subsection{Anti-predictions (falsification criteria)}

\begin{enumerate}
\item Detection of any of: WIMP DM, dark photon, axion DM, supersymmetric particles, fourth-generation fermion, large extra dimensions, $\dot G/G > 10^{-13}$/yr, DM self-interaction, fuzzy DM cores, primordial gravitational-wave $r > 0.036$, KK excitations at LHC, neutrinoless double-beta if $m_1 = 0$.

\item BepiColombo MORE measures $\gamma - 1 = 0 \pm 3\times 10^{-6}$ at the $3\sigma$ level. This rules out DCT and any BD theory with $\omega_0 \le 1.7\times 10^5$.

\item Euclid 2027--2029 finds no $M_{\rm lens}/M_{\rm dyn}$ turnover, or a turnover with peak ratio $\ne 1.30 \pm 0.05$. Falsifies the perturbation-level programme.

\item Future cosmological data (DESI DR3, Euclid DR2) confirm $S_8 \ge 0.83$ at $> 5\sigma$. Pushes DCT out of the perturbation-level programme range.

\item JUNO confirms inverted neutrino mass hierarchy. Falsifies the McKay-derived normal hierarchy.

\item Detection of a non-binary-icosahedral subgroup of $\mathrm{SU}(2)$ producing comparable structural-identity matches to fundamental constants. Challenges the uniqueness of the 600-cell + $2I$ selection.

\item Precision recomputation of $G_{\rm LHY}$ giving anything other than $3701/6300$ (a numerical-error catch).

\end{enumerate}

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

The framework is internally consistent across the cluster in three independent senses. First, the canonical action {Eq.~\eqref{eq:action}} with the post-S35 coupling $\omega(P) = (cP^2 - 3)/2$ produces the same $P_0 = 0.851$, $\omega_0 = 50{,}037$, and $c_{\rm BD} = 138{,}189$ across all eight companion papers; the master identity $2\omega_0 + 3 = c_{\rm BD}\,P_0^2$ holds exactly. Second, the McKay correspondence~\cite{mckay1980,slodowy1980} from $2I$ to $E_8$ used in DCT-SM-01~\cite{dct-sm-01} is the same structural identity that gives the spectral content of DCT-SPI-01~\cite{dct-spi-01}, providing cross-paper convergence on the same finite-group framework. Third, the Cassini bound $\omega_0 > 40{,}000$~\cite{bertotti2003} drives the stiffness--backreaction duality of DCT-SBD-01~\cite{dct-sbd-01}, the PPN predictions of DCT-PPN-01~\cite{dct-ppn-01}, and the Brans--Dicke-class membership of DCT-COS-01~\cite{dct-cos-01}; all three companion papers operate in the same Cassini-allowed region of parameter space.

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

\subsection{Summary of the framework}

DCT v2 is a Brans--Dicke scalar--tensor framework with topological inputs from the 600-cell + $2I$ + McKay $\to E_8$ chain. The framework derives the Standard Model gauge group, the three-generation fermion structure, the proton-to-electron mass ratio at $9.2\times 10^{-8}$, and the CKM mixing angles. It predicts the radial-acceleration relation through Allen--Cahn condensation with the MOND scale $g_\dagger$ derived from the Yukawa mass. It operates at the perturbation level on a $\Lambda$CDM cosmological background after the v1 conformal-frame Hubble-tension resolution was found to be mutually exclusive with the BAO concordance. The post-2025 joint Bayesian preference is $14.12\sigma\,/\,7.01\sigma\,/\,5.40\sigma$. Probability of correctness: $35$--$50\%$ calibrated band.

\subsection{Relationship to existing frameworks}

The framework belongs to the Brans--Dicke class~\cite{bransdicke1961,fujiimaeda2003}, with $\omega_0 = 50{,}037$ at present epoch consistent with the Cassini bound. The disformal coupling at large scales~\cite{bekenstein1993,koivisto2012} provides the perturbation-level deviations from $\Lambda$CDM. The Damour--Nordtvedt cosmological-attractor mechanism~\cite{damournord1993} is consistent with $\omega(P)$ having positive curvature at $P_0$. The McKay-derived gauge structure is in the spirit of grand-unified extensions to $E_8$, but uses the McKay correspondence rather than a direct embedding (avoiding the Distler--Garibaldi obstruction~\cite{distler2010}).

\subsection{Status of derived quantities}

\begin{enumerate}
\item Topological inputs from the 600-cell: VERIFIED. Coordination $z = 12$, vertex-figure $f_v = 20$, vertex count $V = 120$ are mathematical theorems on the 600-cell~\cite{coxeter1973}.
\item Master identity $2\omega_0 + 3 = c_{\rm BD}\,P_0^2$: EXACT for $n = 2$~\cite{audit2026}.
\item Twelve structural identities at log Bayes factor $+27.5$: POST-DICTION; pre-registered extension protocol described in DCT-SPI-01~\cite{dct-spi-01}.
\item Hubble-tension resolution: NOT SOLVED at background level; perturbation-level programme is live. Discussed in DCT-BAO-01~\cite{dct-bao-01}.
\item PPN predictions: derived; BepiColombo MORE 2028 binary test~\cite{dct-ppn-01}.
\item RAR: derived from Allen--Cahn; comparable to MOND on real data~\cite{dct-dm-01}.
\item Standard Model gauge group: derived from McKay~\cite{dct-sm-01}; post-hoc closures explicitly labelled.
\item CMB concordance: structural via conformal-wall theorem~\cite{dct-cmb-01}; $A_L$ Planck-consistent at $0\sigma$.
\end{enumerate}

\subsection{Remaining open questions}

\begin{enumerate}
\item First-principles derivation of $\sin\theta_{13}$ and $1/\alpha$ at $< 1\%$ — DCT\_28 Programs A, B~\cite{dctfoundation}.
\item Resolution of the $\mathbb{Z}_3$ cosmic-string tension at $\sim 25\times$ Planck bound~\cite{dct-sm-01}.
\item Rigorous Bogoliubov-depletion proof of $D = 1/f_v = 0.05$ for $P_0 = 9/10\cdot 19/20$~\cite{dctfoundation}.
\item Vacuum-catastrophe verification script for $\Lambda_P = 2\pi^2\,\rho_{\rm QFT}\,(\ell_P/L)^2$~\cite{audit2026}.
\item Heavy-particle masses ($W$, $Z$, Higgs, $t$) — no clean 600-cell formulae~\cite{dct-sm-01}.
\item Universal-Cause biology framework — clinically validated only via one retrodictive calibration point.
\end{enumerate}

\subsection{Computational implementation}

A reproducible Python implementation of the umbrella framework, with switches for each companion paper's specific predictions and a parametric exposure of $\omega_0$, $P_0$, $c_{\rm BD}$, $f_v$, $z$, is available at the master code repository~\cite{fnd02code}. The script orchestrates the cluster computations and outputs the joint Bayesian preference, the master scorecard, and the per-companion predictions.

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

This is a revised version of zenodo.18703512~\cite{dctfoundation}. The mathematical framework is unchanged: a Brans--Dicke scalar--tensor theory on the 600-cell with topological inputs $z = 12$, $f_v = 20$, $V = 120$, McKay correspondence to $E_8$, and the canonical post-S35 coupling $\omega(P) = (cP^2 - 3)/2$ with $c = 138{,}189$.

The revisions are: (i) reconciliation of the v1 ``no tension exceeding 2$\sigma$'' framing with the explicit acknowledgement of the background BAO falsification, the KiDS-Legacy $S_8$ closure, and the cosmic-chronometer per-$z$ negative; (ii) the conformal-frame Hubble-tension resolution at the background level is presented honestly as mutually exclusive with the BAO concordance, and the perturbation-level programme on a $\Lambda$CDM background is the live framework; (iii) post-2025 cosmological-data references (KiDS-Legacy 2025, DESI DR2, ACT DR6) and joint Bayesian recalculation to $14.12\sigma\,/\,7.01\sigma\,/\,5.40\sigma$; (iv) the ``629+ matched observables'' headline is retracted in favour of the DCT\_27 master scorecard $55/10/5/1/6$ across $80$ graded; (v) the $\delta_{\rm CP} = 2\pi/3$, $A_L = 1.185$, $\omega(P)$ pre-S35 form, and SPARC synthetic-vs-real headline numbers are explicitly retracted to their canonical post-audit values.

The eight companion papers in the DCT cluster~\cite{dct-spi-01,dct-sbd-01,dct-bao-01,dct-cos-01,dct-ppn-01,dct-dm-01,dct-sm-01,dct-cmb-01} carry the per-domain technical detail. The decisive forward tests are BepiColombo MORE 2028 (PPN $\gamma$ at $6.7\sigma$~\cite{iess2021}), Euclid 2027--2029 (cluster $M_{\rm lens}/M_{\rm dyn}$ turnover~\cite{euclid2024}), and LUNAR $\sim 2035$ (Nordtvedt $\eta$ at $20\sigma$~\cite{lunar2035}). Calibrated probability of correctness: $35$--$50\%$.

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\end{document}
