% Self-authored research discussion; compile with pdfLaTeX.
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\title[Complex KDA]{Complex KDA}
\subtitle{Understanding and Enhancing the Expressivity\\of Kimi Delta Attention}
\author{Paper by Julien Siems, Riccardo Grazzi,\\Korbinian P\"oppel, and collaborators}
\date{arXiv:2609.24797v1 \quad|\quad Paper discussion, 8 October 2026}

\begin{document}
\begin{frame}
  \titlepage
  \vspace{-5mm}
  \begin{center}
  \color{AcademicTeal}\large
  Signed gates turn one delta update into a rotation.
  \end{center}
\end{frame}

\begin{frame}{State tracking needs ordered transformations}
  \begin{block}{The task}
  Read group elements $g_1,\ldots,g_t$ and recover every prefix product:
  \[
    s_t=g_t\cdots g_1.
  \]
  \end{block}
  \begin{itemize}
    \item Parity needs a sign flip; modular counting needs a cycle.
    \item Permutation tracking needs noncommuting transformations.
    \item Inputs range over \alert{all group elements}, not only swaps.
  \end{itemize}
  \medskip
  A fixed-size state is useful only if its update can represent the required operations.
  \papersource{Paper, Secs. 1--2 and 5, PDF pp. 1--3 and 6; App. B.1, p. 26.}
\end{frame}

\begin{frame}{CKDA keeps the original gate--delta--readout structure}
  \[
  \begin{aligned}
  H_t&=(I-\beta_t k_tk_t^\top)\Diag(\alpha_t)H_{t-1}
       +\beta_t k_tv_t^\top,\\
  o_t&=H_t^\top q_t,\qquad \norm{k_t}_2=1.
  \end{aligned}
  \]
  \begin{columns}[T]
    \begin{column}{.43\textwidth}
      $H_t\in\R^{n\times d_v}$: matrix state\\[2mm]
      $k_t,q_t\in\R^n$, $v_t\in\R^{d_v}$\\[2mm]
      $\alpha_t\in\R^n$: channel gates
    \end{column}
    \begin{column}{.53\textwidth}
      \begin{enumerate}
        \item Gate the old state.
        \item Correct the key--value prediction.
        \item Query the updated state.
      \end{enumerate}
    \end{column}
  \end{columns}
  \medskip
  CKDA changes the ranges to $\alpha_t\in[-1,1]^n$, $\beta_t\in[0,2]$.
  \papersource{Paper, Eq. (1), PDF p. 3; Sec. 3, p. 4; App. B.2, pp. 26--27.}
\end{frame}

\begin{frame}{Both range extensions are needed for complex eigenvalues}
  \centering
  \begin{tabular}{ccc}
    \toprule
    Channel gates & Update rate & Non-real eigenvalues possible?\\
    \midrule
    $[0,1]$ & $[0,1]$ & No\\
    $[0,1]$ & $[0,2]$ & No\\
    $[-1,1]$ & $[0,1]$ & No\\
    \alert{$[-1,1]$} & \alert{$[0,2]$} & \alert{Yes}\\
    \bottomrule
  \end{tabular}
  \medskip
  \begin{itemize}
    \item Positive diagonal gates give a matrix similar to a symmetric one.
    \item For $\beta\le1$, the delta factor is positive semidefinite.
    \item A scalar gate always leaves the transition symmetric.
  \end{itemize}
  \papersource{Paper, Sec. 3, PDF p. 4; Theorem 8, App. A.3, pp. 22--23.}
\end{frame}

\begin{frame}{Two reflections make a rotation in one update}
  \centering
  \includegraphics[width=\linewidth,height=.29\textheight,keepaspectratio]{figures/fig-01-two-reflections.pdf}
  \[
  k=\frac{1}{\sqrt2}\begin{pmatrix}1\\1\end{pmatrix},\quad
  (I-2kk^\top)\begin{pmatrix}-1&0\\0&1\end{pmatrix}
  =\begin{pmatrix}0&-1\\1&0\end{pmatrix}=R_{\pi/2}.
  \]
  \begin{itemize}
    \item The signed gate supplies a coordinate reflection.
    \item The delta factor supplies the second, freely oriented reflection.
    \item \alert{Real-valued computation}; the name refers to a complex spectrum.
  \end{itemize}
  \papersource{Paper, Fig. 1, PDF p. 2; Eq. (2), p. 4. Matrix expansion is explanatory.}
\end{frame}

\begin{frame}{Negative channel gates open a path to the unit circle}
  \centering
  \includegraphics[width=.91\linewidth,height=.54\textheight,keepaspectratio]{figures/fig-02-spectrum.pdf}
  \[
  \Delta=(1-\alpha)^2\cos^2(2\theta)+4\alpha.
  \qquad
  \Delta<0\ \Longrightarrow\ |\lambda|=\sqrt{-\alpha}.
  \]
  Top: $D=\Diag(\alpha,1)$. Bottom: scalar $D=\alpha I$.
  \papersource{Paper, Fig. 2, PDF p. 2; Sec. 3, p. 4. Here $\beta=2$ and $k=(\cos\theta,\sin\theta)^\top$.}
\end{frame}

\begin{frame}{The transition remains non-expansive and inexpensive}
  \[
  \norm{A_t}_2
  \le
  \underbrace{\norm{I-\beta_tk_tk_t^\top}_2}_{\le1}
  \underbrace{\norm{\Diag(\alpha_t)}_2}_{\le1}
  \le1.
  \]
  \begin{block}{Same structural budget}
  \[
  A_t=D_t-\beta_t k_t(k_t^\top D_t)
  \quad\Rightarrow\quad \text{diagonal + rank one}.
  \]
  State and recurrent work per token remain $O(nd_v)$ per head.
  \end{block}
  The bound controls homogeneous state propagation; repeated additive writes can still increase the state norm.
  \papersource{Paper, Secs. 2--4, PDF pp. 3--6; App. C.6, p. 41. Norm bound expanded for explanation.}
\end{frame}

\begin{frame}{CKDA spans the entire orthogonal DPR1 family}
  \begin{block}{Theorem 1}
  Every real orthogonal $A=D+uv^\top$ with diagonal $D$ can be written
  \[
  A=(I-2kk^\top)S,\qquad \norm{k}_2=1,\quad S_{ii}\in\{-1,+1\}.
  \]
  \end{block}
  \begin{itemize}
    \item A fully asymmetric rank-one correction adds no orthogonal transitions.
    \item Any CKDA transition has \alert{at most one} non-real conjugate pair.
    \item General non-expansive DPR1: at most one such pair \emph{on the unit circle}.
  \end{itemize}
  \papersource{Paper, Theorem 1, PDF p. 5; Theorems 8--9, pp. 22--24. DPR1 = diagonal plus rank one.}
\end{frame}

\begin{frame}{One layer tracks all finite subgroups of $SO(3)$}
  \centering
  \begin{tabular}{lll}
    \toprule
    Target group & Transition dimension & Construction\\
    \midrule
    Cyclic $\mathbb Z_m$ & 2 & Planar rotations\\
    Dihedral $D_m$, including $S_3$ & 2 & Rotations + reflections\\
    $A_4,S_4$ & 3 & Reoriented cube symmetries\\
    $A_5$ & 4 & Lift + many-to-one decoder\\
    \bottomrule
  \end{tabular}
  \medskip
  \begin{itemize}
    \item \textbf{Realization}: one matrix for each group element.
    \item \textbf{Tracking}: multiple hidden matrices may decode to the same element.
    \item Constructions use prescribed initial states and exact algebraic values.
  \end{itemize}
  \papersource{Paper, Theorem 3, PDF pp. 6--7; App. B--C, pp. 25--36. Dimension 4 uses a $4\times4$ matrix state.}
\end{frame}

\begin{frame}{A single layer still cannot track all of $S_5$}
  \begin{block}{Theorem 4: the assumptions matter}
  Non-expansive transitions, at most one unit-circle complex pair per head, and finite reachability under \emph{exact} affine updates.
  \end{block}
  \begin{itemize}
    \item The obstruction allows any finite number of independent heads and an arbitrary joint decoder.
    \item Three CKDA layers suffice for every finite-group word problem.
    \item Regular languages and weighted finite automata (WFAs) over $\mathbb Q$ need $\beta>2$ in this construction.
  \end{itemize}
  \medskip
  \alert{One-layer lower bound + three-layer construction leaves two layers open.}
  \papersource{Paper, Theorems 4--5, PDF p. 7; precision in App. B.4, pp. 28--29; App. D.5, pp. 44--45.}
\end{frame}

\begin{frame}{Absorb cumulative signs into keys and queries}
  \[
  \Diag(\alpha_t)=S_tD_t,\qquad D_t=\Diag(|\alpha_t|),\quad
  P_t=S_1\cdots S_t.
  \]
  \[
  \widetilde H_t=P_tH_t,\quad
  \widetilde k_t=P_tk_t,\quad
  \widetilde q_t=P_tq_t.
  \]
  \begin{block}{Equivalent positive-magnitude recurrence}
  \[
  \widetilde H_t=(I-\beta_t\widetilde k_t\widetilde k_t^\top)
  D_t\widetilde H_{t-1}+\beta_t\widetilde k_tv_t^\top,\quad
  o_t=\widetilde H_t^\top\widetilde q_t.
  \]
  \end{block}
  Reuse KDA kernels; fuse sign application with normalization and gradient writes.
  \papersource{Paper, Lemma 27 and implementation details, App. E, PDF pp. 45--46.}
\end{frame}

\begin{frame}{Fused CKDA retains 96--97\% of KDA kernel throughput}
  \begin{columns}[c]
    \begin{column}{.57\textwidth}
      \centering
      \includegraphics[width=\linewidth,height=.67\textheight,keepaspectratio]{figures/fig-05-throughput.pdf}
    \end{column}
    \begin{column}{.39\textwidth}
      H100, BF16\\
      16 heads, $d_k=d_v=128$\\
      32,768 tokens/step
      \medskip
      \begin{itemize}
        \item Forward + backward recurrence.
        \item Projections and optimizer excluded.
        \item Ungated DeltaProduct$_2$ remains competitive.
      \end{itemize}
    \end{column}
  \end{columns}
  \papersource{Paper, Fig. 5, PDF p. 7; App. E, p. 46. Gauge is an earlier implementation reference, not a matched fusion ablation.}
\end{frame}

\begin{frame}{CKDA extrapolates better on $S_3$ and $S_4$}
  \centering
  \includegraphics[width=\linewidth,height=.46\textheight,keepaspectratio]{figures/fig-06-state-tracking.pdf}
  \begin{itemize}
    \item Train to length 32; report \alert{best of three seeds}.
    \item Full CKDA beats the other KDA ranges; $S_4$ still degrades.
    \item $A_5$ succeeds only in a separate theory-initialized setup.
  \end{itemize}
  \papersource{Paper, Fig. 6 and Sec. 6, PDF p. 8; App. G.2, p. 47. Scaled accuracy maps chance to 0 and perfection to 1.}
\end{frame}

\begin{frame}{Successful state trackers learn the predicted geometry}
  \centering
  \includegraphics[width=\linewidth,height=.43\textheight,keepaspectratio]{figures/fig-07-learned-transitions.pdf}
  \begin{itemize}
    \item A successful head approaches $\beta=2$ and sign-valued gates.
    \item Key PCA: the first three components explain about 95\% of variance.
    \item Complex eigenvalues appear near the unit circle.
  \end{itemize}
  \papersource{Paper, Fig. 7, PDF p. 8. The $A_5$ theory-initialized run also changes architecture and schedule, so initialization is not isolated.}
\end{frame}

\begin{frame}{Periodic continuation tests phase memory without feedback}
  \centering
  \includegraphics[width=.95\linewidth,height=.47\textheight,keepaspectratio]{figures/fig-08-waveform.pdf}
  \begin{itemize}
    \item One fixed groove, 32 phases; 8-frame cue, then zero inputs.
    \item Train to length 136; at 264: \alert{38.1 dB} CKDA vs.\ 2.8 dB Transformer.
    \item CKDA wins among KDA range settings; the nonlinear GRU has lower MSE.
  \end{itemize}
  \papersource{Paper, Fig. 8 and Sec. 6, PDF p. 9; App. G.3, pp. 47--48. Single-layer models; one training seed.}
\end{frame}

\begin{frame}{Small-model gains depend on initialization}
  \centering
  \begin{tabular}{lrr}
    \toprule
    340M / 15B Nemotron-CC tokens & Val.\ PPL $\downarrow$ & Avg.\ accuracy $\uparrow$\\
    \midrule
    Dense Attention & 17.99 & 50.90\%\\
    Standard KDA & \textbf{17.45} & 51.32\%\\
    CKDA, standard init & 17.74 & 51.10\%\\
    CKDA, gate spread & 17.49 & \textbf{52.30\%}\\
    CKDA, spread, no key SiLU & 17.49 & 51.87\%\\
    \bottomrule
  \end{tabular}
  \medskip
  \begin{itemize}
    \item Best CKDA gains \alert{0.98 percentage points} over KDA.
    \item Its validation perplexity is slightly worse.
    \item This average uses ten tasks; the 1.3B evaluation uses nine.
  \end{itemize}
  \papersource{Paper, selected rows of Table 6, PDF p. 50; architecture details in Table 4, p. 49.}
\end{frame}

\begin{frame}{At 1.3B, matched CKDA and KDA controls perform similarly}
  \centering
  \begin{tabular}{lrrr}
    \toprule
    Model & Wiki.\ PPL $\downarrow$ & LAMBADA PPL $\downarrow$ & Avg.\ acc.\ $\uparrow$\\
    \midrule
    KDA, bounded gate & \textbf{15.73} & 10.53 & \textbf{54.09}\\
    CKDA & 15.78 & \textbf{10.08} & 54.06\\
    \midrule
    KDA + attention 3:1 & \textbf{15.04} & 9.93 & 53.92\\
    CKDA + attention 3:1 & 15.34 & \textbf{9.90} & \textbf{54.37}\\
    \bottomrule
  \end{tabular}
  \medskip
  \begin{itemize}
    \item Recurrent: $-0.03$ pp; hybrid: $+0.45$ pp in nine-task accuracy.
    \item The authors' KDA control also uses a bounded safe sigmoid gate.
    \item Their 3:1 full-attention hybrid differs from cited 1:1 sliding-window hybrids.
  \end{itemize}
  \papersource{Paper, Table 2, PDF p. 9; comparison details, pp. 9--10. 100B FineWeb-Edu tokens; accuracy in percent.}
\end{frame}

\begin{frame}{Retrieval results show tradeoffs across tasks}
  \centering
  \begin{tabular}{lrrr}
    \toprule
    Recall (\%) $\uparrow$ & SQuAD & SWDE & FDA\\
    \midrule
    KDA & 38.17 & \textbf{49.59} & 30.40\\
    CKDA & \textbf{39.68} & 48.42 & \textbf{31.49}\\
    \midrule
    KDA + attention 3:1 & \textbf{42.69} & \textbf{64.81} & \textbf{64.79}\\
    CKDA + attention 3:1 & 37.77 & 63.91 & 58.89\\
    \bottomrule
  \end{tabular}
  \medskip
  \begin{itemize}
    \item Hybrid accuracy gains do not extend to these three recall metrics.
    \item RULER is also mixed and highly sensitive to prompt wording.
    \item The paper attributes some low RULER scores to instruction-following failures.
  \end{itemize}
  \papersource{Paper, Table 2, PDF p. 9; RULER discussion and Table 7, pp. 49--50.}
\end{frame}

\begin{frame}{Scaling favors the recurrent family, with KDA close to CKDA}
  \centering
  \includegraphics[width=.85\linewidth,height=.60\textheight,keepaspectratio]{figures/fig-09-scaling.pdf}
  \par\smallskip
  $\Delta L$ relative to the QK-Norm Transformer: more negative is better.
  \par
  Across 47M--1.7B and 6B--50B tokens, KDA often has slightly lower loss.
  \papersource{Paper, Fig. 9, PDF p. 10; full loss grid, Table 12, p. 53. Nemotron-CC, context 4096.}
\end{frame}

\begin{frame}{Language models learn to use the newly allowed dynamics}
  \centering
  \includegraphics[width=.95\linewidth,height=.46\textheight,keepaspectratio]{figures/fig-10-lm-mechanism.pdf}
  \begin{itemize}
    \item Negative gates emerge from positive standard initialization.
    \item $\beta>1$ appears across layers; complex pairs concentrate early.
    \item Functional roles in language remain an open mechanistic question.
  \end{itemize}
  \papersource{Paper, Fig. 10, PDF p. 10. 1.3B / 100B FineWeb-Edu tokens; colors use square-root scaling.}
\end{frame}

\begin{frame}{The contribution is more expressive dynamics at a small cost}
  \begin{enumerate}
    \item \textbf{Mechanism:} signed channel gates + extended delta rate create rotations within a real DPR1 transition.
    \medskip
    \item \textbf{Theory:} full orthogonal DPR1 coverage and stronger group tracking, with explicit precision and reachability assumptions.
    \medskip
    \item \textbf{Evidence:} better symbolic and periodic extrapolation; general language modeling remains close to matched KDA.
  \end{enumerate}
  \medskip
  \alert{Representability, learnability, and practical numerical robustness are distinct.}
  \papersource{Paper, Sec. 7, PDF pp. 10--11; App. B.4, pp. 28--29.}
\end{frame}

\begin{frame}{Paper and discussion resources}
  \textbf{Complex KDA: Understanding and Enhancing the Expressivity of Kimi Delta Attention}
  \medskip

  Julien Siems, Riccardo Grazzi, Korbinian P\"oppel, Jaisidh Singh, Arber Zela, Timur Carstensen, Jenia Jitsev, Frank Hutter, Volkan Cevher, Antonio Orvieto, Aaron Klein.
  \medskip

  \begin{itemize}
    \item \href{https://arxiv.org/abs/2609.24797v1}{arXiv:2609.24797v1}, submitted 21 September 2026.
    \item \href{https://github.com/OpenEuroLLM/ComplexKDA}{Author code: OpenEuroLLM/ComplexKDA}
    \item \href{https://huggingface.co/collections/openeurollm/complexkda}{Author model collection}
  \end{itemize}
  \medskip
  Main discussion: approximately 25 minutes.\\
  The following slides contain derivations, assumptions, and additional results.
  \papersource{Figures are original paper assets or an exact PDF crop. Tables select reported rows; explanatory derivations are labeled.}
\end{frame}

\appendix
\begin{frame}{Backup: separate the sign core from magnitude scaling}
  \[
  A=\widetilde A\Diag(|\alpha|),\qquad
  \widetilde A=(I-\beta kk^\top)S.
  \]
  Decompose $k=k_-+k_+$ into the negative/positive gate subspaces, with
  $\norm{k_-}=\cos\theta$, $\norm{k_+}=\sin\theta$.
  \[
  \left.\widetilde A\right|_{\operatorname{span}(k_-,k_+)}
  =
  \begin{pmatrix}
  \beta\cos^2\theta-1&-\beta\sin\theta\cos\theta\\
  \beta\sin\theta\cos\theta&1-\beta\sin^2\theta
  \end{pmatrix}.
  \]
  \begin{itemize}
    \item At $\beta=2$, this block is a $2\theta$ rotation; its complement has $\pm1$.
    \item Complex pair iff $\beta^2\cos^2(2\theta)<4(\beta-1)$.
    \item Nonuniform magnitudes need a separate spectral argument for full $A$.
  \end{itemize}
  \papersource{Paper, Proposition 2, PDF p. 6; App. A.2--A.3, pp. 21--23. Both key components assumed nonzero.}
\end{frame}

\begin{frame}{Backup: the four-dimensional $A_5$ tracker uses a decoder}
  For a 3D rotation $R_{u,\theta}$, encode
  \[
  a=\begin{pmatrix}\cos(\theta/2)\\ \sin(\theta/2)u\end{pmatrix},
  \qquad
  g(R)=(I_4-2aa^\top)\Diag(-1,1,1,1).
  \]
  \[
  Q=\begin{pmatrix}c&r^\top\\s&C\end{pmatrix}
  \quad\Longrightarrow\quad
  f(Q)=cC-sr^\top-C[r]_\times.
  \]
  \begin{block}{The required invariant}
  \[
  f(g(R)Q)=R\,f(Q),\qquad f(I_4)=I_3.
  \]
  \end{block}
  At most $2|A_5|^2=7200$ hidden matrices; several can encode the same group element.
  \papersource{Paper, Proposition 18 and Corollary 19, App. C.4.2, PDF pp. 34--36. $[r]_\times x=r\times x$.}
\end{frame}

\begin{frame}{Backup: three layers implement clock, buffer, accumulator}
  \begin{enumerate}
    \item \textbf{Clock:} one planar rotation tracks position modulo $2m$.
    \item \textbf{Buffer:} retain the last $2m$ inputs and select block factors.
    \item \textbf{Accumulator:} apply one factor per step; complete the delayed product in the readout.
  \end{enumerate}
  \medskip
  \centering
  \begin{tabular}{lll}
    \toprule
    Target & Sufficient factor budget & Rate condition\\
    \midrule
    Permutation on $n$ coordinates & $n-1$ & $\beta\in[0,2]$\\
    Explicit DFA transition & $2n-2$ & $\beta\in[0,3]$\\
    Rational matrix, with scratch & $7n^2+3n$ & $\beta\ge0$, unbounded\\
    \bottomrule
  \end{tabular}
  \papersource{Paper, App. D, PDF pp. 41--45; $n\ge2$ for the first two budgets. Exact algebraic arithmetic; WFA storage grows polynomially in length.}
\end{frame}

\begin{frame}{Backup: practical gates differ from exact theoretical endpoints}
  \[
  r=\tanh(a/2),\quad
  \alpha=s[\epsilon+(1-\epsilon)|r|],\quad
  \epsilon=e^{-5},\quad \beta=2\sigma(b).
  \]
  \begin{itemize}
    \item $s=+1$ for $r\ge0$ and $-1$ otherwise; $\log|\alpha|$ stays in $[-5,0]$.
    \item Gate is discontinuous at zero; backward detaches $s$ and uses the zero subgradient of $|r|$ at zero.
    \item Standard init: positive gates. Spread init: about half negative.
    \item Finite sigmoid inputs approach but do not attain $\beta=2$, $|\alpha|=1$.
  \end{itemize}
  \medskip
  Removing key SiLU changes an optimization bias, not reachable key directions.
  \papersource{Paper, App. E--G.1, PDF pp. 45--47.}
\end{frame}

\begin{frame}{Backup: RULER gains depend on task and architecture}
  \centering
  \begin{tabular}{lrrrr}
    \toprule
    Accuracy at 4K (\%) & S-NIAH-1 & S-NIAH-2 & S-NIAH-3 & MK-NIAH-1\\
    \midrule
    KDA & 94.0 & \textbf{97.6} & 9.8 & 26.0\\
    CKDA & \textbf{100.0} & 88.6 & \textbf{51.0} & \textbf{37.0}\\
    \midrule
    KDA + attn 3:1 & \textbf{100.0} & \textbf{99.2} & \textbf{95.8} & 63.8\\
    CKDA + attn 3:1 & 81.2 & 83.2 & 86.8 & \textbf{94.4}\\
    \bottomrule
  \end{tabular}
  \medskip
  \begin{itemize}
    \item 500 samples per cell; 4K training context.
    \item Prompt variations can strongly affect instruction following.
    \item Table 7 says hybrid evaluation stops at 4K, yet includes 8K cells. Only 4K hybrid cells are compared here.
  \end{itemize}
  \papersource{Paper, Table 7, PDF p. 50; prompt-sensitivity discussion, p. 49. Caption/data conflict is in v1.}
\end{frame}

\begin{frame}{Backup: the largest scaling cell has mixed rankings}
  \centering
  \begin{tabular}{lrr}
    \toprule
    1.7B / 50B Nemotron-CC tokens & Loss (nats) $\downarrow$ & Avg.\ acc.\ (\%) $\uparrow$\\
    \midrule
    Transformer++ (QK-Norm) & 2.1374 & 55.10\\
    KDA & 2.1046 & 56.87\\
    CKDA & 2.1061 & \textbf{57.21}\\
    KDA + attn 3:1 & \textbf{2.0971} & 56.47\\
    CKDA + attn 3:1 & 2.1002 & 55.82\\
    \bottomrule
  \end{tabular}
  \[
  L(N,D)=E+(AN^{-\alpha_s}+BD^{-\beta_s})^k.
  \]
  Fit holds out this cell. Free-$E$ CKDA error: $+0.0007$ nats.\\
  Shared training trunks and fixed hyperparameters limit uncertainty claims.
  \papersource{Paper, Tables 11--13, PDF pp. 52--54; fitting methodology, p. 51. Some lower-scale accuracy tasks are near chance.}
\end{frame}

\begin{frame}{Backup: why the $S_5$ impossibility needs finite reachability}
  \begin{enumerate}
    \item Finite exact reachability + non-expansion permits an orthogonal reduction that removes additive terms.
    \item The finite hidden matrix group maps onto $S_5$ through the decoder.
    \item Lift a five-cycle and its transposition conjugate to per-head planar rotations.
    \item Their matrix commutator satisfies $C^{10}=I$, while its decoded three-cycle does not.
  \end{enumerate}
  \medskip
  \begin{block}{Arithmetic boundary}
  A finite floating-point format does not imply finite reachability for the exact affine recurrence. Rounding changes the update map.
  \end{block}
  \papersource{Paper, App. B.4, PDF pp. 28--29; App. C.5--C.6, pp. 36--41.}
\end{frame}

\end{document}