Level 10 Math Upgrade v4: Final Notation and Bibliography Polish

papers/project_paper_1_relativity/armada_3_jmp/paper_1c_math_JMP.tex

@@ -50,7 +50,7 @@ We assume $\mathcal{P}$ is \emph{graded}, meaning there exists a surjective rank
 The \emph{layer} at height $t$ is the antichain $L_t = \{v \in V \mid h(v) = t\}$. 
 The poset possesses an emergent \emph{topological dimension} $d$ if the cardinalities of the layers grow asymptotically as
 \begin{equation}
-|L_t| \sim \Theta(t^{d-1}) \quad \text{as} \quad t \to \infty.
+|L_t| = \Theta(t^{d-1}) \quad \text{as} \quad t \to \infty.
 \end{equation}
 \end{definition}
 This definition naturally mirrors the volumetric boundary growth of a $d$-dimensional continuous space, where the cross-sectional area at radial time $t$ scales as $t^{d-1}$.
@@ -91,7 +91,7 @@ The transition kernel $T(u \to v)$ representing the probability that a random wa
 \begin{lemma}[Green's Function Form]
 The retarded Green's function $G_R(x, x')$ satisfies
 \begin{equation}
-\Delta_{\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}
@@ -186,6 +186,16 @@ 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).
 
+\bibitem{Surya2019}
+S.~Surya,
+\newblock \emph{The causal set approach to quantum gravity},
+\newblock Living Reviews in Relativity \textbf{22}, 5 (2019).
+
+\bibitem{Rideout2009}
+D.~Rideout and P.~Wallden,
+\newblock \emph{Spacetime topology from the causal set},
+\newblock Journal of Physics: Conference Series \textbf{174}, 012017 (2009).
+
 \end{thebibliography}
 
 \end{document}