% Reader-authored research discussion; compile with pdfLaTeX.
% Adapted from the arxiv-paper-reader Beamer template.
\documentclass[aspectratio=169,11pt]{beamer}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\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}
\newcommand{\papersource}[1]{\par\smallskip{\scriptsize\color{gray}#1\par}}
\newcommand{\C}{\mathbb C}
\newcommand{\R}{\mathbb R}
\DeclareMathOperator{\diag}{diag}
\DeclareMathOperator{\rank}{rank}
\DeclareMathOperator{\Orb}{Orb}
\newcommand{\norm}[1]{\left\lVert#1\right\rVert}
\setlength{\abovedisplayskip}{6pt}
\setlength{\belowdisplayskip}{6pt}

\title[SFDA]{Semidirect Fourier Delta Attention}
\subtitle{Phase-Controlled Delta Memory\\with Constructive Chunk-WY Kernels}
\author{Paper by Tiantian Zhang (Crystal)\\Columbia University}
\date{arXiv:2607.11897v1\\Research discussion --- 20 October 2026}

\begin{document}

% Main talk: frames 1--21, approximately 25 minutes.
\begin{frame}
  \titlepage
  \vspace{-5mm}
  \begin{center}
    \color{AcademicTeal}
    Phase memory + delta updates + exact chunk-local algebra
  \end{center}
\end{frame}

\begin{frame}{Can phase memory preserve a useful chunk structure?}
  \begin{block}{The design problem}
    Add cyclic state dynamics to KDA while keeping structured products of
    recurrent transitions.
  \end{block}
  \begin{enumerate}
    \item \textbf{Architecture:} replace real decay with phase--decay.
    \item \textbf{Algebra:} explicitly construct a chunk-WY factorization.
    \item \textbf{Evidence:} exact identities and small counter experiments.
  \end{enumerate}
  \medskip
  Fused GPU performance and language-model gains remain open.
  \papersource{Paper, Secs. 1, 4, 7--8, 10; PDF pp. 2--3, 7--10, 16--22.}
\end{frame}

\begin{frame}{Rotate the memory, then correct its prediction}
  \[
  \boxed{S_t=(I-\beta_t k_tk_t^*)\Lambda_tS_{t-1}+\beta_tk_tv_t^*}
  \qquad
  \Lambda_t=\diag(\alpha_t\odot e^{i\theta_t})
  \]
  \begin{align*}
    \bar S_t&=\Lambda_tS_{t-1}
      &&\text{transport and decay},\\
    \widehat v_t&=\bar S_t^*k_t
      &&\text{read the current prediction},\\
    S_t&=\bar S_t+\beta_tk_t(v_t-\widehat v_t)^*
      &&\text{write the residual}.
  \end{align*}
  $S_t\in\C^{d\times d_v}$;\quad $q_t,k_t\in\C^d$;\quad
  $v_t\in\C^{d_v}$;\quad $o_t=S_t^*q_t$.
  \medskip

  \alert{The same key erases and writes.} Here $^*$ is conjugate transpose.
  \papersource{Paper, Sec. 3, Eqs. (13)--(19), PDF p. 5. Three-step form: explanatory rewrite.}
\end{frame}

\begin{frame}{A phase coordinate is a bounded modular counter}
  \begin{columns}[c,onlytextwidth]
    \begin{column}{.46\textwidth}
      \includegraphics[width=\linewidth,height=.68\textheight,keepaspectratio]
        {figures/phase-counter-teaching.pdf}
    \end{column}
    \begin{column}{.51\textwidth}
      With $z_0=1$, $\alpha_t=1$, $\beta_t=0$:
      \[
        z_t=e^{2\pi i a_t/m}z_{t-1}.
      \]
      Hence
      \[
        z_t=e^{2\pi i\sum_{s\le t}a_s/m},
        \quad |z_t|=1.
      \]
      \begin{itemize}
        \item Decode the nearest phase prototype.
        \item A zero initial state cannot rotate into a counter.
        \item No FFT is required.
      \end{itemize}
    \end{column}
  \end{columns}
  \papersource{Paper, Prop. 3.4, PDF p. 7; App. G.2, p. 38.
    Diagram: reader-created teaching illustration.}
\end{frame}

\begin{frame}{Different control families support different claims}
  \begin{tabular}{@{}p{.29\linewidth}p{.66\linewidth}@{}}
    \toprule
    Family & Transition and scope\\
    \midrule
    Real KDA subfamily &
      $(I-\beta kk^\top)\diag(\alpha)$; real decay\\[2mm]
    Default complex SFDA &
      $(I-\beta kk^*)\diag(\alpha e^{i\theta})$; phase memory\\[2mm]
    Structured generalization &
      $\Lambda-ur^*$; cheap reflections or shifts may be added\\[2mm]
    Unrestricted template &
      $(I-\beta kk^*)\widetilde A$; arbitrary symbol-conditioned
      $\widetilde A$\\
    \bottomrule
  \end{tabular}
  \par\medskip
  One-hot finite-state realizability uses the \alert{last row}.
  Reflections and stack shifts require their corresponding controls.
  \papersource{Paper, Table 2, PDF p. 6; Thm. 4.7, p. 9; Sec. 5, pp. 11--13.}
\end{frame}

\begin{frame}{WY factors retain interactions between delta updates}
  Set $u_t=\beta_tk_t$ and $r_t=\Lambda_t^*k_t$, so
  \[
    A_t=\Lambda_t-u_tr_t^*.
  \]
  For two tokens,
  \begin{align*}
    A_2A_1={}&\Lambda_2\Lambda_1
       -\Lambda_2u_1r_1^*-u_2r_2^*\Lambda_1\\
       &+\underbrace{u_2(r_2^*u_1)r_1^*}_{\text{interaction of the two corrections}}.
  \end{align*}
  \begin{block}{What the factors store}
    $\Gamma$: structured control product;\quad
    $Y,W$: correction directions;\\
    $M$: interactions between corrections.
  \end{block}
  $r_t$ is the right WY vector, \emph{not} the attention value.
  \papersource{Paper, Eq. (26), Thms. 4.1--4.2, PDF pp. 7--8.
    Two-step expansion is an explanatory derivation.}
\end{frame}

\begin{frame}{The chunk factorization is constructive}
  For $P_t=A_t\cdots A_1$, initialize $\Gamma_0=I$ and empty $Y_0,W_0,M_0$:
  \begin{align*}
    \Gamma_t&=\Lambda_t\Gamma_{t-1},\\
    Y_t&=[\,\Lambda_tY_{t-1},\ u_t\,],\\
    W_t&=[\,W_{t-1},\ \Gamma_{t-1}^*r_t\,],\\
    M_t&=\begin{bmatrix}
      M_{t-1}&0\\
      -r_t^*Y_{t-1}M_{t-1}&1
    \end{bmatrix}.
  \end{align*}
  \[
    \boxed{P_t=\Gamma_t-Y_tM_tW_t^*}
  \]
  $Y_t,W_t\in\C^{d\times t}$, $M_t\in\C^{t\times t}$.
  No inverse of $\Lambda_t$ is required.
  \papersource{Paper, Thm. 4.2, Eqs. (34)--(39), PDF p. 8; proof in App. A.4, p. 25.}
\end{frame}

\begin{frame}{A complete chunk summary must include new writes}
  With $B_t=\beta_tk_tv_t^*$,
  \[
    B_C^{\rm chunk}=\sum_{i=1}^{C}A_C\cdots A_{i+1}B_i.
  \]
  Compute it by the same recurrence starting from zero:
  \[
    \bar S_0=0,\qquad \bar S_t=A_t\bar S_{t-1}+B_t.
  \]
  Then apply the summary to the incoming state:
  \[
    \boxed{S_{\rm out}=\Gamma_CS_{\rm in}
      -Y_C\bigl(M_C(W_C^*S_{\rm in})\bigr)+B_C^{\rm chunk}}.
  \]
  \begin{itemize}
    \item Each write is transported by all subsequent transitions.
    \item Prefix factors also recover every intra-chunk token output.
  \end{itemize}
  \papersource{Paper, Thms. 4.4--4.5, PDF pp. 8--9; Algorithm 1, p. 14.}
\end{frame}

\begin{frame}{Keep the rank budget local to each chunk}
  \begin{columns}[t,onlytextwidth]
    \begin{column}{.49\textwidth}
      \textbf{Exact local representation}
      \[
        \rank(P_C-\Gamma_C)\le\min(C,d).
      \]
      \begin{enumerate}
        \item Build factors inside a chunk.
        \item Apply them to boundary state $S$.
        \item Start new factors in the next chunk.
      \end{enumerate}
    \end{column}
    \begin{column}{.48\textwidth}
      \textbf{Why rank can grow}
      \[
        A_i=I-e_ie_i^*
      \]
      \[
        A_r\cdots A_1=I-\sum_{i=1}^r e_ie_i^*.
      \]
      The correction has rank $r$ for $r\le d$.
    \end{column}
  \end{columns}
  \medskip
  A globally composed exact summary can grow to rank $d$.
  Small-rank recompression generally introduces approximation.
  \papersource{Paper, Thm. 5.3 and Prop. 5.4, PDF pp. 11--12; Sec. 6, p. 15.}
\end{frame}

\begin{frame}{Reference costs expose the remaining systems work}
  \begin{tabular}{@{}ll@{}}
    \toprule
    Per-chunk operation & Reference cost\\
    \midrule
    Factor storage + write summary & $O(d+dC+C^2+dd_v)$\\[1.5mm]
    Explicit factor construction & $O(dC^2+C^3)$\\[1.5mm]
    Zero-state write recurrence & $O(Cdd_v)$\\[1.5mm]
    Apply to the boundary state & $O(dd_v+dCd_v+C^2d_v)$\\
    \bottomrule
  \end{tabular}
  \medskip
  \begin{itemize}
    \item Fixed $C$ gives linear scaling in sequence length $T$.
    \item If $d_v=d$, the state update still includes a $d^2$ term.
    \item These bounds do not measure all outputs, backward passes,
      device traffic, or wall-clock throughput.
  \end{itemize}
  \papersource{Paper, Thm. 6.2, PDF p. 15; App. B.6 and C, pp. 27--30.
    Width-scaling interpretation added by the reader.}
\end{frame}

\begin{frame}{Nonexpansive transitions can still accumulate writes}
  If $\norm{\Lambda_t}_2\le1$ and $0\le\beta_t\norm{k_t}_2^2\le2$, then
  \[
    \norm{A_t}_2\le1,\qquad
    \norm{S_t}_F\le\norm{S_0}_F+\sum_{i=1}^t\norm{B_i}_F.
  \]
  \begin{block}{Proof idea}
    $I-\beta kk^*$ has eigenvalues $1-\beta\norm{k}^2$ along $k$
    and $1$ on its orthogonal complement.
  \end{block}
  \begin{itemize}
    \item Uniform decay $\norm{\Lambda_t}\le\rho<1$ gives a geometric
      bound on accumulated writes.
    \item Stable state transitions do not automatically imply
      well-conditioned intermediate WY factors.
  \end{itemize}
  \papersource{Paper, Props. 6.3--6.4, Eqs. (74)--(77), PDF pp. 15--16.}
\end{frame}

\begin{frame}{Compact memories require specific control operators}
  \begin{tabular}{@{}p{.28\linewidth}p{.31\linewidth}p{.35\linewidth}@{}}
    \toprule
    Memory & State size & Required mechanism\\
    \midrule
    Product cyclic counters &
      $\ell$ complex coordinates &
      Phase-only; no decay or write\\[2mm]
    Dihedral $D_m$ &
      $2\times2$ real matrix &
      Rotations \emph{and reflections}\\[2mm]
    Registers / resets &
      $d\times d_v$ &
      Erase row $i$, then write row $i$\\[2mm]
    Stack of depth $h$ &
      $h\times d_v$ &
      Zero-fill shift matrices\\
    \bottomrule
  \end{tabular}
  \par\medskip
  Registers already fit KDA.
  Reflection and shift theorems extend the default phase control family.
  \papersource{Paper, Thms. 5.5, 5.7, 5.9, 5.11, PDF pp. 12--13; App. G, pp. 37--38.}
\end{frame}

\begin{frame}{Routing a write also requires protecting old memory}
  Desired update: $x^+=G_\sigma x+w_\sigma r_\sigma$.
  \[
    G_\sigma=R_\sigma\widetilde G_\sigma,
    \qquad w_\sigma=R_\sigma k,
    \qquad \alert{k^\top\widetilde G_\sigma x=0}.
  \]
  Under these conditions,
  \[
    R_\sigma\left[(I-kk^\top)\widetilde G_\sigma x+kr_\sigma\right]
      =G_\sigma x+w_\sigma r_\sigma.
  \]
  \begin{itemize}
    \item $w_\sigma$ must lie in the allowed control orbit of $k$.
    \item The orthogonality condition neutralizes destructive erasure.
    \item Fixed block rotations cannot move a key between blocks.
  \end{itemize}
  \papersource{Paper, Sec. 4.2, PDF p. 10; Thm. F.2 and Prop. F.5, pp. 35--36.}
\end{frame}

\begin{frame}{Constructed phase memory stays exact to length 8192}
  \begin{columns}[c,onlytextwidth]
    \begin{column}{.66\textwidth}
      \includegraphics[width=\linewidth,height=.66\textheight,keepaspectratio]
        {figures/expA_cyclic_counter.png}
    \end{column}
    \begin{column}{.31\textwidth}
      \textbf{Constructed mod-5 counter}
      \begin{itemize}
        \item One real 2D phase block.
        \item Decoder calibrated at 128.
        \item At 8192: 1.000 vs.\ 0.201.
      \end{itemize}
      \medskip
      The baseline is a low-dimensional real-diagonal control model.
    \end{column}
  \end{columns}
  \papersource{Paper, Sec. 7, Fig. 1(a), Table 3, PDF pp. 16--17;
    App. H.2, p. 40. Original author figure.}
\end{frame}

\begin{frame}{Learned phase calibration limits extrapolation}
  \begin{columns}[c,onlytextwidth]
    \begin{column}{.66\textwidth}
      \includegraphics[width=\linewidth,height=.66\textheight,keepaspectratio]
        {figures/expB_learned_counter.png}
    \end{column}
    \begin{column}{.31\textwidth}
      Train at length $48$.
      \medskip

      \begin{tabular}{@{}rr@{}}
        \toprule
        Length & SFDA\\
        \midrule
        96 & 1.000\\
        192 & 0.951\\
        384 & 0.638\\
        768 & 0.422\\
        \bottomrule
      \end{tabular}
      \medskip

      A persistent per-step phase error can accumulate with length.
    \end{column}
  \end{columns}
  \papersource{Paper, Fig. 1(b), Table 4, PDF pp. 17--18; App. H.3, p. 40.
    Original author figure; error accumulation is the author's interpretation.}
\end{frame}

\begin{frame}{The mod-3 study uses a separate evaluation protocol}
  \begin{columns}[t,onlytextwidth]
    \begin{column}{.49\textwidth}
      \textbf{Model and training}
      \begin{itemize}
        \item One layer, one head.
        \item State width $d=16$.
        \item Linear readout; $\sim8.3$k parameters.
        \item AdamW + cosine schedule.
        \item Best in-distribution validation checkpoint.
      \end{itemize}
    \end{column}
    \begin{column}{.48\textwidth}
      \textbf{Comparison and measurement}
      \begin{itemize}
        \item Same architecture and parameter count.
        \item KDA: force $\theta_t=0$.
        \item Train at $T=32$; test to $256$.
        \item Accuracy on last-quarter positions.
        \item Mean over two seeds; chance $1/3$.
      \end{itemize}
    \end{column}
  \end{columns}
  \papersource{Paper, Sec. 8.2, PDF pp. 18--19. These settings are not assigned to the mod-5 study.}
\end{frame}

\begin{frame}{Resets preserve accuracy; uninterrupted counting degrades}
  \centering
  \includegraphics[width=.96\linewidth,height=.55\textheight,keepaspectratio]
    {figures/figure_statetrack.pdf}
  \medskip

  \begin{columns}[t,onlytextwidth]
    \begin{column}{.49\textwidth}
      \textbf{Cyclic $C_3$ at $T=256$}\\
      SFDA $0.341$ vs.\ KDA $0.335$.\\
      Nearly back to chance after $8\times$ extrapolation.
    \end{column}
    \begin{column}{.48\textwidth}
      \textbf{Reset $C_3$ at $T=256$}\\
      SFDA $1.000$ vs.\ KDA $0.680$.\\
      Resets shorten the effective memory horizon.
    \end{column}
  \end{columns}
  \papersource{Paper, Table 6, PDF p. 19; Fig. 2, p. 20.
    Original author figure. Reset interval distribution is not detailed.}
\end{frame}

\begin{frame}{Numerical checks support the algebraic identities}
  \begin{tabular}{@{}lll@{}}
    \toprule
    Identity / property & Table 5 & Table 8\\
    \midrule
    Constructive chunk-WY & $5.7\times10^{-10}$ & $1.9\times10^{-15}$\\[1.5mm]
    Affine chunk transfer & $9.5\times10^{-8}$ & $2.4\times10^{-16}$\\[1.5mm]
    KDA reduction at $\theta=0$ & $0$ & $0$\\[1.5mm]
    Stability $\norm{A_t}_2\le1$ & 2000 samples & max $0.9977$\\
    \bottomrule
  \end{tabular}
  \medskip
  \begin{itemize}
    \item Table 5 reports operator/state residuals.
    \item Table 8 reports worst-case relative errors over its checks.
    \item The two test suites are not specified on a common precision
      and normalization basis.
  \end{itemize}
  \papersource{Paper, Sec. 8.1, Table 5, PDF p. 18;
    App. H.1, Table 8, pp. 39--40. All values above are author-reported.}
\end{frame}

\begin{frame}{Direction reversal and systems scaling remain unresolved}
  \begin{block}{A concrete failure case}
    On the dihedral $D_3$ task, SFDA reaches only about $0.47$
    in-distribution. A full length-extrapolation table is not provided.
  \end{block}
  \begin{itemize}
    \item The exact dihedral construction requires reflection control;
      the experimental implementation of this extension is unclear.
    \item No fused Triton/CUDA throughput or language-model results.
    \item Ratios $7{:}1$ and $15{:}1$ versus KDA $3{:}1$ are future targets.
    \item The source package lists experiment script names but does not
      include those scripts.
  \end{itemize}
  \papersource{Paper, Sec. 8.2, PDF p. 19; Sec. 10, pp. 21--22;
    App. E and H.4, pp. 33--34, 40--41. Source-package observation by the reader.}
\end{frame}

\begin{frame}{The useful contribution couples dynamics to computation}
  \begin{enumerate}
    \item \textbf{Phase supplies periodic dynamics.}\\
      Delta updates supply addressable residual writes.
    \item \textbf{Constructive WY makes the chunk algebra usable.}\\
      Preserve the write summary and the chunk-local rank budget.
    \item \textbf{Learned accuracy depends on phase precision.}\\
      Reset and uninterrupted tasks probe different memory horizons.
  \end{enumerate}
  \begin{block}{The next decisive tests}
    Match real state bytes, isolate reflection control, measure phase drift,
    and benchmark complete fused forward/backward kernels.
  \end{block}
  \papersource{Synthesis of Secs. 3--8 and 10. Proposed next tests are reader analysis.}
\end{frame}

\begin{frame}{Paper and discussion guide}
  Tiantian Zhang (Crystal).\\
  \emph{Semidirect Fourier Delta Attention: Phase-Controlled Delta Memory
    with Constructive Chunk-WY Kernels}.\\
  arXiv:2607.11897v1.
  \medskip

  \href{https://arxiv.org/abs/2607.11897v1}{arxiv.org/abs/2607.11897v1}
  \medskip

  \begin{itemize}
    \item Main update and exact recursion: pp.\ 5--9.
    \item Compact controls, cost, stability: pp.\ 11--16.
    \item Reported counter results: pp.\ 17--20.
    \item Routing theorem: pp.\ 34--37.
  \end{itemize}
  \papersource{All locators refer to the 41-page v1 PDF.
    Paper and author figures: \href{https://creativecommons.org/licenses/by/4.0/}{CC BY 4.0}.
    The phase-circle diagram is a reader-created teaching illustration.}
\end{frame}

% Backup frames 22--26.
\appendix
\begin{frame}{Backup: block-WY closure accounts for cross terms}
  For $A_i=\Gamma_i-Y_iM_iW_i^*$, write
  \[
    A_2A_1=\Gamma_{21}-Y_{21}M_{21}W_{21}^*,
  \]
  with
  \begin{align*}
    \Gamma_{21}&=\Gamma_2\Gamma_1,\\
    Y_{21}&=[\,\Gamma_2Y_1,\ Y_2\,],\\
    W_{21}&=[\,W_1,\ \Gamma_1^*W_2\,],\\
    M_{21}&=\begin{bmatrix}
      M_1&0\\
      -M_2W_2^*Y_1M_1&M_2
    \end{bmatrix}.
  \end{align*}
  The represented rank budget grows from $(r_1,r_2)$ to $r_1+r_2$.
  \papersource{Paper, Thm. 4.1, Eqs. (27)--(33), PDF p. 7.}
\end{frame}

\begin{frame}{Backup: prefix factors produce every token output}
  For a prefix $r\le C$,
  \[
    P_r=\Gamma_r-Y_rM_rW_r^*,\qquad
    S_r=P_rS_0+B_r^{\rm chunk}.
  \]
  Therefore
  \[
    o_r=S_0^*P_r^*q_r+(B_r^{\rm chunk})^*q_r,
  \]
  where
  \[
    P_r^*q_r=\Gamma_r^*q_r-W_rM_r^*Y_r^*q_r.
  \]
  \begin{itemize}
    \item Both the carried state and local writes contribute.
    \item This establishes exact outputs.
    \item An efficient batched implementation must also manage
      prefix storage, repeated work, and backward computation.
  \end{itemize}
  \papersource{Paper, Thm. 4.5, PDF p. 9.
    Adjoint expansion and implementation implications are reader analysis.}
\end{frame}

\begin{frame}{Backup: realifying complex rank one gives real rank two}
  For a complex key, define
  \[
    a=\begin{bmatrix}\Re k\\\Im k\end{bmatrix},
    \qquad J=\begin{bmatrix}0&-I\\I&0\end{bmatrix}.
  \]
  The complete operator satisfies
  \[
    \mathcal R(kk^*)=aa^\top+(Ja)(Ja)^\top.
  \]
  Example: $k=1\in\C$ gives
  \[
    \mathcal R(kk^*)=I_2
    \quad\text{but}\quad aa^\top=\diag(1,0).
  \]
  Complex SFDA needs a complex inner product or paired real updates.
  A real rank-one delta plus block rotations is a distinct valid WY model.
  \papersource{Reader derivation clarifying the whole-update equivalence in Eqs. (2)--(3).
    Paper Prop. 3.3 and Lemma A.1 establish phase-control realification, PDF pp. 6, 24.}
\end{frame}

\begin{frame}{Backup: affine lifting uses one extra coordinate}
  For any affine system $x^+=A_\sigma x+b_\sigma$, use
  \[
    \phi(x)=\begin{bmatrix}x\\1\end{bmatrix},\qquad
    \widetilde A_\sigma=
    \begin{bmatrix}
      A_\sigma&b_\sigma\\
      0&1
    \end{bmatrix},
    \qquad \beta_\sigma=0.
  \]
  Then
  \[
    \widetilde A_\sigma\phi(x)
      =\phi(A_\sigma x+b_\sigma),\qquad D=d+1.
  \]
  \begin{block}{Scope of the open problem}
    The unrestricted template in Problem D.1 admits this standard lift.
    The interesting question needs structured-control, stability, or
    microstep constraints.
  \end{block}
  \papersource{Reader analysis of App. D.1 and D.5, PDF pp. 30--33.
    This construction is not a phase-only SFDA kernel.}
\end{frame}

\begin{frame}{Backup: exact counter constructions use exact endpoints}
  \begin{columns}[t,onlytextwidth]
    \begin{column}{.49\textwidth}
      \textbf{Representation conditions}
      \begin{itemize}
        \item Phase counter: $\beta=0$, $\alpha=1$.
        \item Register replacement: $\beta=1$.
        \item Finite-logit sigmoid gates generally approximate these endpoints.
      \end{itemize}
    \end{column}
    \begin{column}{.48\textwidth}
      \textbf{A phase-error illustration}
      \[
        \Delta\theta_T=T\varepsilon
      \]
      for a constant per-step error $\varepsilon$.
      \[
        |\Delta\theta_T|<\frac{\pi}{m}
      \]
      keeps nearest-prototype decoding inside the correct angular cell.
    \end{column}
  \end{columns}
  \medskip
  Exact expressivity, numerical precision, and learnability are separate
  parts of the design.
  \papersource{Paper, Eqs. (15)--(18), Prop. 3.4, Thm. 5.9, PDF pp. 5, 7, 13.
    Endpoint and phase-error discussion: reader analysis, not a fitted error model.}
\end{frame}

\end{document}