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@@ -330,6 +330,15 @@ initialization over 15B tokens.}
 Figure~\ref{fig:loss} shows both loss curves; GPT-2 residual scaling (lower
 curve) maintains a clear advantage, particularly in the 0.3--0.8B token region.
 
+\begin{figure}[H]
+\centering
+\includegraphics[width=0.95\linewidth]{data/ckpt_comparison.png}
+\caption{Optimizer and initialization comparison.}
+\label{fig:ckpt_comparison}
+\end{figure}
+
+Figure~\ref{fig:ckpt_comparison} compares four configurations. The left panel shows training loss for Muon (Embedding Adam + 1D Adam), Muon (Embedding Muon + 1D Adam), Kaiming init, and Normal init; the center panel zooms in on the two current Muon variants; and the right panel shows gradient norms over optimizer steps. The older Kaiming and Normal initializations converge more slowly and plateau at higher loss. Between the current variants, using Adam for the embedding layer yields lower loss and more stable gradients than using Muon embeddings.
+
 \begin{table}[H]
 \centering
 \caption{Loss at 0.125B-interval milestones, 0--1B tokens.}
@@ -491,6 +500,64 @@ Response length    & $-0.36$ & $-0.48$ & $-0.56$ & $-0.79$ & $+0.58$ & $+0.72$ \
 \end{tabular}
 \end{table}
 
+\subsection{Loss Ratio}
+\label{sec:loss_ratio}
+
+We further define the \textbf{Loss Ratio} as the fraction of conditional
+loss retained after SFT:
+\begin{equation}
+\text{Loss Ratio} = \frac{L_{\text{cond}}^{\text{1K}}}{L_{\text{cond}}^{\text{base}}}.
+\end{equation}
+
+Over $N=3000$ samples:
+\begin{itemize}[nosep]
+  \item Mean $= 1.106$, median $= 1.084$, std $= 0.110$;
+        $90.8\%$ of samples exceed $1.0$.
+  \item Range: $[0.768, 1.962]$; only $9.2\%$ of samples show
+        a decrease ($<1.0$) in conditional loss after SFT.
+\end{itemize}
+
+The predominance of loss ratio $>1$ confirms that the 1K-step SFT
+checkpoint has not converged to a lower-loss region for the
+evaluation samples. Instead, the fine-tuning distribution shift
+increases NLL on most held-out instructions.
+
+Table~\ref{tab:ifd_lr_corr} reports the pairwise correlations.
+Despite the theoretical expectation that IFD\textsubscript{ckpt} and
+Loss Ratio share $L_{\text{cond}}^{\text{1K}}$ in their numerators
+and should therefore correlate strongly~\cite{li2023ifd}, the
+observed correlation is near zero ($r = -0.02$, $\rho = 0.04$).
+This is because the {\em relative} ordering of
+$L_{\text{cond}}^{\text{base}}$ and
+$L_{\text{uncond}}^{\text{ckpt}}$ (which determine the slope
+$k_i = L_{\text{cond},i}^{\text{base}} / L_{\text{uncond},i}^{\text{ckpt}}$
+in the relationship $\text{IFD}_{\text{ckpt},i} = k_i \cdot
+\text{Loss Ratio}_i$) varies widely across samples, breaking the
+proportionality at the sample level.
+
+\begin{table}[H]
+\centering
+\caption{Pairwise correlations between IFD variants and Loss Ratio.}
+\label{tab:ifd_lr_corr}
+\small
+\begin{tabular}{@{}lcc@{}}
+\toprule
+\textbf{Pair} & Pearson $r$ & Spearman $\rho$ \\
+\midrule
+IFD\textsubscript{base} vs.\ IFD\textsubscript{ckpt}  & $+0.97$ & $+0.96$ \\
+IFD\textsubscript{base} vs.\ Loss Ratio       & $-0.15$ & $-0.06$ \\
+IFD\textsubscript{ckpt} vs.\ Loss Ratio       & $-0.02$ & $+0.04$ \\
+\bottomrule
+\end{tabular}
+\end{table}
+
+The near-perfect correlation between IFD\textsubscript{base} and
+IFD\textsubscript{ckpt} ($r = 0.97$) reveals that the IFD ranking is
+highly robust to the choice of evaluation model: samples that the
+base model finds difficult remain difficult after 1K SFT steps.
+This stability justifies using the base-model IFD as a data
+selection signal without re-evaluating after fine-tuning.
+
 % ======================================================================
 \section{Weight Distribution by Component}
 \label{app:weight_dist}