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\newcommand{\SC}{S_{C\mid0}}
\title[HYPIC]{HYPIC}
\subtitle{Accelerating Hybrid-Attention LLM Serving\\with Position-Independent Caching}
\author{Paper authors: Yifei Liu, Juntong Wu, Yang Liu, Junhao Hu,\\Minghao Li, Xiaoxu Chen, and Weihang Chen}
\date{arXiv:2607.01299v2 \;|\; revised July 12, 2026\\Research discussion \;|\; October 16, 2026}

\begin{document}

\begin{frame}
  \titlepage
  \vspace{-5mm}
  \begin{center}
  \term{Cache how a segment transforms memory, then repair its boundary.}
  \end{center}
  \papersource{Source: \href{https://arxiv.org/abs/2607.01299v2}{arxiv.org/abs/2607.01299v2}. Reading presentation; not an author presentation.}
\end{frame}

\begin{frame}{Repeated documents do not imply repeated prefixes}
  \begin{block}{A typical RAG reuse opportunity}
    Request 1: System $\to$ Document A $\to$ \alert{Document B} $\to$ Query 1\\[2mm]
    Request 2: System $\to$ Document C $\to$ \alert{Document B} $\to$ Query 2
  \end{block}
  \begin{itemize}
    \item Prefix caching stops at the first mismatch.
    \item Position-independent caching (PIC) reuses B behind a new prefix.
    \item B's position and contextual hidden states both change.
  \end{itemize}
  \medskip
  Hybrid models complicate reuse: most layers retain a \term{recurrent state}, while a minority retain per-token KV.
  \papersource{Paper, Secs. 1--2, PDF pp. 1--3. Request example is a teaching illustration.}
\end{frame}

\begin{frame}{Three obstacles require three different mechanisms}
  \renewcommand{\arraystretch}{1.35}
  \begin{tabular}{@{}p{0.46\linewidth}p{0.48\linewidth}@{}}
    \toprule
    Obstacle & HYPIC mechanism\\
    \midrule
    End-state addition ignores decay and erasure & Cache each segment's transition $T_C$ and zero-start state $\SC$\\
    Linear layers block arbitrary-token correction & Recompute a short seam at segment beginnings through the entire stack\\
    Cold requests still need prefill & Scatter independent segments across instances, then combine\\
    \bottomrule
  \end{tabular}\par
  \medskip
  \term{The reusable unit is a segment with an explicit composition rule.}
  \papersource{Paper, Sec. 3, PDF pp. 3--5; Sec. 4, PDF pp. 6--9.}
\end{frame}

\begin{frame}{Linear attention separates memory transport from writing}
  \[
    \boxed{S_i=T_iS_{i-1}+u_i},\qquad o_i=q_i^\top S_i
  \]
  \begin{columns}[T,onlytextwidth]
    \begin{column}{0.48\textwidth}
      \begin{itemize}
        \item $S_i\in\mathbb R^{d_k\times d_v}$: memory.
        \item $T_i\in\mathbb R^{d_k\times d_k}$: transport, decay, or erasure.
        \item $u_i\in\mathbb R^{d_k\times d_v}$: new write.
        \item $q_i\in\mathbb R^{d_k}$: read query.
      \end{itemize}
    \end{column}
    \begin{column}{0.48\textwidth}
      \begin{block}{Representative transitions}
        Scalar: $T_i=\gamma I$\\[2mm]
        Diagonal: $T_i=\operatorname{diag}(g_i)$\\[2mm]
        GDN: $T_i=g_i(I-\beta_i k_i k_i^\top)$
      \end{block}
      $T_i,u_i$ depend on the layer's input, not on its recurrent state.
    \end{column}
  \end{columns}
  \papersource{Paper, Eq. (2) and Table 1, PDF p. 3. $g_i$ is scalar for GDN and a vector for diagonal gating.}
\end{frame}

\begin{frame}{Adding segment states retains memory that should decay}
  \[
    S_{C_1C_2\mid0}=\underbrace{T_{C_2}S_{C_1\mid0}}_{\text{transport old memory}}+S_{C_2\mid0}
  \]
  \[
    \Delta=S_{\mathrm{true}}-S_{\mathrm{addition}}
      =(T_{C_2}-I)S_{C_1\mid0}
  \]
  \begin{block}{Constant decay makes the failure explicit}
    $T_i=\gamma I\quad\Rightarrow\quad
    \|\Delta\|/\|S_{C_1\mid0}\|=1-\gamma^{|C_2|}$
  \end{block}
  \centering
  \begin{tabular}{lrrr}
    \toprule
    Suffix length & 256 & 512 & 1024\\
    \midrule
    Error, $\gamma=1-2^{-10}$ & 0.221 & 0.394 & 0.632\\
    \bottomrule
  \end{tabular}
  \papersource{Paper, Eqs. (4)--(5), Table 2, PDF p. 4. Error is structural, not a floating-point artifact.}
\end{frame}

\begin{frame}{A cached segment is an affine map on incoming memory}
  \[
     F_C(S)=T_CS+B_C,\qquad B_C\equiv S_{C\mid0}
  \]
  \[
    (T_B,B_B)\circ(T_A,B_A)
    =\bigl(T_BT_A,\;T_BB_A+B_B\bigr)
  \]
  \begin{itemize}
    \item Compute each tuple once; apply it behind a new prefix.
    \item Composition is associative; segment order still matters.
  \end{itemize}
  \begin{block}{Teaching example: two steps of $S\leftarrow 0.5S+1$}
    $T_C=0.25$, $B_C=1.5$. From prefix state $10$:\\[1mm]
    correct composition $=4$; naive addition $=11.5$.
  \end{block}
  \papersource{Derived from paper Eq. (6), PDF p. 6. Tuple notation and numerical example are teaching additions.}
\end{frame}

\begin{frame}{Reuse is constant in segment length, not in all dimensions}
  \renewcommand{\arraystretch}{1.3}
  \centering
  \begin{tabular}{lccc}
    \toprule
    Transition & Extra $T_C$ storage & Build & Apply to $S$\\
    \midrule
    Scalar & 2 B & $O(|C|)$ & $O(d_kd_v)$\\
    Diagonal & 256 B & $O(|C|d_k)$ & $O(d_kd_v)$\\
    Dense & 32 KiB & $O(|C|d_k^2)$ & $O(d_k^2d_v)$\\
    \bottomrule
  \end{tabular}
  \medskip
  \begin{itemize}
    \item Storage: FP16, $d_k=d_v=128$, per segment/head/layer.
    \item The end-state $\SC$ is additional to the table's storage.
    \item $n$ segments still require $O(n)$ compositions.
    \item Full-attention KV movement and seam work remain.
  \end{itemize}
  \papersource{Paper, Table 3, PDF p. 7. Paper labels 32 KiB as 32 KB.}
\end{frame}

\begin{frame}{Layer-exact algebra coexists with deep contextual drift}
  \begin{block}{The essential condition}
    Composition is exact for the \alert{same layer input hidden states}.\\
    Independent segment prefill changes those inputs at deeper layers.
  \end{block}
  \medskip
  \begin{columns}[T,onlytextwidth]
    \begin{column}{0.44\textwidth}
      \textbf{Layer 0: Qwen3.5-35B}\\[2mm]
      1096 tokens / 4 segments\\[2mm]
      Relative error: $6\times10^{-5}$\\
      Angular error: $0.003^\circ$
    \end{column}
    \begin{column}{0.54\textwidth}
      \begin{tabular}{lcc}
        \toprule
        Deepest linear layer & Relative $L_2$ & Angle\\
        \midrule
        Qwen3.5-35B & 8.92\% & $5.11^\circ$\\
        Ring-flash & 8.69\% & $4.98^\circ$\\
        \bottomrule
      \end{tabular}
    \end{column}
  \end{columns}
  \papersource{Paper, Sec. 4.2, p. 7; Sec. 6.3 and Table 4, p. 12. Deep probe: 512 tokens, two chunks, all full-attention layers disabled. These are measurements, not a general error bound.}
\end{frame}

\begin{frame}{The largest observed splice deviations occur at boundaries}
  \begin{columns}[c,onlytextwidth]
    \begin{column}{0.49\textwidth}
      \centering
      \includegraphics[width=\linewidth,height=0.72\textheight,keepaspectratio]{figures/fig-03.pdf}
    \end{column}
    \begin{column}{0.47\textwidth}
      \begin{itemize}
        \item Token position on the horizontal axis.
        \item Compare full recompute, naive splice, and their deviations.
        \item KV and attention-weighted KV errors peak at segment starts.
      \end{itemize}
      \medskip
      A boundary offers both a useful repair target and an available prefix state.
    \end{column}
  \end{columns}
  \papersource{Paper, Fig. 3 and Sec. 3.2, PDF p. 5. Diagnostic: Qwen3.5-35B-A3B, layer 7, head 0.}
\end{frame}

\begin{frame}{An eight-token seam repairs cross-segment attention}
  \begin{columns}[c,onlytextwidth]
    \begin{column}{0.55\textwidth}
      \centering
      \includegraphics[width=\linewidth,height=0.68\textheight,keepaspectratio]{figures/fig-07.pdf}
    \end{column}
    \begin{column}{0.41\textwidth}
      \begin{itemize}
        \item System prompt: cache all.
        \item Interior segment: recompute first $w$ tokens; reuse the body.
        \item Query: compute end to end.
        \item Default $w=8$.
      \end{itemize}
    \end{column}
  \end{columns}
  \medskip
  Seam queries attend to the assembled prefix under the causal mask.
  \papersource{Paper, Fig. 7 and Sec. 4.3, PDF pp. 7--8. Green: reused KV; yellow: seam; pink: query.}
\end{frame}

\begin{frame}{Seam tokens must also traverse every linear layer}
  \begin{columns}[c,onlytextwidth]
    \begin{column}{0.48\textwidth}
      \centering
      \includegraphics[width=\linewidth,height=0.72\textheight,keepaspectratio]{figures/fig-08.pdf}
    \end{column}
    \begin{column}{0.48\textwidth}
      At each linear layer:
      \begin{enumerate}
        \item Start from the assembled prefix state.
        \item Recur over seam tokens and emit their per-token outputs.
        \item Apply the cached body tuple to jump to the segment end.
      \end{enumerate}
      \medskip
      The body tuple excludes the seam to avoid counting it twice.
    \end{column}
  \end{columns}
  \papersource{Paper, Fig. 8 and Sec. 4.3, PDF p. 8. The reused body still reflects independent-prefill hidden states.}
\end{frame}

\begin{frame}{Shared segment caches become request-specific running state}
  \begin{columns}[T,onlytextwidth]
    \begin{column}{0.48\textwidth}
      \begin{block}{Public pool}
        Segment tuples $(T_C,\SC)$\\
        Local-position K and unchanged V\\
        Trailing causal-convolution state
      \end{block}
    \end{column}
    \begin{column}{0.48\textwidth}
      \begin{block}{Private pool}
        Assembled running state\\
        Globally positioned K copies\\
        Retained for later prefix hits
      \end{block}
    \end{column}
  \end{columns}
  \[
    K_b=R(b-a)K_a
  \]
  \begin{itemize}
    \item Prefix lookup first, PIC lookup for remaining segments second.
    \item Separate capacities and LRU policies for the two pools.
    \item Linear-state re-rotation needs the stated scalar-transition condition.
  \end{itemize}
  \papersource{Paper, Secs. 4.2--4.3, Eqs. (7)--(8), PDF pp. 7--8. Convolution warm-up is covered when $w\geq k-1$.}
\end{frame}

\begin{frame}{Independent cold segments can prefill on different instances}
  \begin{columns}[c,onlytextwidth]
    \begin{column}{0.59\textwidth}
      \centering
      \includegraphics[width=\linewidth,height=0.6\textheight,keepaspectratio]{figures/fig-09.pdf}
    \end{column}
    \begin{column}{0.37\textwidth}
      \begin{itemize}
        \item Scatter only cache misses.
        \item Fetch hit caches from peers.
        \item Combine tuples and repair seams.
        \item Then start normal decoding.
      \end{itemize}
    \end{column}
  \end{columns}
  \[
    O(n|C|)\ \longrightarrow\ O(\lceil n/m\rceil |C|+c)
  \]
  \papersource{Paper, Fig. 9 and Sec. 4.4, PDF pp. 8--9. Simplified scheduling model: $n$ equal-length segments, $m$ workers, combine overhead $c$.}
\end{frame}

\begin{frame}{Three evaluation regimes answer different serving questions}
  \renewcommand{\arraystretch}{1.2}
  \begin{tabular}{@{}p{0.27\linewidth}p{0.67\linewidth}@{}}
    \toprule
    Regime & Workload and measurement\\
    \midrule
    Fully warmed & HotpotQA / TriviaQA: F1; MultiNews / GovReport: ROUGE-L. Compare p50 TTFT.\\
    Production load & Prod-RAG: mean 12k input tokens, 10-minute warm-up, remaining misses. Sweep request rate.\\
    Fully cold & 32k-token prompt, eight segments, TP-1 instances. Scale worker count.\\
    \bottomrule
  \end{tabular}\par
  \medskip
  Four models: Ring-mini, Ring-flash, Qwen3.5-35B, Qwen3.5-122B.\\
  Baselines: Full Recompute, Prefix Cache, Naive Addition.
  \papersource{Paper, Secs. 6.1--6.5, PDF pp. 10--12. Hardware: one 8$\times$H20-3e node, 141 GB/GPU. Figs. 10--11 use TP=1/4/2/4 respectively.}
\end{frame}

\begin{frame}{Warm reuse improves TTFT with model-dependent quality}
  \centering
  \renewcommand{\arraystretch}{1.12}
  \begin{tabular}{lccc}
    \toprule
    Model & Speedup / Full & Speedup / Prefix & Score delta / Full\\
    \midrule
    Ring-mini & 2.94$\times$ & 2.77$\times$ & $-3.44$ points\\
    Ring-flash & 2.97$\times$ & 2.85$\times$ & $-3.29$ points\\
    Qwen3.5-35B & 3.79$\times$ & 3.32$\times$ & $+0.47$ points\\
    Qwen3.5-122B & 4.67$\times$ & 4.05$\times$ & $-0.56$ points\\
    \bottomrule
  \end{tabular}
  \begin{block}{Reported aggregate across 16 model--task cells}
    \centering
    \term{3.25$\times$} mean speedup over Prefix Cache; \term{1.71-point} mean loss.
  \end{block}
  Naive Addition loses 66.9\% of Full Recompute's score, but omits both transitions and seam repair.
  \papersource{Paper, Fig. 10 and Sec. 6.2, PDF pp. 10--11. All segments pre-warmed; p50 TTFT. Points are absolute score differences on a 100-point scale, aggregated across F1 and ROUGE-L.}
\end{frame}

\begin{frame}{More requests fit under the same median-latency target}
  \centering
  \renewcommand{\arraystretch}{1.3}
  \begin{tabular}{lcc}
    \toprule
    Model & QPS / Prefix at 1 s & Peak token throughput / Prefix\\
    \midrule
    Ring-mini & 1.85$\times$ & 1.50$\times$\\
    Ring-flash & 1.49$\times$ & 1.46$\times$\\
    Qwen3.5-35B & 1.58$\times$ & 1.32$\times$\\
    Qwen3.5-122B & 1.71$\times$ & 1.30$\times$\\
    \bottomrule
  \end{tabular}
  \medskip
  \begin{itemize}
    \item Mean sustainable QPS gain: \term{1.66$\times$} over Prefix Cache.
    \item The 1-second threshold is applied to \alert{p50 TTFT}.
    \item Peak token throughput and SLO-constrained QPS are different metrics.
  \end{itemize}
  \papersource{Paper, Fig. 11 and Sec. 6.2, PDF p. 11. Prod-RAG after 10-minute warm-up. No p95/p99 guarantee follows from these median curves.}
\end{frame}

\begin{frame}{Longer segments make reuse more attractive}
  \begin{columns}[T,onlytextwidth]
    \begin{column}{0.49\textwidth}
      \includegraphics[width=\linewidth,height=0.38\textheight,keepaspectratio]{figures/fig-12.pdf}
      \smallskip
      Four segments, 1k $\to$ 4k each:\\
      Full: 141 $\to$ 624 ms\\
      HYPIC: 103 $\to$ 127 ms
    \end{column}
    \begin{column}{0.47\textwidth}
      \includegraphics[width=\linewidth,height=0.38\textheight,keepaspectratio]{figures/fig-13.pdf}
      \smallskip
      Dense-transition construction:\\
      \term{5.2--6.7\%} over main forward\\
      for 256--4096-token segments.
    \end{column}
  \end{columns}
  \medskip
  At fixed 1k-token segments, each extra segment adds about 2.3 ms for HYPIC versus 40.7 ms for Full Recompute.
  \papersource{Paper, Figs. 12--13 and Sec. 6.3, PDF p. 11. Construction probe uses Qwen3.5-35B-A3B; cost is paid on first segment prefill.}
\end{frame}

\begin{frame}{Eight seam tokens are a useful setting in the two tested cells}
  \centering
  \includegraphics[width=0.82\linewidth,height=0.48\textheight,keepaspectratio]{figures/fig-14.pdf}
  \begin{itemize}
    \item At $w=8$: ROUGE-L 0.1164 (35B/MultiNews), 0.1418 (122B/GovReport).
    \item Wider seams add latency without consistent quality gains.
    \item Text and plot disagree on the sweep; no explicit HYPIC-0 point appears.
  \end{itemize}
  \papersource{Paper, Fig. 14 and Sec. 6.4, PDF p. 12. Text: 0/4/8/16/32. Legend: 4/8/16/32/64/128. Details in backup.}
\end{frame}

\begin{frame}{Eight workers cut the measured cold-request TTFT to 0.49 s}
  \centering
  \includegraphics[width=0.78\linewidth,height=0.43\textheight,keepaspectratio]{figures/fig-15.pdf}\par
  \medskip
  \begin{tabular}{lrrrr}
    \toprule
    Configuration & Full, 1 worker & HYPIC, 2 & HYPIC, 4 & HYPIC, 8\\
    TTFT (s) & 2.83 & 1.34 & 0.82 & 0.49\\
    \bottomrule
  \end{tabular}\par
  \medskip
  Communication: 12--15 ms. Combine: about 120 ms.\\
  Most latency reduction comes from parallel scatter prefill.
  \papersource{Paper, Fig. 15(a), Sec. 6.5, PDF p. 12. 32k tokens, eight equal segments, TP-1 instances. Reported 5.7$\times$ uses more workers than the baseline.}
\end{frame}

\begin{frame}{LPT reduces stragglers; immediate transfers limit stalls}
  \begin{block}{Scheduling rule}
    Sort cold segments by decreasing token count. Assign each one to the worker with the smallest accumulated token load.
  \end{block}
  \centering
  \begin{tabular}{lcc}
    \toprule
    Eight uneven segments, four workers & Round-robin & LPT\\
    \midrule
    Scatter forward & 1.10 s & 0.69 s\\
    Total TTFT & 1.26 s & 0.84 s\\
    \bottomrule
  \end{tabular}\par
  \medskip
  Each worker sends a completed segment immediately, overlapping transfer with its next segment's computation.
  \papersource{Paper, Eq. (9), Sec. 4.4, PDF p. 9; Fig. 15(b), Sec. 6.5, PDF p. 12. LPT is about 1.50$\times$ faster than round-robin using displayed TTFT values.}
\end{frame}

\begin{frame}{Contextual accuracy and resource costs shape the tradeoff}
  \begin{itemize}
    \item \term{Accuracy:} deep cached inputs differ from full-context inputs; Ring loses more task score than Qwen in these experiments.
    \item \term{Memory:} store extra transitions and request-specific rotated K; cache capacity affects realized hit rate.
    \item \term{Resources:} cold-request speedup uses additional instances on the tested single-node system.
    \item \term{Evidence:} median load curves and a short isolated drift probe leave tail latency and long-context drift open.
  \end{itemize}
  \begin{block}{Most informative next experiment}
    Keep transitions, remove only seam repair; then measure quality and p95/p99 latency under a fixed cluster and cache budget.
  \end{block}
  \papersource{Analysis based on Secs. 4.2--4.4 and 6.2--6.5, PDF pp. 6--12. Next experiment is a discussion proposal.}
\end{frame}

\begin{frame}{Segment reuse connects algebra to serving execution}
  \begin{enumerate}
    \item \term{Cache a transformation:} $(T_C,S_{C\mid0})$ preserves how a segment changes incoming memory.
    \item \term{Repair a reachable region:} seam tokens carry corrected hidden states through the hybrid stack.
    \item \term{Schedule the same unit:} independent segments allow parallel cold prefill followed by composition.
  \end{enumerate}
  \medskip
  The central distinction: \alert{exact composition for fixed layer inputs; approximate reuse across contextualized deep inputs.}
  \papersource{Yifei Liu et al. HYPIC: Accelerating Hybrid-Attention LLM Serving with Position-Independent Caching. arXiv:2607.01299v2, July 2026. \href{https://arxiv.org/abs/2607.01299v2}{Versioned paper link}.}
\end{frame}

\appendix

\begin{frame}{Backup: unrolling the recurrence explains the cached tuple}
  For a segment of $m$ tokens, with fixed layer inputs:
  \[
    S_m=(T_m\cdots T_1)S_0
      +\sum_{t=1}^m(T_m\cdots T_{t+1})u_t.
  \]
  Cache $T_C$ and $S_{C\mid0}$, which do not depend on $S_0$.\par
  \medskip
  For suffix segments $C_1,\ldots,C_n$ behind prefix state $S_S$:
  \[
    S_{S\,C_1\cdots C_n}
    =\left(\prod_{i=n}^{1}T_{C_i}\right)S_S
    +\sum_{i=1}^{n}\left(\prod_{j=n}^{i+1}T_{C_j}\right)S_{C_i\mid0}.
  \]
  Later transitions multiply on the left. Empty products equal $I$.\\
  Sequential application needs $n$ tuple operations.
  \papersource{Paper, Eq. (6), PDF p. 6. First equation is an explanatory expansion of Eq. (2).}
\end{frame}

\begin{frame}{Backup: two recurrence calls construct the reusable state}
  \begin{columns}[T,onlytextwidth]
    \begin{column}{0.48\textwidth}
      \begin{block}{Content accumulation}
        Initialize $S_0=0$.\\
        Keep normal writes $u_i$.\\[2mm]
        Result: $S_{C\mid0}$.
      \end{block}
    \end{column}
    \begin{column}{0.48\textwidth}
      \begin{block}{Transition accumulation}
        Initialize the accumulator to $I$.\\
        Set all writes to zero.\\[2mm]
        Result: $T_m\cdots T_1=T_C$.
      \end{block}
    \end{column}
  \end{columns}
  \begin{itemize}
    \item SGLang implementation uses FLA kernels, Python, and Triton.
    \item Application markers define segments; token-ID hashes find hits.
    \item A width-$k$ causal convolution needs $k-1$ boundary inputs.
    \item Cache trailing convolution state; $w\geq k-1$ lets the seam warm it up.
  \end{itemize}
  \papersource{Paper, Secs. 4.2--4.3, PDF pp. 7--8; Sec. 5, PDF p. 9. Transition accumulator has shape $d_k\times d_k$.}
\end{frame}

\begin{frame}{Backup: KV and state shifts have different conditions}
  \begin{block}{Full-attention keys}
    $R(a+b)=R(a)R(b)\quad\Rightarrow\quad K_b=R(b-a)K_a.$\\[1mm]
    Rotate cached K; retain V; regenerate Q and seam tokens.
  \end{block}
  \begin{block}{The paper's scalar-transition linear-state case}
    $T_i=\gamma I$ commutes with rotations, giving\\[1mm]
    $S_{C\mid b}=R(b-a)S_{C\mid a}$.
  \end{block}
  The second identity relies on the stated transition structure.\\
  A general dense transition need not commute with RoPE.
  \papersource{Paper, Eqs. (7)--(8), PDF pp. 7--8. In the re-rotation equation, $a,b$ denote starting positions.}
\end{frame}

\begin{frame}{Backup: the seam sweep does not isolate a zero-seam baseline}
  \renewcommand{\arraystretch}{1.3}
  \begin{tabular}{@{}p{0.3\linewidth}p{0.64\linewidth}@{}}
    \toprule
    Evidence in v2 & What it contains\\
    \midrule
    Sec. 6.4 text & Widths $\{0,4,8,16,32\}$\\
    Fig. 14 legend & HYPIC-4/8/16/32/64/128\\
    Naive Addition & No cached transitions and no seam repair\\
    \bottomrule
  \end{tabular}
  \medskip
  \begin{itemize}
    \item The plot supports a useful $w=8$ operating point on two cells.
    \item It does not expose a clear ``correct transitions, $w=0$'' control.
    \item The aggregate Naive Addition gap cannot quantify the seam's isolated contribution.
  \end{itemize}
  \papersource{Paper, Secs. 6.1 and 6.4, Fig. 14, PDF pp. 10 and 12. This discrepancy is present in both the submitted TeX and rendered PDF.}
\end{frame}

\end{document}