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\title[ELF-REG]{ELF-REG: Scaling Continuous Diffusion\\Language Models to Reasoning Tasks}
\subtitle{Better intermediate representations,\\useful answers before the trajectory ends}
\author{Paper authors: Zeyu Michael Li, William Xingxu Chen,\\
Bingshuo Qian, Jiayin Liu, Xiang Cheng\\[2mm]
\scriptsize Duke University / Tsinghua University}
\date{\href{https://arxiv.org/abs/2609.29102v3}{arXiv:2609.29102v3, 8 October 2026}\\
Reading discussion: 14 October 2026}
\begin{document}

\begin{frame}
\titlepage
\end{frame}

\begin{frame}{Can continuous parallel generation learn reasoning?}
\begin{block}{The question}
Can a fully continuous language model produce useful mathematical
solutions and code with few network function evaluations (NFE)?
\end{block}
\begin{itemize}
\item ELF (Embedded Language Flows) denoises all response positions together and discretizes only at the final decode.
\item \alert{Training:} align intermediate features and jointly denoise a global teacher representation.
\item \alert{Inference:} decode a clean prediction from an early trajectory prefix.
\end{itemize}
\papersource{Paper, Secs. 1--3, PDF pp. 2--4. REPA, REG, and early stopping build on prior work.}
\end{frame}

\begin{frame}{Two teacher signals shape one bidirectional student}
\centering
\fig{method_fig}{4.4cm}
\begin{columns}[T,onlytextwidth]
\column{.49\textwidth}
\textbf{REPA: local alignment}\\
Match intermediate student features to clean AR teacher states.
\column{.49\textwidth}
\textbf{REG: a global generated state}\\
Denoise one global representation together with the response.
\end{columns}
\papersource{Original Fig. 1(a), PDF p. 1; Sec. 3.2, p. 4. Teacher and projector are training-only.}
\end{frame}

\begin{frame}{ELF predicts clean states, then decodes once}
\[
z_t=t x+(1-t)\sigma\varepsilon,\quad
\varepsilon\sim\mathcal N(0,I),\quad \sigma=2
\]
\[
v_\theta=\frac{\hat x_\theta-z_t}{1-t},
\qquad z_{j+1}=z_j+\Delta_j v_j
\]
\begin{itemize}
\item $x$: standardized, contextual text states from frozen Qwen3-0.6B-Base; width 1024.
\item Prompt stays fixed; response starts from noise.
\item The previous clean prediction conditions the next update.
\item The shared Transformer also learns token decoding with cross-entropy; one final call emits all tokens.
\end{itemize}
\papersource{Eq. (1), Sec. 3.1, PDF p. 4. Implementation clamps the velocity denominator at 0.05.}
\end{frame}

\begin{frame}{REPA supervises features; REG adds a reconstruction task}
\begin{columns}[T,onlytextwidth]
\column{.49\textwidth}
\textbf{REPA}
\[
1-\cos\!\bigl(p_\phi(u_i^{(\ell)}),h_i\bigr)
\]
\begin{itemize}
\item Teacher: Qwen3-1.7B-Base, layer 20.
\item Student block 4 (B) or 8 (L).
\item Response content positions plus REG; student remains bidirectional.
\end{itemize}
\column{.49\textwidth}
\textbf{REG}
\[
r_t=tr+(1-t)\sigma\varepsilon_r
\]
\begin{itemize}
\item $r$: standardized last content-token state, teacher layer 24.
\item Joint attention and velocity regression.
\item At inference, REG starts from noise and remains decoder context.
\end{itemize}
\end{columns}
\medskip
\[
\mathcal L=\mathcal L_{\mathrm{ELF}}
+0.5\,\mathcal L_{\mathrm{REPA}}+0.1\,\mathcal L_{\mathrm{REG}}
\]
\papersource{Eq. (2), PDF p. 4; Appendix A and Table 7, pp. 18--20.}
\end{frame}

\begin{frame}{Early-stop spends its budget on a longer grid's prefix}
\begin{enumerate}
\item Draw a sorted logit-normal grid for master budget $M=8N$.
\item Run only its first $N-1$ denoiser evaluations.
\item Decode the latest \alert{predicted-clean state} at decoder time 1.
\end{enumerate}
\medskip
\centering
\begin{tabular}{@{}lll@{}}
\toprule
Actual budget & Full-span & Prefix early-stop\\
\midrule
64 NFE & 63 updates over $[0,1]$ & First 63 updates of a 512-NFE grid\\
Decode input & Updated terminal state & Latest clean prediction\\
\bottomrule
\end{tabular}\par
\medskip
\raggedright
The observed exit time averages $t=0.0825$ in the diagnostic runs.\\
$\rho=8$ is an NFE ratio, not a continuous-time fraction.
\papersource{Sec. 3.3, PDF p. 4; Algorithm 1, p. 27; Table 12, p. 29. NFE includes the decoder call.}
\end{frame}

\begin{frame}{Early-stop approximates the remaining flow map}
\[
F_{t,1}(y)=y+\int_t^1v(y_s,s)\,ds,
\qquad D_t(y)=y+(1-t)v(y,t)
\]
\[
e_t(y)=D_t(y)-F_{t,1}(y)
=(1-t)\bigl(v(y,t)-\bar v_{t:1}(y)\bigr)
\]
\begin{block}{Endpoint-error bound}
If $v$ is $C^1$, the flow is unique through time 1,
and $\|a_s\|\le A$ on $[t,1]$, then
\[
\|e_t(y)\|\le \frac{A(1-t)^2}{2}.
\]
\end{block}
Here $y$ contains text and REG. An accurate early exit needs current velocity
to approximate the remaining average velocity $\bar v_{t:1}$.
\papersource{Proposition 1, Eq. (4), PDF p. 5; proof in Appendix C.1, p. 26.}
\end{frame}

\begin{frame}{Matching endpoint tokens is stronger than preserving an answer}
\[
L\|e_t(y_t)\|<m
\quad\Longrightarrow\quad
\mathcal D(\hat y_t)=\mathcal D(y_1)
\]
\begin{itemize}
\item $m$: smallest winning-token logit margin through the first EOS.
\item $L$: Lipschitz constant for \emph{pairwise logit differences}.
\item This sufficient condition preserves the whole endpoint response, including an incorrect one.
\end{itemize}
\begin{block}{Three different questions}
Are the latent states close? Are the token sequences identical?\\
Do the responses produce the same numerical answer?
\end{block}
\papersource{Proposition 2 and Corollary 1, PDF p. 5; Appendix C.2, pp. 26--27.}
\end{frame}

\begin{frame}{The experiments train a separate model for each task}
\centering
\begin{tabular}{@{}llrl@{}}
\toprule
Task & Student & Main NFE & Training corpus\\
\midrule
GSM8K & B / L & 64 & OpenMathInstruct-2 + MetaMathQA\\
MATH-500 & L & 128 & OpenMathInstruct-2, after GSM8K\\
Python code & L & 128 & OpenCodeInstruct\\
MMLU & B & 64 & Auxiliary data + generated rationales\\
\bottomrule
\end{tabular}\par
\medskip
\raggedright
\begin{itemize}
\item B / L: 12 / 32 blocks; hidden width 768 / 1280.
\item Main math/code tables: zero-shot, 16 samples per question, early-stop $\rho=8$.
\item pass@1 is mean single-sample accuracy; pass@$k$ estimates whether at least one of $k$ candidates succeeds.
\item MMLU uses full-span generation. Generation seeds share trained weights.
\end{itemize}
\papersource{Sec. 5, PDF p. 7; Tables 7--8, pp. 19--21; evaluation protocols, p. 22.}
\end{frame}

\begin{frame}{The larger model gains on both math benchmarks}
\centering
\begin{tabular}{@{}llrrrr@{}}
\toprule
Task & Size & NFE & ELF & ELF-REG & Gain (pp)\\
\midrule
GSM8K & B & 64 & 37.71 & 38.11 & +0.40\\
GSM8K & L & 64 & 51.85 & \alert{55.96} & +4.11\\
MATH-500 & L & 128 & 10.55 & \alert{13.39} & +2.84\\
\bottomrule
\end{tabular}\par
\medskip
\raggedright
\begin{itemize}
\item All values: pass@1 (\%), averaged over 16 generation seeds.
\item ELF-REG-B at 16 NFE: 34.42\%; FMLM+ (Init) at 64 NFE: 33.60\%.
\item ELF-REG-L on GSM8K: 52.89\% at 16 NFE, 55.04\% at 32 NFE.
\item B's final gap is small; its training-efficiency benefit is clearer.
\end{itemize}
\papersource{Table 2, PDF p. 8; Table 14, p. 30. Cross-paper data and protocols differ.}
\end{frame}

\begin{frame}{Code gains appear across all four main benchmarks}
\centering
{\small
\begin{tabular}{@{}lrrrrr@{}}
\toprule
Model & NFE & MBPP-500 & MBPP-378 & HumanEval & HE+\\
\midrule
Open-dCoder & 128 & 16.70 & 23.90 & 20.80 & 17.60\\
oDLM & 128 & 15.70 & 22.99 & 17.87 & 16.40\\
PlaidQ+CFG & 257 & 15.53 & 24.58 & 22.04 & 19.70\\
ELF-L & 128 & 17.19 & 26.75 & 19.66 & 18.75\\
\textbf{ELF-REG-L} & 128 & \textbf{19.10} & \textbf{28.92} & \textbf{22.56} & \textbf{21.46}\\
\bottomrule
\end{tabular}\par}
\medskip
\raggedright
\begin{itemize}
\item pass@1 (\%); both MBPP variants use base tests.
\item ELF uses EvalPlus sanitization and a single-function alias rule.
\item ELF allows the remaining 1024-token canvas for its response; PlaidQ reports a 128-token completion cap.
\end{itemize}
\papersource{Table 1, PDF p. 7; Appendix B.3 and B.5, pp. 22--25. HE+ = HumanEval+.}
\end{frame}

\begin{frame}{Function-name handling changes the MBPP ranking}
\centering
\begin{tabular}{@{}lrrrr@{}}
\toprule
ELF-REG-L, 128 NFE & MBPP-500 & MBPP-378 & HumanEval & HE+\\
\midrule
Sanitizer + alias & 19.10 & 28.92 & 22.56 & 21.46\\
Sanitizer only & 15.95 & 24.21 & 22.26 & 21.15\\
\bottomrule
\end{tabular}\par
\medskip
\begin{itemize}
\item Same generated programs, 16 seeds; only the scoring pipeline changes.
\item Without aliases, Open-dCoder leads MBPP-500 (16.70); PlaidQ+CFG leads MBPP-378 (24.58).
\item ELF-REG-L still improves over ELF-L on every benchmark.
\end{itemize}
\begin{block}{Interpretation}
The internal improvement survives; cross-paper rankings depend on evaluation choices.
\end{block}
\papersource{Table 15, PDF p. 31. External methods retain their reported evaluation protocols.}
\end{frame}

\begin{frame}{Few-NFE generation can retain strong candidate coverage}
\centering
\begin{tabular}{@{}llrr@{}}
\toprule
Benchmark & Model & NFE & pass@10 (\%)\\
\midrule
MBPP-378 & ELF-REG-L & 8 & \alert{46.39}\\
 & PlaidQ-D16 & 17 & 40.49\\
\midrule
HumanEval & ELF-REG-L & 16 & \alert{41.21}\\
 & PlaidQ+CFG & 257 & 39.87\\
\bottomrule
\end{tabular}\par
\medskip
\raggedright
\begin{itemize}
\item ELF uses the same task checkpoint at every NFE; PlaidQ-D16 has dedicated few-step training.
\item MATH-500 pass@16 at 64 NFE: 44.00\% $\to$ 52.20\%.
\item Exception: MBPP-500 pass@10 at 128 NFE falls slightly, 40.50\% $\to$ 39.92\%, despite higher pass@1.
\end{itemize}
\papersource{Table 3, PDF p. 8; Tables 17--18, p. 32. Cross-paper code protocols differ.}
\end{frame}

\begin{frame}{Fewer epochs translate into smaller GPU-hour savings}
\begin{columns}[c,onlytextwidth]
\column{.55\textwidth}
\fig{gsm_learning_curves}{4.8cm}
\column{.42\textwidth}
ELF-L reaches 51.48\% GSM8K accuracy in:
\begin{itemize}
\item Baseline: 14 epochs.
\item ELF-REG: 8 epochs.
\end{itemize}
\medskip
\alert{43\% fewer epochs}\\
\alert{31.4\% fewer GPU-hours}\\
324.7 $\to$ 222.6 H200 hours.
\medskip

The teacher adds work to each training step.
\end{columns}
\papersource{Original Fig. 3, PDF p. 7; Table 9, p. 21. Curves use 4 generation seeds and 64 NFE.}
\end{frame}

\begin{frame}{REG's training benefit mostly survives its removal}
\centering
\begin{tabular}{@{}lr@{}}
\toprule
ELF-B, epoch 6 & GSM8K pass@1 (\%)\\
\midrule
Baseline & 21.74\\
REPA only & 24.56\\
REPA+REG & \alert{28.92}\\
Same trained weights, REG removed at inference & \alert{28.66}\\
REPA + one learnable token & 25.60\\
\bottomrule
\end{tabular}\par
\medskip
\raggedright
The benefit is consistent with improved shared representations during training.
An extra token alone does not reproduce the gain.
\papersource{Table 5, PDF p. 10; 64-NFE early-stop, 16 generation seeds.}
\end{frame}

\begin{frame}{Prefix early-stop beats full-span at matched NFE}
\centering
\begin{tabular}{@{}lrrrrr@{}}
\toprule
ELF-REG-B & 4 & 8 & 16 & 32 & 64\\
\midrule
Full-span & 7.25 & 15.75 & 25.27 & 30.95 & 34.19\\
Prefix, $\rho=8$ & \alert{11.82} & \alert{26.20} & \alert{34.42} & \alert{37.15} & \alert{38.11}\\
\bottomrule
\end{tabular}\par
\medskip
\raggedright
\begin{itemize}
\item Same trained checkpoint, same actual NFE, 16 generation seeds.
\item The comparison changes both grid coverage and decoder input.
\item $\rho=16$ is slightly stronger at 16--64 NFE, slightly weaker at 4--8 NFE.
\item The choice $\rho=8$ balances performance across budgets.
\end{itemize}
\papersource{Table 6, PDF p. 10; ratio sweep in Table 37, p. 42. Values are GSM8K pass@1 (\%).}
\end{frame}

\begin{frame}{The trajectory keeps changing after accuracy plateaus}
\centering
\fig{gsm_decoder_stability_and_endpoint_error}{4.25cm}
\raggedright
\begin{itemize}
\item Dashed lines mark 64 NFE; complete-response agreement is still low.
\item ELF-REG-B has 16.1\% lower text endpoint error at this exit.
\item Panel (a) uses uniform time; (b)--(e) use a logit-normal grid.
\end{itemize}
\papersource{Original Fig. 2, PDF p. 6. Four seeds; endpoint error/agreement at NFE 512 are fixed by construction.}
\end{frame}

\begin{frame}{Most answers agree even when most responses differ}
\centering
\begin{tabular}{@{}lrr@{}}
\toprule
64-NFE exit vs.\ its 512-NFE endpoint & ELF-B & ELF-REG-B\\
\midrule
Numerical-answer agreement (\%) & 80.27 & \alert{87.11}\\
Complete-response agreement (\%) & 1.46 & \alert{8.61}\\
Incorrect $\to$ correct, count & 79 & 70\\
Correct $\to$ incorrect, count & 78 & 36\\
Net accuracy change (pp) & +0.02 & +0.64\\
\bottomrule
\end{tabular}\par
\medskip
\raggedright
Paired comparison: 1,319 questions $\times$ 4 seeds = 5,276 pairs.\\
Later denoising often changes wording while leaving the task answer unchanged.
\papersource{Table 11, PDF p. 29. Net changes computed from counts before rounding.}
\end{frame}

\begin{frame}{Re-noising replaces Euler updates with fresh corruption}
Keep the same checkpoint, NFE, and final prediction time $t_\star$.
\[
t_i=t_\star\left(\frac{i}{N-2}\right)^{p},
\quad i=0,\ldots,N-2,\qquad p=0.5
\]
\[
z_{i+1}=t_{i+1}\hat x_i
+(1-t_{i+1})\eta\sigma\xi_i,\qquad
\xi_i\sim\mathcal N(0,I),\quad \eta=1.2
\]
\begin{itemize}
\item Use a power time schedule and fresh noise after each nonterminal prediction.
\item Preserve the previous clean prediction as self-conditioning.
\item Apply the same rule to REG; keep the prompt fixed.
\end{itemize}
The Euler endpoint-error identity does not cover this update rule.
\papersource{Sec. 5.1, PDF p. 9; Appendix D.1.1, pp. 33--34.}
\end{frame}

\begin{frame}{Re-noising improves pass@1, with a coverage tradeoff}
\centering
\begin{tabular}{@{}lrrr@{}}
\toprule
ELF-REG-L & NFE & Default sampler & Re-noising\\
\midrule
GSM8K & 64 & 55.91 & \alert{60.37}\\
MATH-500 & 128 & 13.15 & \alert{20.15}\\
MBPP-378 & 128 & 27.84 & \alert{38.76}\\
HumanEval & 128 & 23.93 & \alert{33.08}\\
\bottomrule
\end{tabular}\par
\medskip
\raggedright
\begin{itemize}
\item Matched \alert{4-seed} comparisons, separate from 16-seed headline results.
\item ELF-REG pass@1 improves on every tested benchmark at NFE $\ge16$, but decreases at NFE 4.
\item GSM8K at 64 NFE: pass@4 drops from 72.48\% to 71.11\%.
\end{itemize}
\papersource{Table 4, PDF p. 9; Table 24, p. 34. Table entries are pass@1 (\%).}
\end{frame}

\begin{frame}{The efficiency claim needs explicit accounting}
\begin{itemize}
\item \textbf{System cost.} Reported ELF sizes exclude the frozen 0.6B prompt encoder; it remains active at inference. The 1.7B teacher is training-only.
\medskip
\item \textbf{Compute.} NFE excludes prompt encoding and does not equal latency across architectures. Task-token budgets exclude foundation pretraining.
\medskip
\item \textbf{Scope.} Separate task models, Python-only code, fixed response canvas; changing canvas length requires finetuning.
\medskip
\item \textbf{Evaluation.} Output caps and code cleanup differ across methods. Code SCCFG was tuned on MBPP/HumanEval; training was not repeated across seeds.
\end{itemize}
\papersource{Appendix B.2--B.5, PDF pp. 20--25; Appendix F, p. 46.}
\end{frame}

\begin{frame}{Useful task answers can emerge before text stabilizes}
\begin{enumerate}
\item \textbf{Representation supervision helps.}\\
REPA+REG improves the ELF family and accelerates learning.
\medskip
\item \textbf{A clean prediction is a useful exit.}\\
The same task checkpoint supports multiple NFE budgets without few-step training.
\medskip
\item \textbf{Judge the answer, the trajectory, and the cost separately.}\\
Whole-response agreement, task accuracy, candidate coverage, and runtime measure different properties.
\end{enumerate}
\medskip
\papersource{Primary source: Li et al., \emph{ELF-REG: Scaling Continuous Diffusion Language Models to Reasoning Tasks}, \href{https://arxiv.org/abs/2609.29102v3}{arXiv:2609.29102v3}, 8 October 2026. Main discussion ends here; optional backup slides follow.}
\end{frame}

\appendix

\begin{frame}{Backup: guidance is learned into one denoiser call}
\[
c_t=\max(1-t,0.05),\qquad
v_\theta=\frac{\hat x_\theta-z_t}{c_t}
\]
\[
v_{\mathrm{target}}=\operatorname{sg}\left[
\frac{x-z_t}{c_t}
+b\left(1-\frac1w\right)(v_{\mathrm{sc}}-v_{\mathrm{no\text{-}sc}})
\right]
\]
\begin{itemize}
\item $b$: self-conditioning indicator; $w$: guidance scale.
\item Two detached auxiliary predictions define the guided training target; REG uses the analogous target.
\item $x_t^\star=\operatorname{sg}(z_t+c_t v_{\mathrm{target}})$ gives
$\|v_\theta-v_{\mathrm{target}}\|^2=c_t^{-2}\|\hat x_\theta-x_t^\star\|^2$.
\item Ordinary CFG is 1; SCCFG is 3 for GSM8K and 2 for code/MATH.
\end{itemize}
\papersource{Appendix A.3, Eqs. (7)--(9), PDF pp. 18--19; Table 7, p. 20.}
\end{frame}

\begin{frame}{Backup: the implemented sampler has augmented state}
\[
d_j=D_\theta(y_j,q_j,t_j),\quad
v_j=\frac{d_j-y_j}{c_j},\quad
y_{j+1}=y_j+\Delta_jv_j,\quad q_{j+1}=d_j
\]
\[
d_j-y_{M-1}
=\underbrace{(c_j-(1-t_j))v_j}_{\text{denominator-clamp correction}}
+\underbrace{\sum_{\ell=j}^{M-2}\Delta_\ell(v_j-v_\ell)}_{\text{remaining velocity changes}}
\]
\begin{itemize}
\item $y_j$: text and REG states; $q_j$: previous clean prediction.
\item Both the state and self-conditioning determine the remaining trajectory.
\item The norm covers the full response canvas, including post-EOS positions that can affect earlier logits through attention.
\end{itemize}
\papersource{Appendix C.3, Eqs. (12)--(14), PDF pp. 27--28.}
\end{frame}

\begin{frame}{Backup: coupling bounds distributional disagreement}
\[
W_2(P_t,P_1)\le\sqrt{\E\|e_t(y_t)\|^2}
\]
\[
\operatorname{TV}(Q_t,Q_1)
\le \Pr[L\|e_t(y_t)\|\ge m]
\le \Pr[m\le L\eta]+\frac{\E\|e_t(y_t)\|^2}{\eta^2}
\]
\begin{itemize}
\item $P_t,P_1$: latent distributions; $Q_t,Q_1$: decoded distributions.
\item Use the same initial noise to couple early and final predictions.
\item $\eta>0$ trades small-margin probability against large-error probability.
\item Assumes finite second moments and the decoder conditions almost surely. Agreement is with the model's own endpoint.
\end{itemize}
\papersource{Corollary 1, PDF p. 5; Appendix C.2, pp. 26--27.}
\end{frame}

\begin{frame}{Backup: matched epochs can still cost more to train}
\centering
{\small
\begin{tabular}{@{}llrrr@{}}
\toprule
Task & Model & Epochs & Task tokens (B)$^\ast$ & H200 GPU-hours\\
\midrule
GSM8K & ELF-L & 15 & 8.370 & 347.9\\
 & ELF-REG-L & 12 & 6.696 & 333.9\\
Code & ELF-L & 12 & 14.958 & 317.3\\
 & ELF-REG-L & 12 & 14.958 & 407.9\\
MATH & ELF-L & 8 & 17.810 & 449.6\\
 & ELF-REG-L & 8 & 17.810 & 563.1\\
MMLU & ELF-B & 30 & 1.259 & 10.9\\
 & ELF-REG-B & 30 & 1.259 & 16.7\\
\bottomrule
\end{tabular}\par}
\medskip
\raggedright
The target accuracy determines whether faster learning offsets the teacher cost.
\papersource{Table 8, PDF p. 21. $^\ast$MATH totals include GSM8K initialization; foundation pretraining is excluded throughout. MATH epochs count finetuning only.}
\end{frame}

\begin{frame}{Backup: re-noising drives most of the sampler gain}
\begin{columns}[c,onlytextwidth]
\column{.46\textwidth}
\centering
\fig{fig5_components}{5.65cm}
\column{.51\textwidth}
GSM8K at 64 NFE, pass@1 (\%):
\begin{itemize}
\item Default sampler: 55.91.
\item Power schedule only: 56.94.
\item Re-noising only: 60.01.
\item Both changes: \alert{60.37}.
\end{itemize}
\medskip
Excess noise ($\eta=2$) hurts.\\
The best time exponent depends on NFE.
\end{columns}
\papersource{Original Fig. 5, left panel, PDF p. 37; Table 30, p. 36. ELF-REG-L, four generation seeds.}
\end{frame}

\begin{frame}{Backup: better encoded targets are easier to recover}
\begin{columns}[c,onlytextwidth]
\column{.64\textwidth}
\centering
\fig{clean_encoder_learning_and_recovery}{5.6cm}
\column{.33\textwidth}
\begin{itemize}
\item Top: learning curves with Qwen3 vs.\ t5-small.
\item Bottom: recovery from \emph{supplied gold answers} after corruption.
\item Encoder changes also change tokenizer and preprocessing.
\end{itemize}
\end{columns}
\papersource{Original Fig. 6, Appendix D.3.1, PDF p. 38. Recovery does not measure solving from the prompt alone.}
\end{frame}

\begin{frame}{Backup: teacher alignment does not impose AR decoding}
\centering
\fig{shifted_teacher_layerwise_influence_all_times}{3.7cm}
\raggedright
\begin{itemize}
\item Shift-1 aligns student position $i$ to teacher position $i-1$.
\item Forward perturbation influence increases at supervised block 4; final-block ordering depends on corruption time.
\item Default shift-0 settles earlier across position bins; settling is not monotonically left-to-right (Fig. 9).
\end{itemize}
\papersource{Original Fig. 8, PDF p. 39; Table 34, p. 40; Fig. 9, p. 42. Student attention stays bidirectional.}
\end{frame}

\begin{frame}{Backup: coarse-sampling errors have damaging directions}
\begin{itemize}
\item Inject coarse-trajectory errors into a 64-NFE reference: noisy text plus its self-conditioning; leave REG untouched.
\item Random errors of matched per-position norm are much less harmful.
\item ELF-REG better tolerates its own errors; cross-model ordering depends on the error source.
\end{itemize}
\medskip
\centering
\begin{tabular}{@{}lrr@{}}
\toprule
Reduced full-span NFE & ELF-B retention & ELF-REG-B retention\\
\midrule
16 & 65.93\% & 71.34\%\\
32 & 81.95\% & 85.06\%\\
\bottomrule
\end{tabular}\par
\papersource{Fig. 10 and Table 38, PDF p. 43. The table is a separate retention test on 4,693 pairs both models solve at full-span NFE 64; not accuracy on all test questions.}
\end{frame}

\begin{frame}{Backup: the best exit depends on the generation process}
\begin{block}{FMLM+ uses progressive token commitment}
At 16 NFE, the paper's reproduction obtains 25.38\% GSM8K pass@1
with full-span sampling, but 16.83\% by truncating a 64-NFE master schedule.
The authors attribute the loss to an unfinished token-commit schedule.
\end{block}
\begin{itemize}
\item Early-stop can exploit ELF's complete clean prediction; the same exit rule need not suit a hybrid sampler.
\item Separate MMLU models also benefit from REPA+REG under full-span generation: 45.34\% $\to$ 46.71\% at 64 NFE.
\end{itemize}
\papersource{Tables 21--22, PDF p. 33. FMLM+ comparison uses 4 local generation seeds; MMLU uses 16.}
\end{frame}

\end{document}