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diff --git a/main.tex b/main.tex
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--- a/main.tex
+++ b/main.tex
@@ -157,11 +157,20 @@ Figure~\ref{fig:ifd} shows the distribution.
 \begin{figure}[H]
 \centering
 \includegraphics[width=0.80\linewidth]{data/ifd_compare_clean.png}
-\caption{IFD comparison: base model vs.\ trained checkpoint. The
+\caption{IFD scatter: base model vs.\ trained checkpoint. The
 diagonal line marks $\mathrm{IFD}_{\text{base}} = \mathrm{IFD}_{\text{ckpt}}$.}
 \label{fig:ifd}
 \end{figure}
 
+\begin{figure}[H]
+\centering
+\includegraphics[width=0.80\linewidth]{data/ifd_density_dist.png}
+\caption{IFD density distribution: base model and SFT checkpoint.}
+\label{fig:ifd_density}
+\end{figure}
+
+Figure~\ref{fig:ifd_density} shows the corresponding density
+estimates, confirming the systematic leftward shift after SFT.
 The pretrained base model (15B tokens) has mean IFD $0.9625$;
 $29.8\%$ of samples exceed $1.0$. After 1K SFT steps, mean IFD drops
 to $0.7539$, with only $0.4\%$ of samples above $1.0$. The average
@@ -177,6 +186,7 @@ modeling. Detailed analysis is provided in Appendix~\ref{app:ifd}.
 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}.
+Table~\ref{tab:model_config} summarizes the configuration.
 
 \begin{table}[H]
 \centering
@@ -466,6 +476,39 @@ a residual signal persists: instructions that require complex reasoning
 tend to remain non-trivially harder than simple rewrite or extraction
 tasks even after SFT.
 
+\subsection{A Note on IFD Bias from Response Length}
+\label{sec:ifd_bias}
+
+Both $L_{\text{cond}}$ and $L_{\text{uncond}}$ are reported as per-token
+average losses. For a response of length $T$, the unconditional loss is
+$L_{\text{uncond}} = \frac{1}{T} \sum_{t=1}^T \log P(x_t)$.
+Since the variance of this average scales as $1/T$, shorter responses
+exhibit much larger fluctuations in $L_{\text{uncond}}$---a mathematical
+necessity, not a signal of instruction difficulty. Consequently, IFD,
+being a ratio of two such averages, inherits a systematic length bias:
+short responses inflate IFD variance.
+
+Figure~\ref{fig:length_bias} illustrates this artifact. Responses
+with $<20$ tokens (e.g., ``Paris,'' ``42'') show wildly scattered
+$L_{\text{uncond}}$ values, producing spurious high or low IFD scores.
+Longer responses ($>50$ tokens) converge toward the model's intrinsic
+mean loss, yielding stable IFD estimates.
+
+\begin{figure}[H]
+\centering
+\includegraphics[width=0.75\linewidth]{data/length_vs_loss_ifd.png}
+\caption{Response length vs.\ unconditional loss and IFD. Short
+responses produce high-variance $L_{\text{uncond}}$ estimates,
+inflating IFD noise.}
+\label{fig:length_bias}
+\end{figure}
+
+This bias is well-known in the IFD literature but often omitted.
+For practitioners, a simple mitigation is to filter samples with
+$<20$ response tokens before computing IFD. In our Alpaca-style
+dataset, this removes approximately $5$--$8\%$ of samples and
+substantially reduces false positives in the high-IFD tail.
+
 % ======================================================================
 \begin{thebibliography}{99}