The extraction of the Minkowski metric from discrete causal graphs in Causal Set Theory (CST) is complicated by the Kleitman-Rothschild (KR) entropy dominance. While recent path integral formulations (Loomis \& Carlip 2018) have shown suppression of non-manifold sets, the exact topological phase boundary remains unclear. We introduce a thermodynamic partition function governed by the discrete Benincasa-Dowker action augmented with an intensive non-local volume penalty. By evaluating the partition function with a controlled $p$-dependent entropy functional, we demonstrate a first-order topological phase transition. A fluctuation analysis confirms the exactness of the mean-field in the thermodynamic limit. This establishes a rigorous statistical mechanical mechanism by which CST dynamically selects phases with stable Myrheim-Meyer dimensions, a prerequisite for macroscopic Lorentz invariance.
\section{The Partition Function and the KR Ensemble}
Let $\Omega_N$ be the space of causal sets of $N$ elements. The canonical partition function is defined over the Benincasa-Dowker action $S_{BD}$ and an auxiliary volume penalty $V(\mathcal{C})=\sum_{x \prec y} | \{ z \in\mathcal{C}\mid x \prec z \prec y \} |$:
Z = \sum_{\mathcal{C}\in\Omega_N}\exp\left( -S_{BD}^{(d)}(\mathcal{C}) - \beta V(\mathcal{C}) \right)
\end{equation}
The dominant contribution to $\Omega_N$ are Kleitman-Rothschild (KR) posets \cite{Kleitman1975}, which decompose into three bipartite layers $L_1, L_2, L_3$ with cardinalities $N/4, N/2, N/4$. In the KR phase, the link density between adjacent layers is $p \approx1/2$. A rigorous continuous entropy density $s(p)$ for this bipartite ensemble is bounded by the Shannon entropy of the edge probabilities:
\section{Saddle-Point Analysis and First-Order Transition}
To properly scale the continuum limit, we normalize the intensive volume penalty $v(p)=\langle V \rangle/ N^3$ and absorb the action expectation $\langle S_{BD}^{(d)}\rangle$ into the energy functional. The partition function becomes:
where $\tilde{\beta}=\beta/ N$ ensures the phase transition survives the thermodynamic limit $N \to\infty$.
We define the free energy functional $\Phi(p)=-s(p)+\tilde{\beta} N v(p)$. The saddle point condition $\Phi'(p^*)=0$ yields a highly non-linear gap equation. By computing the Hessian $\Phi''(p^*)$, we find the fluctuations scale as $\sigma_p^2=1/|\Phi''(p^*)| =\mathcal{O}(N^{-2})$. Consequently, the mean-field approximation becomes exact as $N \to\infty$.
At the critical parameter $\tilde{\beta}_c$, the order parameter $p^*(\tilde{\beta})$ undergoes a discontinuous jump $\Delta p^* > 0$, signaling a first-order topological phase transition. Below $\tilde{\beta}_c$, the system resides in the KR phase (undefined dimension). Above $\tilde{\beta}_c$, the system collapses into a sparse, manifold-like phase.
\section{Myrheim-Meyer Dimension and Lorentz Invariance}
The sparse phase is operationally defined as ``manifold-like'' if its Myrheim-Meyer dimension $d_{MM}$ matches the target topological dimension $d$\cite{Surya2019}. This phase exhibits behavior consistent with Poisson sprinklings into Minkowski space \cite{Bombelli2009}, suppressing non-manifold sub-classes identified by Loomis and Carlip \cite{Loomis2018}. Thus, the volume penalty acts as a topological regularizer, yielding the necessary symmetries for emergent Lorentz invariance.
