diff --git a/papers/project_paper_1_relativity/armada_3_jmp/paper_1c_math_JMP.tex b/papers/project_paper_1_relativity/armada_3_jmp/paper_1c_math_JMP.tex
index 738b862f..7741d2e6 100644
--- a/papers/project_paper_1_relativity/armada_3_jmp/paper_1c_math_JMP.tex
+++ b/papers/project_paper_1_relativity/armada_3_jmp/paper_1c_math_JMP.tex
@@ -28,7 +28,7 @@
 \maketitle
 
 \begin{abstract}
-We investigate the behavior of random walks and discrete diffusion processes on locally finite directed acyclic graphs (DAGs), formalized as graded partially ordered sets (posets). We generalize Pólya's classical recurrence theorem to the directed setting by defining recurrence operationally: the persistent intersection of a random walk with a distinguished maximal chain. By formalizing the discrete d'Alembertian operator on graded posets and analyzing its associated retarded Green's function, we prove that the requirement for chain-recurrent classical correlations mathematically bounds the emergent topological dimension of the poset to $d \le 2$. Furthermore, invoking the Kleitman-Rothschild dominance theorem for the asymptotic combinatorics of generic posets, we show that unstructured discrete topologies undergo a combinatorial explosion in graph conductance, guaranteeing macroscopic spectral gaps and strict transience ($d \to \infty$). This result establishes a rigorous graph-theoretic constraint: stable, localized discrete topological structures that require the persistence of historical information cannot exceed two emergent spatial dimensions.
+We investigate the behavior of random walks and discrete diffusion processes on locally finite directed acyclic graphs (DAGs), formalized as graded partially ordered sets (posets). We generalize Pólya's classical recurrence theorem to the directed setting by defining recurrence operationally: the persistent intersection of a random walk with a distinguished maximal chain. By formalizing the discrete directed Laplacian (transition generator) on graded posets and analyzing its associated retarded Green's function, we prove that the requirement for chain-recurrent classical correlations mathematically bounds the emergent topological dimension of the poset to $d \le 2$. Furthermore, invoking the Kleitman-Rothschild dominance theorem for the asymptotic combinatorics of generic posets, we show that unstructured discrete topologies undergo a combinatorial explosion in graph conductance, guaranteeing macroscopic spectral gaps and strict transience ($d \to \infty$). This result establishes a rigorous graph-theoretic constraint: stable, localized discrete topological structures that require the persistence of historical information cannot exceed two emergent spatial dimensions.
 \end{abstract}
 
 \section{Introduction}
@@ -37,9 +37,9 @@ The study of random walks on graphs provides profound insights into the topologi
 
 In discrete models of mathematical physics, particularly causal set theory and related approaches to quantum gravity~\cite{Bombelli1987,Sorkin2003}, one frequently encounters locally finite Directed Acyclic Graphs (DAGs) acting as fundamental evolving networks. On a DAG, a random walk is constrained by causality; it must follow directed edges and, containing no cycles, can never return to its precise vertex of origin. Consequently, the traditional formulation of recurrence---returning to a starting point---must be generalized.
 
-In this paper, we establish a generalized recurrence threshold for DAGs. We define recurrence as the persistent temporal intersection of a diffusion state with a localized structural trajectory (a distinguished maximal chain). The propagation of the walk is governed by the discrete d'Alembertian operator acting on the graded graph. By computing the trace of the Green's function along the chain, we prove that the necessary condition for recurrent classical correlations bounds the topological dimension of the graph.
+In this paper, we establish a generalized recurrence threshold for DAGs. We define recurrence as the persistent temporal intersection of a diffusion state with a localized structural trajectory (a distinguished maximal chain). The propagation of the walk is governed by the discrete Laplacian acting on the graded graph. By computing the trace of the Green's function along the chain, we prove that the necessary condition for recurrent classical correlations bounds the topological dimension of the graph.
 
-\section{Formalism: Graded Posets and the Discrete d'Alembertian}
+\section{Formalism: Graded Posets and the Discrete Laplacian}
 
 \subsection{Locally Finite Graded Posets}
 
@@ -58,12 +58,15 @@ This definition naturally mirrors the volumetric boundary growth of a $d$-dimens
 \begin{definition}[Spatial Homogeneity Condition]
 A graded poset satisfies the \emph{spatial homogeneity condition} if, for any origin vertex $v_0$, the forward causal future $J^+(v_0) \cap L_t$ asymptotically covers the entire layer $L_t$ as $t \to \infty$, such that a random walk rapidly mixes and the transition probabilities uniformly distribute across the layer volume.
 \end{definition}
+\begin{remark}
+This condition restricts our core theorem to a specific class of highly symmetric, regular graded posets (discrete analogues of isotropic expanding spacetimes). Generic DAGs do not naturally satisfy this condition due to lightcone localization.
+\end{remark}
 
 \begin{definition}[Distinguished Maximal Chain]
 A \emph{distinguished maximal chain} $\gamma$ is a sequence of vertices $\gamma = \{v_0, v_1, v_2, \dots\}$ such that $v_t \in L_t$ and $(v_t, v_{t+1}) \in E$ for all $t \ge 0$. This chain represents a localized, persistent geometric observer or structural reference trajectory.
 \end{definition}
 
-\subsection{Difference Operators and the d'Alembertian}
+\subsection{Difference Operators and the Discrete Laplacian}
 
 To study diffusion and wave propagation on $\mathcal{P}$, we define the incidence kinematics. Let $\mathcal{H}_V$ and $\mathcal{H}_E$ be the Hilbert spaces of square-integrable functions on the vertices and edges, respectively.
 The forward difference operator $\nabla^+: \mathcal{H}_V \to \mathcal{H}_E$ and the backward difference operator $\nabla^-: \mathcal{H}_E \to \mathcal{H}_V$ are defined as:
@@ -72,23 +75,23 @@ The forward difference operator $\nabla^+: \mathcal{H}_V \to \mathcal{H}_E$ and
 (\nabla^- g)(v) &= \sum_{u: (u,v)\in E} g(u,v) - \sum_{w: (v,w)\in E} g(v,w).
 \end{align}
 
-\begin{definition}[Discrete d'Alembertian]
-The discrete d'Alembertian operator $\square_{\mathcal{P}}: \mathcal{H}_V \to \mathcal{H}_V$ is constructed from the directed divergence of the gradient ($\propto \nabla^- \nabla^+$) and is given by the weighted composition:
+\begin{definition}[Discrete Laplacian]
+The normalized discrete directed Laplacian operator $\Delta_{\mathcal{P}}: \mathcal{H}_V \to \mathcal{H}_V$ is constructed from the directed divergence of the gradient ($\propto \nabla^- \nabla^+$) and is given by the weighted composition:
 \begin{equation}
-\square_{\mathcal{P}} f(v) = \frac{1}{\deg_{\mathrm{out}}(v)} \sum_{w: (v,w)\in E} f(w) - f(v).
+\Delta_{\mathcal{P}} f(v) = \frac{1}{\deg_{\mathrm{out}}(v)} \sum_{w: (v,w)\in E} f(w) - f(v).
 \end{equation}
 \end{definition}
 
-For a transition probability distribution $P_t(v)$ of a random walk, the forward evolution equation (the discrete diffusion equation) is governed by the dual operator such that $\square_{\mathcal{P}}^* P_t = 0$.
+For a transition probability distribution $P_t(v)$ of a random walk, the forward evolution equation (the discrete diffusion equation) is governed by the dual operator such that $\Delta_{\mathcal{P}}^* P_t = 0$.
 
 \section{The Retarded Green's Function}
 
-The transition kernel $T(u \to v)$ representing the probability that a random walk starting at $u$ reaches $v$ can be extracted from the retarded Green's function of the d'Alembertian operator.
+The transition kernel $T(u \to v)$ representing the probability that a random walk starting at $u$ reaches $v$ can be extracted from the retarded Green's function of the Laplacian operator.
 
 \begin{lemma}[Green's Function Form]
 The retarded Green's function $G_R(x, x')$ satisfies
 \begin{equation}
-\square_{\mathcal{P}} G_R(x, x') = \delta(x, x').
+\Delta_{\mathcal{P}} G_R(x, x') = \delta(x, x').
 \end{equation}
 Because the graph is directed and graded, $G_R(x, x') = 0$ unless $h(x) \le h(x')$. 
 \end{lemma}
@@ -134,9 +137,9 @@ The asymptotic combinatorics of generic posets are heavily constrained by the Kl
 As the number of vertices $N \to \infty$, almost all posets on $N$ elements are dominated by a 3-layer structure.
 \end{proposition}
 
-In this canonical K-R structure, the middle layer contains roughly $N/2$ vertices, while the top and bottom layers contain $N/4$ vertices each. The number of covering relations (edges) between layers is maximal, meaning the graph conductance $\Phi$ is strictly bounded away from zero. 
+In this canonical K-R limit, the middle layer contains roughly $N/2$ vertices, while the top and bottom layers contain $N/4$ vertices each. As $N \to \infty$, the degree of any vertex scales as $\mathcal{O}(N)$. Consequently, generic posets explicitly fail the local-finiteness assumption required for stable diffusion. The number of covering relations (edges) between layers is maximal, meaning the graph conductance $\Phi$ is strictly bounded away from zero. 
 
-By Cheeger's inequality, a macroscopic graph conductance $\Phi$ implies a macroscopic spectral gap $\Delta$ in the discrete Laplacian (or d'Alembertian). A macroscopic spectral gap forces extreme multi-path mixing; a random walk on an unstructured K-R poset will visit the maximal antichain in a single step, scattering its probability distribution uniformly over $\mathcal{O}(2^N)$ microstates. 
+By Cheeger's inequality, a macroscopic graph conductance $\Phi$ implies a macroscopic spectral gap $\Delta$ in the discrete Laplacian. A macroscopic spectral gap forces extreme multi-path mixing; a random walk on an unstructured K-R poset will visit the maximal antichain in a single step, scattering its probability distribution uniformly over $\mathcal{O}(2^N)$ microstates. 
 
 Consequently, for unstructured posets, the effective layer volume $|L_t|$ grows exponentially, mapping to an emergent topological dimension $d \to \infty$. 
 
@@ -146,7 +149,7 @@ Because unstructured generic posets map to $d \to \infty$, random walks on typic
 
 \section{Conclusion}
 
-We have demonstrated that Pólya's threshold extends to directed acyclic graphs when recurrence is defined with respect to a distinguished continuous chain. The evaluation of the Green's function for the discrete d'Alembertian explicitly links the probability of chain-intersection to the topological dimensionality of the poset layers. 
+We have demonstrated that Pólya's threshold extends to directed acyclic graphs when recurrence is defined with respect to a distinguished continuous chain. The evaluation of the Green's function for the discrete Laplacian explicitly links the probability of chain-intersection to the topological dimensionality of the poset layers. 
 
 Our results indicate that if a discrete topological system requires recurrent classical correlations---where a diffusion state must persistently return localized information to a structural point of reference---the underlying DAG is mathematically constrained to a topological dimension of $d \le 2$. In higher dimensions ($d \ge 3$), the rapid algebraic expansion of the phase space ensures that all random walks are transient diffusion states. Furthermore, the Kleitman-Rothschild dominance theorem establishes that unstructured posets naturally undergo an exponential layer explosion, guaranteeing macroscopic graph conductance and strict transience. Thus, low-dimensional recurrent structures must be dynamically or topologically protected against generic combinatorial expansion.