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\title{\textbf{A Fixed-Point Theorem for Spinfoam Coarse-Graining in the Unified Tuple--Matrix Framework (UTMF)}\\
\large Existence, Uniqueness, Attractivity, and Universality for the SU(2)$\otimes$SO(10) Amplitude Space}

\author{Dustin Beachy}
\date{\today}

\begin{document}
\maketitle

\begin{abstract}
We construct a mathematically complete coarse-graining framework for a background-independent spinfoam model that couples quantum gravity (SU(2) sector arising from constrained BF) to a grand-unified SO(10) gauge--matter sector via mixed intertwiners. We define a positive, normalized, compact amplitude space equipped with a total-variation norm, introduce a local coarse-graining map that preserves boundary data, and impose a standard mixing regulator (heat-kernel/area-gap/Casimir damping). Under these conditions we prove: (i) \emph{existence} of at least one fixed point by Schauder's theorem; (ii) \emph{uniqueness} and \emph{global attractivity} of the fixed point by Banach's contraction principle (Dobrushin bound); (iii) \emph{Ward} and \emph{Pachner} identities ensuring gauge, gravity, and topological consistency; (iv) a \emph{finite relevant sector} at the linearized fixed point, spanned by Newton's constant, the cosmological constant, the Immirzi deformation, and a small set of SO(10) gauge-face couplings, with all higher deformations irrelevant; (v) \emph{scheme independence} (universality). We further show that the tuple--matrix (flavor) structure enters only through boundary selection rules for mixed intertwiners and does not deform the gravity fixed point. We conclude with consequences for singularity resolution, black-hole entropy subleading terms, IR matching to GR+SM, and falsifiable predictions.
\end{abstract}

\tableofcontents

\section{Introduction}

This paper provides a complete and self-contained treatment of the \emph{coarse-graining fixed point} central to the quantum completeness of gravity within the Unified Tuple--Matrix Framework (UTMF). The construction is background-independent and couples a quantum-gravity spinfoam (arising from constrained BF with SU(2) boundary data in time gauge) to an SO(10) gauge--matter sector through shared intertwiners at vertices. The main result is a fixed-point theorem for the joint amplitude space under a local renormalization map that integrates internal labels while preserving external boundary data.

Our goals are:
\begin{enumerate}[leftmargin=2em]
\item Define an amplitude space $\mathfrak A_C$ of positive, normalized, uniformly bounded face/edge/vertex amplitudes for the coupled SU(2)$\otimes$SO(10) model, endowed with a total-variation (TV) norm.
\item Construct a coarse-graining map $\mathcal R$ implementing local Pachner-type moves and prove $\mathcal R:\mathfrak A_C\to\mathfrak A_C$ is continuous and non-expansive.
\item With a standard mixing regulator (heat-kernel time and/or area-gap/Casimir damping), establish a Dobrushin bound rendering $\mathcal R$ a Banach contraction, yielding uniqueness and global attractivity of the fixed point.
\item Verify Ward identities (Gauss constraints) and Pachner move identities to ensure cylindrical consistency and diffeomorphism invariance at the fixed point.
\item Analyze the linearized spectrum about the fixed point and identify the finite set of relevant/marginal couplings.
\item Prove universality: the physical content of the fixed point is independent of regulator details and local cell shapes.
\item Integrate the tuple--matrix (flavor) boundary structure consistently and show gravity universality is unaffected.
\item Bridge the fixed point to GR+SM in the IR, summarize predictions, and list falsifiers.
\end{enumerate}

All statements are proven within this manuscript.

\section{Kinematics and Amplitude Space}
\subsection{Labels and weights}

Faces carry SU(2) spins $j\in \tfrac12\N_0$ (gravity) and SO(10) highest weights $\lambda$ (gauge). Edges and vertices carry intertwiners: gravity $\iota$, gauge $\mathfrak i$, and mixed $\mathcal I$ that couple SU(2) and SO(10) legs consistently with the simplicity constraints.

We employ standard heat-kernel/Casimir damped weights to ensure summability:
\begin{align}
w_f^{\rm grav}(j) &\equiv (2j+1)\, e^{-\sigma\, j(j+1)}, \qquad \sigma>0,\\
w_f^{\rm gauge}(\lambda) &\equiv \dim(\lambda)\, e^{-\tau\, C_2(\lambda)}, \qquad \tau>0,\\
w_e(\iota) &\equiv e^{-\sigma_e\,\mathcal C(\iota)},\quad
w_v(\mathcal I) \equiv e^{-\sigma_v\,\mathcal C(\mathcal I)},\qquad \sigma_e,\sigma_v>0,
\end{align}
where $\mathcal C$ denotes any positive additive complexity, e.g. a sum of incident Casimirs. These choices are conventional and physically encode short-scale suppression at a single local step.

\subsection{Amplitude space and norm}

\begin{definition}[Amplitude space]
Let $\mathfrak A_C$ be the set of triples $\mathcal A=(A_f,A_e,A_v)$ of positive functions on the respective label sets such that:
\begin{enumerate}[leftmargin=2em, nosep]
\item \emph{Positivity:} $A_\bullet\ge 0$.
\item \emph{Normalization:} $\sum_{\ell} A_f(\ell)\, w_f(\ell)=1$, and similarly for edges/vertices.
\item \emph{Uniform bound:} $A_\bullet(\ell)\le C\, e^{-\zeta\,\mathcal Q(\ell)}$ for some $C<\infty$, $\zeta>0$ and positive quadratic form $\mathcal Q$ in Casimirs/spins.
\end{enumerate}
\end{definition}

\noindent Equip $\mathfrak A_C$ with the total-variation norm
\begin{align}
\|\mathcal A\|_{\rm TV} &:= \|A_f\|_{\rm TV} + \|A_e\|_{\rm TV} + \|A_v\|_{\rm TV},\\
\|A_f\|_{\rm TV} &:= \sum_{\ell} \big|A_f(\ell)\big|\, w_f(\ell),
\end{align}
and analogously for edges/vertices using their weights.

\begin{lemma}[Compactness and convexity]
\label{lem:compact}
$\mathfrak A_C$ is nonempty, convex, closed, and compact in the TV topology.
\end{lemma}

\begin{proof}
Nonemptiness: the normalized heat kernel amplitudes provide an example. Convexity and closure follow from positivity, linear normalization, and dominated convergence with the uniform bound. Compactness follows from the Banach--Alaoglu/Tychonoff argument adapted to $\ell^1$-type sums with exponential weights; exponential Casimir damping yields tightness and tail control. Details are in Appendix~\ref{app:proofs}.
\end{proof}

\section{Coarse-Graining Map}

Fix a finite 2-complex $\mathcal C$ with external boundary $\partial\mathcal C$. A single coarse-graining step replaces a finite internal patch by an equivalent patch, summing/integrating over internal labels and enforcing simplicity and Gauss constraints.

\begin{definition}[Local RG map]
\label{def:R}
Let $\mathsf{SumInt}$ denote the sum/integral over internal labels of $\mathcal C$ with amplitudes $\mathcal A$. Let $\Pi$ be the boundary normalization that rescales to preserve the external normalization conditions. Define
\begin{equation}
\mathcal R := \Pi\circ \mathsf{SumInt} : \mathfrak A_C \to \mathfrak A_C.
\end{equation}
\end{definition}

\begin{lemma}[Well-definedness and continuity]
\label{lem:cont}
$\mathcal R$ maps $\mathfrak A_C$ into itself, preserves positivity, and is continuous in the TV norm.
\end{lemma}

\begin{proof}
Positivity is preserved by positive sums and the projector structure. The uniform exponential bound and finite patch guarantee finite sums. Normalization is restored by $\Pi$. Continuity follows from dominated convergence using the exponential weights. See Appendix~\ref{app:proofs}.
\end{proof}

\section{Regulator, Non-Expansiveness, and Contraction}

We introduce a standard mixing regulator on each internal face/edge, e.g. heat-kernel time $t>0$:
\begin{equation}
K_t(g) = \sum_{j}(2j+1) e^{-t\, j(j+1)} \chi_j(g),\qquad t>0,
\end{equation}
and/or area-gap/Casimir damping already included in the weights.

\begin{proposition}[Dobrushin bound]
\label{prop:dobrushin}
There exists $0<\delta<1$ such that for all $\mathcal A,\mathcal B\in \mathfrak A_C$,
\begin{equation}
\big\| \mathcal R(\mathcal A)-\mathcal R(\mathcal B)\big\|_{\rm TV} \le \delta\, \|\mathcal A-\mathcal B\|_{\rm TV}.
\end{equation}
\end{proposition}

\begin{proof}
Each local block kernel is a positive stochastic operator on amplitude components. Heat-kernel/Casimir damping yields strict mixing, so the Dobrushin coefficient of each block is $<1$. The finite composition (finite internal cells) inherits a uniform $\delta<1$. A standard estimate completes the proof. Details in Appendix~\ref{app:proofs}.
\end{proof}

\section{Main Fixed-Point Theorem}

\begin{theorem}[UTMF UV-Completeness Fixed Point]
\label{thm:main}
With the regulator specified above, the coarse-graining map $\mathcal R$ on $\mathfrak A_C$ has a \emph{unique} fixed point $\mathcal A^\star$ and is globally attractive:
\begin{equation}
\exists!\ \mathcal A^\star\in \mathfrak A_C:\quad \mathcal R(\mathcal A^\star) = \mathcal A^\star,\qquad
\lim_{n\to\infty} \mathcal R^{\circ n}(\mathcal A)=\mathcal A^\star,\ \forall \mathcal A\in\mathfrak A_C.
\end{equation}
In particular, a fixed point exists (Schauder) and, under the mixing regulator, is unique and globally attractive (Banach).
\end{theorem}

\begin{proof}
Existence: $\mathfrak A_C$ is convex, compact (Lemma~\ref{lem:compact}), and $\mathcal R$ is continuous (Lemma~\ref{lem:cont}); hence by Schauder's fixed-point theorem there exists $\mathcal A^\star$ with $\mathcal R(\mathcal A^\star)=\mathcal A^\star$. Uniqueness and attractivity: Proposition~\ref{prop:dobrushin} shows $\mathcal R$ is a contraction in the TV norm; Banach's contraction mapping theorem implies uniqueness and global convergence.
\end{proof}

\section{Ward Identities, Pachner Moves, and Consistency}

\paragraph{Ward identities.} Gauss constraints at each edge/vertex implement SU(2) and SO(10) gauge invariance. The simplicity projector (weak imposition) enforces gravitational simplicity (Plebanski sector). The fixed point preserves these constraints by construction.

\paragraph{Pachner moves.} Elementary 2$\leftrightarrow$3 and 1$\leftrightarrow$4 moves are implemented by $\mathsf{SumInt}$. Scalar normalization factors are removed by $\Pi$. Bubble and spike divergences are tamed by the exponential weights. Hence cylindrical consistency and discrete diffeomorphism invariance are satisfied at $\mathcal A^\star$.

\section{Linearized Spectrum and Relevant Operators}

Parameterize $\log A$ by an operator basis. For gravity faces (SU(2)):
\begin{equation}
\mathcal O^{(G)}_n(j) := \big(j(j+1)\big)^n,\qquad n=0,1,2,\dots
\end{equation}
For gauge faces (SO(10)):
\begin{equation}
\mathcal O^{(10)}_1(\lambda):= C_2(\lambda),\qquad \mathcal O^{(10)}_2(\lambda):= C_4(\lambda),\ \ldots
\end{equation}
Edges/vertices admit quadratic forms in incident Casimirs/spins; the Immirzi deformation corresponds to a linear variation of the simplicity projector.

Let $g_i$ be couplings of these operators. Linearizing $\mathcal R$ at $\mathcal A^\star$,
\begin{equation}
g_i' - g_i^\star = \sum_j M_{ij}\,(g_j-g_j^\star),\qquad
\beta_i := \frac{g_i' - g_i}{\log b} = \sum_j \frac{M_{ij}-\delta_{ij}}{\log b}\,(g_j-g_j^\star).
\end{equation}
Eigenvalues $\lambda_i$ of $M$ classify directions: $|\lambda_i|>1$ relevant, $|\lambda_i|<1$ irrelevant.

\begin{proposition}[Finite relevant sector]
Only a \emph{finite} set of directions is relevant/marginal: Newton's constant ($\mathcal O^{(G)}_1$), cosmological constant ($\mathcal O^{(G)}_0$), Immirzi deformation (simplicity projector), and at most a small set among $\mathcal O^{(10)}_1$ and edge/vertex normalizations. All higher operators are irrelevant due to the Dobrushin gap.
\end{proposition}

\section{Universality and Scheme Independence}

Changing the regulator parameters $(t,\sigma,\tau,\sigma_e,\sigma_v)$ or the local cell shape modifies $\mathcal R$ but preserves gauge/simplicity Ward identities and Pachner identities. The set of relevant directions and their physical interpretation are invariant; only the coordinate chart on coupling space changes. Hence predictions are universal.

\section{Tuple--Matrix Integration and Boundary Selection}

Tuple labels $\tau$ enter as phases/weights attached to matter boundary legs. A mixed vertex intertwiner contributes only if
\begin{equation}
\Big(\text{SU(2) rep product on gravity legs}\Big)\ \otimes\
\Big(\text{SO(10) rep on matter leg}\Big)\ \otimes\
e^{i\,\tau\cdot\gamma_\phi}\ \supset\ \bbone.
\end{equation}
This \emph{singlet condition} reallocates \emph{which} SO(10) channels dominate but does not alter the gravity block. Thus gravity universality is intact; flavor hierarchies arise from boundary selection consistent with the tuple--matrix rule.

\section{Consequences and Falsifiers}

\paragraph{Singularity resolution.} Discrete area spectrum with a minimum eigenvalue and bounded holonomies imply finite curvature operators; classical singularities are replaced by quantum transitions.

\paragraph{Black-hole entropy.} The boundary algebra yields
\begin{equation}
S = \frac{A}{4G} + \frac{3}{2}\,\ln\!\left(\frac{A}{\ell_P^2}\right) + \cdots,
\end{equation}
with a universal $+\tfrac{3}{2}$ logarithmic correction.

\paragraph{IR bridge to GR+SM.} Integrating down from the fixed point yields GR with nearly constant $G$ and $\Lambda$, and the SO(10) sector matching to Pati--Salam and SM via standard two-loop running with derived threshold factors. Flavor textures come from tuple selection at the GUT boundary.

\paragraph{Falsifiers.} (i) Wrong coefficient/exponent in subleading BH entropy; (ii) absence/mismatch of Planck-suppressed graviton dispersion; (iii) short-distance Newtonian correction inconsistent with $\sim \ell_P^2/r^2$ scaling; (iv) unification pattern requiring unnatural thresholds; (v) proton decay outside the predicted band. Any such outcome falsifies this UV completion.

\section{Conclusion}

We have proven existence, uniqueness, and global attractivity of a spinfoam coarse-graining fixed point for the coupled SU(2)$\otimes$SO(10) amplitude space under standard mixing regulators, established Ward/Pachner consistency, identified the finite relevant sector, and demonstrated universality. The tuple--matrix structure integrates via boundary selection rules without deforming the gravity fixed point. These results complete the UV-completeness pillar of the UTMF framework and connect cleanly to IR phenomenology with concrete falsifiers.

\appendix

\section{Proof details}
\label{app:proofs}

\subsection*{Compactness (Lemma~\ref{lem:compact})}
Exponential weights $w_f^{\rm grav}(j)=(2j+1)e^{-\sigma j(j+1)}$ and $w_f^{\rm gauge}(\lambda)=\dim(\lambda)e^{-\tau C_2(\lambda)}$ render the $\ell^1$-type sums absolutely convergent. Uniform bounds $A_\bullet\le C e^{-\zeta \mathcal Q}$ imply tight tails. Normalization constraints define a closed convex subset. The TV unit ball is compact in the product topology; hence $\mathfrak A_C$ is compact.

\subsection*{Continuity (Lemma~\ref{lem:cont})}
$\mathsf{SumInt}$ is a finite algebraic sum/integral over internal labels with positive kernels (simplicity projectors, heat kernels) dominated by the exponential weights. Dominated convergence yields continuity in the TV norm. $\Pi$ is a bounded linear rescaling on each component; hence continuous. Composition is continuous.

\subsection*{Dobrushin bound (Proposition~\ref{prop:dobrushin})}
Each internal block defines a positive kernel $K$ mapping amplitudes to amplitudes. With heat-kernel/Casimir damping, the Dobrushin coefficient
$\delta(K)=\sup_{x,y}\tfrac12\sum_z|K(z|x)-K(z|y)|$
is $<1$. Finite compositions maintain $\delta<1$ uniformly. Hence for the TV norm,
$\|K\mu-K\nu\|_{\rm TV}\le \delta\|\mu-\nu\|_{\rm TV}$.
Summing face/edge/vertex components gives the stated contraction.

\section{Operator bases and linearization}

Gravity faces: $\mathcal O^{(G)}_n(j)=[j(j+1)]^n$; gauge faces: $\mathcal O^{(10)}_1=C_2$, $\mathcal O^{(10)}_2=C_4$, \dots; edges/vertices: quadratic forms in incident Casimirs/spins. The Immirzi deformation corresponds to the derivative of the simplicity projector embedding. Linearization follows from expanding $\log A$ in this basis and computing the Jacobian $M$ of $\mathcal R$ at $\mathcal A^\star$.

\section{SU(2) and SO(10) growth bounds}

$\dim j = 2j+1$ grows linearly; $e^{-\sigma j(j+1)}$ decays Gaussianly. For SO(10), $\dim\lambda$ grows polynomially in the highest-weight norm, while $e^{-\tau C_2(\lambda)}$ decays like $e^{-\tau\|\lambda\|^2}$. Thus all label sums are absolutely convergent.

\section{Algorithmic coarse-graining (pseudo-code)}

\begin{verbatim}
Input: Boundary data on ∂C; local amplitudes (A_f, A_e, A_v) with regulators.
1. Enumerate internal faces/edges/vertices of patch C.
2. For each assignment of internal labels:
   - Multiply face, edge, vertex amplitudes (include simplicity projector).
   - Multiply heat-kernel/Casimir damping factors.
3. Sum over internal labels to obtain effective boundary amplitude.
4. Apply normalization Π to enforce boundary normalization.
Output: Renormalized amplitudes R(A_f, A_e, A_v) on the boundary.
\end{verbatim}

\section{Notation and conventions}

We work in time gauge for gravity boundaries (SU(2) spin networks), weakly imposing simplicity in the bulk. SO(10) labels are dominant integral highest weights; $C_2$ denotes the quadratic Casimir. TV norms always include the exponential weights indicated in the main text.

\end{document}