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  \hspace{4mm}{\scriptsize\color{gray}SparseGDN \;|\; Paper discussion}
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\title[SparseGDN]{SparseGDN: Chunk Sparse in Linear Attention}
\author{Paper authors: Jiedong Zhuang, Peng Zhao, Jialin Tang, Haoji Hu}
\date{arXiv:2610.41420v1 \;|\; 30 October 2026}
\begin{document}

\begin{frame}[plain]
  \vspace*{1mm}
  {\LARGE\bfseries\color{AcademicBlue}SparseGDN: Chunk Sparse in Linear Attention\par}
  \vspace{4mm}
  Jiedong Zhuang, Peng Zhao, Jialin Tang, Haoji Hu\\
  Zhejiang University and ByteDance
  \begin{block}{Central idea}
    Approximate local Delta solves, select persistent memory writes,
    and reconstruct outputs for every token from compact states.
  \end{block}
  \href{https://arxiv.org/abs/2610.41420v1}{arXiv:2610.41420v1}, submitted 30 October 2026
  \vspace{3mm}

  {\small Publication not found as of 2026-11-03
  (\href{https://arxiv.org/abs/2610.41420}{arXiv record}).\\
  Public code not found as of 2026-11-03.\\[3pt]
  Research group discussion \;|\; 3 November 2026}
\end{frame}

\begin{frame}{Linear-time GDN prefill still performs expensive dense work}
  \begin{center}
  \renewcommand{\arraystretch}{1.4}
  \begin{tabular}{lp{0.64\linewidth}}
    \toprule
    Stage & Dense work at every chunk and value head\\
    \midrule
    Intra & Local Delta solve; transformed values $U$ and correction $W$\\
    $H$ & Read correction $U-WS$; recurrent state update and snapshots\\
    $O$ & Historical-state reads and causal local interactions\\
    \bottomrule
  \end{tabular}
  \end{center}
  \begin{block}{Question}
    Can fewer recurrent events and less state traffic accelerate
    pretrained GDNs while retaining long-context task quality?
  \end{block}
  Reported speedups cover the \alert{GDN core}, including online selection.
  \papersource{Paper, Secs. 1--2.2, PDF pp. 2--4; timing scope, Sec. 3.1, p. 12.}
\end{frame}

\begin{frame}{State changes are small for many heads and large for a few}
  \centering
  \fig[0.49\textheight]{observation1.pdf}
  \begin{itemize}
    \item Median relative change: 6.56\%; 63.8\% of transitions are below 10\%.
    \item 21 of 32 heads have median change below 10\%.
    \item Motivates different recurrent budgets across head pairs.
  \end{itemize}
  \papersource{Paper, Fig. 2a, PDF p. 2. One 8K RULER input, Qwen3.5-4B layer 16. Color: relative state change.}
\end{frame}

\begin{frame}{Local inverse coefficients concentrate near the causal diagonal}
  \centering
  \fig[0.64\textheight]{observation.pdf}
  \par
  Keep diagonal and adjacent 16-token blocks inside each 64-token chunk.
  \papersource{Paper, Fig. 2b, PDF p. 2. Left: mean $|M^{-1}-I|$ over 123 chunks and 32 heads; right: retained support.}
\end{frame}

\begin{frame}{Dense GDN corrects values before writing residual information}
  \[
    U'_c=U_c-W_cS_c
  \]
  \[
    S_{c+1}=2^{\gamma_c}S_c
      +K_c^\top\left(U'_c\odot2^{\gamma_c-g_c}\right)
  \]
  \[
    O_c=\frac1{\sqrt d}
      \left[2^{g_c}\odot(Q_cS_c)+A_cU'_c\right]
  \]
  \begin{itemize}
    \item $S_c\in\mathbb R^{128\times128}$: state entering chunk $c$.
    \item $g_c$: cumulative log-base-two retention; $\gamma_c=g_{c,63}$.
    \item $A_c$: causal query--key interactions with relative decay.
  \end{itemize}
  \papersource{Paper, Sec. 2.1, PDF p. 4, Eqs. (1)--(4). Head index omitted; $\odot$ scales token rows.}
\end{frame}

\begin{frame}{Five stages turn an event mask into work actually avoided}
  \centering
  \fig[0.61\textheight]{intra+H+O.pdf}
  \begin{itemize}
    \item Intra computes $U$, compact coefficients, and scores for all chunks.
    \item Selected events alone materialize $W$ and execute recurrent read/write.
  \end{itemize}
  \papersource{Paper, Fig. 3, PDF p. 3; approximation versus execution boundary, Sec. 2.2, p. 4.}
\end{frame}

\begin{frame}{Seven inverse blocks replace a full 64-token Delta solve}
  \[
    M U=\diag(\beta)V,\qquad
    M W=\diag(\beta\odot2^g)K
  \]
  \[
    M^{-1}\approx C=
    \begin{pmatrix}
      C_{00}&0&0&0\\
      C_{10}&C_{11}&0&0\\
      0&C_{21}&C_{22}&0\\
      0&0&C_{32}&C_{33}
    \end{pmatrix}
  \]
  \[
    C_{ii}=M_{ii}^{-1},\qquad
    C_{i,i-1}=-C_{ii}M_{i,i-1}C_{i-1,i-1}
  \]
  \begin{itemize}
    \item Retained blocks are exact inverse blocks in exact arithmetic.
    \item 1,248 / 2,016 = 61.9\% of causal off-diagonal entries remain.
  \end{itemize}
  \papersource{Paper, Sec. 2.3, PDF p. 5, Eqs. (5)--(8). Each block is $16\times16$; omitted distant blocks cause approximation.}
\end{frame}

\begin{frame}{Tail-16 ranks chunks before recurrent propagation}
  \[
    s_{c,h}=\left(\sum_{t=48}^{63}\sum_{v=0}^{d_v-1}
      U_{c,h,t,v}^{2}\right)^{1/2}
  \]
  \[
    q_c=\frac12\sum_{h\,\mathrm{in\ pair}}
      \frac{s_{c,h}}{\|s_{\cdot,h}\|_2}
  \]
  \begin{itemize}
    \item Late tokens undergo less attenuation before the chunk boundary.
    \item Tail-16 fits the final value tile already resident on chip.
    \item Per-head normalization balances the two heads for chunk ranking.
  \end{itemize}
  \begin{block}{Ranking proxy}
    The true write also depends on keys, decay, and the correction $WS$.
  \end{block}
  \papersource{Paper, Secs. 2.3--2.4.1, PDF pp. 5--7, Eqs. (9)--(10).}
\end{frame}

\begin{frame}{Budget upgrades favor persistent and visible memory}
  \[
    L_h=\sum_{j=1}^{C-2}r_h^j,\qquad
    G_h=\sqrt{\eta_h/\bar\eta}
  \]
  \[
    \Delta_{a\to b}=
      \sum_{c\in\mathcal C_b\setminus\mathcal C_a}
      \sum_{h\,\mathrm{in\ pair}}s_{c,h}L_hG_h
  \]
  \begin{itemize}
    \item $r_h$: mean end retention over the last at most 16 chunks.
    \item $L_h$: geometric persistence proxy over a $C-2$ chunk horizon.
    \item $\eta_h$: sampled energy of SiLU output gates and RMSNorm scales.
    \item Upgrade gains use raw tail magnitudes across head pairs.
  \end{itemize}
  \papersource{Paper, Sec. 2.4.2, PDF pp. 7--8, Eqs. (11)--(12). $C=\lceil T/64\rceil$. Later Delta updates are outside the lifetime proxy.}
\end{frame}

\begin{frame}{The selector adapts both pair budgets and chunk identities}
  \centering
  \fig[0.46\textheight]{selector1.pdf}
  \[
    (\rho_L,\rho_M,\rho_H)=(0.08,0.30,0.95)
  \]
  \begin{itemize}
    \item First and last chunks are anchors; other chunks follow $q_c$.
    \item Assign high tiers, then medium tiers, using upgrade-gain separation.
    \item Nominal mean keep: 28.44--31.31\%; measured mean: about 29\%.
  \end{itemize}
  \papersource{Paper, Fig. 5 and Sec. 2.4, PDF pp. 7--8. Shown 9-low/4-medium/3-high counts are illustrative.}
\end{frame}

\begin{frame}{Deferred materialization removes work before the event scan}
  \[
    W_i=\overline C_{ii}K_i+
        \overline C_{i,i-1}K_{i-1},\qquad
    \overline C_{ij}=C_{ij}\diag(\beta_j\odot2^{g_j})
  \]
  \begin{center}
  \renewcommand{\arraystretch}{1.35}
  \begin{tabular}{lll}
    \toprule
    & Dense & SparseGDN\\
    \midrule
    Materialize $W$ & Every chunk/head & Selected events only\\
    Recurrent layout & Dense chunk sequence & CSR chain per head pair\\
    Intermediate states & Every boundary & Referenced event snapshots\\
    \bottomrule
  \end{tabular}
  \end{center}
  \begin{itemize}
    \item CSR rows prioritize high-, medium-, then low-retention chains.
    \item Preallocated capacity $E_{\max}=16k_H$ avoids host synchronization.
  \end{itemize}
  \papersource{Paper, Sec. 2.5, PDF pp. 8--9, Eqs. (15)--(16). The first block has no preceding-block term.}
\end{frame}

\begin{frame}{Skipping an event preserves decay while omitting read/write}
  \begin{block}{A skipped chunk}
    \[
      U'_{c,h}=U_{c,h},\qquad
      S_{c+1,h}=2^{\gamma_{c,h}}S_{c,h}
    \]
  \end{block}
  \[
    P_{0,h}=0,\qquad P_{c+1,h}=\sum_{j=0}^{c}\gamma_{j,h}
  \]
  \[
    S_{c,h}=2^{P_{c,h}-P_{e+1,h}}S_{e+1,h}
    \quad\text{for consecutive retained events }e<c
  \]
  \begin{itemize}
    \item Scan retained events and bridge each gap with one accumulated decay.
    \item Overwrite $U$ with $U'$ only at retained events.
    \item Return the final state separately for autoregressive decoding.
  \end{itemize}
  \papersource{Paper, Sec. 2.6, PDF pp. 9--10, Eqs. (17)--(21). Zero-initialized prefill.}
\end{frame}

\begin{frame}{A skipped even chunk still contributes to its neighbor's output}
  Group $(c,c+1)=(2r,2r+1)$ reuses the entry state $S_c$:
  \[
    \begin{aligned}
      O_{c+1}=\frac1{\sqrt d}\bigl[&
      2^{g_{c+1}+\gamma_c}\odot(Q_{c+1}S_c)\\
      &+A_{c+1,c}U'_c+A_{c+1,c+1}U'_{c+1}\bigr].
    \end{aligned}
  \]
  \[
    O_{c+1}-O^{\mathrm{chunk}}_{c+1}
      =\frac{1-z_c}{\sqrt d}A_{c+1,c}U'_c
  \]
  \begin{itemize}
    \item If $z_c=0$, this local contribution bypasses persistent state.
    \item It also stays outside the next chunk's recurrent read correction.
    \item Group-local reads are part of the sparse operator definition.
  \end{itemize}
  \papersource{Paper, Sec. 2.7, PDF pp. 10--11, Eqs. (22)--(25). Comparison fixes Intra results and the event mask.}
\end{frame}

\begin{frame}{Three evaluation scopes answer different questions}
  \begin{center}
  \renewcommand{\arraystretch}{1.35}
  \begin{tabular}{l>{\raggedright\arraybackslash}p{0.69\linewidth}}
    \toprule
    Quality & 13 RULER tasks, six lengths 4K--128K; 100 examples/task/length; 7,800 examples/model\\
    Core latency & Intra+$H$+$O$ versus Selector+Intra+$H$+$O$; fixed workload at each length; 24-layer total\\
    Peak memory & Complete-model BF16 prefill, batch 1, cache enabled; last-token logits only; up to 256K\\
    \bottomrule
  \end{tabular}
  \end{center}
  \begin{itemize}
    \item Qwen3.5-4B/9B: 24 GDN layers, 16 Q/K heads, 32 value heads.
    \item BF16 activations and matrix inputs; FP32 state accumulation/statistics.
    \item Core timings exclude projections, attention layers, MLPs, and decoding.
  \end{itemize}
  \papersource{Paper, Sec. 3.1, PDF p. 12; Sec. 3.7, pp. 19--20. Exact GPU model is unspecified in final text.}
\end{frame}

\begin{frame}{Default retention keeps aggregate RULER quality near baseline}
  \begin{center}
  \renewcommand{\arraystretch}{1.5}
  \begin{tabular}{lrrrr}
    \toprule
    Model & Native & Sparse & $\Delta$ (pp) & Keep (\%)\\
    \midrule
    4B & 96.142 & 96.159 & $+0.017$ & 29.044\\
    9B & 96.617 & 96.502 & $-0.115$ & 29.043\\
    \bottomrule
  \end{tabular}
  \end{center}
  \begin{itemize}
    \item 4B: largest absolute length-level change is $+0.341$ pp at 16K.
    \item 9B: largest drop is $-0.692$ pp at 8K.
    \item For the other five 9B lengths, $|\Delta|\le0.103$ pp.
  \end{itemize}
  \begin{block}{Units}
    Scores are percentages; arithmetic score differences are percentage
    points (pp), computed before rounding.
  \end{block}
  \papersource{Paper, Table 1, PDF p. 12; Sec. 3.2, p. 13. Aggregate averages the six lengths.}
\end{frame}

\begin{frame}{Core prefill speedup grows with context length}
  \centering
  \fig[0.63\textheight]{qwen4b.pdf}
  \begin{itemize}
    \item 4B, 128K: 447.97 $\to$ 230.61 ms; $1.943\times$ speedup.
    \item 9B, 128K: 453.34 $\to$ 235.02 ms; $1.929\times$ speedup.
  \end{itemize}
  \papersource{Paper, Fig. 1, PDF p. 1; Table 1, p. 12. Batch 1; all 24 GDN layers; online selection included.}
\end{frame}

\begin{frame}{Each stage benefits, while every token still needs an output}
  \begin{center}
  \renewcommand{\arraystretch}{1.5}
  \begin{tabular}{lrrr}
    \toprule
    4B at 128K & Native (ms) & Sparse (ms) & Ratio\\
    \midrule
    Intra / Intra+Selector & 157.04 & 63.23 & $2.484\times$\\
    $H$ & 161.18 & 80.61 & $1.999\times$\\
    $O$ & 129.74 & 86.66 & $1.497\times$\\
    \bottomrule
  \end{tabular}
  \end{center}
  \begin{itemize}
    \item $W$ construction moves from native Intra to selected-event $H$.
    \item $H$ saves corrections and writes; $O$ saves state traffic.
    \item Local interactions and token outputs remain dense in $O$.
  \end{itemize}
  \papersource{Paper, Fig. 7, PDF p. 13. Per-stage medians; their sum need not equal a median total.}
\end{frame}

\begin{frame}{Batched execution remains faster at every completed paired point}
  \centering
  \fig[0.59\textheight]{batch.pdf}
  \begin{itemize}
    \item Batch 2/4/8: $1.560\times$--$1.988\times$ GDN-core speedup.
    \item Crosses indicate Native OOM; their plotted height is arbitrary.
  \end{itemize}
  \papersource{Paper, Fig. 8 and Table 2, PDF p. 14. Sparse completion is independently verified for only two OOM points.}
\end{frame}

\begin{frame}{The retention budget sets a measurable speed--quality trade-off}
  \centering
  \fig[0.60\textheight]{pareto.pdf}
  \begin{itemize}
    \item About 9.8\% keep: $\sim2.1\times$ speedup, $\sim2$ pp score loss.
    \item About 39.3\% keep: $1.618\times$, with small positive observed deltas.
  \end{itemize}
  \papersource{Paper, Table 3, PDF p. 15; Fig. 9, p. 16. Speedup: geomean over six lengths. Vertical differences: pp.}
\end{frame}

\begin{frame}{Matched-budget random selection loses useful information}
  \begin{center}
  \renewcommand{\arraystretch}{1.3}
  \begin{tabular}{lrr}
    \toprule
    4K RULER score (\%) & 4B & 9B\\
    \midrule
    Native & 96.641 & 96.821\\
    Block-adjacent + all & 96.410 & 96.846\\
    Full64 + adaptive & 96.669 & 96.787\\
    Block-adjacent + random & 95.067 & 95.621\\
    SparseGDN & 96.377 & 96.846\\
    \bottomrule
  \end{tabular}
  \end{center}
  \begin{itemize}
    \item Adaptive beats random by 1.310 / 1.226 pp at about 29\% keep.
    \item Across six lengths: advantages of 2.006 / 1.614 pp.
  \end{itemize}
  \papersource{Paper, Table 4, PDF p. 17; Sec. 3.5, pp. 15--16. Custom variants share group-local reads; adaptive masks are reselected.}
\end{frame}

\begin{frame}{Dynamic allocation improves the accuracy--retention trade-off}
  \begin{center}
  \renewcommand{\arraystretch}{1.5}
  \begin{tabular}{lrr}
    \toprule
    9B, all six lengths & Score (\%) & Keep (\%)\\
    \midrule
    Fixed tier counts 9/4/3 & 96.183 & 29.816\\
    Dynamic tier counts & 96.502 & 29.043\\
    \bottomrule
  \end{tabular}
  \end{center}
  \begin{block}{Paired score gain}
    $+0.319$ pp; 95\% bootstrap CI $[0.114,0.532]$ pp.
  \end{block}
  \begin{itemize}
    \item Pair identities remain input-dependent in the fixed-count baseline.
    \item Realized keep rates are reported separately and are not matched.
    \item Tail-16 balances ranking quality with the final on-chip tile.
  \end{itemize}
  \papersource{Paper, Table 5, PDF p. 17; Tail-window analysis, pp. 17--18. 20,000 stratified paired resamples.}
\end{frame}

\begin{frame}{Less materialization and state traffic contribute to speed}
  \begin{center}
  \renewcommand{\arraystretch}{1.3}
  \begin{tabular}{llrr}
    \toprule
    Optimization & Scope & 4B & 9B\\
    \midrule
    Deferred-$W$ & Intra+$H$+$O$ & $1.336\times$ & $1.338\times$\\
    CSR event scan & $H$ & $1.149\times$ & $1.153\times$\\
    Compact states & $H$ & $1.398\times$ & $1.402\times$\\
    Tier scheduling & $H$ & $1.029\times$ & $1.030\times$\\
    Mapped $O$ & $O$ & $1.239\times$ & $1.226\times$\\
    Compact + mapped & Intra+$H$+$O$ & $1.226\times$ & $1.224\times$\\
    \bottomrule
  \end{tabular}
  \end{center}
  \begin{itemize}
    \item Geometric means across 4K--128K; each row has its own reference.
    \item Same masks and bitwise-identical last-token prefill logits on timing inputs.
    \item Isolated ratios should not be multiplied; separate Selector is excluded.
  \end{itemize}
  \papersource{Paper, Sec. 3.6, PDF pp. 18--19; Table 6, p. 19.}
\end{frame}

\begin{frame}{Compact execution reduces complete-model peak allocation}
  \centering
  \fig[0.57\textheight]{peak.pdf}
  \begin{itemize}
    \item At 128K, both models save 26.222 GB of absolute peak memory.
    \item Sparse completes at 256K; Native OOMs during its first warmup.
  \end{itemize}
  \papersource{Paper, Fig. 11 and Table 7, PDF p. 20. BF16, batch 1, cache enabled, last-token logits. OOM crosses are markers.}
\end{frame}

\begin{frame}{SparseGDN couples memory selection with the execution layout}
  \begin{enumerate}
    \item Approximate local Delta solves, then spend recurrent budgets
          on input-dependent head-pair--chunk events.
    \item Preserve gap decay and group-local reads while storing fewer
          state snapshots and constructing fewer correction matrices.
    \item At about 29\% keep, retain near-baseline aggregate RULER scores
          with $1.50\times$--$1.94\times$ GDN-core prefill speedup.
  \end{enumerate}
  \begin{block}{What further deployment evidence needs to resolve}
    Hand-designed proxies may miss rare transitions. Broader generation
    workloads, full-model latency, and explicit accelerator details matter.
  \end{block}
  {\small\href{https://arxiv.org/abs/2610.41420v1}{Paper and version}
  \quad Publication and public code not found as of 2026-11-03.}
  \papersource{Paper, Sec. 4, PDF p. 20; Sec. 5, p. 21.}
\end{frame}

\appendix

\begin{frame}{Backup: the Delta solve produces two distinct right-hand sides}
  \[
    M=I+A_\Delta,\quad
    A_\Delta[t,s]=\beta_t\langle K[t],K[s]\rangle
       2^{g_t-g_s}\ind[s<t]
  \]
  \[
    U\approx C\diag(\beta)V,\qquad
    W\approx C\diag(\beta\odot2^g)K
  \]
  \[
    A_c[i,j]=\langle Q_c[i],K_c[j]\rangle
       2^{g_{c,i}-g_{c,j}}\ind[j\le i]
  \]
  \begin{itemize}
    \item $\beta$ enters both $M$ and the right-hand sides.
    \item $A_\Delta$ uses key--key interactions for the local solve.
    \item $A_c$ uses query--key interactions for output reconstruction.
    \item Four triangular blocks contribute 480 off-diagonal entries;
          three adjacent blocks contribute 768.
  \end{itemize}
  \papersource{Paper, Secs. 2.1 and 2.3, PDF pp. 4--5. Entry-count decomposition is a teaching calculation.}
\end{frame}

\begin{frame}{Backup: tier counts target 30\% retention}
  \[
    B(x,k)=\frac{k(n-k)}n
      (\overline x_{1:k}-\overline x_{k+1:n})^2
  \]
  \begin{center}
  \renewcommand{\arraystretch}{1.3}
  \begin{tabular}{cr}
    \toprule
    High-tier count $n_H$ & Candidate medium counts $n_M$\\
    \midrule
    1 & 11, 12, 13\\
    2 & 7, 8, 9\\
    3 & 3, 4, 5\\
    4 & 0\\
    \bottomrule
  \end{tabular}
  \end{center}
  \[
    \rho_{\mathrm{nominal}}=
    \frac{n_H\rho_H+n_M\rho_M+(16-n_H-n_M)\rho_L}{16}
  \]
  \papersource{Paper, Sec. 2.4.3, PDF p. 8. First/last valid chunks are anchors; keep counts use ties-to-even rounding.}
\end{frame}

\begin{frame}{Backup: gate visibility before recurrence}
  \[
    \eta_h=\frac1{4C}\sum_{c=0}^{C-1}\sum_{a=0}^{3}
       \left[\nu_{v_{c,a}}
        \SiLU(Z_{t_{c,a},h,v_{c,a}})\right]^2
  \]
  \[
    t_{c,a}=64c+8+16a,\qquad
    v_{c,a}=73(4c+a)\bmod d_v
  \]
  \begin{itemize}
    \item $Z$ is the pre-activation output gate projected from layer input.
    \item $\nu$ is the learned RMSNorm scale; out-of-range samples are zero.
    \item Sample one token per 16-token block and one rotating coordinate.
    \item Normalize gate energy relative to all value heads in the layer.
  \end{itemize}
  \papersource{Paper, Sec. 2.4, PDF pp. 7--8. The selector factor is distinct from the model's output gate itself.}
\end{frame}

\begin{frame}{Backup: group-local output differs from a naive sparse recurrence}
  \[
    A_{c+1,c}[i,j]=\langle Q_{c+1}[i],K_c[j]\rangle
       2^{g_{c+1,i}+\gamma_c-g_{c,j}}
  \]
  \[
    S_{c+1}=2^{\gamma_c}S_c+
      z_cK_c^\top(U'_c\odot2^{\gamma_c-g_c})
  \]
  \[
    U'_{c+1}=\begin{cases}
      U_{c+1}-W_{c+1}S_{c+1}, & z_{c+1}=1,\\
      U_{c+1}, & z_{c+1}=0.
    \end{cases}
  \]
  \begin{block}{Example: skip chunk 4, keep chunk 5}
    $S_5=2^{\gamma_4}S_4$ and $U'_5=U_5-W_5S_5$, while
    $O_5$ still includes $A_{5,4}U_4/\sqrt d$.
  \end{block}
  \papersource{Paper, Sec. 2.7, PDF p. 11, Eqs. (24)--(25). The extra local read is not separately ablated for quality.}
\end{frame}

\begin{frame}{Backup: Tail-16 balances proxy recall with score cost}
  \centering
  \fig[0.59\textheight]{tail.pdf}
  \begin{itemize}
    \item Tail-64 improves recall, with $2.59\times$ standalone scan cost.
    \item Production Tail-16 is fused into Intra's final on-chip tile.
  \end{itemize}
  \papersource{Paper, Fig. 10, PDF p. 18; Sec. 3.5, pp. 17--18. Left differences: pp. Right scores are illustrative.}
\end{frame}

\begin{frame}{Backup: selector diagnostics differ in evidence strength}
  \begin{center}
  \renewcommand{\arraystretch}{1.3}
  \begin{tabular}{lrr}
    \toprule
    Change in percentage points & 4B & 9B\\
    \midrule
    Add lifetime: importance recall & $+32.037$ & $+31.484$\\
    Tail vs random: importance recall & $+14.171$ & $+13.646$\\
    Remove visibility: RULER score & $-0.487$ & $-0.410$\\
    \bottomrule
  \end{tabular}
  \end{center}
  \begin{itemize}
    \item Recall: 130 stratified 8K requests/model, visibility disabled.
    \item Tail vs random fixes tiers and exact budgets.
    \item Visibility: full 8K set, 1,300 examples/model; model gate unchanged.
  \end{itemize}
  Visibility-removal 95\% CIs: $[-1.128,+0.128]$ and
  $[-1.077,+0.205]$ pp; both include zero.
  \papersource{Paper, Sec. 3.5, PDF p. 16. Dense-decode importance is a diagnostic proxy.}
\end{frame}

\begin{frame}{Backup: OOM entries have two distinct meanings}
  \begin{center}
  \renewcommand{\arraystretch}{1.3}
  \begin{tabular}{llrr}
    \toprule
    Model & Batch & 64K & 128K\\
    \midrule
    4B & 2 & $1.945\times$ & $1.988\times$\\
    4B & 4 & $1.833\times$ & OOM$^\dagger$\\
    4B & 8 & OOM$^\dagger$ & OOM\\
    9B & 2 & $1.931\times$ & $1.939\times$\\
    9B & 4 & $1.830\times$ & OOM\\
    9B & 8 & OOM & OOM\\
    \bottomrule
  \end{tabular}
  \end{center}
  \begin{itemize}
    \item OOM$^\dagger$: Native OOM; Sparse independently completes.
    \item 4B batch-4/128K: 843.411 ms; batch-8/64K: 835.367 ms.
    \item Plain OOM: Sparse completion not independently verified.
  \end{itemize}
  \papersource{Paper, Table 2 and Sec. 3.3, PDF p. 14. Timings are for the GDN core of the whole batch.}
\end{frame}

\begin{frame}{Backup: peak versus incremental memory}
  \[
    M_{\mathrm{incr}}=M_{\mathrm{peak}}-M_{\mathrm{entry}}
  \]
  \begin{center}
  \renewcommand{\arraystretch}{1.2}
  \begin{tabular}{llrrr}
    \toprule
    Model & Length & Native peak & Sparse peak & Incr. reduction\\
    \midrule
    4B & 128K & 53.116 & 26.894 & 59.56\%\\
    9B & 128K & 64.066 & 37.844 & 57.97\%\\
    4B & 256K & OOM & 44.637 & ---\\
    9B & 256K & OOM & 56.794 & ---\\
    \bottomrule
  \end{tabular}
  \end{center}
  \begin{itemize}
    \item Peaks are in GB; entry allocation includes persistent model weights.
    \item Reduction percentages refer to the forward increment.
    \item 256K establishes prefill feasibility; quality is tested through 128K.
  \end{itemize}
  \papersource{Paper, Sec. 3.7, PDF pp. 19--20; Table 7, p. 20. One warmup, three measured forwards; maximum peak reported.}
\end{frame}

\end{document}