% Self-authored reading seminar; built from the arxiv-paper-reader template.
% Compile with: python3 build_slides.py slides.en.tex
\documentclass[aspectratio=169,11pt]{beamer}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{lmodern}
\usepackage{amsmath,amssymb,graphicx,booktabs}

\definecolor{AcademicBlue}{RGB}{26,57,86}
\definecolor{AcademicTeal}{RGB}{0,105,112}
\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}
\setbeamerfont{block title}{size=\normalsize}
\newcommand{\papersource}[1]{\par\smallskip{\scriptsize\color{gray}#1\par}}
\newcommand{\FRAC}{FRAC}
\newcommand{\gread}{g_{\mathrm{read}}}
\newcommand{\gwrite}{g_{\mathrm{write}}}

\title[FRAC]{Fractional State Space Transition\\for Long Sequence Modeling}
\subtitle{A reading seminar on FRAC}
\author{\small Paper authors: Ivan Kobyzev, Abbas Ghaddar, Ali Nasiri-Sarvi,\\
Lifeng Shang, Yufei Cui\\[2mm]
\footnotesize Huawei Noah's Ark Lab, Montreal Research Center}
\date{\small arXiv:2609.36314v1 \;|\; 28 September 2026\\
Reading prepared: 7 October 2026}

\begin{document}

\begin{frame}
  \titlepage
  \vspace{-4mm}
  \begin{center}
  \small\color{AcademicTeal}
  A finite bank of exponential modes provides a practical long-memory prior.
  \end{center}
\end{frame}

\begin{frame}[t]{A bounded state still needs a memory law}
\begin{block}{Research question}
How can a recurrent model retain useful history across widely separated timescales?
\end{block}
\[
\dot h(t)=-\lambda h(t)+u(t)
\quad\Longrightarrow\quad
h(t)=e^{-\lambda t}h(0)\quad\text{when }u=0.
\]
\begin{itemize}
\item A single exponential has one characteristic timescale, $1/\lambda$.
\item FRAC organizes multiple decay modes using fractional dynamics.
\item The goal is broader temporal coverage with a fixed-size decoding state.
\end{itemize}
\papersource{Paper, Secs. 1--2.2, PDF pp. 1--2. Related-work distinctions: Sec. 5, p. 9.}
\end{frame}

\begin{frame}[t]{Fractional dynamics induce two different tail laws}
\[
{}^CD_t^\alpha h(t)=-\lambda h(t)+u(t),\qquad 0<\alpha<1,\quad\lambda>0
\]
\[
h(t)=\underbrace{E_\alpha(-\lambda t^\alpha)h(0)}_{\text{initial-state relaxation}}
+\underbrace{\int_0^t g_{\alpha,\lambda}(t-\xi)u(\xi)\,d\xi}_{\text{response to past inputs}}
\]
\begin{columns}[T]
\begin{column}{0.49\textwidth}
\begin{block}{Homogeneous kernel}
\[
E_\alpha(-\lambda t^\alpha)\sim
\frac{t^{-\alpha}}{\lambda\Gamma(1-\alpha)}
\]
\end{block}
\end{column}
\begin{column}{0.49\textwidth}
\begin{block}{Impulse-response kernel}
\[
g_{\alpha,\lambda}(t)\sim
\frac{\alpha\,t^{-\alpha-1}}{\lambda^2\Gamma(1-\alpha)}
\]
\end{block}
\end{column}
\end{columns}
\medskip
These are large-time asymptotics. Direct fractional solvers retain history.
\papersource{Paper, Sec. 2.3, Eqs. (4)--(8), PDF p. 3. $E$ denotes a Mittag--Leffler function.}
\end{frame}

\begin{frame}[t]{A geometric mode bank approximates fractional kernels}
\[
E_\alpha(-\lambda t^\alpha)
=\int_0^\infty e^{-t/\tau}R_{\alpha,\lambda}(\tau)\,d\tau
\approx\sum_{m=1}^M c_me^{-t/\tau_m}
\]
\[
\tau_m=\tau_0q^{m-1},\qquad q>1,\qquad c_m>0.
\]
\begin{itemize}
\item Positive diffusive density: fast and slow exponentials share the memory load.
\item Log-space quadrature gives $c_m\propto\tau_m^{-\alpha}$ at large timescales.
\item On a finite horizon $[0,T]$, Theorem 2 guarantees a suitable finite bank.
\end{itemize}
\begin{block}{Different error norms are essential}
Uniform error for the homogeneous kernel; $L^1$ error for the impulse kernel, which is singular at $t=0$.
\end{block}
\papersource{Paper, Theorems 1--2, Eqs. (9)--(12), PDF p. 4; Appendix B, pp. 17--19.}
\end{frame}

\begin{frame}[t]{The mode bank has an exact local ZOH update}
\setlength{\abovedisplayskip}{8pt}
\setlength{\belowdisplayskip}{8pt}
\[
\dot s_m=-\frac{s_m}{\tau_m}+\frac{u}{\lambda\tau_m},
\qquad \tilde h_M=\sum_m c_ms_m
\]
Hold the input and controls fixed during an interval of length $\Delta_t$:
\[
s_{t,m}=\rho_{t,m}s_{t-1,m}+\beta_{t,m}u_t,
\quad \rho_{t,m}=e^{-\Delta_t/\tau_m},
\quad \beta_{t,m}=\frac{1-\rho_{t,m}}{\lambda}.
\]
\medskip
\textbf{Token-dependent timescale control}
\[
\tilde\tau_{t,m}=\frac{\tau_m}{\lambda_t^{1/\alpha_t}},
\qquad \tilde\rho_{t,m}=
\exp\!\left(-\frac{\Delta_t\lambda_t^{1/\alpha_t}}{\tau_m}\right).
\]
The product $\Delta_t\lambda_t^{1/\alpha_t}$ advances an internal clock.
\papersource{Paper, Proposition 3 and Eqs. (13)--(20), PDF p. 5; Appendix A, Remark 4, p. 17.}
\end{frame}

\begin{frame}[t]{All experiments modify the theoretical write amplitude}
\begin{columns}[T]
\begin{column}{0.49\textwidth}
\begin{block}{Exact ZOH derivation}
\[
\tilde\beta_{t,m}=\frac{1-\tilde\rho_{t,m}}{\lambda_t}
\]
\end{block}
\end{column}
\begin{column}{0.49\textwidth}
\begin{block}{Algorithm 1: experimental model}
\[
\tilde\beta_{t,m}=1-\tilde\rho_{t,m}
\]
\end{block}
\end{column}
\end{columns}
\medskip
\begin{itemize}
\item Authors remove the explicit $1/\lambda_t$ write-amplitude factor for training stability.
\item Constant $\lambda$: related by constant state/readout rescaling.
\item Token-varying $\lambda_t$: this is a substantive modeling choice.
\end{itemize}
\begin{block}{Interpretation}
The selective layer generalizes a fixed-system fractional construction; the theory supplies its memory prior.
\end{block}
\papersource{Paper, Eq. (20), PDF p. 5; Algorithm 1 and Remarks 7--8, Appendix D, p. 21.}
\end{frame}

\begin{frame}[t]{Read and write routing share a timescale prior}
\begin{align*}
c_{t,m}&=\operatorname{softmax}_m[-\alpha_t\log\tau_m+\gread(u_t)_m],\\
b_{t,m}&=\operatorname{softmax}_m[-\alpha_t\log\tau_m+\gwrite(u_t)_m].
\end{align*}
\begin{itemize}
\item The prior favors $\tau_m^{-\alpha_t}$; smaller $\alpha_t$ flattens this distribution.
\item Independent content projections select where to write and read.
\item The read prior has an asymptotic basis; matching the write prior is empirical.
\end{itemize}
\begin{block}{Actual recurrent rule used in the experiments}
\[
s_{t,m}=\tilde\rho_{t,m}s_{t-1,m}
+(1-\tilde\rho_{t,m})b_{t,m}u_t,
\qquad h_t=\sum_m c_{t,m}s_{t,m}.
\]
\end{block}
\papersource{Paper, Sec. 3.2, Eq. (23), PDF p. 6; Algorithm 1, p. 21.}
\end{frame}

\begin{frame}[t]{The mixer separates clock control from content}
\begin{columns}[T]
\begin{column}{0.58\textwidth}
\centering
\includegraphics[width=\linewidth,height=5.0cm,keepaspectratio]{figures/fig-01-architecture.pdf}
\end{column}
\begin{column}{0.40\textwidth}
\begin{itemize}
\item Pre-convolution branch: $\Delta_t$, $\alpha_t$, $\lambda_t$.
\item Content branch: causal depthwise convolution and SiLU.
\item Direct feedthrough: $h_t+Du_t$.
\item Gated normalization, output projection, residual connection.
\end{itemize}
\end{column}
\end{columns}
\medskip
$M=16$, geometric $\tau\in[1,2^{17}]$; $\Delta\in[10^{-4},1]$, $\lambda\in[0.25,4]$.
\papersource{Original Fig. 1, Sec. 3.3, PDF p. 6; language-model configuration, Appendix E.3, p. 23.}
\end{frame}

\begin{frame}[t]{Affine recurrence supports scan and bounded-state decoding}
\[
\mathbf s_t=\Lambda_t\mathbf s_{t-1}+\mathbf v_t,
\qquad\Lambda_t=\operatorname{Diag}(\tilde\rho_{t,1},\ldots,\tilde\rho_{t,M}).
\]
\begin{itemize}
\item Associative affine composition enables chunked parallel training and prefill.
\item Decode uses the same one-step rule and caches $S_t\in\mathbb R^{H\times M\times d_h}$.
\item Direct recurrence work: $O(LHMd_h)$; state per layer: $O(HMd_h)$.
\end{itemize}
\medskip
\centering
\begin{tabular}{lrr}
\toprule
Principal state per layer & Scalars & Relative to FRAC \\
\midrule
FRAC & 65,536 & 1$\times$ \\
Mamba2 / Mamba3 & 524,288 & 8$\times$ \\
GDN & 294,912 & 4.5$\times$ \\
\bottomrule
\end{tabular}
\papersource{Paper, Eq. (24), p. 6; Appendix E.3, p. 23. Counts exclude convolution buffers and auxiliary caches.}
\end{frame}

\begin{frame}[t]{The evaluation asks three different questions}
\begin{block}{Does the prior transfer across sequence lengths?}
Heavy-tail probe: train at 512, test up to 128K.\\
Language-model NIAH: train at 4K, test up to 64K.
\end{block}
\begin{block}{Does the model preserve useful general capabilities?}
MADLab, LongBench, LM Harness, and real-world retrieval truncated to 2K.
\end{block}
\begin{block}{What are the costs and design dependencies?}
Prefill / decode / training throughput; 390M model ablations; finite-bank fitting.
\end{block}
DNA uses a separate protocol: train and test at each chosen context length.
\papersource{Paper, Sec. 4, pp. 6--9; Appendices E--G, pp. 21--27.}
\end{frame}

\begin{frame}[t]{FRAC extrapolates the heavy-tail task more effectively}
\begin{columns}[T]
\begin{column}{0.60\textwidth}
\includegraphics[width=\linewidth,height=4.7cm,keepaspectratio]{figures/fig-02-heavy-tail.pdf}
\end{column}
\begin{column}{0.38\textwidth}
One layer, $\sim$200K parameters; train only at 512.
\medskip
\[
\text{label}=\operatorname{sign}\!\sum_k\frac{v_k}{(L-k)^{0.1}}
\]
\medskip
128K accuracy, mean $\pm$ SD:
\begin{itemize}
\item FRAC: $63.8\pm1.9$
\item GDN: $58.4\pm3.8$
\item Attention: $50.0\pm3.7$
\end{itemize}
\end{column}
\end{columns}
\medskip
The 128K gain over GDN is \alert{5.4 percentage points} across 10 runs.
\papersource{Original Fig. 2, p. 7; Appendix E.1, Tables 5--6, p. 22. Accuracy in percent.}
\end{frame}

\begin{frame}[t]{MADLab shows preserved core abilities, with modest gains}
\setlength{\tabcolsep}{10pt}
\centering
\begin{tabular}{lrrrrr}
\toprule
Model & Comp. & F-ICR & Mem. & Copy & Avg. \\
\midrule
Mamba2 & 88.3 & 7.2 & 99.9 & 99.9 & 73.8 \\
GDN & 88.9 & 9.7 & 98.8 & 99.9 & 74.3 \\
Mamba3-SISO & 89.1 & 10.1 & 100.0 & 100.0 & 74.8 \\
\FRAC & \textbf{90.4} & \textbf{11.5} & 99.9 & 99.9 & \textbf{75.4} \\
\bottomrule
\end{tabular}
\medskip
\begin{itemize}
\item $\sim$500K parameters; two sequence mixers and two SwiGLU layers.
\item Compression and fuzzy recall account for the average gain.
\item Fuzzy recall remains difficult: even the best accuracy is only 11.5\%.
\item Five-seed means after task-specific hyperparameter selection.
\end{itemize}
\papersource{Paper, Table 1, PDF p. 7; Appendix E.2, pp. 22--23. Standard deviations omitted here for readability.}
\end{frame}

\begin{frame}[t]{Language models share data and token budgets}
\begin{itemize}
\item Approximately \alert{1.3B parameters}, pretrained from scratch on \alert{100B tokens}.
\item Deduplicated FineWeb-Edu; Llama-2 tokenizer with a 32K vocabulary.
\item Fully packed \alert{4K training sequences}; hidden size 2048, 16 heads.
\item AdamW, learning rate $6\times10^{-4}$, 10\% warm-up, batch 1M tokens.
\end{itemize}
\medskip
\centering
\begin{tabular}{lc}
\toprule
Model & Layers \\
\midrule
FRAC / Mamba2 / Mamba3-SISO & 48 \\
Mamba3-MIMO & 44 \\
GDN & 22 \\
Transformer (Qwen3 architecture) & 26 \\
\bottomrule
\end{tabular}
\papersource{Paper, Sec. 4.2, p. 7; Appendix E.3, pp. 23--24. Depth varies to match parameter counts.}
\end{frame}

\begin{frame}[t]{NIAH improves beyond 4K, but retrieval still degrades}
\centering
\includegraphics[width=\linewidth,height=3.9cm,keepaspectratio]{figures/fig-03-niah.pdf}
\medskip
\begin{itemize}
\item S-NIAH-1: repetitive filler; S-NIAH-2: number; S-NIAH-3: UUID.
\item FRAC's advantage is clear beyond the training range, especially at 8K--16K.
\item At 64K, S-NIAH-2/3 accuracy is still around 10\% or lower.
\end{itemize}
\papersource{Original Fig. 3, PDF p. 8; task definitions, Appendix E.3, p. 24. Last bullet is an approximate visual reading.}
\end{frame}

\begin{frame}[t]{LongBench improves by 1.9 score points over GDN}
\begin{columns}[T]
\begin{column}{0.48\textwidth}
\centering
\begin{tabular}{lr}
\toprule
Model & Average score \\
\midrule
Transformer & 14.7 \\
Mamba2 & 12.0 \\
GDN & 16.0 \\
Mamba3-SISO & 14.4 \\
Mamba3-MIMO & 11.4 \\
\FRAC & \textbf{17.9} \\
\bottomrule
\end{tabular}
\end{column}
\begin{column}{0.48\textwidth}
\begin{itemize}
\item Best on \alert{8 of 14} tasks.
\item $+1.9$ points over GDN; about $+11.9\%$ relative using rounded averages.
\item Mixed metrics: F1, ROUGE, accuracy, and edit similarity.
\item Eight tasks have mean input length below 8K.
\end{itemize}
\end{column}
\end{columns}
\medskip
\begin{block}{Reading the reported gain}
The text says ``1.9\%''; the table supports a 1.9-point difference in benchmark score.
\end{block}
\papersource{Paper, Table 2, p. 8; task lengths and metrics, Table 7, p. 24.}
\end{frame}

\begin{frame}[t]{LongBench gains coexist with a large code-completion loss}
\centering
\begin{tabular}{llrrr}
\toprule
Task & Metric & FRAC & GDN & Transformer \\
\midrule
GovReport & ROUGE & \textbf{13.8} & 7.9 & 7.2 \\
SAMSum & ROUGE & \textbf{28.2} & 13.2 & 11.3 \\
NarrativeQA & F1 & \textbf{13.2} & 11.5 & 8.7 \\
2WikiMQA & F1 & 12.2 & 14.8 & 15.5 \\
LCC & Edit similarity & 11.0 & 18.7 & \textbf{26.0} \\
RepoBench-P & Edit similarity & 19.5 & 16.9 & 25.6 \\
\bottomrule
\end{tabular}
\medskip
\begin{itemize}
\item Strong gains in several summarization and QA tasks.
\item LCC loses 15.0 points to the Transformer; its mean input is only 1,235 tokens.
\item Longer retention and precise local continuation are different capabilities.
\end{itemize}
\papersource{Paper, selected columns of Table 2, PDF p. 8; Table 7, p. 24. All metrics are higher-is-better.}
\end{frame}

\begin{frame}[t]{Short-context quality stays competitive}
\centering
\begin{tabular}{lrrr}
\toprule
Model & Wiki. PPL$\downarrow$ & Harness avg.$\uparrow$ & Retrieval avg.$\uparrow$ \\
\midrule
Transformer & \textbf{15.3} & \textbf{55.3} & \textbf{38.5} \\
GDN & 16.4 & 54.2 & 30.9 \\
Mamba3-SISO & 16.1 & 54.6 & 35.1 \\
Mamba3-MIMO & 15.9 & \textbf{55.3} & 36.8 \\
\FRAC & 16.5 & 55.1 & 36.4 \\
\bottomrule
\end{tabular}
\medskip
\begin{itemize}
\item Harness average: eight accuracy metrics, excluding both perplexity columns.
\item FRAC trails the best Harness average by 0.2 percentage points.
\item Real-world retrieval uses \alert{2K-truncated inputs}; FRAC ranks third.
\end{itemize}
\papersource{Paper, selected results from Tables 3--4, PDF p. 8; evaluation protocol, p. 24.}
\end{frame}

\begin{frame}[t]{DNA modeling benefits from longer training contexts}
\begin{columns}[T]
\begin{column}{0.52\textwidth}
\centering
\includegraphics[width=\linewidth,height=5.1cm,keepaspectratio]{figures/fig-04-dna.pdf}
\end{column}
\begin{column}{0.45\textwidth}
\begin{itemize}
\item HG38; character-level autoregressive modeling.
\item About 7M parameters, 8 layers, hidden size 256.
\item Lower test perplexity at long contexts, with a plateau near 32K--64K.
\item Each model is trained and tested at the same chosen length.
\end{itemize}
\end{column}
\end{columns}
\papersource{Original Fig. 4, PDF p. 9; Appendix E.4, p. 25. Mamba3 here is the SISO variant.}
\end{frame}

\begin{frame}[t]{FRAC leads 16K prefill; Mamba2 trains faster}
\centering
\begin{tabular}{lrrrr}
\toprule
Model & Prefill 4K & Prefill 16K & Decode & Train 4K \\
\midrule
Transformer & \textbf{79,173} & 56,297 & \textbf{62} & 218,184 \\
Mamba2 & 41,060 & 67,571 & 34 & \textbf{278,620} \\
GDN & 35,481 & 58,384 & 20 & 215,357 \\
Mamba3-SISO & 44,998 & 74,314 & 27 & 238,924 \\
Mamba3-MIMO & 30,710 & 46,707 & 25 & 200,994 \\
\FRAC & 49,946 & \textbf{77,490} & 33 & 241,598 \\
\bottomrule
\end{tabular}
\medskip
\begin{itemize}
\item All values: tokens/s. Prefill is single-card, batch size 1.
\item At 16K: FRAC is 14.7\% faster than Mamba2 and 37.6\% faster than Transformer.
\item Decode averages 64 generated tokens; Transformer uses a 16K prefix.
\item Training uses the pretraining setup; FRAC is 13.3\% below Mamba2.
\end{itemize}
\papersource{Paper, Table 8, Appendix F, PDF p. 25. Accelerator model is unspecified; training uses 16 cards.}
\end{frame}

\begin{frame}[t]{The structured prior matters beyond having many states}
\setlength{\tabcolsep}{8pt}
\begin{center}
\begin{tabular}{lrrr}
\toprule
Variant & ProofPile & PG19 & GovReport \\
\midrule
\FRAC & \textbf{47.2} & \textbf{28.4} & \textbf{11.3} \\
Fixed $\alpha=1$ & 61.3 & 36.2 & 19.7 \\
Without write prior & 68.5 & 41.6 & 18.2 \\
Pure multiscale bank & 76.3 & 47.1 & 17.2 \\
Without softmax read/write & 80.4 & 36.9 & 15.5 \\
Freely learned timescales & 54.1 & 34.3 & 13.5 \\
Without direct feedthrough $D$ & 70.9 & 46.4 & 23.6 \\
\bottomrule
\end{tabular}
\end{center}
\begin{itemize}
\item 390M parameters, 30B tokens, 4K training, 16K evaluation; PPL$\downarrow$.
\item Pure multiscale bank changes $\alpha$, both priors, and timescale initialization together.
\end{itemize}
\papersource{Paper, Table 9 and ablation definitions, Appendix G, PDF pp. 26--27.}
\end{frame}

\begin{frame}[t]{Sixteen modes balance fit quality and model cost}
\begin{columns}[T]
\begin{column}{0.48\textwidth}
\begin{block}{Independent homogeneous-kernel fit}
\centering
\begin{tabular}{cr}
\toprule
$M$ & Mean maximum error \\
\midrule
8 & $1.34\times10^{-3}$ \\
16 & $6.82\times10^{-4}$ \\
32 & $6.45\times10^{-4}$ \\
\bottomrule
\end{tabular}
\end{block}
\end{column}
\begin{column}{0.48\textwidth}
\begin{itemize}
\item Fit positive simplex weights separately for each $\alpha\in[0.10,0.99]$.
\item Normalized time $s\in[4,6553.6]$; geometric $\tau\in[1,2^{17}]$.
\item This measures bank expressivity, not learned token-wise kernel error.
\end{itemize}
\end{column}
\end{columns}
\medskip
In the LM ablation, $M=8$ worsens all three PPLs. $M=32$ improves only GovReport,
with about 5\% more parameters and 7\% slower computation.
\papersource{Paper, Tables 9--10 and their protocols, Appendix G, PDF pp. 26--27.}
\end{frame}

\begin{frame}[t]{The useful interpretation has three boundaries}
\begin{enumerate}
\item \textbf{Finite temporal coverage.} The theorem concerns a finite horizon. Under frozen controls, a finite mixture ultimately has an exponential tail.
\medskip
\item \textbf{Retention versus retrieval.} Better long-range scores coexist with poor LCC results and low accuracy at the hardest NIAH lengths.
\medskip
\item \textbf{Theory versus architecture.} Learned routing, token-varying controls, and the modified write factor extend the fixed-system construction.
\end{enumerate}
\medskip
Future direction proposed by the authors: combine fractional transitions with delta-rule updates.
\papersource{Paper, Sec. 6, p. 10; Appendix D, p. 21; Appendix H.1, p. 27. Finite-mixture tail statement: reading analysis.}
\end{frame}

\begin{frame}[t]{Memory geometry is a useful design axis for SSMs}
\begin{block}{Mechanism}
Fractional kernels $\rightarrow$ geometric exponential bank $\rightarrow$ selective read/write $\rightarrow$ efficient affine scan.
\end{block}
\begin{block}{Evidence}
Stronger long-range extrapolation and LongBench scores; short-context quality remains competitive.
\end{block}
\begin{block}{Practical lesson}
Choose temporal coverage and routing together, and reproduce the actual Algorithm 1 write rule.
\end{block}
\papersource{Kobyzev et al., Fractional State Space Transition for Long Sequence Modeling, 2026.\\
\href{https://arxiv.org/abs/2609.36314v1}{arXiv:2609.36314v1}.}
\end{frame}

\appendix

\begin{frame}[t]{Backup: why the two kernel norms suffice}
\[
H_{\alpha,\lambda}(\tau)=\frac{R_{\alpha,\lambda}(\tau)}{\lambda\tau},
\qquad
c_m=h\tau_mR_{\alpha,\lambda}(\tau_m),\quad
d_m=h\tau_mH_{\alpha,\lambda}(\tau_m).
\]
Let $E_M$ and $g_M$ denote the finite sums. Then
\begin{align*}
|h(t)-\tilde h_M(t)|
&\le |h(0)|\,|E_\alpha(-\lambda t^\alpha)-E_M(t)|\\
&\quad+\|u\|_\infty\int_0^T|g_{\alpha,\lambda}(r)-g_M(r)|\,dr\\
&\le\varepsilon\bigl(|h(0)|+\|u\|_\infty\bigr).
\end{align*}
\begin{itemize}
\item Fixed $\alpha,\lambda$; bounded input; finite $T$; $s_m(0)=h(0)$.
\item The impulse kernel is singular at zero but integrable, so $L^1$ is sufficient.
\item The theorem does not supply an explicit guarantee for the chosen $M=16$.
\end{itemize}
\papersource{Paper, Theorem 2 and Proposition 3, pp. 4--5; Appendices B--C, pp. 17--21.}
\end{frame}

\begin{frame}[t]{Backup: dimensions and one-token computation}
\[
u_t\in\mathbb R^{H\times d_h},\quad
\Delta_t,\alpha_t,\lambda_t\in\mathbb R^H,\quad
S_t\in\mathbb R^{H\times M\times d_h}.
\]
\begin{enumerate}
\item Form $a_t=\Delta_t\lambda_t^{1/\alpha_t}$, then $\tilde\rho_{t,m}=e^{-a_t/\tau_m}$.
\medskip
\item Form $c_t=\mathrm{softmax}(-\alpha_t\log\tau+\gread(u_t))$ and independent $b_t$.
\medskip
\item Update $S_t=\tilde\rho_t\odot S_{t-1}+(1-\tilde\rho_t)b_t\odot u_t$.
\medskip
\item Read $h_t=\sum_m c_{t,:,m}\odot S_{t,:,m,:}$; mixer adds $Du_t$.
\end{enumerate}
\medskip
Mode weights are broadcast over feature coordinates. The content projections use
$g(u)=\sigma(p)Wu$, with separate read and write parameters.
\papersource{Paper, Algorithm 1, Appendix D, PDF p. 21; direct feedthrough, Sec. 3.3, p. 6.}
\end{frame}

\begin{frame}[t]{Backup: the effective memory is content dependent}
For zero initial state, unrolling the actual recurrence gives
\[
h_t=\sum_{k\le t}K_{t,k}u_k,
\quad
K_{t,k}=\sum_m c_{t,m}(1-\tilde\rho_{k,m})b_{k,m}
\prod_{j=k+1}^{t}\tilde\rho_{j,m}.
\]
\begin{itemize}
\item Current read routing, past write routing, and intervening clocks all matter.
\item The selective model generally has no fixed kernel depending only on $t-k$.
\end{itemize}
\begin{block}{Stable within-chunk computation}
\[
\Gamma_{t,k}=\sum_m c_{t,m}\kappa_{k,m}
e^{\ell_{t,m}-\ell_{k,m}},
\qquad \ell_{t,m}=\sum_{j=t_{\rm start}}^t\log\tilde\rho_{j,m}.
\]
\end{block}
\papersource{Top equation: teaching derivation from Algorithm 1. Stable implementation: Appendix F, PDF p. 26.}
\end{frame}

\end{document}