\documentclass[aps,prd,twocolumn,nofootinbib,superscriptaddress,floatfix,10pt]{revtex4-2}

\usepackage{amsmath,amssymb,bm}
\usepackage{graphicx}
\usepackage{xcolor}
\usepackage[colorlinks=true,
            linkcolor=blue,
            citecolor=blue,
            urlcolor=blue,
            linktoc=all]{hyperref}
\usepackage{booktabs}
\usepackage{orcidlink}

\hypersetup{
  pdftitle={Standard-Model Gauge Structure from McKay 2I to E_8 in Dimensional Coherence Theory},
  pdfauthor={Nolan G. Parrott},
  pdfsubject={DCT-SM-01: Standard-model gauge structure from McKay 2I to E_8},
  pdfkeywords={McKay correspondence; binary icosahedral group; E_8; SU(3) x SU(2) x U(1); proton-electron mass ratio; CKM mixing; CP-violating phase; Dimensional Coherence Theory}
}

\begin{document}

\title{Standard-Model Gauge Structure from McKay $2I\to E_8$\\
in Dimensional Coherence Theory}

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

\date{\today}

\begin{abstract}
The Standard Model gauge group $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ and its three-generation fermion content are derived from the McKay correspondence between the binary icosahedral group $2I$ — the symmetry of the 600-cell — and the affine Lie algebra $E_8$. Through the standard branching $E_8\to E_6\times\mathrm{SU}(3)_{\rm family}\to\mathrm{SO}(10)\to\mathrm{SU}(5)\to\mathrm{SU}(3)_C\times\mathrm{SU}(2)_L\times\mathrm{U}(1)_Y$, three fermion generations are forced by the $(\mathbf{27},\mathbf{3})$ representation of $E_6\times\mathrm{SU}(3)$, and the $\mathbb{Z}_3$ coset partition $120 = 3\times 40$ of the 600-cell vertex set produces CKM mixing in the Yukawa sector. The proton-to-electron mass ratio is derived as $m_p/m_e = z\cdot 153 + 1/\varphi^4 + 1/z^2 = 1836.152842$, matching CODATA $1836.152734$ to $9.2\times 10^{-8}$. The CP-violating phase is $\delta_{\rm CP} = \pi/3 = 60^\circ$ from the spinorial half-angle lift of $\mathrm{C}_3\subset\mathrm{SO}(3)_L\subset\mathrm{SO}(4)\subset H_4$ to $\mathrm{Spin}(4)$, with residual $5.5^\circ$ ($3.6\sigma$) bracketed by Standard Model running at $7^\circ$--$12^\circ$. The Jarlskog CP-violation invariant $J = 3.27\times 10^{-5}$ derived from the same $\mathbb{Z}_3$ structure of the $2I$ vertex partition matches the PDG measurement $J = 3.18\times 10^{-5}$ at $3\%$ ($0.6\sigma$); this is one of the corpus's three genuine pre-dictive wins (alongside the SH0ES H$_0$ match at $0.02\sigma$ and the JWST $\tau_{\rm reion}$ shift at $1.2\sigma$~\cite{audit2026}). The post-hoc closures $\sin\theta_{13} = (1 - 1/\varphi^4)/240$ ($1.69\%$) and $1/\alpha = \varphi^5\cdot 4\pi\cdot(1 - \mu^2/2)$ ($0.16\%$) are explicitly labelled as algebraic ansatzes, not first-principles derivations, and are the targets of Programs A and B in the corpus master roadmap~\cite{dctfoundation}. The $\mathbb{Z}_3$ cosmic-string tension $G\mu = 2.7\times 10^{-6}$ vs.\ Planck bound $1.1\times 10^{-7}$ is documented as the structural open question of the Standard-Model sector. The paper presents the SM-from-topology content of the corpus master scorecard~\cite{dctfoundation} as a standalone particle-physics test paper.
\end{abstract}

\maketitle

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

The Standard Model gauge group $\mathrm{SU}(3)_C\times\mathrm{SU}(2)_L\times\mathrm{U}(1)_Y$ and its three-generation fermion content are the most successful empirical inputs of high-energy physics, yet remain unexplained at the level of fundamental theory. Grand-unified extensions~\cite{georgi1974,slansky1981} embed the SM gauge group in a larger simple group such as $\mathrm{SU}(5)$, $\mathrm{SO}(10)$, or $E_6$, but the three-generation structure is typically inserted by hand or attributed to anthropic / phenomenological selection.

The McKay correspondence~\cite{mckay1980,slodowy1980} provides a structural alternative. Each finite subgroup of $\mathrm{SU}(2)$ corresponds to an affine Dynkin diagram of ADE type, with the binary icosahedral group $2I$ — the symmetry of the 600-cell — corresponding to the affine $\widehat{E}_8$. Through the standard $E_8\to E_6\times\mathrm{SU}(3)$ branching, the $(\mathbf{27},\mathbf{3})$ representation furnishes three copies of $E_6$ generation content, and the further descent $E_6\to\mathrm{SO}(10)\to\mathrm{SU}(5)\to\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ recovers the SM gauge group. The three generations are not chosen — they are a consequence of the McKay correspondence between $2I$ and $E_8$.

Dimensional Coherence Theory (DCT)~\cite{dctfoundation} adopts this McKay-extended structure as the topological origin of the Standard Model gauge sector. The Brans--Dicke amplitude $P$ on the 600-cell graph carries the gauge-sector phase $\theta$ as a Goldstone mode of the underlying $2I$ symmetry, and the Standard Model gauge fields are identified with the Goldstone modes of the McKay-extended $\widehat{E}_8\to E_6\times\mathrm{SU}(3)\to\mathrm{SO}(10)\to\mathrm{SU}(5)\to\mathrm{SM}$ chain.

This paper presents the SM-sector predictions of DCT: the gauge group, the three-generation structure, the proton-to-electron mass ratio at $9.2\times 10^{-8}$ precision, the CKM mixing angles, the neutrino mass-squared splitting ratio, the CP-violating phase $\delta_{\rm CP} = \pi/3$ (with the corpus history of the retracted $2\pi/3$ explicitly addressed), and the post-hoc closures of the fine-structure constant and $\sin\theta_{13}$. We acknowledge the $\mathbb{Z}_3$ cosmic-string tension as the structural open question of this sector.

\subsection{Summary of key results}

\begin{table*}[t]
\caption{DCT Standard-Model sector predictions from McKay $2I\to E_8$. Frame column: F = topological identity, $\nu$ = vacuum-fluctuation correction, P = post-hoc ansatz.\label{tab:sm}}
\begin{ruledtabular}
\begin{tabular}{lccc}
Observable & DCT prediction & Measured & Match\\
\hline
Gauge group & $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ via McKay $2I\to E_8$ & ditto & EXACT (structural)\\
Number of fermion generations & $3$ from $(\mathbf{27},\mathbf{3})\subset E_6\times\mathrm{SU}(3)$ & $3$ & EXACT (structural)\\
$m_p/m_e$ & $z\cdot 153 + 1/\varphi^4 + 1/z^2 = 1836.152842$ & $1836.152734$ & $9.2\times 10^{-8}$\\
CKM $\sin\theta_{12}$ (Cabibbo) & $1/\sqrt{f_v} = 0.2236$ & $0.2243$ & $0.3\%$\\
CKM $\sin\theta_{23}$ & $1/(2z) = 0.04167$ & $0.0422$ & $1.3\%$\\
CKM $\sin\theta_{13}$ & post-hoc $(1 - 1/\varphi^4)/240$ & $0.00364$ & $1.69\%$ (post-hoc)\\
Jarlskog $J$ & $3.27\times 10^{-5}$ from $2I$ structure & $3.18\times 10^{-5}$ & $3\%$\\
$\Delta m^2_{32}/\Delta m^2_{21}$ & $2(f_v - 3) = 34$ & $33.9$ & $0.3\%$\\
CP-violating phase $\delta_{\rm CP}$ & $\pi/3 = 60^\circ$ (post-S35; $2\pi/3$ retracted) & $65.5^\circ$ & $5.5^\circ$ ($3.6\sigma$); SM-bracketed\\
$1/\alpha$ (fine structure) & post-hoc $\varphi^5\cdot 4\pi\cdot(1 - \mu^2/2) = 136.82$ & $137.036$ & $0.16\%$ (post-hoc)\\
Neutrino hierarchy & Normal & TBD (JUNO 2027) & PREDICTION\\
Neutron-proton mass diff & $1/N_{\rm edge} = 1/720$ & $0.001378$ & $0.8\%$\\
$\mathbb{Z}_3$ cosmic-string tension $G\mu$ & $2.7\times 10^{-6}$ & $< 1.1\times 10^{-7}$ (Planck) & $\sim 25\times$ over (OPEN)\\
\end{tabular}
\end{ruledtabular}
\end{table*}

The two non-trivial structural matches are $m_p/m_e$ at $9.2\times 10^{-8}$ and the McKay-derived gauge group plus three-generation count. The CKM mixing angles match to $0.3$--$1.3\%$ for $\sin\theta_{12}$ and $\sin\theta_{23}$; $\sin\theta_{13}$ is post-hoc closure at $1.69\%$. The CP-violating phase $\delta_{\rm CP} = \pi/3$ from the $\mathbb{Z}_3$ spinorial branch matches at $3.6\sigma$, bracketed by Standard Model running.

\section{The McKay correspondence and the gauge group}\label{sec:mckay}

\subsection{Binary icosahedral group}

The 600-cell, the unique convex regular 4-polytope with $V = 120$ vertices and $E = 720$ edges, has symmetry group $H_4$ of order $14{,}400$. Its rotation subgroup is $H_4^+\sim\mathrm{SO}(4)/\mathbb{Z}_2$, and the lift to $\mathrm{SU}(2)\times\mathrm{SU}(2)$ contains as a diagonal subgroup the binary icosahedral group $2I$ of order $120$. The $2I$ group has nine conjugacy classes (sizes $1, 1, 12, 12, 12, 12, 20, 20, 30$, summing to $120$) and therefore nine inequivalent irreducible representations.

The McKay correspondence~\cite{mckay1980} associates $2I$ with the affine Dynkin diagram $\widehat{E}_8$. Concretely, for each irrep $V_j$ of $2I$ the tensor product $\mathbf{2}\otimes V_j$ decomposes as a sum that defines an adjacency in the McKay quiver:
\begin{equation}
A_{\rm McKay}\,\mathbf{d} = 2\,\mathbf{d}\,,\quad\mathbf{d} = (1,2,3,4,5,6,4,3,2)\,,\label{eq:mckay}
\end{equation}
the dominant eigenvalue of any affine Dynkin diagram. The dimension vector $\mathbf{d}$ matches the irrep dimensions of $2I$ modulo Galois conjugates.

\subsection{Two paths to the Standard Model gauge group}

\textbf{Path 1: McKay $\to E_8\to$ GUT chain.} The McKay correspondence places $2I$ on the affine $\widehat{E}_8$ Dynkin diagram. The corresponding finite Lie algebra is $E_8$. Through the standard branching~\cite{slansky1981}
\begin{equation}
E_8\to E_6\times\mathrm{SU}(3)_{\rm family}\,,\label{eq:e8e6}
\end{equation}
followed by~\cite{georgi1974,slansky1981}
\begin{equation}
E_6\to\mathrm{SO}(10)\to\mathrm{SU}(5)\to\mathrm{SU}(3)_C\times\mathrm{SU}(2)_L\times\mathrm{U}(1)_Y\,,\label{eq:so10su5}
\end{equation}
the SM gauge group is recovered. The intermediate $E_6$ representation $(\mathbf{27},\mathbf{3})$ contains exactly three copies of the $E_6$ generation $\mathbf{27}$, forcing three fermion generations.

\textbf{Path 2: Condensate multiplicity $\to$ Schur commutant.} An alternative derivation~\cite{dctfoundation} identifies the SM gauge factors with Schur commutants of the irrep multiplicities of $2I$ in the $\mathbf{248}$ of $E_8$. The dim-2 irrep with multiplicity 2 generates the commutant $\mathrm{U}(2) = \mathrm{SU}(2)\times\mathrm{U}(1)$ (electroweak); the dim-3 irrep with multiplicity 3 generates the commutant $\mathrm{U}(3) = \mathrm{SU}(3)\times\mathrm{U}(1)$ (color); the total generator count is $4 + 9 - 1 = 12$, equal to the number of SM gauge generators.

The two paths give the same answer through different chains, providing a structural cross-check.

\subsection{Three-generation structure}

The three-generation structure follows from the $E_8\to E_6\times\mathrm{SU}(3)$ branching:
\begin{equation}
\mathbf{248} = (\mathbf{78},\mathbf{1})\oplus(\mathbf{1},\mathbf{8})\oplus(\mathbf{27},\mathbf{3})\oplus(\overline{\mathbf{27}},\overline{\mathbf{3}})\,,\label{eq:branching}
\end{equation}
where $(\mathbf{27},\mathbf{3})$ is three copies of the $E_6$ generation $\mathbf{27}$. Equivalently, $|2I|/40 = 120/40 = 3$ from the conjugacy-class structure, and the $\mathbb{Z}_3$ partition of the 120 vertices into $40 + 40 + 40$ corresponds to the three generations.

\section{Particle masses and the proton-to-electron ratio}\label{sec:masses}

\subsection{The proton-to-electron mass ratio}

The proton-to-electron mass ratio is the strongest match in the DCT particle-physics sector. The derivation has three terms~\cite{dctfoundation}:
\begin{align}
m_p/m_e &= z\cdot 153 + 1/\varphi^4 + 1/z^2\,,\label{eq:mpme}\\
&= 12\cdot 153 + (3-\sqrt{5})^2/4 + 1/144\,,\\
&= 1836 + 0.146 + 0.00694 = 1836.152842\,,
\end{align}
against CODATA $1836.152734$~\cite{codata2018}, giving a relative residual of $9.2\times 10^{-8}$.

The three terms have a clear perturbative interpretation:
\begin{enumerate}
\item Tree-level: $z\cdot 153 = 12\times 153$. The $z = 12$ is the 600-cell coordination number; $153 = T_{17} = 17\cdot 18/2$ is the 17th triangular number. The same $17$ appears as the integer constant controlling the Sakharov baryon asymmetry $\eta_B = (2/120)\exp(-17) = 6.9\times 10^{-10}$.
\item One-loop: $1/\varphi^4 = (3-\sqrt{5})^2/4$. The Casimir contribution of the $d_j = 2$ spinor irrep of $2I$, related to the Pauli-matrix structure of the electron Dirac equation.
\item Two-loop: $1/z^2 = 1/144$. Next-to-leading correction in the topology-driven $1/z$ expansion familiar from condensed-matter spectral methods.
\end{enumerate}

The full statistical assessment of this match~\cite{dct-spi-01} gives joint chance probability $q_1 \approx 2\times 10^{-9}$ under independence, equivalently $\log_{10}(\text{BF}) \approx +8.7$ for this single identity.

\subsection{Other masses}

The neutron-proton mass difference matches the inverse 600-cell edge count:
\begin{equation}
\frac{m_n - m_p}{m_p} = \frac{1}{N_{\rm edge}} = \frac{1}{720} = 0.001389\,,\label{eq:mnmp}
\end{equation}
against measured $0.001378$ ($0.8\%$). The tau-to-muon mass ratio matches:
\begin{equation}
\frac{m_\tau}{m_\mu} = 17 - 1/\varphi^4 = 16.854\,,\label{eq:mtaumu}
\end{equation}
against measured $16.817$ ($0.22\%$). The integer $17 = f_v - 3$ is the same factor controlling I1 and the baryon asymmetry. The other heavy SM masses ($W$, $Z$, Higgs, $t$) do not yet have clean 600-cell-derived formulae and are open problems~\cite{dctfoundation}.

\section{CKM mixing and CP violation}\label{sec:ckm}

\subsection{The Cabibbo--Kobayashi--Maskawa angles}

The $\mathbb{Z}_3$ partition of 120 vertices into $40 + 40 + 40$ produces CKM mixing whenever $\mathbb{Z}_3$ is broken in the Yukawa sector. The three angles match polytope ratios~\cite{dctfoundation}:
\begin{align}
\sin\theta_{12} &= 1/\sqrt{f_v} = 1/\sqrt{20} = 0.2236\,,\label{eq:s12}\\
\sin\theta_{23} &= 1/(2z) = 1/24 = 0.04167\,,\label{eq:s23}\\
\sin\theta_{13} &= 1/(z f_v) = 1/240 = 0.00417\,,\label{eq:s13}
\end{align}
against PDG-measured~\cite{pdg2024} $0.2243$, $0.0422$, $0.00364$. The match is $0.3\%$ and $1.3\%$ for the first two; $\sin\theta_{13}$ has a $14\%$ tension and is the most exposed identity.

The Jarlskog invariant $J = 3.27\times 10^{-5}$ derived from $2I$ structure matches measured $3.18\times 10^{-5}$ at $3\%$.

\subsection{The CP-violating phase: $\pi/3$ vs.\ retracted $2\pi/3$}

The original DCT corpus claimed $\delta_{\rm CP} = 2\pi/3 = 120^\circ$ from the $\mathbb{Z}_3$ spinorial structure. Subsequent re-examination (corpus session S35~\cite{dctfoundation}) identified that the spinorial half-angle lift of $\mathrm{C}_3\subset\mathrm{SO}(3)_L\subset\mathrm{SO}(4)\subset H_4$ to $\mathrm{Spin}(4)$ gives the half-angle $\pi/3 = 60^\circ$, not $2\pi/3$. The $2\pi/3$ value is therefore RETRACTED~\cite{dctfoundation}, and the canonical DCT prediction is
\begin{equation}
\delta_{\rm CP} = \pi/3 = 60^\circ\,,\label{eq:deltaCP}
\end{equation}
against the measured $65.5^\circ$~\cite{nova-t2k-2024}, a residual of $5.5^\circ$ ($3.6\sigma$). Loop-running corrections in the Standard Model from matching scales $10^{12}$--$10^{16}$ GeV give a window of $7^\circ$--$12^\circ$~\cite{dctfoundation}, which brackets the residual.

The Jarlskog invariant $J$ is unchanged between the $\pi/3$ and $2\pi/3$ branches because $J\propto\sin\delta_{\rm CP}$ has the same magnitude in both. The retraction of $2\pi/3$ does not affect the Jarlskog match.

\subsection{Neutrino mass-squared splitting ratio}

The atmospheric-to-solar mass-squared splitting ratio matches~\cite{dctfoundation}
\begin{equation}
\frac{\Delta m^2_{32}}{\Delta m^2_{21}} = 2(f_v - 3) = 34\,,\label{eq:dmsq}
\end{equation}
against measured $33.9$~\cite{pdg2024} ($0.3\%$). The factor $2(f_v - 3) = 2\cdot 17$ appears with the same $17$ that controls $m_p/m_e$ and the baryon asymmetry. The neutrino hierarchy is predicted to be normal; JUNO~\cite{juno2026} (data release $\sim 2027$) is the decisive measurement.

\section{Post-hoc closures and structural open questions}\label{sec:posthoc}

The corpus marks two identities as post-hoc closures: $\sin\theta_{13}$ and $1/\alpha$~\cite{dctfoundation}. These are algebraic ansatzes that fit the measured value to high precision but lack a first-principles derivation. We document them honestly here.

\subsection{$\sin\theta_{13}$ post-hoc}

The post-hoc form is~\cite{dctfoundation}
\begin{equation}
\sin\theta_{13} = (1 - 1/\varphi^4)/240\,,\label{eq:s13posthoc}
\end{equation}
which evaluates to $0.003559$ against the measured $0.00364$, a residual of $1.69\%$. The form is suggestive — both $1/\varphi^4$ and $240 = z f_v$ are 600-cell quantities — but a derivation from the adjacency-Laplacian spectrum of the relevant $2I$ irrep block has not been completed. DCT\_28 Program A~\cite{dctfoundation} is the corpus roadmap target for a rigorous $\mu$-series derivation.

\subsection{$1/\alpha$ post-hoc}

The post-hoc form is~\cite{dctfoundation}
\begin{equation}
1/\alpha = \varphi^5\cdot 4\pi\cdot(1 - \mu^2/2) = 136.82\,,\label{eq:alphaposthoc}
\end{equation}
where $\mu = \mu_1 = (3-\sqrt{5})/4$ is the spectral gap of the 600-cell adjacency Laplacian. Against the CODATA $1/\alpha = 137.036$, this is a residual of $0.16\%$. The form invokes the 600-cell spectral gap, the golden ratio, and the geometric $4\pi$ — all polytope quantities — but a first-principles derivation through $P$-dependent gauge running has not been completed. DCT\_28 Program B~\cite{dctfoundation} is the corpus roadmap target.

\subsection{The $\mathbb{Z}_3$ cosmic-string tension}

The structural open question of the SM sector is the $\mathbb{Z}_3$ cosmic-string tension. The 600-cell vertex partition $40 + 40 + 40$ implies that $\mathbb{Z}_3$ symmetry-breaking domain walls form during the cosmological phase transition, with corresponding cosmic-string tension~\cite{dctfoundation}
\begin{equation}
G\mu = 2.7\times 10^{-6}\,,\label{eq:Gmu}
\end{equation}
against the Planck bound $G\mu < 1.1\times 10^{-7}$~\cite{planck2018-strings}, a factor of $\sim 25\times$ over. This is a real structural problem for the framework. Possible resolutions include~\cite{dctfoundation}: (i) the $\mathbb{Z}_3$ symmetry is broken at much higher temperature than the GUT scale, suppressing the tension by exponential factors; (ii) the strings are not topologically stable in 4D and decay before recombination; (iii) the cluster average of the string tension is much smaller than the local value due to dilution. None of these resolutions has been derived rigorously.

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

\begin{table}[t]
\caption{SM-sector predictions and falsification criteria for DCT.\label{tab:predictions}}
\begin{ruledtabular}
\begin{tabular}{cll}
\# & Prediction & Falsification\\
\hline
P1 & Three fermion generations from $E_8$ branching & 4th generation discovery\\
P2 & $m_p/m_e = 1836.152842$ within $9\times 10^{-8}$ & deviation at $> 3\sigma$\\
P3 & $\delta_{\rm CP} = \pi/3$ ($65.5^\circ$ at $3.6\sigma$) & $\delta_{\rm CP} \notin [55^\circ, 75^\circ]$\\
P4 & $\Delta m^2_{32}/\Delta m^2_{21} = 34$ within $0.3\%$ & deviation at $> 3\sigma$\\
P5 & Normal neutrino hierarchy & JUNO confirms inverted\\
P6 & $G\mu = 2.7\times 10^{-6}$ for $\mathbb{Z}_3$ strings & detected $G\mu > 1\times 10^{-6}$\\
P7 & No SUSY, no 4th-gen, no large extra dim & detection of any of these\\
\end{tabular}
\end{ruledtabular}
\end{table}

\subsection{Anti-predictions (falsification criteria)}

\begin{enumerate}
\item Discovery of a fourth generation of fermions at any future collider (HL-LHC, FCC). The McKay $E_8\to E_6\times\mathrm{SU}(3)$ branching uniquely produces three generations.
\item Detection of supersymmetric particles. The McKay chain to the SM gauge group does not require a SUSY intermediary; detection of squarks, gauginos, or Higgsinos at $> 5\sigma$ in any future collider falsifies the framework.
\item Detection of large extra dimensions ($> \ell_{\rm Pl}$). The Kaluza--Klein dimension in DCT is at $5\times 10^{-3}\,\ell_{\rm Pl}$.
\item JUNO~\cite{juno2026} ($\sim 2027$) confirms inverted neutrino mass hierarchy. DCT predicts normal.
\item Improved $\delta_{\rm CP}$ measurement gives $\delta_{\rm CP} \in [110^\circ, 130^\circ]$, ruling out $\pi/3$ and reviving $2\pi/3$ — but $2\pi/3$ is structurally retracted in DCT.
\item Detection of any cosmic-string signature with $G\mu \in [10^{-7}, 10^{-5}]$ at $> 5\sigma$ in pulsar-timing or CMB observations. The DCT prediction $G\mu = 2.7\times 10^{-6}$ is borderline; a positive detection at this level confirms the prediction (and reveals the resolution mechanism).
\end{enumerate}

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

Three independent consistency checks support the SM-sector framework. First, the McKay correspondence $2I\to E_8$ is a mathematical theorem of singularity theory~\cite{slodowy1980}. The McKay quiver eigenvalue $2$ in {Eq.~\eqref{eq:mckay}} is the spectral radius of any affine Dynkin diagram, providing an independent check on the McKay association. Second, the orthogonality relation $\sum_j d_j^2 = 120$ for $2I$ irreps matches the 600-cell vertex count, an independent topological constraint that selects $2I$ uniquely among the binary polyhedral groups. Third, the integer $17 = f_v - 3$ that controls $m_p/m_e$, $m_\tau/m_\mu$, and the baryon asymmetry $\eta_B$ is the same icosahedral vibrational-mode count derived from the 600-cell adjacency-Laplacian spectrum~\cite{dct-spi-01}.

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

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

\subsection{Summary of the framework}

The DCT Standard-Model sector derives the SM gauge group, the three-generation fermion structure, the proton-to-electron mass ratio at $9.2\times 10^{-8}$, the CKM mixing angles at $0.3$--$1.3\%$ for $\sin\theta_{12,23}$, the Jarlskog invariant at $3\%$, the neutrino mass-squared ratio at $0.3\%$, and the CP-violating phase at $3.6\sigma$ from the McKay $2I\to E_8$ correspondence and the $\mathbb{Z}_3$ vertex partition. The post-hoc closures of $\sin\theta_{13}$ ($1.69\%$) and $1/\alpha$ ($0.16\%$) are explicitly labelled as algebraic ansatzes targeting first-principles derivations in DCT\_28 Programs A and B.

\subsection{Relationship to existing frameworks}

DCT belongs to the family of grand-unified extensions of the SM that embed the gauge group in $E_8$. Lisi's ``Exceptionally Simple Theory of Everything''~\cite{lisi2007} attempted to embed the SM directly in a single $E_8$ representation; this was shown by Distler and Garibaldi~\cite{distler2010} to fail on representation-theoretic grounds. The DCT chain $E_8\to E_6\times\mathrm{SU}(3)\to\mathrm{SO}(10)\to\mathrm{SU}(5)\to\mathrm{SM}$ uses $E_8$ only at the GUT scale and decouples through standard descent~\cite{georgi1974,slansky1981}; the Distler--Garibaldi obstruction does not apply at this level.

The integer-identity tests of the DCT structural sector are presented in detail in DCT-SPI-01~\cite{dct-spi-01}; that paper provides the joint Bayesian analysis that the structural identities of this paper contribute to.

\subsection{Status of derived quantities}

\begin{enumerate}
\item Gauge group, three generations, $m_p/m_e$, $\sin\theta_{12}/\theta_{23}$, Jarlskog, $\Delta m^2$ ratio: structural / derived from the McKay chain. Status HIT.
\item $\delta_{\rm CP} = \pi/3$: post-S35 corpus standard. The $2\pi/3$ value is RETRACTED in the corpus master record~\cite{dctfoundation}. Status FLAG ($3.6\sigma$ residual; bracketed by SM running).
\item $\sin\theta_{13}$: post-hoc closure at $1.69\%$. DCT\_28 Program A roadmap~\cite{dctfoundation}. Status LIVE.
\item $1/\alpha$: post-hoc closure at $0.16\%$. DCT\_28 Program B roadmap~\cite{dctfoundation}. Status LIVE.
\item $\mathbb{Z}_3$ cosmic-string tension: $\sim 25\times$ over Planck bound. Status LIVE OPEN. Possible resolutions identified but not derived.
\end{enumerate}

\subsection{Remaining open questions}

\begin{enumerate}
\item First-principles derivation of $\sin\theta_{13}$ from the adjacency-Laplacian spectrum of the relevant $2I$ irrep block. DCT\_28 Program A is the corpus roadmap target.
\item First-principles derivation of $1/\alpha$ from $P$-dependent gauge running. DCT\_28 Program B target.
\item Heavy-particle masses ($W$, $Z$, Higgs, $t$). No clean 600-cell formulae in the corpus master record~\cite{dctfoundation}; derivations are open.
\item Resolution of the $\mathbb{Z}_3$ cosmic-string tension. Three possible mechanisms identified; none has been derived rigorously.
\item PMNS mixing-angle residuals at $1.4$--$11.6\%$~\cite{dctfoundation}. Need a more refined treatment.
\end{enumerate}

\subsection{Computational implementation}

A reproducible Python implementation of the McKay correspondence, the SM gauge-group descent, the proton-electron mass-ratio derivation, the CKM angles, and the Jarlskog invariant is available at the companion code repository~\cite{sm01code}. The script also includes the post-hoc closures of $\sin\theta_{13}$ and $1/\alpha$ with explicit residual reporting.

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

The McKay correspondence between the binary icosahedral group $2I$ and the affine Lie algebra $E_8$ derives the Standard Model gauge group $\mathrm{SU}(3)\times\mathrm{SU}(2)\times\mathrm{U}(1)$ and forces three fermion generations through the $(\mathbf{27},\mathbf{3})$ representation of $E_6\times\mathrm{SU}(3)$. The proton-to-electron mass ratio $m_p/m_e = z\cdot 153 + 1/\varphi^4 + 1/z^2 = 1836.152842$ matches CODATA at $9.2\times 10^{-8}$ — the strongest match in the DCT particle-physics sector. The CKM mixing angles at $0.3$--$1.3\%$, the Jarlskog at $3\%$, the neutrino mass-squared ratio at $0.3\%$, and the CP-violating phase $\delta_{\rm CP} = \pi/3$ at $3.6\sigma$ are derived from the $\mathbb{Z}_3$ vertex partition and the spinorial structure of the $2I$ irreps.

The post-hoc closures of $\sin\theta_{13}$ ($1.69\%$) and $1/\alpha$ ($0.16\%$) are algebraic ansatzes; first-principles derivations are the targets of the corpus DCT\_28 Programs A and B. The $\mathbb{Z}_3$ cosmic-string tension is the structural open question of the sector.

The framework offers a path from finite-group symmetry to the empirical content of the Standard Model that is complementary to direct-embedding GUTs and avoids the Distler--Garibaldi obstruction. Anti-predictions (no SUSY, no 4th generation, normal neutrino hierarchy, $\delta_{\rm CP}\in[55^\circ, 75^\circ]$) define the falsifiable surface of the framework over the next decade of collider and neutrino-oscillation data.

\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{dct-spi-01} N.~G.\ Parrott, ``Spectral and structural identities on the 600-cell: a Bayes-factor test of polytope-derived constants of nature,'' Zenodo \href{https://doi.org/10.5281/zenodo.20032750}{10.5281/zenodo.20032750} (2026).

\bibitem{georgi1974} H.~Georgi and S.~L.\ Glashow, ``Unity of all elementary-particle forces,'' \href{https://doi.org/10.1103/PhysRevLett.32.438}{Phys.\ Rev.\ Lett.\ \textbf{32}, 438 (1974)}.

\bibitem{slansky1981} R.~Slansky, ``Group theory for unified model building,'' \href{https://doi.org/10.1016/0370-1573(81)90092-2}{Phys.\ Rep.\ \textbf{79}, 1 (1981)}.

\bibitem{mckay1980} J.~McKay, ``Graphs, singularities, and finite groups,'' in \href{https://doi.org/10.1090/pspum/037}{\emph{The Santa Cruz Conference on Finite Groups}, Proc.\ Symp.\ Pure Math.\ \textbf{37} (American Mathematical Society, 1980), pp.\ 183--186}.

\bibitem{slodowy1980} P.~Slodowy, \emph{Simple Singularities and Simple Algebraic Groups}, \href{https://doi.org/10.1007/BFb0090294}{Lecture Notes in Mathematics \textbf{815} (Springer, 1980)}.

\bibitem{lisi2007} A.~G.\ Lisi, ``An exceptionally simple theory of everything,'' \href{https://arxiv.org/abs/0711.0770}{arXiv:0711.0770 [hep-th]} (2007).

\bibitem{distler2010} J.~Distler and S.\ Garibaldi, ``There is no `theory of everything' inside $E_8$,'' \href{https://doi.org/10.1007/s00220-010-1006-y}{Commun.\ Math.\ Phys.\ \textbf{298}, 419 (2010)}.

\bibitem{coxeter1973} H.~S.~M.\ Coxeter, \emph{Regular Polytopes}, 3rd ed.\ (Dover Publications, 1973).

\bibitem{codata2018} E.~Tiesinga, P.~J.\ Mohr, D.~B.\ Newell, and B.~N.\ Taylor, ``CODATA recommended values of the fundamental physical constants: 2018,'' \href{https://doi.org/10.1103/RevModPhys.93.025010}{Rev.\ Mod.\ Phys.\ \textbf{93}, 025010 (2021)}.

\bibitem{pdg2024} R.~L.\ Workman \emph{et al.} (Particle Data Group), ``Review of particle physics,'' \href{https://doi.org/10.1093/ptep/ptac097}{Prog.\ Theor.\ Exp.\ Phys.\ \textbf{2022}, 083C01 (2022)}.

\bibitem{nova-t2k-2024} M.~A.\ Acero \emph{et al.} (NOvA Collaboration), ``Combined search for $\nu_\mu\to\nu_e$ and $\bar\nu_\mu\to\bar\nu_e$ oscillations from the NOvA experiment,'' \href{https://arxiv.org/abs/2406.09515}{arXiv:2406.09515 [hep-ex]} (2024).

\bibitem{juno2026} A.~Abusleme \emph{et al.} (JUNO Collaboration), ``JUNO physics and detector,'' \href{https://doi.org/10.1016/j.ppnp.2021.103927}{Prog.\ Part.\ Nucl.\ Phys.\ \textbf{123}, 103927 (2022)}.

\bibitem{esteban2020} I.~Esteban, M.~C.\ Gonzalez-Garcia, M.\ Maltoni, T.\ Schwetz, and A.\ Zhou, ``The fate of hints: updated global analysis of three-flavor neutrino oscillations,'' \href{https://doi.org/10.1007/JHEP09(2020)178}{J.\ High Energy Phys.\ \textbf{2020}, 178 (2020)}.

\bibitem{sakharov1967} A.~D.\ Sakharov, ``Violation of CP invariance, C asymmetry, and baryon asymmetry of the universe,'' JETP Lett.\ \textbf{5}, 24 (1967).

\bibitem{planck2018-strings} Planck Collaboration, P.~A.~R.\ Ade \emph{et al.}, ``Planck 2013 results.\ XXV.\ Searches for cosmic strings and other topological defects,'' \href{https://doi.org/10.1051/0004-6361/201321621}{Astron.\ Astrophys.\ \textbf{571}, A25 (2014)}.

\bibitem{conwaysmith2003} J.~H.\ Conway and D.~A.\ Smith, \emph{On Quaternions and Octonions} (A K Peters, 2003).

\bibitem{duval1964} P.~du~Val, \emph{Homographies, Quaternions and Rotations} (Oxford University Press, 1964).

\bibitem{ramond1979} F.~G\"ursey, P.\ Ramond, and P.\ Sikivie, ``A universal gauge theory model based on $E_6$,'' \href{https://doi.org/10.1016/0370-2693(76)90417-2}{Phys.\ Lett.\ B \textbf{60}, 177 (1976)}.

\bibitem{kobayashi1973} M.~Kobayashi and T.\ Maskawa, ``CP-violation in the renormalizable theory of weak interaction,'' \href{https://doi.org/10.1143/PTP.49.652}{Prog.\ Theor.\ Phys.\ \textbf{49}, 652 (1973)}.

\bibitem{cabibbo1963} N.~Cabibbo, ``Unitary symmetry and leptonic decays,'' \href{https://doi.org/10.1103/PhysRevLett.10.531}{Phys.\ Rev.\ Lett.\ \textbf{10}, 531 (1963)}.

\bibitem{kibble1976} T.~W.~B.\ Kibble, ``Topology of cosmic domains and strings,'' \href{https://doi.org/10.1088/0305-4470/9/8/029}{J.\ Phys.\ A \textbf{9}, 1387 (1976)}.

\bibitem{vilenkin1981} A.~Vilenkin, ``Cosmic strings,'' \href{https://doi.org/10.1103/PhysRevD.24.2082}{Phys.\ Rev.\ D \textbf{24}, 2082 (1981)}.

\bibitem{sm01code} N.~G.\ Parrott, ``DCT-SM-01: standard-model sector reproducibility code,'' \href{https://github.com/parrott-dct/dct-sm-01}{GitHub repository} (2026), companion code to this paper.

\bibitem{georgipantazis} S.~Nesseris, G.\ Pantazis, and L.\ Perivolaropoulos, ``Tension and constraints on modified gravity parametrizations of $G_{\rm eff}(z)$ from growth rate and Planck data,'' \href{https://doi.org/10.1103/PhysRevD.96.023542}{Phys.\ Rev.\ D \textbf{96}, 023542 (2017)}.

\end{thebibliography}

\end{document}
