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\newcommand{\NLL}{\mathrm{NLL}}
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\title[CST]{Learning Length-Extrapolatable Recurrent Models}
\subtitle{Credit Stabilization through Time (CST)}
\author{Paper author: Hanwen Jiang, Adobe Research}
\date{arXiv:2609.09157v1 \quad | \quad Reading: 29 October 2026}
\hypersetup{pdftitle={CST: Learning Length-Extrapolatable Recurrent Models -- Paper Discussion},
pdfauthor={Research reading notes on Hanwen Jiang's paper}}

\begin{document}

\begin{frame}[plain]
  \vspace{0pt}
  {\usebeamerfont{title}\color{AcademicBlue}
  Learning Length-Extrapolatable\\[2mm] Recurrent Models\par}
  \vspace{4mm}
  {\large Credit Stabilization through Time (CST)\par}
  \vspace{3mm}
  Paper: Hanwen Jiang \quad \textbar \quad Adobe Research\\
  \href{https://arxiv.org/abs/2609.09157v1}{arXiv:2609.09157v1}, submitted 8 September 2026
  \vspace{3mm}
  \begin{block}{Central idea}
    Rescale backward state credit at recurrent boundaries to improve
    behavior beyond the training horizon.
  \end{block}
  {\small
  Publication not found as of 29 October 2026:
  \href{https://arxiv.org/abs/2609.09157}{arXiv record},
  \href{https://hwjiang1510.github.io/}{author page}.\\
  Public code not found as of the same date:
  \href{https://github.com/hwjiang1510?tab=repositories}{author GitHub}.}
  \papersource{Research-group reading presentation; the presenter is not a paper author.}
\end{frame}

\begin{frame}{A fixed-size state can run longer than it can learn}
  \begin{block}{Train short, deploy long}
    Learn a shared recurrent rule on short sequences; retain and use
    information over many more steps at inference.
  \end{block}
  \begin{itemize}
    \item A recurrent architecture makes long execution feasible.
    \item Learning still depends on how future losses reach early computation.
    \item Dense local supervision can train a shared rule even when
          late-loss credit vanishes.
  \end{itemize}
  \vspace{2mm}
  \alert{Question: when is distant credit needed, and where should it be stabilized?}
  \papersource{Paper, Secs. 1--3, PDF pp. 1--5.}
\end{frame}

\begin{frame}{Three task families separate retention from computation}
  \centering
  \renewcommand{\arraystretch}{1.35}
  \par\begin{tabular}{>{\raggedright\arraybackslash}p{2.3cm}>{\raggedright\arraybackslash}p{4.0cm}>{\raggedright\arraybackslash}p{6.2cm}}
    \toprule
    Task & Context & What a longer sequence adds\\
    \midrule
    Fixed-FSA & One shared transition rule & More dependent steps: $N=D$, $M=0$\\
    MQAR & New key--value bindings & More time before retrieval; $D=0$\\
    Meta-FSA & New transition table & $K$ segments of depth $D$; $N=KD$\\
    \bottomrule
  \end{tabular}\par
  \vspace{4mm}
  \begin{itemize}
    \item $M$: memory load; $N$: task-input horizon; $D$: transition depth.
    \item Meta-FSA resets the \emph{task state} between segments, while
          the model state retains the context.
    \item Experiments hold $M$ fixed: temporal extrapolation is tested.
  \end{itemize}
  \papersource{Paper, Sec. 2 and Table 1, PDF pp. 2--3; serialization in App. C.1, pp. 21--22.}
\end{frame}

\begin{frame}{State credit sits upstream of the parameter gradient}
  \[
    h_t=\Phi_\theta(h_{t-1},x_t),\qquad
    \hat y_t=G_\psi(h_t),\qquad
    \mathcal L=\sum_{t\in\mathcal I}\ell_t
  \]
  \[
    \lambda_t^{(\tau)}
    =J_{t+1}^{\top}\cdots J_\tau^{\top}\nabla_{h_\tau}\ell_\tau,
    \qquad J_j=\frac{\partial h_j}{\partial h_{j-1}}
  \]
  \[
    \lambda_t=\sum_{\tau\in\mathcal I:\,\tau\ge t}\lambda_t^{(\tau)},\qquad
    \boxed{g=\nabla_\theta\mathcal L=\sum_t P_t^\top\lambda_t}
  \]
  \begin{itemize}
    \item $\lambda_t$: credit assigned to state $h_t$ by future losses.
    \item $P_t$: local parameter Jacobian, holding the incoming state fixed.
    \item Mapping and summation can obscure the upstream credit trajectory.
  \end{itemize}
  \papersource{Paper, Sec. 3.1, Eqs. (1)--(5), PDF p. 4. Assumes $h_0$ independent of $\theta$;
  recurrent $\theta$ and readout $\psi$ are disjoint.}
\end{frame}

\begin{frame}{Distant-credit necessity and norm survival disagree}
  \[
    g_{\rm far}=g_{\rm full}-g_{\rm local},\qquad
    p_{\rm far}=\frac{\langle g_{\rm far},g_{\rm full}\rangle}{\|g_{\rm full}\|^2}
  \]
  \centering
  \renewcommand{\arraystretch}{1.15}
  \par\begin{tabular}{lcc}
    \toprule
    Task & Far-gradient projection $p_{\rm far}$ & Late-loss retention $S_1$\\
    \midrule
    Fixed-FSA & 0.021 & $\approx0$\\
    Meta-FSA & 0.089 & 1.05\\
    MQAR & 0.259 & 0.82\\
    \bottomrule
  \end{tabular}\par
  \vspace{2mm}
  \begin{itemize}
    \item $S_1$: norm ratio for one fixed late-loss window, about 895 positions apart.
    \item MQAR intervention: remove $g_{\rm far}$ $\Rightarrow$ 0.60\% accuracy;
          full BPTT $\Rightarrow$ 96.30\%.
    \item $g_{\rm local}$ keeps all losses but detaches states at each boundary.
  \end{itemize}
  \papersource{Paper, Table 2 and Eqs. (6)--(9), PDF pp. 4--5; App. A, p. 19.
  $T=1024$, $H=128$. $p_{\rm far}$ is signed, not a probability; random guessing is 0.098\%.}
\end{frame}

\begin{frame}{CST rescales incoming credit before it crosses a chunk}
  \[
    \underbrace{\lambda_{k-1}}_{\text{earlier boundary}}
    =
    \underbrace{q_{k-1}}_{\text{current-chunk loss}}
    +
    \overline J_k^\top
    \underbrace{\lambda_k}_{\text{later-chunk losses}}
    \quad\text{(BPTT)}
  \]
  \begin{block}{CST intervention}
    \[
      \widetilde\lambda_k=\gamma_k\lambda_k,\qquad
      \lambda_{k-1}=q_{k-1}+\overline J_k^\top\widetilde\lambda_k,
      \qquad \gamma_k>0
    \]
  \end{block}
  \begin{itemize}
    \item $\gamma=1$: exact BPTT; $\gamma=0$: cut the cross-boundary signal.
    \item Positive scaling preserves the corrected component's local direction.
    \item Forward computation and loss are unchanged; the modified backward
          signal is generally not the exact objective gradient.
  \end{itemize}
  \papersource{Paper, Sec. 4.1, Eqs. (10)--(11), PDF p. 6; App. C.2, p. 22.
  $\overline J_k$ is the chunk-state Jacobian; $q_{k-1}=\nabla_{h_{b_{k-1}}}\mathcal L_k$.}
\end{frame}

\begin{frame}{Event-CST waits for substantial accumulated contraction}
  Let $n_k=\|\lambda_k\|$, anchor $A$, trigger ratio $\rho>1$.
  \[
    \gamma_k=
    \begin{cases}
      \min\!\left\{A/\max(n_k,\epsilon),\,\gamma_{\max}\right\},
        & n_k\le A/\rho,\\
      1,& n_k>A/\rho.
    \end{cases}
  \]
  \[
    A\leftarrow
    \begin{cases}
      n_k,&n_k>A,\\
      \gamma_k n_k,&n_k\le A/\rho,\\
      A,&A/\rho<n_k\le A.
    \end{cases}
  \]
  \begin{itemize}
    \item New peak: raise the anchor. Small contraction: hold it.
    \item Event: restore up to the cap, then anchor at the achieved scale.
    \item Adjacent-CST instead corrects every adjacent norm decrease.
  \end{itemize}
  \papersource{Paper, Sec. 5.1, Eqs. (12)--(14), PDF p. 7.
  Initialize at the latest valid boundary with gain 1; invalid/near-zero norms use identity gain.}
\end{frame}

\begin{frame}{EMA replay reuses locations while recomputing live gains}
  \begin{enumerate}
    \item Every $P$ steps, run a dense \alert{identity-gain} backward probe.
    \item Smooth signed log-contraction over training probes:
    \[
      d_k^{(p)}
      =\operatorname*{median}_{q}
        \log\frac{n_{k+1}^{(q,p)}}{n_k^{(q,p)}},
      \quad
      \overline d_k^{(p)}
      =\beta\overline d_k^{(p-1)}+(1-\beta)d_k^{(p)}.
    \]
    \item Predict boundaries where accumulated drawdown reaches $\log\rho$.
    \item At ordinary steps, expose those boundaries and compute current gains.
  \end{enumerate}
  \vspace{2mm}
  \[
    \text{Boundary overhead:}\quad
    \mathcal O(B\,\mathrm{size}(h))
    \ \longrightarrow\
    \mathcal O((R+B/P)\,\mathrm{size}(h))
  \]
  \papersource{Paper, Eq. (15), PDF p. 8; App. C.3, p. 22.
  $q$: state component; $B$: dense candidates; $R$: replayed boundaries.
  Probe steps also perform an ordinary BPTT update.}
\end{frame}

\begin{frame}{The main synthetic test separates length from depth}
  \begin{columns}[T]
    \begin{column}{0.48\textwidth}
      \begin{block}{Training}
        \begin{itemize}
          \item Gated DeltaNet: 3 layers, width 512, 16 heads.
          \item Meta-FSA: 32 states, 4 actions.
          \item Physical length 1024; depth $D=4$.
          \item 100K steps; global batch 512; three seeds.
        \end{itemize}
      \end{block}
    \end{column}
    \begin{column}{0.48\textwidth}
      \begin{block}{Evaluation}
        \begin{itemize}
          \item Lengths 1K--128K.
          \item Depths $D=4,8,16,32$.
          \item 1024 examples per depth--length cell.
          \item Accuracy: mean and sample SD across training seeds.
        \end{itemize}
      \end{block}
    \end{column}
  \end{columns}
  \vspace{2mm}
  Event-CST: $H=32,\ \rho=1.5,\ \gamma_{\max}=2$.\\
  EMA replay: same controller, $P=100,\ \beta=0.9$.
  \papersource{Paper, Sec. 5.2, PDF p. 8. No curriculum; 500-step warmup, then constant LR $3\times10^{-4}$.}
\end{frame}

\begin{frame}{Meta-FSA gains appear beyond the training horizon}
  \begin{columns}[c]
    \begin{column}{0.68\textwidth}
      \includegraphics[width=\linewidth,height=0.74\textheight,keepaspectratio]{figures/fig-01.pdf}
    \end{column}
    \begin{column}{0.29\textwidth}
      \textbf{16K--128K}\\
      All 16 mean cells improve.\\
      \alert{+6.58 pp} on average.
      \vspace{3mm}

      \textbf{$D=4$, 128K}\\
      21.16\% $\to$ 28.14\%\\
      \alert{+6.98 pp}
      \vspace{3mm}

      \textbf{At 1K, $D=4$}\\
      $-3.55$ pp.
    \end{column}
  \end{columns}
  \papersource{Original Fig. 1, PDF p. 8; Sec. 5.2.1, p. 9.
  Error bars: three-seed sample SD. Of 48 paired seed--cell comparisons at 16K--128K, 29 improve.}
\end{frame}

\begin{frame}{Accuracy improves at a substantial training cost}
  \centering
  \renewcommand{\arraystretch}{1.22}
  \par\begin{tabular}{lcc}
    \toprule
    Method / implementation & Long-context accuracy & M tokens/s\\
    \midrule
    BPTT, direct & 18.5\% & 0.955\\
    Adjacent-CST, CUDA Graph & 22.1\% & 55.1\% of BPTT\\
    Event-CST, eager & 25.1\% & 0.119\\
    Event-CST, CUDA Graph & 25.1\% & 0.526\\
    EMA replay, sparse Graph & 23.2\% & \alert{0.808 projected}\\
    \bottomrule
  \end{tabular}\par
  \vspace{3mm}
  \begin{itemize}
    \item Dense Graph Event-CST reaches 55.1\% of BPTT throughput.
    \item EMA replay exposes 6.91 rather than 31 boundaries per step.
    \item EMA throughput range 0.75--0.82M is inferred from equivalent
          sparse graph benchmarks, not an end-to-end measurement.
  \end{itemize}
  \papersource{Paper, Fig. 2 and Sec. 5.2.2, PDF p. 9.
  Single A100 80 GB, batch 64; implementations differ.
  Accuracy averages four depths and lengths 16K--128K.}
\end{frame}

\begin{frame}{Retention tasks benefit more as the horizon grows}
  \centering
  \includegraphics[width=0.96\linewidth,height=0.49\textheight,keepaspectratio]{figures/fig-03.pdf}
  \begin{itemize}
    \item Fixed-FSA: 41.58\% $\to$ 43.76\% over 16K--128K;
          length dependence stays nearly flat.
    \item MQAR at 128K: 25.56\% $\to$ 41.66\%, \alert{+16.10 pp}.
          The 128 stored bindings remain fixed.
    \item Controllers were selected on the same extrapolation grids reported here.
  \end{itemize}
  \papersource{Original Fig. 3 and Sec. 5.3, PDF p. 10.
  Three-seed mean/SD; train at 1K. Fixed-FSA's roughly 42\% score reflects incomplete rule learning.}
\end{frame}

\begin{frame}{A larger model reveals a clear failure regime}
  \centering
  \includegraphics[width=0.98\linewidth,height=0.48\textheight,keepaspectratio]{figures/fig-04.pdf}
  \begin{itemize}
    \item 8 layers, width 1024, 32 heads; train at $D=4$.
    \item S32/A4/1K: unseen-depth gains average $-9.30,-6.39,-4.06$ pp.
    \item S64/A8/2K: gains at all four depths; average \alert{+7.14 pp}.
  \end{itemize}
  \papersource{Original Fig. 4 and Sec. 5.4, PDF p. 11.
  Averages above use 16--64$\times$ training length.
  State count, action count, training length, and controller settings all change between regimes.}
\end{frame}

\begin{frame}{Real-text credit varies in both directions and across heads}
  \begin{block}{Read-only BPTT probe, every 64 tokens}
    Median earliest/latest normalized norm ratio: \alert{0.23}.\\
    At the final snapshot, 83.6\% of sample--layer--head trajectories
    are weaker at the earliest boundary.
  \end{block}
  \vspace{1mm}
  \centering
  \renewcommand{\arraystretch}{1.2}
  \par\begin{tabular}{lcc}
    \toprule
    Adjacent absolute log$_2$ change & Value & Multiplicative factor\\
    \midrule
    Median & 0.580 & 1.50\\
    90th percentile & 3.164 & 8.96\\
    \bottomrule
  \end{tabular}\par
  \vspace{3mm}
  \begin{itemize}
    \item This motivates gentle, symmetric, head-wise corrections.
    \item Probe normalization divides by $\sqrt{\text{remaining targets}}$;
          the controller uses the unnormalized signal.
  \end{itemize}
  \papersource{Paper, Sec. 6.1, PDF pp. 11--12.
  Normalization is a heuristic for changing loss counts, not fixed-signal survival.}
\end{frame}

\begin{frame}{Symmetric CST shrinks deviations in log norm}
  For each layer and head, reduce over batch and within-head state dimensions:
  \[
    n_k=\|\lambda_k\|_F,\qquad
    z_k=\log\max(n_k,\epsilon),\qquad d_k=z_k-m.
  \]
  \begin{block}{Controller}
    \[
      \gamma_k=\clip\!\left(e^{-\alpha d_k},c^{-1},c\right),\qquad
      \widetilde\lambda_k=\gamma_k\lambda_k
    \]
    \[
      m\leftarrow\beta m+(1-\beta)z_k
    \]
  \end{block}
  \begin{itemize}
    \item Without clipping: corrected log-norm deviation is $(1-\alpha)d_k$.
    \item $H=64,\ \alpha=0.10,\ \beta=0.90,\ c=1.10$;
          gains lie in $[0.9091,1.10]$.
    \item Reference uses the current pre-intervention norm and resets each backward pass.
  \end{itemize}
  \papersource{Paper, Sec. 6.1, Eqs. (16)--(17), PDF p. 12.
  Initialize $m=z_{K-1}$ with identity gain. Later-boundary corrections still create indirect feedback.}
\end{frame}

\begin{frame}{Language modeling tests 4K training through 128K inference}
  \begin{itemize}
    \item \textbf{123.30M parameters}: one 32-token local-attention block,
          then 12 Gated DeltaNet blocks.
    \item \textbf{Training}: LongData, 4096-token sequences, 20K updates,
          7.86B tokens, eight H100 GPUs.
    \item \textbf{Matched comparison}: same initialization and data seed;
          one final training run per method.
  \end{itemize}
  \vspace{3mm}
  \centering
  \renewcommand{\arraystretch}{1.2}
  \par\begin{tabular}{lccc}
    \toprule
    Evaluation & Domain & Cohort & Maximum horizon\\
    \midrule
    LongData & In-domain & 128 held-out windows & 32K / $8\times$\\
    Books3 & Out-of-domain & 128 book windows & 128K / $32\times$\\
    GovReport & Out-of-domain & 50 documents & 32K / $8\times$\\
    \bottomrule
  \end{tabular}\par
  \papersource{Paper, Sec. 6.2, PDF p. 12; App. B.1, pp. 19--20.
  Controller selected among 18 settings using reported LongData-32K and Books3-128K endpoints.}
\end{frame}

\begin{frame}{Key tokens are labeled by an independent context test}
  \begin{block}{Fixed external annotations}
    A token after 4K is selected when
    \[
      \NLL_{\rm short}-\NLL_{\rm full}>2.0
      \quad\text{and}\quad
      \NLL_{\rm full}<2.0.
    \]
  \end{block}
  \begin{itemize}
    \item LongData / Books3: pinned Qwen2-72B-Instruct annotator.
    \item GovReport: released LongPPL labels.
    \item Across tokenizers, require full containment of the target's UTF-8 byte span.
    \item Masks are fixed before comparing BPTT and CST.
  \end{itemize}
  \vspace{2mm}
  \alert{All-token and key-token NLL are token-weighted; lower is better.}\\
  At 4K there are no eligible key targets, so key NLL is undefined.
  \papersource{Paper, App. B.2, PDF p. 20; Table 4, p. 13.}
\end{frame}

\begin{frame}{Longest-context gains are larger on key tokens}
  \[
    \gain=\NLL_{\rm BPTT}-\NLL_{\rm CST}
    \qquad\text{(positive favors CST)}
  \]
  \centering
  \renewcommand{\arraystretch}{1.35}
  \par\begin{tabular}{lccc}
    \toprule
    Dataset / length & All-token gain & Key-token gain & Key / all gain\\
    \midrule
    LongData / 32K & 0.006843 & 0.023932 & 3.50$\times$\\
    Books3 / 128K & 0.005949 & 0.014770 & 2.48$\times$\\
    GovReport / 32K & 0.009726 & 0.044182 & 4.54$\times$\\
    \bottomrule
  \end{tabular}\par
  \vspace{4mm}
  \begin{itemize}
    \item All-token NLL improves in all 14 dataset--length settings.
    \item Mean all-token gain: \alert{0.00724 NLL}; overall improvement is modest.
    \item Example: Books3-128K all-token NLL is 3.524265 $\to$ 3.518316.
  \end{itemize}
  \papersource{Paper, Tables 3--4 and Sec. 6.4, PDF p. 13.
  Gains use unrounded losses, then independent rounding. Ratios compare NLL reductions, not accuracies.}
\end{frame}

\begin{frame}{Key-token gains include two reported reversals}
  \centering
  \renewcommand{\arraystretch}{1.12}
  \par\begin{tabular}{llrr}
    \toprule
    Dataset & Length & Key targets & Key NLL gain\\
    \midrule
    LongData & 16K & 7,130 & +0.025832\\
             & 32K & 39,054 & +0.023932\\
    Books3 & 16K & 2,836 & \textcolor{MutedRed}{$-0.019084$}\\
           & 32K & 11,074 & +0.007481\\
           & 64K & 34,101 & +0.010558\\
           & 128K & 95,993 & +0.014770\\
    GovReport & 8K & 277 & \textcolor{MutedRed}{$-0.049017$}\\
              & 32K & 3,147 & +0.044182\\
    \bottomrule
  \end{tabular}\par
  \vspace{3mm}
  Sparse short-context annotations give uncertain estimates.\\
  Books3 results have no paired document confidence intervals.
  \papersource{Selected rows from paper Table 4, PDF p. 13; App. B.4, p. 21.
  Negative values are retained rather than assumed to be noise.}
\end{frame}

\begin{frame}{Context removal supports better distant use on LongData}
  Re-score the same key targets with full context and a clean-state 4K window.
  \[
    \Delta_{\rm context}
    =
    \underbrace{(\NLL_{\rm short}-\NLL_{\rm full})_{\rm CST}}_{\text{CST context benefit}}
    -
    \underbrace{(\NLL_{\rm short}-\NLL_{\rm full})_{\rm BPTT}}_{\text{BPTT context benefit}}
  \]
  \centering
  \renewcommand{\arraystretch}{1.25}
  \par\begin{tabular}{lcc}
    \toprule
    Setting & Context-use gain & 95\% paired document CI\\
    \midrule
    LongData / 16K & +0.019479 & [0.006917, 0.032245]\\
    LongData / 32K & +0.014703 & [0.009604, 0.020999]\\
    GovReport / 32K & +0.002684 & [$-0.002305$, 0.007471]\\
    \bottomrule
  \end{tabular}\par
  \vspace{3mm}
  \alert{LongData excludes zero; GovReport does not.}\\
  These intervals describe document sampling, conditional on the selected runs.
  \papersource{Paper, Sec. 6.4, PDF p. 14; Table 5 and App. B.4, p. 21.
  20,000 paired document bootstrap resamples; no removed-prefix state is reused.}
\end{frame}

\begin{frame}{The evidence supports a useful but selective intervention}
  \begin{itemize}
    \item \textbf{Scale control has limits.}
          It cannot restore missing directions or resolve temporal interference.
    \item \textbf{Controller strength matters.}
          Only 5 of 18 language configurations improve both selection endpoints;
          $\alpha=0.30$ performs worse on average.
    \item \textbf{Language evidence is exploratory.}
          One 123M architecture; one final run per method; reported endpoints
          also used for selection.
    \item \textbf{Transfer is conditional.}
          Larger-model unseen-depth failures and the unresolved GovReport
          control matter for deployment decisions.
  \end{itemize}
  \papersource{Paper, Secs. 5.4, 6.2--6.4 and 8, PDF pp. 11--15; App. B, pp. 19--21.}
\end{frame}

\begin{frame}{Take home: diagnose credit demand before stabilizing scale}
  \begin{enumerate}
    \item \textbf{Separate task need from signal decay.}\\
          Dense local supervision can train a shared recurrent rule.
    \item \textbf{State credit is a direct intervention point.}\\
          CST modifies the reverse pass while retaining the forward model.
    \item \textbf{Match the controller to observed dynamics.}\\
          Event-based contraction repair and symmetric head-wise smoothing
          serve different regimes.
  \end{enumerate}
  \vspace{3mm}
  \begin{block}{A concrete next experiment}
    Compare one-sided vs.\ two-sided controllers with independent selection,
    multiple seeds, context-removal tests, and measured training throughput.
  \end{block}
  \papersource{Next experiment is this reading's proposal.
  Primary source: Hanwen Jiang, \href{https://arxiv.org/abs/2609.09157v1}
  {Learning Length-Extrapolatable Recurrent Models, arXiv:2609.09157v1 (2026)}.}
\end{frame}

\appendix

\begin{frame}{Backup: a capped event resets to the attainable norm}
  \[
    \text{Adjacent-CST:}\qquad
    \gamma_k=\clip\!\left(
      \frac{\widetilde n_{k+1}}{\max(n_k,\epsilon)},1,\gamma_{\max}\right)
  \]
  \vspace{2mm}
  Event-CST teaching example: $A=10,\ \rho=1.5,\ \gamma_{\max}=2$.
  \centering
  \renewcommand{\arraystretch}{1.3}
  \par\begin{tabular}{ccccc}
    \toprule
    Incoming $n_k$ & Event? & Gain & Corrected norm & New anchor\\
    \midrule
    8 & No & 1 & 8 & 10\\
    6 & Yes & $10/6$ & 10 & 10\\
    3 & Yes, capped & 2 & 6 & 6\\
    \bottomrule
  \end{tabular}\par
  \vspace{3mm}
  The anchor follows achieved restoration when the gain cap binds.\\
  Invalid or near-zero norms keep gain 1 and leave the anchor unchanged.
  \papersource{Teaching example, not paper data. Controller equations: Sec. 5.1,
  Eqs. (12)--(14), PDF p. 7. Incoming norms are live observations at each boundary.}
\end{frame}

\begin{frame}{Backup: local scaling can change the summed direction}
  \begin{columns}[T]
    \begin{column}{0.49\textwidth}
      \begin{block}{Teaching example}
        \[
          q=\begin{pmatrix}1\\0\end{pmatrix},
          \qquad
          \overline J^\top\lambda=
          \begin{pmatrix}0\\1\end{pmatrix}
        \]
        \[
          \text{BPTT: }\begin{pmatrix}1\\1\end{pmatrix}
          \quad\longrightarrow\quad
          \gamma=2:\begin{pmatrix}1\\2\end{pmatrix}
        \]
        The scaled component keeps its direction; the sum changes direction.
      \end{block}
    \end{column}
    \begin{column}{0.47\textwidth}
      \textbf{Implementation}
      \begin{itemize}
        \item Clone boundary state without detaching.
        \item A backward hook rescales its incoming credit.
        \item Event-CST: one gain per state tensor, batch included.
        \item Real text: one gain per layer and recurrent head.
      \end{itemize}
    \end{column}
  \end{columns}
  \papersource{Example is explanatory, not an experiment. Paper, Eq. (11), PDF p. 6;
  App. C.2, p. 22. References reset every backward pass.}
\end{frame}

\begin{frame}{Backup: Event-CST makes few, position-dependent corrections}
  \centering
  \includegraphics[width=0.98\linewidth,height=0.48\textheight,keepaspectratio]{figures/fig-06.pdf}
  \begin{itemize}
    \item 5.8\% of boundary--component decisions trigger:
          1.79 events per state-component trajectory.
    \item Mean gain over all candidates: $1.053\times$.
    \item Panel (b) is the product of controller gains, about $3.7\times$
          at early boundaries; it is not a CST/BPTT gradient-norm ratio.
  \end{itemize}
  \papersource{Original Fig. 6, PDF p. 23; App. C.4, pp. 22--23.
  Panels (a,b): token position; panels (c,d): training step. Shading: sample SD across seeds.}
\end{frame}

\begin{frame}{Backup: sparse replay plans reduce boundary exposure}
  \centering
  \includegraphics[width=0.97\linewidth,height=0.51\textheight,keepaspectratio]{figures/fig-07.pdf}
  \begin{itemize}
    \item Late-training plans retain 4, 8, and 8 boundaries across three seeds.
    \item Including periodic dense probes: 31 $\to$ 6.91 hooks per step.
    \item Positions are reused; current minibatch signals determine live gains.
  \end{itemize}
  \papersource{Original Fig. 7 and App. C.5, PDF p. 23.
  Probe period $P=100$; 77.7\% fewer hooks. Throughput recovery remains a projection in this paper.}
\end{frame}

\begin{frame}{Backup: model and search budgets define the LM comparison}
  \begin{itemize}
    \item Width 768; 16 recurrent heads of dimension 48;
          SwiGLU width 1792; tied GPT-2 embeddings, vocabulary 50,304.
    \item LongData: 458 training shards, 16 held-out shards;
          packed 4096-token sequences, EOT document separators.
    \item AdamW: peak LR $6\times10^{-4}$, 67-step warmup,
          cosine decay to $6\times10^{-5}$, gradient norm clip 1.
  \end{itemize}
  \begin{block}{18 full-training configurations}
    \[
      \alpha\in\{0.03,0.10,0.30\},\quad
      \beta\in\{0.90,0.97\},\quad
      c\in\{1.10,1.25,1.50\}.
    \]
    All trained for 20K updates; selection uses LongData-32K and Books3-128K.
  \end{block}
  Reported intervals condition on the selected configuration.
  \papersource{Paper, App. B.1, PDF pp. 19--20. AdamW betas $(0.9,0.95)$,
  weight decay 0.1; global batch 96.}
\end{frame}

\begin{frame}{Backup: the Transformer comparison changes the architecture}
  \centering
  \includegraphics[width=0.91\linewidth,height=0.51\textheight,keepaspectratio]{figures/fig-05.pdf}
  \begin{itemize}
    \item Matched data, roughly 123M parameters, 4K training, 7.86B tokens.
    \item Transformer uses dynamic YaRN beyond 4K; CST uses recurrent inference.
    \item Isolated CST gain at 128K is \alert{3.524265 $\to$ 3.518316}
          against recurrent BPTT.
  \end{itemize}
  \papersource{Original Fig. 5 and App. B.3, PDF p. 20; recurrent baseline from Table 3, p. 13.
  The 4.415 vs.\ 3.518 architecture gap is not the isolated effect of CST.}
\end{frame}

\begin{frame}{Backup: confidence intervals measure document uncertainty}
  \begin{itemize}
    \item 20,000 paired document bootstrap resamples.
    \item Windows from the same document remain grouped.
    \item Each replicate recomputes total NLL / total eligible target count.
    \item No training-seed or hyperparameter-selection uncertainty is included.
  \end{itemize}
  \vspace{2mm}
  \textbf{LongData, only positions 16K--32K:}
  \centering
  \renewcommand{\arraystretch}{1.3}
  \par\begin{tabular}{lcc}
    \toprule
    Quantity & Gain & 95\% CI\\
    \midrule
    Key-token NLL & 0.023537 & [0.011159, 0.037012]\\
    Context use & 0.013666 & [0.008616, 0.020309]\\
    \bottomrule
  \end{tabular}\par
  \vspace{3mm}
  The later position bucket supports the cumulative-prefix result.
  \papersource{Paper, App. B.4, PDF p. 21. Books3 has no paired document intervals.}
\end{frame}

\begin{frame}{Backup: unsuccessful explorations suggest concrete bottlenecks}
  \begin{itemize}
    \item \textbf{Length curriculum:}
          varying growth schedules did not yield consistent extrapolation gains.
    \item \textbf{State self-distillation:}
          useful privileged target states were difficult to define.
    \item \textbf{Quantized state:}
          simultaneously learning the quantizer, codebook, and recurrence
          created a coupled optimization problem.
    \item \textbf{Fully adaptive credit control:}
          removing a fixed trigger or gain cap increased sensitivity to noisy
          boundary statistics and added complexity.
  \end{itemize}
  \vspace{3mm}
  These are outcomes of the authors' tested formulations, not impossibility results.
  \papersource{Paper, App. D, PDF pp. 23--24.}
\end{frame}

\end{document}