Level 10 Math Upgrade v9: Ultimate JMP proofreader code overwrite

papers/project_paper_1_relativity/armada_3_jmp/paper_1c_math_JMP.tex

@@ -47,7 +47,7 @@ It is crucial to note that this paper explicitly isolates the structural, graph-
 
 \subsection{Locally Finite Graded Posets}
 
-Let $\mathcal{P} = (V, \preceq)$ be a locally finite partially ordered set. The transitive reduction of $\mathcal{P}$ yields a directed acyclic graph $G = (V, E)$, where $(u,v) \in E$ if $u \prec v$ and there is no $w$ such that $u \prec w \prec v$.
+Let $\mathcal{P} = (V, \text{\raisebox{-0.2ex}{$\preceq$}})$ be a locally finite partially ordered set. The transitive reduction of $\mathcal{P}$ yields a directed acyclic graph $G = (V, E)$, where $(u,v) \in E$ if $u \prec v$ and there is no $w$ such that $u \prec w \prec v$.
 
 \begin{definition}[Graded Poset and Layer Volumes]
 We assume $\mathcal{P}$ is \emph{graded}, meaning there exists a surjective rank or height function $h: V \to \mathbb{Z}_{\ge 0}$ such that for all covering edges $(u, v) \in E$, $h(v) = h(u) + 1$. 
@@ -86,7 +86,7 @@ The normalized discrete directed Laplacian operator $\Delta_{\mathcal{P}}: \math
 \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 $P_{t+1}(v) - P_t(v) = \Delta_{\mathcal{P}}^* P_t(v)$.
+For a transition probability distribution $P_t(v)$ of a random walk, the forward evolution equation (the discrete diffusion equation) is governed by the adjoint operator such that $P_{t+1}(v) - P_t(v) = \Delta_{\mathcal{P}}^* P_t(v)$.
 
 \section{The Retarded Green's Function}
 
@@ -153,7 +153,7 @@ In this canonical K-R limit, the middle layer contains roughly $N/2$ vertices, w
 \end{equation}
 is strictly bounded away from zero because the numerator $|E(S, \bar{S})|$ scales identically with the volume denominator $\sum \deg_{\mathrm{out}}$ due to the maximal cross-layer connectivity. 
 
-By Cheeger's inequality ($\Delta \ge \Phi^2/2$), a macroscopic graph conductance $\Phi > 0$ strictly implies a macroscopic spectral gap $\Delta$ in the discrete Laplacian. A macroscopic spectral gap forces extreme multi-path mixing with a mixing time $\tau_{\mathrm{mix}} = \mathcal{O}(1)$; a random walk on an unstructured K-R poset will visit the maximal antichain in a single step, scattering its probability distribution uniformly over the $\exp(\mathcal{O}(N^2))$ available microstates. 
+By Cheeger's inequality ($\lambda_1 \ge \Phi^2/2$), a macroscopic graph conductance $\Phi > 0$ strictly implies a macroscopic spectral gap $\lambda_1$ in the discrete Laplacian. A macroscopic spectral gap forces extreme multi-path mixing with a mixing time $\tau_{\mathrm{mix}} = \mathcal{O}(1)$; a random walk on an unstructured K-R poset will visit the maximal antichain in a single step, scattering its probability distribution uniformly over the $\exp(\mathcal{O}(N^2))$ available microstates. 
 
 Consequently, for unstructured posets, the effective layer volume $|L_t|$ grows exponentially, mapping to an emergent topological dimension $d \to \infty$. 
 
@@ -198,7 +198,7 @@ R.~D. Sorkin,
 \bibitem{Kleitman1975}
 D.~J. Kleitman and B.~L. Rothschild,
 \newblock \emph{Asymptotic enumeration of partial orders on a finite set},
-\newblock Transactions of the American Mathematical Society \textbf{205}, 205--220 (1975).
+\textbf{205}, 205--220 (1975).
 
 \bibitem{Surya2019}
 S.~Surya,