% Independent reading of arXiv:2605.31163v3; compile with pdfLaTeX.
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\title[Memory by Design]{Memory by Design:\\Probabilistic Sequence Layers}
\subtitle{Dense uncertainty as a recurrent memory state}
\author[Independent paper reading]{Paper: Matthew Dowling, Hyungju Jeon,\\Cristina Savin, Il Memming Park\\[5pt]\small Independent paper reading}
\date{arXiv:2605.31163v3 \quad Revised 18 August 2026\\\small Reading: 14 September 2026 \;|\; 25--30 minutes + backup}
\begin{document}
\begin{frame}\titlepage\end{frame}

\begin{frame}{A memory update must decide where to write}
\begin{block}{Question}
When two keys overlap, can a new association be written without repeatedly damaging an established one?
\end{block}
\begin{itemize}
\item A mean state stores \alert{what is currently remembered}.
\item A covariance state records \alert{which directions are resolved}.
\item Bayesian Layer (BL) uses both to shape the next write.
\end{itemize}
\medskip
Evidence: improved recall in reported settings, with extra state and language-model perplexity costs.
\papersource{Paper, Secs. 1--2, pp. 1--5; Table 2, p. 9; Tables 6--7, p. 29.}
\end{frame}

\begin{frame}{The design model separates writing from reading}
\centering\fig{1.0}{fig-01}
\begin{itemize}
\item Input selects an auxiliary probabilistic memory model.
\item Filtering updates a belief; a separate query produces the output.
\item The input sequence itself has \alert{no likelihood in this design model}.
\end{itemize}
\papersource{Original Fig. 1, p. 2; Sec. 2, pp. 2--3.}
\end{frame}

\begin{frame}{A Gaussian belief makes memory updates tractable}
\[
\operatorname{vec}(\mat S_t)\sim
\mathcal N\bigl(\operatorname{vec}(\mat M_t),\mat I_m\otimes\mat P_t\bigr)
\]
\begin{columns}[T]
\column{.49\textwidth}
\begin{block}{Persistent state}
$\mat M_t\in\mathbb R^{D\times m}$: stored associations.\\[5pt]
$\mat P_t\in\mathbb R^{D\times D}$: shared key-space covariance.
\end{block}
\column{.49\textwidth}
\begin{block}{Input-dependent quantities}
Key/query: $\mat k_t,\mat q_t\in\mathbb R^D$.\\[5pt]
Value: $\mat v_t\in\mathbb R^m$.\\[5pt]
Dynamics $\mat A_t$; variances $\ell_t^2,r_t^2$.
\end{block}
\end{columns}
\[
\mat S_t\mid\mat S_{t-1}\sim\mathcal N(\mat A_t\mat S_{t-1},\ell_t^2\mat I),
\qquad
\mat v_t\mid\mat S_t\sim\mathcal N(\mat S_t^\top\mat k_t,r_t^2\mat I_m)
\]
\papersource{Sec. 2.1, Definition 1, pp. 3--4. Gaussian matrix notation follows the paper.}
\end{frame}

\begin{frame}{The innovation follows an uncertainty-shaped key}
\begin{align*}
\bar{\mat M}_t&=\mat A_t\mat M_{t-1}, &
\bar{\mat P}_t&=\mat A_t\mat P_{t-1}\mat A_t^\top+\ell_t^2\mat I,\\
\mat u_t&=\bar{\mat P}_t\mat k_t, &
\beta_t&=(r_t^2+\mat k_t^\top\mat u_t)^{-1}.
\end{align*}
\[
\mat M_t=\bar{\mat M}_t+\beta_t\mat u_t(\mat v_t-\bar{\mat M}_t^\top\mat k_t)^\top,
\qquad
\mat P_t=\bar{\mat P}_t-\beta_t\mat u_t\mat u_t^\top.
\]
\begin{itemize}
\item Covariance determines \alert{direction, gain, and future uncertainty}.
\item Readout: $\mat y_t=\mat M_t^\top\mat q_t$; query does not drive the write.
\item $\beta_t$ is a precision; the scalar write fraction is $g_t=\beta_t\mat k_t^\top\mat u_t$.
\end{itemize}
\papersource{Proposition 1, Eqs. (9)--(13), p. 4; error-correction form, Appendix B, p. 15.}
\end{frame}

\begin{frame}{Resolved directions receive less interference}
\centering\fig{1.0}{fig-02}
\[
\mat k_A=\rho\mat k_B+\sqrt{1-\rho^2}\mat k_\perp
\]
\[
\mat u_t\approx\rho\bar\sigma_B^2\mat k_B+
\sqrt{1-\rho^2}\bar\sigma_\perp^2\mat k_\perp,
\qquad\bar\sigma_B^2\ll\bar\sigma_\perp^2
\]
\raggedright
After repeated B writes, a colliding A write is directed toward the unresolved residual. This approximation assumes near-diagonal covariance in the B/residual basis.
\papersource{Original Fig. 2; Eq. (15), Sec. 2.3, p. 5.}
\end{frame}

\begin{frame}{Resetting covariance recovers the Delta-rule core}
Reset $\bar{\mat P}_t\leftarrow\lambda_t\mat I$ at every step:
\[
\mat M_t=\mat A_t\mat M_{t-1}
+\eta_t\mat k_t\bigl(\mat v_t-(\mat A_t\mat M_{t-1})^\top\mat k_t\bigr)^\top,
\quad
\eta_t=\frac{\lambda_t}{r_t^2+\lambda_t\|\mat k_t\|^2}.
\]
\begin{center}
\begin{tabular}{ll}
\toprule Core recurrence & Choice of dynamics\\\midrule
DeltaNet & $\mat A_t=\mat I$\\
Gated DeltaNet & $\mat A_t=\alpha_t\mat I$\\
KDA / Kimi Linear & $\mat A_t=\operatorname{diag}(\boldsymbol\alpha_t)$\\
\bottomrule
\end{tabular}
\end{center}
The residual correction remains; covariance history is discarded. Full architectures also contain components outside this correspondence.
\papersource{Table 1; Proposition 2, Eq. (17), p. 6.}
\end{frame}

\begin{frame}{Additive cores use a different observation model}
\[
\boldsymbol\phi_t\sim\mathcal N(0,\lambda_t\mat I),\qquad
\mat v_t\mid\boldsymbol\phi_t\sim\mathcal N(\boldsymbol\phi_t,r_t^2\mat I),
\]
\[
\mat S_t=\mat A_t\mat S_{t-1}+\mat k_t\boldsymbol\phi_t^\top
\quad\Longrightarrow\quad
\mat M_t=\mat A_t\mat M_{t-1}+\omega_t\mat k_t\mat v_t^\top.
\]
\begin{itemize}
\item $\omega_t=\lambda_t/(\lambda_t+r_t^2)$; observation measures the \emph{new write coefficient}.
\item Dynamics recover Linear Attention, RetNet, diagonal GLA, and the SSD/Mamba-2 core.
\item The \alert{mean recurrence closes without covariance feedback}. This does not imply zero posterior uncertainty.
\end{itemize}
\papersource{Sec. 3.2, pp. 6--7; Appendix A, pp. 14--15. Last qualification: reading-side derivation, backup.}
\end{frame}

\begin{frame}{Repeated writes predict a positive gain floor}
Assume unit keys, $\mat A_t=\mat I$, constant $r,\ell>0$, and sustained writes at A.
\[
x_{t+1}=\frac{r^2x_t}{r^2+x_t}+\ell^2,\qquad x_t=\bar\sigma_{A,t}^2
\]
\[
x_\infty=\frac{\ell^2+\sqrt{\ell^4+4r^2\ell^2}}2,\qquad
g_\infty=\frac{x_\infty}{r^2+x_\infty}
\]
\begin{itemize}
\item With $r^2=\ell^2=0.05$, $g_\infty\approx0.618$; the innovation residual still vanishes.
\item Cross-direction variance increment tends to $(1-\rho^2)\ell^2$.
\item These are conditional asymptotic results, not universal retention guarantees.
\end{itemize}
\papersource{Appendix E.1--E.2, pp. 17--19; Eq. (42), p. 19.}
\end{frame}

\begin{frame}{Dense covariance adds state and arithmetic}
\begin{columns}[T]
\column{.49\textwidth}
\begin{block}{One head, $G$ noise groups}
State: $O(Dm+GD^2)$.\\[5pt]
Per token: $O(Dm+GD^2)$ with structured dynamics.\\[5pt]
Linear in sequence length; quadratic in key dimension.
\end{block}
\column{.49\textwidth}
\begin{block}{Implementation}
Covariance pass produces gains.\\[5pt]
Affine mean update uses chunkwise machinery.\\[5pt]
Chunk boundaries remain sequential; filter arithmetic uses fp32.
\end{block}
\end{columns}
\medskip
Retrofit: four heads with dense $256\times256$ covariance blocks in each of four replaced layers. No throughput or peak-memory comparison table is reported.
\papersource{Sec. 2.2, p. 4; Appendix D, pp. 16--17; Discussion, pp. 10--11.}
\end{frame}

\begin{frame}{The evaluations test different levels of the claim}
\begin{center}
\begin{tabular}{lll}
\toprule Setting & Question & Repetitions\\\midrule
Deterministic collision & Does the algebra predict recall? & Exact trajectory\\[3pt]
Learned collision & Does the mechanism survive training? & 3 seeds\\[3pt]
Random-key recall & Does it survive unstructured crowding? & 3 seeds\\[3pt]
MQAR variants & Can tokens form associations? & 1 retained run\\[3pt]
WikiText from scratch & Does recall help token prediction? & 3 seeds\\[3pt]
340M retrofit & Can pretrained retrieval improve? & 1 seed\\
\bottomrule
\end{tabular}
\end{center}
\medskip
Matching backbone, state budget, data tokens, and actual hardware cost are different controls.
\papersource{Table 3, p. 17; experimental details, Appendices E--G.}
\end{frame}

\begin{frame}{The stated collision recurrence preserves B}
$D=16$, $m=6$, $\rho=0.92$; 12 seed steps, 40 B writes, 60 A writes.\\
$\mat P_0=3\mat I$, $r^2=\ell^2=0.05$, $\mat A=\mat I$.
\begin{center}
\begin{tabular}{lrrr}
\toprule Pairwise probability $p^{AB}$ & BL & Reset/Delta & Linear Attn.\\\midrule
After B writes ($t=52$) & 0.73106 & 0.73106 & $\approx1$\\
After A writes ($t=112$) & 0.72309 & 0.35138 & 0.00054\\
\bottomrule
\end{tabular}
\end{center}
\begin{itemize}
\item Pairwise probability is not full-vocabulary accuracy.
\item Independent calculation agrees with the appendix values.
\item \alert{v3 discrepancy:} Fig. 3(a)'s BL endpoint appears near 0.60; it does not match the stated 0.72309 plateau.
\end{itemize}
\papersource{Appendix E.3--E.4, pp. 20--22. Table independently recomputed from the reported setup; Fig. 3(a), p. 7.}
\end{frame}

\begin{frame}{Learned recall survives stronger interference in BL}
\centering
\includegraphics[width=.91\linewidth,height=.08\textheight,keepaspectratio]{figures/fig-03-legend.pdf}\\[4pt]
\includegraphics[width=.76\linewidth,height=.48\textheight,keepaspectratio]{figures/fig-03-cd.pdf}\par
\raggedright
\begin{itemize}
\item Train: $n_f\le8$, $\rho\in[0.60,0.80]$; addresses bypass learned projections.
\item Left: accuracy to $n_f=256$ at $\rho=0.80$; right: margin at $n_f=64$.
\item v3 metric/reset descriptions conflict; read panel labels literally.
\end{itemize}
\papersource{Original Fig. 3(c,d), p. 7, cropped with shared legend; Sec. 4.1, p. 8; Appendix E.5, pp. 23--26.}
\end{frame}

\begin{frame}{Random-key crowding exposes a finite memory budget}
\begin{columns}[c]
\column{.64\textwidth}
\centering\fig{1.0}{fig-07}
\column{.34\textwidth}
\begin{itemize}
\item Frozen projection: 64 raw dimensions to $D=16$.
\item Train: $n/D\le4$; test: $6,8,12,16$.
\item BL reduces loss; all methods degrade at high density.
\end{itemize}
\end{columns}
\medskip
The rank argument concerns exact linear value reconstruction. It is not a general accuracy bound for the entire multilayer classifier.
\papersource{Original Fig. 7, p. 26; Appendix E.5.4, p. 25. Qualification: reading-side analysis.}
\end{frame}

\begin{frame}{MQAR gains depend on the convolution configuration}
\centering\fig{.97}{fig-04}
\raggedright
\begin{itemize}
\item Base: BL approaches the best curves at larger widths.
\item Update at 16 writes: BL $\approx88\%$; GDN-conv/Mamba2 $\approx60\%$.
\item Conv-free GDN remains higher ($\approx98\%$) and also leads BL on Block-MQAR at 16 pairs.
\end{itemize}
\papersource{Original Fig. 4, p. 8; Sec. 4.2, pp. 8--9; Appendix F, pp. 26--27. Approximate curve readings.}
\end{frame}

\begin{frame}{Recall improves while overall perplexity rises}
\begin{center}
\begin{tabular}{lrrr}
\toprule Model & Overall PPL $\downarrow$ & AR-hit PPL $\downarrow$ & Other PPL $\downarrow$\\\midrule
BL & $20.01\pm0.15$ & $\mathbf{18.15\pm1.08}$ & $20.03\pm0.15$\\
GDN & $\mathbf{18.75\pm0.07}$ & $23.72\pm1.76$ & $\mathbf{18.69\pm0.08}$\\
Mamba-2 & $21.36\pm0.27$ & $424.8\pm120.4$ & $20.58\pm0.19$\\
\bottomrule
\end{tabular}
\end{center}
\begin{itemize}
\item WikiText-103; matched recurrent-state budgets; 520M training tokens; 3 seeds.
\item AR-hit: pair count $\le16$ in the 130M-token corpus; most recent repeat within 1,024 tokens in the same window.
\item BL vs. GDN: \alert{23.48\% lower AR-hit PPL}, but \alert{6.72\% higher overall PPL}.
\end{itemize}
\papersource{Table 2, p. 9; Appendix G.1, pp. 29--30. Mean $\pm$ SD; changes recomputed from table means.}
\end{frame}

\begin{frame}{The farthest AR-hit bucket has substantial seed variation}
\begin{center}
\begin{tabular}{lrr}
\toprule Distance to previous pair & BL PPL $\downarrow$ & GDN PPL $\downarrow$\\\midrule
1--64 tokens & $7.21\pm0.52$ & $10.09\pm1.07$\\[4pt]
65--256 tokens & $13.91\pm1.00$ & $19.57\pm2.07$\\[4pt]
257--1,024 tokens & $92.1\pm8.9$ & $99.9\pm0.7$\\
\bottomrule
\end{tabular}
\end{center}
\begin{itemize}
\item BL has the lowest mean in every distance bucket.
\item At 257--1,024 tokens, the gap is not resolved relative to seed variation.
\item These are BPE-token perplexities on AR-hit positions, not word-level WikiText scores.
\end{itemize}
\papersource{Table 2, p. 9; the authors' qualification in Appendix G.1, p. 30.}
\end{frame}

\begin{frame}{Four Bayesian layers are distilled into a 340M GDN}
\begin{block}{Retrofit}
24-layer GDN; replace indices $\{5,11,17,23\}$.\\
Four heads, head dimension 256, hidden width 1,024.
\end{block}
\begin{enumerate}
\item Phase A: parallel layer matching, $\approx100$M tokens per layer.
\item Phase B: joint teacher-logit KL and boundary MSE, $\approx300$M tokens.
\end{enumerate}
\medskip
Train at length 2,048; one seed; reported data exposure $\approx400$M tokens. GDN-FT continues the same four-layer mask for 400M tokens with next-token CE.
\papersource{Sec. 4.3, pp. 9--10; Appendix G, pp. 27--28. The objectives differ between BL and GDN-FT.}
\end{frame}

\begin{frame}{NIAH gains persist against continued training}
\begin{center}
\begin{tabular}{rrrrr}
\toprule Context & GDN-zero & GDN-FT & BL & BL$-$FT (points)\\\midrule
2,048 & 49.52 & 48.46 & \textbf{50.81} & +2.35\\[3pt]
4,096 & 31.08 & 31.58 & \textbf{34.02} & +2.44\\[3pt]
8,192 & 16.81 & 16.06 & \textbf{20.00} & +3.94\\[3pt]
16,384 & 7.85 & 8.36 & \textbf{8.81} & +0.45\\
\bottomrule
\end{tabular}
\end{center}
\begin{itemize}
\item Mean over six NIAH tasks; CWE excluded. Not the full RULER suite.
\item At 16k, absolute retrieval remains poor in all three models.
\item At 2k, \texttt{single\_2} regresses from 96.50 to 89.00 vs. GDN-zero.
\end{itemize}
\papersource{Table 6 (Appendix Table A1), p. 29. Higher is better; differences recomputed.}
\end{frame}

\begin{frame}{Retrieval gains carry a persistent perplexity penalty}
\begin{center}
\begin{tabular}{lrrr}
\toprule Corpus at 8k & GDN-zero & BL & Relative PPL rise\\\midrule
WikiText-103 & 14.57 & 14.97 & 2.75\%\\[3pt]
SlimPajama-heldout & 12.15 & 12.43 & 2.30\%\\[3pt]
PG-19 validation & 19.29 & 19.81 & 2.70\%\\
\bottomrule
\end{tabular}
\end{center}
\begin{itemize}
\item The penalty persists from 2k to 16k; no overall PPL crossover.
\item Paper summary: 2.5--2.7\%; SlimPajama table values imply 2.30--2.38\% across lengths.
\item GDN-FT stays close to GDN-zero. Tokenizers differ from the from-scratch experiment.
\end{itemize}
\papersource{Table 7 (Appendix Table A2), p. 29. Lower PPL is better; percentages use rounded table entries.}
\end{frame}

\begin{frame}{Gains vary with task and needle position}
\centering\fig{.99}{fig-05}
\raggedright
\begin{itemize}
\item Columns are needle-depth deciles, not network layers; green favors BL over GDN-zero.
\item Middle positions at 4k and recent positions at 8k improve; early 2k can regress.
\item 208 of 240 task/length/decile cells have fewer than 30 BL samples.
\end{itemize}
\papersource{Original Fig. 5, p. 10; sample counts, Appendix G, p. 29. Matched-control heatmap: Fig. 8, p. 28.}
\end{frame}

\begin{frame}{The contribution is an uncertainty-adaptive write rule}
\begin{enumerate}
\item \textbf{Mechanism:} covariance history shapes write direction and gain, with analytically testable collision dynamics.
\item \textbf{Evidence:} stronger recall in reported tasks; overall language-model PPL can worsen.
\item \textbf{Scope:} dense-state cost, small-scale evidence, single-seed retrofit, and v3 reporting inconsistencies remain relevant.
\end{enumerate}
\begin{block}{Discussion question}
How much recall gain survives when covariance structure, actual throughput, and training objective are controlled together?
\end{block}
\papersource{Paper, Sec. 5, pp. 10--11; Tables 2, 6--7. Discussion question: reading-side analysis.}
\end{frame}

\begin{frame}{Source and reading scope}
\textbf{Matthew Dowling, Hyungju Jeon, Cristina Savin, Il Memming Park.}\\[5pt]
\emph{Memory by Design: Probabilistic Sequence Layers}.\\
arXiv:2605.31163v3, revised 18 August 2026.\\[5pt]
\href{https://arxiv.org/abs/2605.31163v3}{https://arxiv.org/abs/2605.31163v3}
\medskip
\begin{itemize}
\item Read the versioned author TeX, all main sections, and Appendices A--G.1; cross-checked the 30-page PDF.
\item Figures are original vector assets; Fig. 3(c,d) is a documented crop with its shared legend.
\item Deterministic recurrence and table differences checked independently. No training or benchmark evaluation rerun.
\end{itemize}
\papersource{Prepared 14 September 2026. Detailed caveats accompany the Chinese reading notes.}
\end{frame}

\appendix
\begin{frame}{Backup: Kronecker closure keeps covariance compact}
\[
\widetilde{\mat A}_t=\mat I_m\otimes\mat A_t,\quad
\mat C_t=\mat I_m\otimes\mat k_t^\top,\quad \mat R_t=r_t^2\mat I_m
\]
\[
\widetilde{\mat A}_t(\mat I_m\otimes\mat P_{t-1})\widetilde{\mat A}_t^\top
+\ell_t^2\mat I_{Dm}=\mat I_m\otimes\bar{\mat P}_t
\]
\[
\text{Posterior covariance}=\mat I_m\otimes
\left(\bar{\mat P}_t-
\frac{\bar{\mat P}_t\mat k_t\mat k_t^\top\bar{\mat P}_t}
{r_t^2+\mat k_t^\top\bar{\mat P}_t\mat k_t}\right).
\]
\begin{itemize}
\item Shared column dynamics/noise preserve this structure exactly.
\item Grouped observation noise requires one covariance per group.
\end{itemize}
\papersource{Appendix C, pp. 15--16; grouped noise, Appendix D, p. 17.}
\end{frame}

\begin{frame}{Backup: a two-dimensional example of write steering}
\[
\mat k_B=\begin{pmatrix}1\\0\end{pmatrix},\qquad
\mat k_A=\begin{pmatrix}0.8\\0.6\end{pmatrix},\qquad
\bar{\mat P}=\begin{pmatrix}0.05&0\\0&2\end{pmatrix}
\]
\[
\mat u=\bar{\mat P}\mat k_A=\begin{pmatrix}0.04\\1.2\end{pmatrix}.
\]
\begin{itemize}
\item B is well resolved; the orthogonal component is uncertain.
\item A triggers the write, but its direction largely avoids B.
\item This example uses covariance principal axes. In arbitrary fixed coordinates, off-diagonal entries represent that geometry.
\end{itemize}
\papersource{Teaching example constructed for this reading, not a paper experiment; illustrates Eq. (15), p. 5.}
\end{frame}

\begin{frame}{Backup: distillation matches layers, then model outputs}
\[
\mathcal L_A^{(\ell)}=
\frac{\|y_\ell^{BL}-y_\ell^{GDN}\|_2^2}
{\operatorname{Var}(y_\ell^{GDN})+10^{-6}}
+0.05\bigl(1-\cos(y_\ell^{BL},y_\ell^{GDN})\bigr)
\]
\[
\mathcal L_B=\lambda_{KL}\operatorname{KL}(p_{GDN}\Vert p_{BL})+
\frac{\lambda_b}{|\mathcal I|}\sum_{\ell\in\mathcal I}
\frac{\|y_\ell^{BL}-y_\ell^{GDN}\|_2^2}
{\operatorname{Var}(y_\ell^{GDN})+10^{-6}}
\]
\begin{itemize}
\item $y_\ell$: matched layer-boundary outputs; frozen teacher GDN.
\item Phase B: $(\lambda_{KL},\lambda_b)=(0.8,0.2)$ then $(0.9,0.1)$.
\item GDN-FT uses next-token CE. The equal-token control does not isolate architecture from objective choice.
\end{itemize}
\papersource{Appendix G, pp. 27--28. Loss normalization reproduced as specified in the paper.}
\end{frame}

\begin{frame}{Backup: reporting differences affect reproduction}
\begin{itemize}
\item \textbf{Fig. 3(a):} visible BL endpoint differs from the appendix's independently reproducible 0.72309.
\item \textbf{Fig. 3(c):} axis is accuracy; main text describes margin. The appendix also uses a different softmax definition.
\item \textbf{Learned reset:} main text describes a learned scalar; Appendix E.5.5 fixes reset process variance to 0.05.
\item \textbf{MQAR:} distinguish GDN from GDN-conv. Imported Base curves under the modified generator need clarification.
\end{itemize}
\medskip
These differences limit quantitative claims from the affected experiments; the recurrence derivation can be checked separately.
\papersource{Fig. 3, p. 7; Secs. 4.1--4.2, pp. 8--9; Appendices E.3--E.5 and F, pp. 20--27.}
\end{frame}

\begin{frame}{Backup: two algebraic qualifications}
\textbf{Latent-input posterior uncertainty need not be zero.}
\[
c_t=\frac{\lambda_t r_t^2}{\lambda_t+r_t^2}>0,\quad
\mat C_t=\mat A_t\mat C_{t-1}\mat A_t^\top+c_t\mat k_t\mat k_t^\top.
\]
The mean update does not need $\mat C_t$, so the layer may omit it.
\medskip

\textbf{Use the exact gain formula for asymptotic expansion.}
\[
\ell^2\gg r^2:\quad g_\infty=1-\frac{r^2}{\ell^2}
+O\!\left(\frac{r^4}{\ell^4}\right).
\]
Appendix E.2 prints a factor $1/2$ in the leading correction. Its exact formula implies the expansion above.
\papersource{Reading-side derivations from Sec. 3.2 / Appendix A and Eq. (42), p. 19.}
\end{frame}
\end{document}