% Original discussion slides based on arXiv:2607.02303v1.
% Built from the arxiv-paper-reader Beamer template. No author code is executed.
\documentclass[aspectratio=169,11pt]{beamer}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{amsmath,amssymb,graphicx,booktabs}
\usepackage{pgfplots}
\pgfplotsset{compat=1.18}

\definecolor{AcademicBlue}{RGB}{26,57,86}
\definecolor{AcademicTeal}{RGB}{0,105,112}
\definecolor{RecencyOrange}{RGB}{199,91,42}
\definecolor{BaselineGray}{RGB}{110,114,120}
\setbeamercolor{structure}{fg=AcademicBlue}
\setbeamercolor{frametitle}{fg=AcademicBlue,bg=white}
\setbeamercolor{title}{fg=AcademicBlue}
\setbeamercolor{alerted text}{fg=AcademicTeal}
\setbeamercolor{block title}{fg=white,bg=AcademicBlue}
\setbeamercolor{block body}{fg=black,bg=AcademicBlue!5}
\setbeamertemplate{navigation symbols}{}
\setbeamertemplate{itemize items}[circle]
\setbeamertemplate{footline}{%
  \hfill\usebeamerfont{page number in head/foot}%
  \insertframenumber\,/\,\inserttotalframenumber\hspace{4mm}\vspace{3mm}}
\setbeamersize{text margin left=8mm,text margin right=8mm}
\setbeamerfont{frametitle}{size=\Large,series=\bfseries}
\setbeamerfont{title}{size=\LARGE,series=\bfseries}
\newcommand{\papersource}[1]{\par\smallskip{\scriptsize\color{gray}#1\par}}
\newcommand{\norm}[1]{\lVert #1\rVert}
\newcommand{\score}{\beta_t\norm{e_t}}
\newcommand{\HOLA}{\textsc{Hola}}
\newcommand{\spacer}{\vspace{2mm}}
\setlength{\tabcolsep}{6pt}
\renewcommand{\arraystretch}{1.15}

\title[HOLA paper discussion]{A Hippocampus for\\Linear Attention}
\subtitle{An Exact Memory for What the Recurrent State Forgets}
\author{Paper: Wanyun Cui\\\small Shanghai University of Finance and Economics}
\date{arXiv:2607.02303v1 \;|\; 2 July 2026\\
  \small Research discussion prepared 22 September 2026}

\begin{document}

% 1
\begin{frame}
  \titlepage
  \vspace{-6mm}
  \begin{center}\small\color{AcademicTeal}
    A bounded exact KV memory complements a compressed recurrent state.
  \end{center}
\end{frame}

% 2
\begin{frame}{A fixed recurrent state can lose specific facts}
  \begin{columns}[T,onlytextwidth]
    \begin{column}{0.47\textwidth}
      \begin{block}{Recurrent memory}
        A fixed matrix $S\in\mathbb{R}^{d_k\times d_v}$ compresses the prefix.
      \end{block}
      \begin{itemize}
        \item Bounded state as context grows.
        \item Shared key directions cause interference.
        \item Distant exact associations can be lost.
      \end{itemize}
    \end{column}
    \begin{column}{0.49\textwidth}
      \begin{block}{Full attention}
        Every causal token remains available as a KV pair.
      \end{block}
      \begin{itemize}
        \item KV storage grows with length.
        \item Exact extraction remains stronger on some tasks.
        \item HOLA adds a small selective KV store to GDN.
      \end{itemize}
    \end{column}
  \end{columns}
  \spacer
  \alert{Research question: which facts should bypass state compression?}
  \papersource{Paper, Secs. 1--3.1, PDF pp. 1--4. Hippocampus/neocortex are design analogies.}
\end{frame}

% 3
\begin{frame}{Two memories provide complementary readouts}
  \begin{center}
    \includegraphics[width=0.96\linewidth,height=0.48\textheight,keepaspectratio]{figures/fig-02-method.pdf}
  \end{center}
  \[
    o_t=S_t^{\top}q_t+\lambda_t g_t(q_t)
  \]
  State: a linear estimator.\quad Cache: a bounded non-parametric read.
  \par\smallskip
  \alert{Exact KV storage does not guarantee correct recall.}
  \papersource{Original Fig. 2, PDF p. 4; Definition 2 / Eq. (3), p. 5. Column-vector convention in these slides.}
\end{frame}

% 4
\begin{frame}{The delta rule already computes a useful write signal}
  \[
    e_t=v_t-\alpha_t S_{t-1}^{\top}k_t,
    \qquad S_t=\alpha_t S_{t-1}+\beta_t k_t e_t^{\top}
  \]
  \begin{itemize}
    \item $e_t$: value prediction residual after decay $\alpha_t$.
    \item $\beta_t\in[0,1]$: how strongly the correction is written.
    \item State-path keys satisfy $\norm{k_t}_2=1$.
  \end{itemize}
  \begin{block}{Cache score: the committed residual update}
    \[
      m_t=\norm{\beta_t k_t e_t^{\top}}_F
          =\beta_t\norm{k_t}_2\norm{e_t}_2=\score
    \]
  \end{block}
  This measures the correction, not the full state change including decay.
  \papersource{Paper, Eqs. (1), (5)--(6), PDF pp. 4--5; GDN instantiation, Sec. 3.5, p. 6.}
\end{frame}

% 5
\begin{frame}{Keep high-write tokens regardless of their age}
  \begin{block}{Persistent cache}
    Keep the top-$w$ causal KV pairs by their insertion-time score $m_i$.
    Default: $w=64$ per layer.
  \end{block}
  \begin{center}
  \begin{tabular}{lrrr}
    \toprule
    Teaching example & $\norm{e}$ & $\beta$ & $m=\beta\norm{e}$\\
    \midrule
    Token A & 10 & 0.1 & 1.0\\
    Token B & 2 & 0.8 & \textbf{1.6}\\
    \bottomrule
  \end{tabular}
  \end{center}
  \begin{itemize}
    \item Fixed scores allow online or blockwise top-$w$ merging.
    \item Scores are not refreshed when queries or memory contents change.
    \item Large updates are a heuristic for future usefulness, not a guarantee.
  \end{itemize}
  \papersource{Paper, Sec. 3.3, PDF pp. 5--6. Numerical example is explanatory, not a paper experiment.}
\end{frame}

% 6
\begin{frame}{The visible set has at most 321 entries}
  \[
    \mathcal V_t=\text{persistent cache}
      \;\cup\;\text{causal current-block tokens}
      \;\cup\;\text{null sink}
  \]
  \begin{columns}[T,onlytextwidth]
    \begin{column}{0.30\textwidth}
      \begin{block}{64 persistent}
        Highest-score KV pairs retained across blocks.
      \end{block}
    \end{column}
    \begin{column}{0.37\textwidth}
      \begin{block}{Up to 256 current}
        Tokens from the current processing block, causally masked.
      \end{block}
    \end{column}
    \begin{column}{0.28\textwidth}
      \begin{block}{1 sink}
        An additional null entry for read stability.
      \end{block}
    \end{column}
  \end{columns}
  \spacer
  \begin{itemize}
    \item $w+C+1=64+256+1=321$ bounds the visible set.
    \item Only $w$ is the persistent distant-token budget.
    \item The paper specifies cache semantics, not full block-level code.
  \end{itemize}
  \papersource{Paper, Sec. 3.3, PDF p. 6; Sec. 5, p. 10; Appendix A, p. 12.}
\end{frame}

% 7
\begin{frame}{An exact cache needs a sufficiently sharp read}
  \[
    \tilde q_t=\mathrm{RMSNorm}_{\gamma}(q_t),\qquad
    \tilde k_j=\mathrm{RMSNorm}_{\gamma}(k_j)
  \]
  \[
    o_t^{\mathrm{cache}}=\sum_{j\in\mathcal V_t}
      \mathrm{softmax}_j\!\left(\frac{\tilde q_t^{\top}\tilde k_j}{\sqrt d}\right)v_j
  \]
  \begin{center}
  \begin{tabular}{lll}
    \toprule
    Cache normalization & Approximate norm & Logit scale\\
    \midrule
    Unit L2 & $1$ & $\tau\cos\theta/\sqrt d$\\
    RMSNorm, $\gamma\approx1$ & $\sqrt d$ & $\sqrt d\cos\theta$\\
    \bottomrule
  \end{tabular}
  \end{center}
  Learn Q/K scales; freeze $\tau=1$. Keep the state path unit-normalized.
  \papersource{Paper, Eq. (7) and Sec. 3.4, PDF p. 6. Scale approximation ignores epsilon and learned anisotropy.}
\end{frame}

% 8
\begin{frame}{Read normalization is separated from update stability}
  The state update contains the operator $I-\beta kk^{\top}$.
  \[
    \lambda_{\parallel k}=1-\beta\norm{k}^{2},
    \qquad\lambda_{\perp k}=1
  \]
  \begin{columns}[T,onlytextwidth]
    \begin{column}{0.47\textwidth}
      \begin{block}{Unit-norm state key}
        $\lambda_{\parallel k}=1-\beta\in[0,1]$.

        GDN additionally applies decay $\alpha_t$.
      \end{block}
    \end{column}
    \begin{column}{0.49\textwidth}
      \begin{block}{RMS-sized state key}
        $\lambda_{\parallel k}=1-\beta d$ can have large magnitude.

        Example: $d=256$, $\beta=0.5$ gives $-127$.
      \end{block}
    \end{column}
  \end{columns}
  \spacer
  \alert{Sharpen the cache read while preserving the state update's normalization.}
  \papersource{Paper, Sec. 3.4, PDF p. 6. Eigenvalue calculation and numerical example are explanatory.}
\end{frame}

% 9
\begin{frame}{The strongest comparison shares the same GDN backbone}
  \begin{center}
  \begin{tabular}{ll}
    \toprule
    Main scale & 340M; 24 layers; width 1024; 4 heads, dim 256\\
    Training & 15B SlimPajama tokens; context 2048\\
    Recipe & AdamW; peak LR $4\times10^{-4}$; $8\times$A800\\
    Cache & $w=64$, $C=256$, RMSNorm, frozen $\tau=1$\\
    Internal controls & GDN anchor; HOLA+recency; HOLA\\
    Uncertainty & Main-scale results use a single seed\\
    \bottomrule
  \end{tabular}
  \end{center}
  \begin{itemize}
    \item Recency changes only which tokens the matched cache retains.
    \item Transformer++, GLA/GSA, DeltaNet/KDA and one GDN row are borrowed from recipe-matched papers.
  \end{itemize}
  \papersource{Paper, Sec. 4.1 / Table 1, PDF p. 7; Sec. 5, p. 10; Appendix A, p. 12.}
\end{frame}

% 10
\begin{frame}{HOLA lowers perplexity in the reported 340M comparison}
  \begin{center}
  \begin{tabular}{lrrl}
    \toprule
    Model & Wiki PPL $\downarrow$ & LAMBADA PPL $\downarrow$ & Provenance\\
    \midrule
    Transformer++ & 26.88 & 42.15 & Borrowed\\
    KDA & 26.18 & 31.37 & Borrowed\\
    GDN anchor & 27.32 & 30.95 & Internal\\
    HOLA+recency & 25.04 & 32.33 & Internal\\
    HOLA & \textbf{22.92} & \textbf{30.26} & Internal\\
    \bottomrule
  \end{tabular}
  \end{center}
  \begin{block}{Against the internal GDN anchor}
    Wiki: $27.32\rightarrow22.92$, a \textbf{16.1\% relative reduction}.\\
    LAMBADA: $30.95\rightarrow30.26$, a 2.23\% relative reduction.
  \end{block}
  These are perplexity changes, not accuracy percentage points.
  \papersource{Selected rows from Table 1, PDF p. 7. Relative changes calculated from reported values.}
\end{frame}

% 11
\begin{frame}{Retrieval improves; commonsense scores remain close}
  \begin{center}
  \begin{tabular}{lrrr}
    \toprule
    Model & FDA $\uparrow$ & SWDE $\uparrow$ & Six-task mean $\uparrow$\\
    \midrule
    Transformer++ (borrowed) & \textbf{46.1} & 25.9 & 42.34\\
    GDN anchor & 11.7 & 29.0 & 42.54\\
    HOLA+recency & 16.9 & 29.9 & \textbf{43.17}\\
    HOLA & 20.1 & \textbf{35.9} & 42.85\\
    \bottomrule
  \end{tabular}
  \end{center}
  \begin{itemize}
    \item FDA: +71.8\% relative to GDN, but still below full attention.
    \item SWDE: +23.8\% relative to GDN; above this full-attention row.
    \item Small commonsense differences have no reported error bars.
  \end{itemize}
  \papersource{Table 1, PDF p. 7. Six-task mean mixes accuracy and acc\_norm; retrieval scorer details are not fully specified.}
\end{frame}

% 12
\begin{frame}{Matched recency isolates the value of token selection}
  \begin{block}{Hold architecture and memory budget fixed}
    Same $w$, chunk, sharpened read, gate and kernel.\\
    Change eviction: most recent tokens $\rightarrow$ largest $\beta\norm{e}$.
  \end{block}
  \begin{center}
  \begin{tabular}{lrr}
    \toprule
    340M result & Recency & HOLA\\
    \midrule
    Wiki PPL $\downarrow$ & 25.04 & \textbf{22.92}\\
    S-NIAH-1 at 32k $\uparrow$ & 0.24 & \textbf{0.58}\\
    Commonsense six-task mean $\uparrow$ & 43.17 & 42.85\\
    \bottomrule
  \end{tabular}
  \end{center}
  Recency already helps Wiki PPL versus GDN; selection adds further gains.
  \papersource{Sec. 4.5, PDF pp. 8--9; Table 1 and Fig. 1. Recency quality evaluation uses full recomputation; this is not a decode-speed comparison.}
\end{frame}

% 13
\begin{frame}{Single-needle recall remains stronger out to 32k}
  \begin{columns}[c,onlytextwidth]
    \begin{column}{0.69\textwidth}
      \begin{tikzpicture}
      \begin{axis}[
        width=\linewidth,height=5.25cm,
        xmin=0.8,xmax=5.2,ymin=0,ymax=1.06,
        xtick={1,2,3,4,5},xticklabels={2k,4k,8k,16k,32k},
        ytick={0,0.2,0.4,0.6,0.8,1.0},
        xlabel={Context length (trained at 2k)},ylabel={S-NIAH-1 recall},
        tick label style={font=\small},label style={font=\small},
        ymajorgrids=true,grid style={gray!20},
        legend style={font=\scriptsize,draw=none,fill=white,at={(0.5,1.04)},anchor=south,legend columns=2,column sep=4pt},
        legend cell align=left]
        \addplot[blue!65,densely dotted,thick,mark=diamond*] coordinates {(1,1)(2,0)(3,0)(4,0)(5,0)};
        \addlegendentry{Transformer++}
        \addplot[BaselineGray,dashed,thick,mark=square*] coordinates {(1,1)(2,1)(3,0.83)(4,0.30)(5,0.14)};
        \addlegendentry{GDN}
        \addplot[RecencyOrange,dash dot,thick,mark=triangle*] coordinates {(1,1)(2,0.93)(3,0.74)(4,0.50)(5,0.24)};
        \addlegendentry{HOLA+recency}
        \addplot[AcademicTeal,very thick,mark=*] coordinates {(1,1)(2,0.99)(3,0.98)(4,0.94)(5,0.58)};
        \addlegendentry{HOLA}
      \end{axis}
      \end{tikzpicture}
    \end{column}
    \begin{column}{0.28\textwidth}
      \textbf{At 32k}\\[2mm]
      HOLA: \alert{0.58}\\
      Recency: 0.24\\
      GDN: 0.14\\[3mm]
      +44 percentage points versus GDN.\\[3mm]
      Still far from perfect recall.
    \end{column}
  \end{columns}
  \papersource{Redrawn from Fig. 1b, PDF p. 1, using exact coordinates in fig\_teaser.tex. RULER: Mistral tokenizer, limit 100. This Transformer++ checkpoint is not adapted for length extrapolation.}
\end{frame}

% 14
\begin{frame}{Harder RULER tasks reveal clear limits}
  \begin{center}
  \begin{tabular}{lrrr}
    \toprule
    Task / context & GDN & Recency & HOLA\\
    \midrule
    S-NIAH-2 / 2k & 0.38 & 0.83 & \textbf{1.00}\\
    Multi-query / 2k & \textbf{0.24} & 0.22 & 0.18\\
    Multi-key-1 / 4k & 0.17 & 0.13 & \textbf{0.30}\\
    S-NIAH-2 / 8k & 0.09 & 0.04 & \textbf{0.35}\\
    S-NIAH-3 / 8k & \textbf{0.07} & 0.02 & 0.05\\
    Multi-query / 8k & 0.02 & 0.01 & \textbf{0.08}\\
    \bottomrule
  \end{tabular}
  \end{center}
  \begin{itemize}
    \item HOLA wins many cells, but not every task or length.
    \item Several hard tasks remain near the floor at 8k.
    \item Reported needle results do not cover the full RULER task suite.
  \end{itemize}
  \papersource{Selected cells from Table 3, PDF p. 9. Accuracy, higher is better. Setup mentions VT but reports no VT result.}
\end{frame}

% 15
\begin{frame}{Perplexity gains repeat across three tested scales}
  \begin{center}
  \begin{tabular}{lrrrr}
    \toprule
    Scale & Train tokens & GDN PPL & HOLA PPL & Reduction\\
    \midrule
    46M & 0.50B & 71.0 & 59.5 & 16.20\%\\
    170M & 6.22B & 35.98 & 30.51 & 15.20\%\\
    340M & 15.0B & 27.32 & 22.92 & 16.11\%\\
    \bottomrule
  \end{tabular}
  \end{center}
  \begin{itemize}
    \item Within each scale: matched GDN/HOLA backbone and recipe.
    \item 46M: FineWeb-Edu, context 4096, width 512, 12 layers.
    \item 170M/340M: SlimPajama, context 2048, width 1024.
  \end{itemize}
  \spacer
  \alert{This is cross-scale consistency, not a parameter-only scaling law.}
  \papersource{Table 2, PDF p. 8; Appendix A / Table 6, p. 12. Relative reductions calculated from displayed values.}
\end{frame}

% 16
\begin{frame}{At 46M, committed residual beats residual alone}
  Flat cache read; same backbone and cache budget for cached variants.
  \begin{center}
  \begin{tabular}{lrrr}
    \toprule
    Eviction score & Far $\uparrow$ & 4 keys $\uparrow$ & Wiki PPL $\downarrow$\\
    \midrule
    GDN, no cache & 0.55 & 0.41 & 70.21\\
    Cumulative attention & 0.43 & 0.55 & 71.45\\
    $\norm{e}$ & 0.22 & 0.48 & 70.50\\
    $\beta\norm{v}$ & 0.42 & 0.54 & 70.76\\
    $\beta\norm{e}$ & \textbf{0.67} & \textbf{0.56} & \textbf{70.10}\\
    \bottomrule
  \end{tabular}
  \end{center}
  \begin{itemize}
    \item Residual magnitude alone is a poor far-passkey signal here.
    \item The PPL gain remains small until the read is changed.
  \end{itemize}
  \papersource{Table 4, PDF p. 9. Far: teacher-forced next-token accuracy, passkey at depth 0.1 in 4k. This is not the 340M RULER protocol.}
\end{frame}

% 17
\begin{frame}{Read normalization is the largest change in this ablation}
  Fixed eviction: $\beta\norm{e}$.\quad 46M component study.
  \begin{center}
  \begin{tabular}{lrrrrr}
    \toprule
    Model / cache read & PPL $\downarrow$ & Far & 4 keys & 8 keys & 16 keys\\
    \midrule
    GDN, no cache & 70.21 & 0.55 & 0.41 & 0.35 & 0.31\\
    Unit L2 & 70.10 & 0.67 & 0.56 & 0.41 & 0.31\\
    RMSNorm-$\gamma$ & \textbf{59.5} & \textbf{0.75} & \textbf{0.77} & \textbf{0.67} & \textbf{0.41}\\
    \bottomrule
  \end{tabular}
  \end{center}
  \begin{itemize}
    \item Wiki PPL falls by 10.60 when changing the cache read.
    \item Multi-key recall improves, while far-passkey accuracy is preserved.
    \item RMSNorm row: three-seed mean; variance is not reported.
  \end{itemize}
  \papersource{Table 5, PDF p. 10. Recall columns: higher is better. Other rows reuse flat-read anchors. No fixed-high-temperature row is reported.}
\end{frame}

% 18
\begin{frame}{Memory overhead is bounded; speed gains are unmeasured}
  \begin{columns}[T,onlytextwidth]
    \begin{column}{0.48\textwidth}
      \begin{block}{Additional model parameters}
        $L(2d+2H)=12{,}480$ trainable scalars.\\[2mm]
        Less than 0.004\% at 340M.\\[2mm]
        Plus 96 frozen temperature scalars.
      \end{block}
    \end{column}
    \begin{column}{0.48\textwidth}
      \begin{block}{Additional decoding state}
        Up to $w+C=320$ KV pairs per layer.\\[2mm]
        About 31 MB in bf16 at the stated configuration.
      \end{block}
    \end{column}
  \end{columns}
  \begin{center}
  \begin{tabular}{lcc}
    \toprule
    Peak GPU allocation, batch 1 & GDN & HOLA\\
    \midrule
    At 32k and at 128k, weights included & 0.72 GB & 0.75 GB\\
    \bottomrule
  \end{tabular}
  \end{center}
  128k tests memory usage only. No throughput or latency table is reported.
  \papersource{Paper, Sec. 3.5, PDF p. 6. Reported rounded peaks imply about 4.2\% extra; paper says about 5\%.}
\end{frame}

% 19
\begin{frame}{The evidence supports a selective complement to GDN}
  \begin{itemize}
    \setlength{\itemsep}{4mm}
    \item \textbf{Supported:} lower PPL and better reported retrieval at the tested scales, with a strong matched-recency control.
    \item \textbf{Not established:} reliable commonsense gains, optimal eviction, or superiority to a matched learned eviction module.
    \item \textbf{Remaining limits:} a small persistent cache, single-seed main results, and weak hard multi-needle recall.
    \item \textbf{Deployment questions:} full causal implementation details, larger models, and measured speed/quality trade-offs.
  \end{itemize}
  \papersource{Paper, Secs. 3--5, PDF pp. 4--10. Evidence synthesis and deployment questions are this reading's analysis.}
\end{frame}

% 20
\begin{frame}{Design the cache's write and read together}
  \begin{enumerate}
    \setlength{\itemsep}{5mm}
    \item \textbf{Preserve a small set of exact associations.}\\
      A fixed recurrent state alone loses some details.
    \item \textbf{Use the update to guide retention.}\\
      $\beta\norm{e}$ is available without an extra eviction network.
    \item \textbf{Make retained associations retrievable.}\\
      A separate RMSNorm path makes the cache read more selective.
  \end{enumerate}
  \spacer
  \alert{Most defensible claim: bounded exact memory helps, and both selection and read normalization matter.}
  \papersource{Synthesis of Eqs. (3), (6), (7), Table 1, and Tables 4--5. No claim of perfect or universally optimal recall.}
\end{frame}

% 21
\begin{frame}{Reference and reading scope}
  \textbf{Wanyun Cui.}\\[2mm]
  \emph{A Hippocampus for Linear Attention: An Exact Memory for What the Recurrent State Forgets.}\\[3mm]
  arXiv:2607.02303v1, 2 July 2026.\\[2mm]
  \url{https://arxiv.org/abs/2607.02303v1}
  \spacer
  \begin{itemize}
    \item Read all substantive sections and Appendix A from author TeX.
    \item Cross-checked the 12-page versioned PDF and figure coordinates.
    \item Original Figure 2 is cropped; Figure 1b is redrawn from its data.
    \item These are independent discussion slides; experiments were not reproduced.
  \end{itemize}
  \papersource{Prepared with the arxiv-paper-reader workflow. Chinese notes, editable TeX, figures, and local build helper accompany this PDF.}
\end{frame}

\appendix

% 22
\begin{frame}{Backup: training recipes differ across scales}
  \begin{center}
  \begin{tabular}{lrrlrr}
    \toprule
    Scale & Width & Layers & Corpus & Tokens & Context\\
    \midrule
    46M & 512 & 12 & FineWeb-Edu & 0.5B & 4096\\
    170M & 1024 & 12 & SlimPajama & 6.22B & 2048\\
    340M & 1024 & 24 & SlimPajama & 15.0B & 2048\\
    \bottomrule
  \end{tabular}
  \end{center}
  \begin{itemize}
    \item 170M/340M: 4 heads, head dimension 256, tied embeddings, vocabulary 32000, Mistral tokenizer.
    \item 340M: AdamW, weight decay 0.01, cosine schedule, warmup 1000, clip 1.0, batch 0.5M tokens, one epoch.
    \item HOLA: $w=64$, $C=256$, gate init $-4$, RMSNorm, frozen $\tau=1$, SDPA cache kernel.
  \end{itemize}
  \papersource{Paper, Sec. 4.1, PDF p. 7; Appendix A / Table 6, p. 12.}
\end{frame}

% 23
\begin{frame}{Backup: full RULER snapshot at 2k and 4k}
  \begin{center}
  \begin{tabular}{llrrrrrr}
    \toprule
    Context & Model & S1 & S2 & S3 & MK1 & MV & MQ\\
    \midrule
    2k & Transformer++ & 1.00 & 1.00 & 0.79 & 0.71 & 0.34 & 0.33\\
       & GDN & 1.00 & 0.38 & 0.26 & 0.17 & 0.17 & 0.24\\
       & Recency & 1.00 & 0.83 & 0.89 & 0.26 & 0.21 & 0.22\\
       & HOLA & 1.00 & 1.00 & 0.96 & 0.25 & 0.28 & 0.18\\
    \midrule
    4k & Transformer++ & 0.00 & 0.00 & 0.00 & 0.00 & 0.00 & 0.00\\
       & GDN & 1.00 & 0.52 & 0.23 & 0.17 & 0.14 & 0.19\\
       & Recency & 0.93 & 0.27 & 0.38 & 0.13 & 0.17 & 0.17\\
       & HOLA & 0.99 & 0.89 & 0.43 & 0.30 & 0.28 & 0.26\\
    \bottomrule
  \end{tabular}
  \end{center}
  At 2k S3, HOLA (0.96) exceeds Transformer++ (0.79).
  \papersource{Paper, Table 3, PDF p. 9. Accuracy, higher is better. S1/S2/S3: single needle; MK1/MV/MQ: multi-key/value/query.}
\end{frame}

% 24
\begin{frame}{Backup: numerical details need careful qualification}
  \begin{itemize}
    \setlength{\itemsep}{4mm}
    \item \textbf{Logit examples use unexplained dimensions.}\\
      $6.6/\sqrt d\approx0.83$ suggests $d\approx64$, while $\sqrt d\approx11$ suggests another dimension. The main model uses $d=256$.
    \item \textbf{The 46M GDN anchors differ.}\\
      Table 2 reports 71.0; Tables 4--5 report 70.21. The paper does not explain the difference.
    \item \textbf{A tiny gain has a rounding discrepancy.}\\
      BoolQ $0.548\rightarrow0.584$ is +3.6 points from displayed values; the prose says +3.5.
    \item \textbf{Mechanism claims exceed the shown ablation.}\\
      Table 5 has no fixed-high-temperature row or attention-entropy diagnostic.
  \end{itemize}
  \papersource{Paper, Sec. 3.4, p. 6; Sec. 4.2 / Table 2, p. 8; Tables 4--5, pp. 9--10. These are reading/audit observations.}
\end{frame}

% 25
\begin{frame}{Backup: a conceptual causal implementation}
  \begin{enumerate}
    \setlength{\itemsep}{3mm}
    \item Compute Q/K/V, decay $\alpha_t$, and write gate $\beta_t$.
    \item Predict with the decayed state; form $e_t$ and update $S_t$.
    \item Fix $m_t=\beta_t\norm{e_t}$; maintain bounded top-$w$ KV.
    \item Read the cache plus the causal current-block prefix and sink.
    \item Combine the state read and gated cache read.
  \end{enumerate}
  \begin{block}{Details requiring implementation-level verification}
    Block update timing, deduplication, score ties, per-head selection, top-$w$ gradients, and the exact cache-gate parameterization.
  \end{block}
  \papersource{Explanatory procedure based on Sec. 3, PDF pp. 4--6. The downloaded source is paper TeX, not an executable model implementation.}
\end{frame}

\end{document}