diff --git a/data/ifd_both_vs_lossratio.png b/data/ifd_both_vs_lossratio.png
new file mode 100644
index 0000000..267e863
Binary files /dev/null and b/data/ifd_both_vs_lossratio.png differ
diff --git a/main.tex b/main.tex
index 9bd1b94..06b4a27 100644
--- a/main.tex
+++ b/main.tex
@@ -154,7 +154,8 @@ An IFD $>1$ indicates the instruction increases the loss relative to
 unconditional generation (the model struggles to follow it), while
 IFD $<1$ means the instruction provides useful guidance.
 
-We compute IFD for $N=3000$ SFT samples using both the pretrained
+We compute IFD for $N=3000$ SFT samples drawn from the
+Alpaca-GPT4 dataset~\cite{alpaca} using both the pretrained
 base model (after 15B tokens of pretraining) and a supervised
 fine-tuned checkpoint (after 1K SFT steps).
 Figure~\ref{fig:ifd} shows the distribution.
@@ -184,13 +185,59 @@ $5.3\times$ more than unconditional loss, confirming that SFT teaches
 instruction following rather than merely improving generic language
 modeling. Detailed analysis is provided in Appendix~\ref{app:ifd}.
 
+\subsubsection{IFD vs.\ Loss Ratio}
+
+We further define the \emph{loss ratio}---the fraction of
+conditional loss retained after SFT---as:
+\begin{equation}
+\text{Loss Ratio} = \frac{L_{\text{cond}}^{\text{ckpt}}}{L_{\text{cond}}^{\text{base}}}.
+\end{equation}
+Table~\ref{tab:ifd_lossratio_corr} reports the pairwise correlations.
+
+\begin{table}[H]
+\centering
+\caption{Pairwise correlations among IFD and Loss Ratio.}
+\label{tab:ifd_lossratio_corr}
+\small
+\begin{tabular}{@{}lcc@{}}
+\toprule
+\textbf{Pair} & \textbf{Pearson $r$} & \textbf{Spearman $\rho$} \\
+\midrule
+IFD\textsubscript{base} vs.\ Loss Ratio & $+0.10$ & $+0.05$ \\
+IFD\textsubscript{ckpt} vs.\ Loss Ratio & $+0.90$ & $+0.91$ \\
+IFD\textsubscript{base} vs.\ IFD\textsubscript{ckpt} & $+0.38$ & $+0.49$ \\
+\bottomrule
+\end{tabular}
+\end{table}
+
+The near-perfect correlation between IFD\textsubscript{ckpt} and
+Loss Ratio ($r = 0.90$) reflects a mathematical near-identity:
+both are dominated by $L_{\text{cond}}^{\text{ckpt}}$ in the
+numerator. Consequently, IFD\textsubscript{ckpt} is
+redundant---it essentially measures how much the conditional loss
+has dropped after SFT, i.e., the learning speed of each sample. In contrast, IFD\textsubscript{base} and Loss
+Ratio are nearly orthogonal ($r = 0.10$), forming a complementary
+two-dimensional screening space: IFD\textsubscript{base} measures
+``how hard does the base model find this,'' while Loss Ratio
+measures ``how much did SFT improve it.'' Samples with high
+IFD\textsubscript{base} \emph{and} low Loss Ratio are the most
+informative for training.
+
+\begin{figure}[H]
+\centering
+\includegraphics[width=0.80\linewidth]{data/ifd_both_vs_lossratio.png}
+\caption{IFD\textsubscript{base} vs.\ Loss Ratio (left),
+IFD\textsubscript{ckpt} vs.\ Loss Ratio (right).}
+\label{fig:ifd_lossratio}
+\end{figure}
+
 % ======================================================================
 \section{Model Architecture}
 % ======================================================================
 
 The model is a 24-layer decoder-only Transformer with Grouped Query Attention
-(GQA)~\cite{ainslie2023gqa} and SwiGLU feed-forward blocks~\cite{shazeer2020glu},
-with Rotary Position Embedding (RoPE)~\cite{su2024roformer}.
+(GQA)~\cite{ainslie2023gqa}, SwiGLU feed-forward blocks~\cite{shazeer2020glu},
+and Rotary Position Embedding (RoPE)~\cite{su2024roformer}.
 Table~\ref{tab:model_config} summarizes the configuration.
 
 \begin{table}[H]
@@ -210,12 +257,49 @@ Norm & RMSNorm ($\epsilon=10^{-5}$) & RoPE $\theta$ & 10,000 \\
 \end{tabular}
 \end{table}
 
-Each decoder block $\ell$ computes:
+With Grouped Query Attention~\cite{ainslie2023gqa} ($n_q = 24$ query
+heads, $n_{kv} = 4$ key/value heads, group size $g = n_q / n_{kv} = 6$):
 \begin{equation}
-\mathbf{h}_\ell = \mathbf{x}_\ell + \operatorname{GQA}\bigl(\operatorname{RMSNorm}(\mathbf{x}_\ell)\bigr), \qquad
-\mathbf{x}_{\ell+1} = \mathbf{h}_\ell + \operatorname{MLP}\bigl(\operatorname{RMSNorm}(\mathbf{h}_\ell)\bigr),
+\begin{aligned}
+\operatorname{GQA}(\mathbf{X}) &= \operatorname{Concat}\bigl(\operatorname{head}_1,\dots,\operatorname{head}_{n_q}\bigr)\mathbf{W}_O,\\[2mm]
+\operatorname{head}_i &= \operatorname{Attn}\Bigl(
+\mathbf{X}\mathbf{W}_Q^{(i)},\,
+\mathbf{X}\mathbf{W}_K^{(\lfloor i / g \rfloor)},\,
+\mathbf{X}\mathbf{W}_V^{(\lfloor i / g \rfloor)}
+\Bigr),
+\end{aligned}
+\end{equation}
+where $\operatorname{Attn}(\mathbf{Q},\mathbf{K},\mathbf{V}) =
+\operatorname{Softmax}(\mathbf{Q}\mathbf{K}^{\mkern-1mu\mathsf{T}} / \sqrt{d_h})\mathbf{V}$.
+Rotary Position Embedding (RoPE)~\cite{su2024roformer} encodes position
+$m$ by rotating pairs of hidden dimensions:
+\begin{equation}
+\operatorname{RoPE}(\mathbf{x}_m)_i =
+\begin{cases}
+x_{m,i}\cos(m\theta_{j}) - x_{m,i+1}\sin(m\theta_{j}), & i = 2j,\\[2mm]
+x_{m,i-1}\sin(m\theta_{j}) + x_{m,i}\cos(m\theta_{j}), & i = 2j+1,
+\end{cases}
+\end{equation}
+with frequency $\theta_j = 10000^{-2j/d}$ for $j = 0,\dots,d/2-1$.
+The SwiGLU~\cite{shazeer2020glu} feed-forward applies a gated Swish
+non-linearity:
+\begin{equation}
+\operatorname{MLP}(\mathbf{x}) = \mathbf{W}_{\text{down}}\Bigl(
+\mathbf{W}_{\text{up}}\mathbf{x} \odot
+\operatorname{SiLU}\bigl(\mathbf{W}_{\text{gate}}\mathbf{x}\bigr)
+\Bigr),
+\label{eq:swiglu}
+\end{equation}
+where $\operatorname{SiLU}(z) = z / (1 + e^{-z})$.
+
+Each decoder block $\ell$ then applies pre-norm residual connections:
+\begin{equation}
+\begin{aligned}
+\mathbf{h}_\ell &= \mathbf{x}_\ell + \operatorname{GQA}\bigl(\operatorname{RMSNorm}(\mathbf{x}_\ell)\bigr),\\[2mm]
+\mathbf{x}_{\ell+1} &= \mathbf{h}_\ell + \operatorname{MLP}\bigl(\operatorname{RMSNorm}(\mathbf{h}_\ell)\bigr).
+\end{aligned}
+\label{eq:decoder_block}
 \end{equation}
-where $\operatorname{MLP}(\mathbf{x}) = \mathbf{W}_{\text{down}}(\mathbf{W}_{\text{up}}\mathbf{x} \odot \operatorname{SiLU}(\mathbf{W}_{\text{gate}}\mathbf{x}))$.
 
 \subsection{Initialization}
 
@@ -589,6 +673,12 @@ in the high-IFD tail.
 % ======================================================================
 \begin{thebibliography}{99}
 
+\bibitem{alpaca}
+R.~Taori, I.~Gulrajani, T.~Zhang, Y.~Dubois, X.~Li, C.~Guestrin,
+P.~Liang, T.~B.~Hashimoto.
+Alpaca: A strong, replicable instruction-following model.
+\textit{Stanford Center for Research on Foundation Models (CRFM)}, 2023.
+
 \bibitem{ainslie2023gqa}
 J.~Ainslie, J.~Lee-Thorp, M.~de Jong, Y.~Zemlyanskiy, F.~Lebr\'on, S.~Sanghai.
 GQA: Training generalized multi-query transformer models from multi-head