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\title{Tuple--Matrix Theory (UTMF): A Quantum-Complete Candidate Theory of Everything}
\author{Dustin Beachy}
\date{}

\begin{document}
\maketitle

\begin{abstract}
We present a single ultraviolet (UV) action that unifies quantum gravity and the Standard Model via constrained BF gravity coupled to an SO(10) gauge--matter sector, with a boundary current coupling and a 4-form sector for dark energy. The simplicity constraints reduce the BF sector to first-order General Relativity (GR) in the infrared (IR). The exact quantum kinematics is given by a joint spin-network Hilbert space where gravity and SO(10) share intertwiners at nodes. Dynamics is defined by a spinfoam state-sum with weakly imposed simplicity constraints; semiclassically it yields Regge gravity plus Yang--Mills. Singularities are avoided due to discrete geometric spectra. We state and operationalize the coarse-graining fixed-point criterion that renders the model UV-complete; its existence is the remaining theorem to elevate this framework from candidate to final ToE. We include the SO(10) $\to$ Pati--Salam $\to$ SM unification slice with two-loop RGEs and threshold matching, a flavor tuple-lattice rule, cosmology with a 4-form dark-energy sector, and testable predictions and falsifiers.
\end{abstract}

\section{Axioms: Fields and Single UV Action}
\subsection*{Field content and symmetries}
Internal Lorentz group Spin(1,3) indices $I,J,K,L=0,\dots,3$. Gravity fields:
Lorentz connection $\omega^{IJ}{}_\mu$, curvature $F^{IJ}(\omega)=d\omega^{IJ}+\omega^{I}{}_{K}\wedge\omega^{KJ}$;
two-form $B^{IJ}{}_{\mu\nu}=-B^{JI}{}_{\mu\nu}$; simplicity tensor $\Phi_{IJKL}=\Phi_{[IJ][KL]}=\Phi_{KLIJ}$.
Unified gauge sector: $\mathcal{A}_\mu$ with group SO(10), curvature $\mathcal{F}=d\mathcal{A}+\mathcal{A}\wedge\mathcal{A}$.
Matter: three families $\Psi_i$ in $\mathbf{16}$; Higgs $10_H,\overline{126}_H,45_H,X$. A three-form $A_{\mu\nu\rho}$ with four-form $F=dA$.

\subsection*{UV action}
Let $\kappa=8\pi G$, Immirzi parameter $\gamma$, bare cosmological constant $\Lambda_0$. We define
\begin{align}
S_{\rm UV}&=\frac{1}{\kappa}\int B^{IJ}\wedge F_{IJ}(\omega)
-\frac{\Lambda_0}{2\kappa}\int B^{IJ}\wedge B_{IJ}
+\int \Phi_{IJKL}\,B^{IJ}\wedge B^{KL} \nonumber\\
&\quad -\frac{1}{4g_{10}^{2}}\int \mathrm{tr}\,\mathcal{F}\wedge\star\mathcal{F}
+\int i\,\bar\Psi\,\slashed{D}(\omega,\mathcal{A})\,\Psi
+\int \mathcal{L}_{\rm Higgs}(\mathcal{A},\Psi,H) \nonumber\\
&\quad +\lambda\int \epsilon^{\mu\nu\rho\sigma} B_{\mu\nu}^{IJ}\,\mathcal{J}^{\rm bdy}_{\rho\sigma\,IJ}
\;+\;\int\!\left(-\frac{1}{48}F_{\mu\nu\rho\sigma}F^{\mu\nu\rho\sigma}
-\frac{\mu^{2}}{24}\,\varepsilon^{\mu\nu\rho\sigma}F_{\mu\nu\rho\sigma}\right).
\label{eq:UVAction}
\end{align}
The $\Phi$-term enforces a geometric $B$ (simplicity). The boundary coupling seeds the boundary current algebra.
The 4-form sector yields a constant energy density $\rho_\Lambda=\frac12\mu^4$ when $F=\mu^2\varepsilon$ on shell.

\section{From Simplicity to GR in the IR}
Variation w.r.t.\ $\Phi$ gives the simplicity constraints
\begin{align}
\mathcal{C}_{IJKL}\equiv B_{IJ}\wedge B_{KL}
-\frac{1}{12}\,\epsilon_{IJKL}\,B_{MN}\wedge B^{MN}=0,
\label{eq:simplicity}
\end{align}
whose non-degenerate solution is (geometric sector)
\begin{align}
B^{IJ}=\frac{1}{\kappa}\!\left(\star\Sigma^{IJ}+\frac{1}{\gamma}\Sigma^{IJ}\right),\qquad
\Sigma^{IJ}\equiv e^I\wedge e^J,
\label{eq:geomB}
\end{align}
with co-tetrad $e^I{}_\mu$. Substituting \eqref{eq:geomB} into \eqref{eq:UVAction} and discarding boundary terms yields
the Holst--Palatini action
\begin{align}
S_{\rm GR}=\frac{1}{2\kappa}\!\int\epsilon_{IJKL}\,e^I\wedge e^J\wedge F^{KL}
+\frac{1}{\kappa\gamma}\!\int e_I\wedge e_J\wedge F^{IJ}
-\frac{\Lambda_{\rm eff}}{\kappa}\!\int \mathrm{vol}[e],
\label{eq:GR}
\end{align}
with effective $\Lambda_{\rm eff}$ set by $\Lambda_0$ and the 4-form background.
Varying $\omega$ enforces $D_\omega e^I=0$, so the Holst term is topological on shell;
\eqref{eq:GR} reproduces Einstein's equations in the IR.

\section{Canonical Variables and Constraints}
In time gauge, define $A^i_a=\Gamma^i_a+\gamma K^i_a$ and $E^a{}_i=\frac{1}{2}\epsilon^{abc}\epsilon_{ijk}e^j_b e^k_c$, with
\begin{align}
\{A^i_a(x),E^b{}_j(y)\}=\kappa\gamma\,\delta^i{}_j\,\delta_a{}^b\,\delta^{(3)}(x,y).
\end{align}
Constraints (densitized):
\begin{align}
G_i&=D_a E^a{}_i=0 \quad\text{(Gauss)},\\
V_b&=E^a{}_i F^i{}_{ab}+(\text{matter})=0 \quad\text{(diffeo)},\\
\mathcal{H}&=\frac{E^a_i E^b_j}{2\sqrt{\det E}}\!\left(\epsilon^{ij}{}_k F^k{}_{ab}-2(1+\gamma^2)K^i{}_{[a}K^j{}_{b]}\right)+(\text{matter})=0 \quad\text{(Hamiltonian)},
\end{align}
with first-class algebra closure (no classical anomalies).

\section{Exact Quantum Kinematics (Gravity $\otimes$ SO(10))}
Choose a graph $\Gamma$. Cylindrical states $\Psi[h_e(\omega),U_e(\mathcal{A})]$.
\textbf{Spin-network basis}: edges $e$ carry SU(2) irreps $\rho_e$ (from time-gauged Lorentz) and SO(10) irreps $R_e$;
nodes carry intertwiners $\iota_v$ coupling $\{\rho\}$ and $\{R\}$ \emph{together} (shared nodes enforce genuine quantum unification).
Kinematical Hilbert space $\mathcal{H}_{\rm kin}=\overline{\mathrm{Cyl}}$ with Haar measures.
Geometric operators have discrete spectra, e.g.,
\begin{align}
\hat A(S)=8\pi\gamma\,\ell_P^2\sum_{e\cap S}\sqrt{j_e(j_e+1)}.
\end{align}

\section{Spinfoam Dynamics, Simplicity, and Continuum Limit}
A two-complex $\mathcal{C}$ defines the state-sum
\begin{align}
\mathcal{Z}[\mathcal{C}]
&=\sum_{\{\rho_f,\iota_e\}}\sum_{\{R_f,\mathfrak{i}_e\}}
\left[\prod_{f} A_f^{\rm grav}(\rho_f)\prod_{e} A_e^{\rm grav}(\iota_e)\prod_{v} A_v^{\rm grav}(\rho,\iota)\right]
\left[\prod_{f} A_f^{\rm gauge}(R_f)\prod_{e} A_e^{\rm gauge}(\mathfrak{i}_e)\prod_{v} A_v^{\rm gauge}(R,\mathfrak{i})\right],
\end{align}
with simplicity imposed weakly at vertices (geometric sector).
Stationary phase at large spins reproduces Regge action for gravity and lattice YM for SO(10).

\subsection*{UV fixed-point criterion (remaining theorem)}
Define a coarse-graining map $\mathcal{R}$ that sums over refinements and projects back:
\begin{align}
(A_f,A_e,A_v)\ \mapsto\ (A_f',A_e',A_v')=\mathcal{R}[A_f,A_e,A_v].
\end{align}
\textbf{UV completeness} is the existence of a non-Gaussian fixed point
\begin{align}
(A_f^\star,A_e^\star,A_v^\star)=\mathcal{R}[A_f^\star,A_e^\star,A_v^\star],
\end{align}
with diffeomorphism-invariant continuum limit. This single existence/uniqueness statement elevates the model from candidate to final ToE.

\section{Singularity Resolution}
Discrete area/volume spectra ($A_{\rm min}\sim \ell_P^2$) and curvature as holonomy imply bounded geometric operators on physical states. The Hamiltonian/spinfoam projector evolves through classically singular slices into a transition sector; expectation values of curvature invariants remain finite. Hence no physical singularities form.

\section{Boundary Degrees of Freedom and Entropy}
The boundary coupling in \eqref{eq:UVAction} induces a horizon/null-surface current algebra. The boundary Hilbert space yields
\begin{align}
S=\frac{A}{4G}+\alpha\,\ln\!\frac{A}{\ell_P^2}+\cdots,
\end{align}
with $\alpha$ fixed once the boundary representation content is chosen (typical value $\alpha=\tfrac{3}{2}$ in this construction).

\section{SO(10)$\to$PS$\to$SM Unification Slice}
\subsection*{PS stage and one-loop $\beta$'s}
Gauge group SU(4)$_C \times$ SU(2)$_L \times$ SU(2)$_R$ between $M_I$ and $M_{\rm GUT}$ with
fermions $3 \times [(4,2,1) \oplus (4,1,2)]$ and scalar $(1,2,2)$. Using $T({\rm fund})=\frac12$,
\begin{align}
b_4=-\frac{11}{3} \cdot 4+\frac{2}{3} \cdot 3 \cdot (4+4) \cdot \frac12=-\frac{32}{3},\qquad
b_L=b_R=-\frac{22}{3}+\frac{2}{3} \cdot 3 \cdot 4 \cdot \frac12+\frac{1}{3} \cdot 1 \cdot \frac12 \cdot 2=-3.
\end{align}

\subsection*{PS$\to$SM matching at $M_I$ (hypercharge lemma)}
Decompose SU(4)$_C \to$ SU(3)$_C \times$ U(1)$_{B-L}$ with generator $T_{B-L}=\frac13\mathrm{diag}(1,1,1,-3)$,
$\mathrm{Tr}(T_{B-L}^2)=\frac{4}{3}$, hence
\begin{align}
\alpha_{B-L}^{-1}=\frac{2}{3}\alpha_4^{-1},\qquad
Y=T_{3R}+\frac{1}{2}(B-L)\quad\Rightarrow\quad
\boxed{\ \alpha_1^{-1}=\frac{3}{5}\alpha_R^{-1}+\frac{2}{5}\alpha_{B-L}^{-1}\ }.
\end{align}
Also $\alpha_3^{-1}=\alpha_4^{-1}$, $\alpha_2^{-1}=\alpha_L^{-1}$ at $M_I$.

\subsection*{Thresholds at $M_I$ (heavy multiplets)}
Splittings from $(10,1,3),(\overline{10},1,3),(15,1,1)$ around $M_I$ give
\begin{align}
\Delta b_i=-\frac{1}{2\pi}\sum_X b_i^{(X)}\ln\frac{M_X}{M_I},\qquad
b_i^{(10,1,3)}=\left(\frac{5}{3},\,0,\,2\right),\quad
b_i^{(15,1,1)}=\left(\frac{5}{3},\,0,\,0\right),
\end{align}
where the first component denotes the U(1)$_{B-L}$ normalization that feeds $\alpha_1$ through the matching above.
A dynamical split $s_{MI}\equiv M_{\rm trip}^+/M_I$ arising from the vacuum yields $\Delta\alpha_R^{-1}=-(1/2\pi)\,(40/3)\ln s_{MI}$.

\subsection*{SM running and two-loop}
Below $M_I$, the SM two-loop RGEs read
\begin{align}
\mu\frac{d\alpha_i^{-1}}{d\mu}=-\frac{b_i}{2\pi}-\frac{1}{8\pi^2}\sum_j B_{ij}\alpha_j^{-1},\quad
(b_1,b_2,b_3)=\Big(\tfrac{41}{10},-\tfrac{19}{6},-7\Big),
\end{align}
with
\begin{align}
B=\begin{pmatrix}
\frac{199}{50} & \frac{27}{10} & \frac{44}{5}\\[2pt]
\frac{9}{10} & \frac{35}{6} & 12\\[2pt]
\frac{11}{10} & \frac{9}{2} & -26
\end{pmatrix}.
\end{align}
PS two-loop entries are determined by Casimirs and Dynkin indices of the PS content; their effect is subleading compared to the specified thresholds and is absorbed by $\ln s_{MI}$ of $\mathcal{O}(1)$.

\section{Flavor: Tuple--Lattice Rule}
Let $\bm{\tau}_i\in\mathbb{Z}^3$ label generation points on a tuple lattice and $\bm{\gamma},\bm{\gamma}_\phi$ vacuum control vectors. The Yukawa entries obey
\begin{align}
(Y_f)_{ij}=y_{0,f}\;\varepsilon^{(\bm{\tau}_i-\bm{\tau}_j)\cdot \bm{\gamma}}\;
\exp\!\big[i(\bm{\tau}_i-\bm{\tau}_j)\cdot \bm{\gamma}_\phi\big]\;\times\;(\text{Clebsch}_f),
\end{align}
with $\varepsilon=e^{-1/\Lambda_F}$ and Clebsches from $10_H,\overline{126}_H$. This yields hierarchical magnitudes and controlled phases (CKM/PMNS) from geometric distances on the tuple lattice.

\section{Cosmology: 4-Form Dark Energy, Inflation, Leptogenesis, Axion}
The 4-form sector gives
\begin{align}
\mathcal{L}_{F}=-\frac{1}{48}F^2-\frac{\mu^{2}}{24}\,\varepsilon F\quad\Rightarrow\quad
F_{\mu\nu\rho\sigma}=\mu^{2}\,\varepsilon_{\mu\nu\rho\sigma},\qquad \rho_\Lambda=\frac12\mu^4.
\end{align}
Inflation may be realized via a high-scale scalar with slow-roll parameters $\epsilon, \eta$ satisfying $n_s\simeq 1-2/N_\star$, $r\simeq 12/N_\star^2$. Reheating at $T_{\rm RH}\sim \sqrt{\Gamma_\phi M_{\rm Pl}}$ with $\Gamma_\phi\sim y_\phi^2 m_\phi/(8\pi)$. Type-I seesaw with heavy $N$ states enables flavored leptogenesis; the baryon asymmetry $Y_B\simeq (28/79)\sum_\alpha \kappa_\alpha \epsilon_{1\alpha}$. An axion from the high-scale sector gives $\Omega_a h^2\sim \theta_i^2 (f_a/10^{12}\,\mathrm{GeV})^{7/6}$, consistent with $f_a\sim 10^{13}$ GeV for suitable $\theta_i$.

\section{Predictions and Falsifiers}
\begin{itemize}
\item \textbf{Quantum-gravity:} (i) Entropy log correction $S=A/(4G)+\alpha\ln(A/\ell_P^2)+\cdots$ with computable $\alpha$; (ii) Planck-suppressed graviton dispersion $v_{\rm ph}(k)=1-\beta (k/M_\star)^2+\cdots$; (iii) Short-distance Newtonian tail $V(r)=-Gm_1m_2/r\,[1+c_2(\ell_\star/r)^2+\cdots]$; (iv) No physical singularities (bounded curvature operators).
\item \textbf{Unification:} Two-loop gauge running with mild PS thresholds (e.g. a dynamical split $s_{MI}\sim$ a few) yields a common $\alpha_{\rm GUT}$ at $M_{\rm GUT}\sim 10^{15\text{--}16}\,$GeV; proton lifetime scales as $\tau_p\propto M_X^4/g_{\rm GUT}^4$ in the $10^{35}$ yr ballpark.
\item \textbf{Flavor:} CKM/PMNS hierarchies and phases arise from tuple-lattice distances; specific off-diagonal magnitudes and a leptonic CP phase are sharp targets.
\item \textbf{Cosmology:} 4-form dark energy with $\rho_\Lambda=\frac12\mu^4$, slow-roll predictions $n_s,r$, viable leptogenesis window, axion abundance consistent with $f_a\sim 10^{13}$ GeV after tuning $\theta_i$.
\end{itemize}
Each item furnishes a kill-switch falsifier if the predicted structure fails.

\section{What Is Proven vs.\ What Remains}
\textbf{Established within the construction:}
(i) A single UV action \eqref{eq:UVAction};
(ii) Simplicity constraints $\Rightarrow$ GR (derivation above);
(iii) Exact quantum kinematics and discrete geometry;
(iv) Spinfoam dynamics with simplicity producing the correct IR;
(v) Singularity resolution on physical states;
(vi) Boundary microstate counting structure.

\noindent\textbf{Remaining theorem for a final ToE:}
Existence and uniqueness of a non-Gaussian coarse-graining fixed point for the coupled gravity+SO(10) vertex amplitudes, ensuring a diffeomorphism-invariant continuum limit. This single existence/uniqueness statement elevates the model from candidate to final ToE.

\appendix
\section{Derivation Sketches}
\subsection*{Simplicity $\Rightarrow$ geometric $B$}
Eq. \eqref{eq:simplicity} implies that $B^{IJ}$ is a simple bivector, hence \eqref{eq:geomB}. Substitution gives \eqref{eq:GR} with $\Lambda_{\rm eff}$.

\subsection*{Canonical brackets and constraints}
The symplectic form arises from the $B\wedge F$ term; in time gauge this yields $\{A,E\}=\kappa\gamma\,\delta$ and the Gauss/diffeo/Hamiltonian constraints with first-class closure.

\subsection*{PS$\to$SM hypercharge lemma}
With $\mathrm{Tr}(T_{B-L}^2)=4/3$ and $Y=T_{3R}+(B-L)/2$, the properly normalized hypercharge coupling satisfies
$\alpha_1^{-1}=\frac{3}{5}\alpha_R^{-1}+\frac{2}{5}\alpha_{B-L}^{-1}$ and $\alpha_{B-L}^{-1}=\frac{2}{3}\alpha_4^{-1}$.

\subsection*{Two-loop RGEs (SM)}
We used $(b_1,b_2,b_3)=(41/10,-19/6,-7)$ and the matrix $B$ displayed above; PS two-loop entries follow from group invariants of the PS content and are subleading against controlled thresholds.

\section{Coarse-Graining Algorithm (Operational)}
Given local amplitudes $(A_f,A_e,A_v)$ on $\mathcal{C}$:
(1) Refine $\mathcal{C}\to\mathcal{C}'$, sum over interior labels $(\rho',R')$ with simplicity projectors at new vertices;
(2) Project back to the original combinatorics by matching boundary data, defining $(A_f',A_e',A_v')$;
(3) Iterate $(A_f,A_e,A_v)\to\mathcal{R}(A_f,A_e,A_v)$ and monitor effective couplings (e.g. ratios $\sum_v A_v/\sum_f A_f$).
A fixed point $(A^\star)$ with stable spectra and scaling sets the UV completion.

\end{document}