diff --git a/arxiv-style/equations.tex b/arxiv-style/equations.tex
index 7b64203..c63fa43 100644
--- a/arxiv-style/equations.tex
+++ b/arxiv-style/equations.tex
@@ -1,23 +1,9 @@
 \documentclass[10pt, twocolumn]{article}
 \usepackage{amsmath, amssymb}
 \usepackage{bm}
-\usepackage{booktabs}
 \usepackage[margin=1in]{geometry}
 \usepackage{microtype}
 
-% Custom operators
-\DeclareMathOperator*{\argmin}{arg\,min}
-\DeclareMathOperator*{\argmax}{arg\,max}
-\DeclareMathOperator{\CE}{CE}
-\DeclareMathOperator{\SNR}{SNR}
-
-% Bold math symbols
-\newcommand{\bX}{\bm{X}}
-\newcommand{\bS}{\bm{S}}
-\newcommand{\bR}{\bm{R}}
-\newcommand{\br}{\bm{r}}
-\newcommand{\by}{\bm{y}}
-
 \title{Equations: Mask-DDPM Methodology}
 \author{}
 \date{}
@@ -26,53 +12,53 @@
 \maketitle
 
 \section{Problem Formulation}
-Each training instance is a fixed-length window of length $L$, comprising continuous channels $\bX \in \mathbb{R}^{L \times d_c}$ and discrete channels $\bY = \{y^{(j)}_{1:L}\}_{j=1}^{d_d}$, where each discrete variable satisfies $y^{(j)}_t \in \mathcal{V}_j$ for a finite vocabulary $\mathcal{V}_j$.
+Each training instance is a fixed-length window of length $L$, comprising continuous channels $\bm{X} \in \mathbb{R}^{L \times d_c}$ and discrete channels $\bm{Y} = \{y^{(j)}_{1:L}\}_{j=1}^{d_d}$, where each discrete variable satisfies $y^{(j)}_t \in \mathcal{V}_j$ for a finite vocabulary $\mathcal{V}_j$.
 
 \section{Transformer Trend Module for Continuous Dynamics}
 We posit an additive decomposition of the continuous signal:
 \begin{equation}
-\bX = \bS + \bR,
+\bm{X} = \bm{S} + \bm{R},
 \label{eq:additive_decomp}
 \end{equation}
-where $\bS \in \mathbb{R}^{L \times d_c}$ captures the smooth temporal trend and $\bR \in \mathbb{R}^{L \times d_c}$ represents distributional residuals.
+where $\bm{S} \in \mathbb{R}^{L \times d_c}$ captures the smooth temporal trend and $\bm{R} \in \mathbb{R}^{L \times d_c}$ represents distributional residuals.
 
 The causal Transformer trend extractor $f_{\phi}$ predicts the next-step trend via:
 \begin{equation}
-\hat{\bS}_{t+1} = f_{\phi}(\bX_{1:t}), \quad t = 1, \dots, L-1.
+\hat{\bm{S}}_{t+1} = f_{\phi}(\bm{X}_{1:t}), \quad t = 1, \dots, L-1.
 \label{eq:trend_prediction}
 \end{equation}
 Training minimizes the mean-squared error:
 \begin{equation}
-\mathcal{L}_{\text{trend}}(\phi) = \frac{1}{(L-1)d_c} \sum_{t=1}^{L-1} \bigl\| \hat{\bS}_{t+1} - \bX_{t+1} \bigr\|_2^2.
+\mathcal{L}_{\text{trend}}(\phi) = \frac{1}{(L-1)d_c} \sum_{t=1}^{L-1} \bigl\| \hat{\bm{S}}_{t+1} - \bm{X}_{t+1} \bigr\|_2^2.
 \label{eq:trend_loss}
 \end{equation}
-At inference, the residual target is defined as $\bR = \bX - \hat{\bS}$.
+At inference, the residual target is defined as $\bm{R} = \bm{X} - \hat{\bm{S}}$.
 
 \section{DDPM for Continuous Residual Generation}
 Let $K$ denote diffusion steps with noise schedule $\{\beta_k\}_{k=1}^K$, $\alpha_k = 1 - \beta_k$, and $\bar{\alpha}_k = \prod_{i=1}^k \alpha_i$. The forward corruption process is:
 \begin{align}
-q(\br_k \mid \br_0) &= \mathcal{N}\bigl( \sqrt{\bar{\alpha}_k}\,\br_0,\; (1 - \bar{\alpha}_k)\mathbf{I} \bigr), \\
-\br_k &= \sqrt{\bar{\alpha}_k}\,\br_0 + \sqrt{1 - \bar{\alpha}_k}\,\boldsymbol{\epsilon}, \quad \boldsymbol{\epsilon} \sim \mathcal{N}(\mathbf{0}, \mathbf{I}),
+q(\bm{r}_k \mid \bm{r}_0) &= \mathcal{N}\bigl( \sqrt{\bar{\alpha}_k}\,\bm{r}_0,\; (1 - \bar{\alpha}_k)\mathbf{I} \bigr), \\
+\bm{r}_k &= \sqrt{\bar{\alpha}_k}\,\bm{r}_0 + \sqrt{1 - \bar{\alpha}_k}\,\boldsymbol{\epsilon}, \quad \boldsymbol{\epsilon} \sim \mathcal{N}(\mathbf{0}, \mathbf{I}),
 \label{eq:forward_process}
 \end{align}
-where $\br_0 \equiv \bR$.
+where $\bm{r}_0 \equiv \bm{R}$.
 
 The reverse process is parameterized as:
 \begin{equation}
-p_{\theta}(\br_{k-1} \mid \br_k, \hat{\bS}) = \mathcal{N}\bigl( \boldsymbol{\mu}_{\theta}(\br_k, k, \hat{\bS}),\; \boldsymbol{\Sigma}(k) \bigr).
+p_{\theta}(\bm{r}_{k-1} \mid \bm{r}_k, \hat{\bm{S}}) = \mathcal{N}\bigl( \boldsymbol{\mu}_{\theta}(\bm{r}_k, k, \hat{\bm{S}}),\; \boldsymbol{\Sigma}(k) \bigr).
 \label{eq:reverse_process}
 \end{equation}
 Training employs the $\epsilon$-prediction objective:
 \begin{equation}
-\mathcal{L}_{\text{cont}}(\theta) = \mathbb{E}_{k,\br_0,\boldsymbol{\epsilon}} \left[ \bigl\| \boldsymbol{\epsilon} - \boldsymbol{\epsilon}_{\theta}(\br_k, k, \hat{\bS}) \bigr\|_2^2 \right].
+\mathcal{L}_{\text{cont}}(\theta) = \mathbb{E}_{k,\bm{r}_0,\boldsymbol{\epsilon}} \left[ \bigl\| \boldsymbol{\epsilon} - \boldsymbol{\epsilon}_{\theta}(\bm{r}_k, k, \hat{\bm{S}}) \bigr\|_2^2 \right].
 \label{eq:ddpm_loss}
 \end{equation}
 Optionally, SNR-based reweighting yields:
 \begin{equation}
-\mathcal{L}^{\text{snr}}_{\text{cont}}(\theta) = \mathbb{E}_{k,\br_0,\boldsymbol{\epsilon}} \left[ w_k \bigl\| \boldsymbol{\epsilon} - \boldsymbol{\epsilon}_{\theta}(\br_k, k, \hat{\bS}) \bigr\|_2^2 \right],
+\mathcal{L}^{\text{snr}}_{\text{cont}}(\theta) = \mathbb{E}_{k,\bm{r}_0,\boldsymbol{\epsilon}} \left[ w_k \bigl\| \boldsymbol{\epsilon} - \boldsymbol{\epsilon}_{\theta}(\bm{r}_k, k, \hat{\bm{S}}) \bigr\|_2^2 \right],
 \label{eq:snr_loss}
 \end{equation}
-where $w_k = \min(\SNR_k, \gamma) / \SNR_k$ and $\SNR_k = \bar{\alpha}_k / (1 - \bar{\alpha}_k)$. The final continuous output is reconstructed as $\hat{\bX} = \hat{\bS} + \hat{\bR}$.
+where $w_k = \min(\mathrm{SNR}_k, \gamma) / \mathrm{SNR}_k$ and $\mathrm{SNR}_k = \bar{\alpha}_k / (1 - \bar{\alpha}_k)$. The final continuous output is reconstructed as $\hat{\bm{X}} = \hat{\bm{S}} + \hat{\bm{R}}$.
 
 \section{Masked Diffusion for Discrete Variables}
 For discrete channel $j$, the forward masking process follows schedule $\{m_k\}_{k=1}^K$:
@@ -88,12 +74,12 @@ applied independently across variables and timesteps.
 
 The denoiser $h_{\psi}$ predicts categorical distributions conditioned on continuous context:
 \begin{equation}
-p_{\psi}\bigl( y^{(j)}_0 \mid y_k, k, \hat{\bS}, \hat{\bX} \bigr) = h_{\psi}(y_k, k, \hat{\bS}, \hat{\bX}).
+p_{\psi}\bigl( y^{(j)}_0 \mid y_k, k, \hat{\bm{S}}, \hat{\bm{X}} \bigr) = h_{\psi}(y_k, k, \hat{\bm{S}}, \hat{\bm{X}}).
 \label{eq:discrete_denoising}
 \end{equation}
 Training minimizes the categorical cross-entropy:
 \begin{equation}
-\mathcal{L}_{\text{disc}}(\psi) = \mathbb{E}_{k} \left[ \frac{1}{|\mathcal{M}|} \sum_{(j,t) \in \mathcal{M}} \CE\bigl( h_{\psi}(y_k, k, \hat{\bS}, \hat{\bX})_{j,t},\; y^{(j)}_{0,t} \bigr) \right],
+\mathcal{L}_{\text{disc}}(\psi) = \mathbb{E}_{k} \left[ \frac{1}{|\mathcal{M}|} \sum_{(j,t) \in \mathcal{M}} \mathrm{CE}\bigl( h_{\psi}(y_k, k, \hat{\bm{S}}, \hat{\bm{X}})_{j,t},\; y^{(j)}_{0,t} \bigr) \right],
 \label{eq:discrete_loss}
 \end{equation}
 where $\mathcal{M}$ denotes masked positions at step $k$.
@@ -104,6 +90,6 @@ The combined objective balances continuous and discrete learning:
 \mathcal{L} = \lambda \, \mathcal{L}_{\text{cont}} + (1 - \lambda) \, \mathcal{L}_{\text{disc}}, \quad \lambda \in [0,1].
 \label{eq:joint_objective}
 \end{equation}
-Type-aware routing enforces deterministic reconstruction $\hat{x}^{(i)} = g_i(\hat{\bX}, \hat{\bY})$ for derived variables.
+Type-aware routing enforces deterministic reconstruction $\hat{x}^{(i)} = g_i(\hat{\bm{X}}, \hat{\bm{Y}})$ for derived variables.
 
 \end{document}
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