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\title[Mamba-3 and oscillation]{Understanding Mamba-3\\Through Oscillatory Dynamics}
\subtitle{Paper reading and research discussion}
\author{Paper authors: Shutian Liang and Yue Song\\[1mm]\small College of AI, Tsinghua University}
\date{arXiv:2610.13341v1 \quad $\cdot$ \quad Reading: 19 October 2026}
\hypersetup{pdftitle={Understanding Mamba-3 Through Oscillatory Dynamics -- Paper Reading},pdfauthor={Paper reading notes},pdfsubject={Based on Liang and Song, arXiv:2610.13341v1}}

\begin{document}

\begin{frame}
\titlepage
\vspace{-7mm}
\begin{center}\color{AcademicTeal}
Decay controls retention; phase shapes accumulation and access.
\end{center}
\papersource{Reading presentation based on the paper; not an author presentation. \href{https://arxiv.org/abs/2610.13341v1}{arxiv.org/abs/2610.13341v1}}
\end{frame}

\begin{frame}[t]{Rotation changes the direction of recurrent memory}
\figfull{fig-01-geometry.pdf}{4.7cm}
\par\medskip\raggedright
\alert{Question:} What does learned rotation contribute beyond retaining state magnitude?
\papersource{Paper, Secs. 1--2, Fig. 1, PDF pp. 1--3. Fixed coefficients and no input in the trajectory illustration.}
\end{frame}

\begin{frame}[t]{A rotary pair separates retention from phase}
\[
h_t=\bar A_t h_{t-1}+\bar B_t x_t,\qquad y_t=C_t^\top h_t
\]
\begin{columns}[T,onlytextwidth]
\begin{column}{0.48\textwidth}
\begin{block}{Mamba-2: shared decay within a head}
\[
\bar A_t=\alpha_t I
\]
Homogeneous propagation preserves direction and scales magnitude.
\end{block}
\end{column}
\begin{column}{0.49\textwidth}
\begin{block}{Mamba-3: a two-dimensional rotary pair}
\[
\bar A_t=\rho_tR(\theta_t)
\]
$\rho_t$: radial retention.\\
$\theta_t$: input-dependent phase increment.
\end{block}
\end{column}
\end{columns}
\medskip
\begin{itemize}
\item SISO and MIMO share the decay--rotation structure.
\item Their write/read parameterizations differ; interventions evaluate both.
\end{itemize}
\papersource{Paper, Sec. 2.1, Eqs. (2), (4)--(5), PDF pp. 2--3.}
\end{frame}

\begin{frame}[t]{Each frozen interval has an exact continuous flow}
\[
z_{t+1}=\rho_tR(\theta_t)z_t,\qquad z_t=(H_t,P_t)^\top
\]
\[
a_t=-\frac{\log\rho_t}{\Delta_t},\quad
\nu_t=\frac{\theta_t}{\Delta_t},\quad
J=\begin{bmatrix}0&-1\\1&0\end{bmatrix},\quad J^2=-I
\]
\begin{block}{Exact matrix exponential}
\[
A_t^{\rm rot}=-a_tI+\nu_tJ
\quad\Longrightarrow\quad
e^{A_t^{\rm rot}\Delta_t}=e^{-a_t\Delta_t}R(\nu_t\Delta_t)
=\rho_tR(\theta_t).
\]
\end{block}
\begin{itemize}
\item Assumptions: $0<\rho_t<1$, $\Delta_t>0$, coefficients fixed within the interval.
\item This step analyzes homogeneous propagation, with input injection separated.
\end{itemize}
\papersource{Paper, Sec. 3, Eqs. (9)--(13), PDF p. 4; App. C.1, PDF p. 21.}
\end{frame}

\begin{frame}[t]{Nonzero phase gives an underdamped realization}
\[
\dot H=-aH-\nu P,\qquad \dot P=\nu H-aP
\]
\[
\boxed{\ddot H+2a\dot H+(a^2+\nu^2)H=0}
\]
\[
\omega_0=\sqrt{a^2+\nu^2},\qquad
\zeta=\frac{a}{\sqrt{a^2+\nu^2}}<1,\qquad \omega_d=|\nu|
\]
\begin{itemize}
\item $a$ determines the decay envelope; $|\nu|$ determines oscillation frequency.
\item The sign of $\nu$ determines rotation direction.
\item For $\nu\ne0$, recover the second coordinate as $P=-(\dot H+aH)/\nu$.
\end{itemize}
\papersource{Paper, Sec. 3, Eqs. (14)--(17), PDF p. 4. Equivalence holds within a frozen interval and requires $\nu\ne0$.}
\end{frame}

\begin{frame}[t]{A shared second-order structure isolates damping regime}
\[
A_t(\kappa_t)=
\begin{bmatrix}0&1\\-(a_t^2-\kappa_t)&-2a_t\end{bmatrix},
\qquad \lambda_\pm=-a_t\pm\sqrt{\kappa_t}
\]
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{lll}
\toprule
Regime & Constraint & Eigenvalues \\
\midrule
Underdamped & $\kappa_t=-\nu_t^2<0$ & Complex conjugate \\
Critical & $\kappa_t=0$ & Repeated real \\
Overdamped & $0<\kappa_t<a_t^2$ & Distinct negative real \\
\bottomrule
\end{tabular}
\end{center}
\begin{itemize}
\item All use $\bar A_t=\exp(\Delta_tA_t)$ and coupled two-dimensional states.
\item State dimension, input/readout and input-update rule are held unchanged.
\end{itemize}
\papersource{Paper, Sec. 4.1, Eqs. (18)--(21), PDF p. 5; App. D.1, PDF p. 24.}
\end{frame}

\begin{frame}[t]{Underdamping preserves relative two-dimensional geometry}
\textbf{Theory setting:} homogeneous dynamics, fixed coefficients and fixed $\omega_0$.
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{lll}
\toprule
Regime & Dominant decay rate & Long-run geometry \\
\midrule
Underdamped & $a<\omega_0$ & Two equally decaying modes \\
Critical & $\omega_0$ & Rescaled operator tends to rank one \\
Overdamped & $a-\sqrt{a^2-\omega_0^2}$ & Slow mode dominates \\
\bottomrule
\end{tabular}
\end{center}
\[
z_n=e^{-an\Delta}R(n\nu\Delta)z_0
\]
\alert{In rotation coordinates:} a common scale factor preserves relative angles.
\par\medskip
Critical damping has the fastest asymptotic exponential attenuation at fixed $\omega_0$.
\papersource{Paper, Sec. 4.2, Eqs. (22)--(27), PDF p. 5; App. C.2, PDF pp. 22--23. Overdamped statement assumes a nonzero slow-mode component.}
\end{frame}

\begin{frame}[t]{Input-dependent underdamping wins on state tracking}
\begin{center}
\renewcommand{\arraystretch}{1.13}
\begin{tabular}{lrrr}
\toprule
Model & Parity & Mod 3 & Arithmetic$^*$ \\
\midrule
Mamba-2 & 3.91 & 0.04 & 36.76 \\
Mamba-3, no rotation & 3.03 & 0.78 & 48.95 \\
Mamba-3, static rotation & 4.05 & 0.85 & 25.49 \\
Overdamped & 3.27 & 0.58 & 81.60 \\
Critically damped & 3.22 & 0.32 & 1.97 \\
\textbf{Underdamped} & \textbf{100.00} & \textbf{95.26} & \textbf{95.68} \\
\bottomrule
\end{tabular}
\end{center}
\begin{itemize}
\item Training curriculum: maximum length 40 $\rightarrow$ 160; test length \textbf{224}.
\item Shared $d_{\rm model}=d_{\rm state}=64$; validation selects among eight learning rates.
\item Static rotation performs poorly: phase must respond to the incoming symbol.
\end{itemize}
\papersource{Paper, Table 2, PDF p. 6; App. D.1, PDF p. 24. Scaled Accuracy (\%) $\uparrow$; v1 does not spell out its scaling formula. $^*$Without brackets.}
\end{frame}

\begin{frame}[t]{Selected state pairs learn conditional modular rotations}
\figfull{fig-05-modular-state.pdf}{3.9cm}
\par\medskip\raggedright
\begin{itemize}
\item Parity: input 0 $\rightarrow 0.50^\circ$; input 1 $\rightarrow 179.51^\circ$.
\item Mod 3: input 0 $\rightarrow 0.21^\circ$; input 1 $\rightarrow 119.78^\circ$.
\item These selected pairs resemble an ideal count-of-ones phase code: $360^\circ/m$ per 1.
\end{itemize}
\papersource{Paper, App. E.1, Fig. 5, PDF p. 27. Selected state pairs, not a claim about every channel.}
\end{frame}

\begin{frame}[t]{Context-dependent prediction is more phase-sensitive}
\figfull{fig-02-mimo-detail.pdf}{4.7cm}
\par\smallskip\raggedright
\begin{itemize}
\item Remove phase in a contiguous layer window; color is accuracy change in pp.
\item Modest effects on Relations; broad losses on LAMBADA. MIMO panels shown; SISO has the same qualitative pattern.
\end{itemize}
\papersource{Paper, Sec. 5.1, Fig. 2 (MIMO panels), PDF pp. 6--7. Vector crop; original axes retained and original identical colorbars repeated. Official 1.5B checkpoints.}
\end{frame}

\begin{frame}[t]{From-scratch training tests what rotation helps models learn}
\begin{center}
\renewcommand{\arraystretch}{1.18}
\begin{tabular}{lcc}
\toprule
Configuration & 187M & 444M \\
\midrule
Layers / model width & 12 / 768 & 24 / 1024 \\
State dimension / head dimension & 128 / 64 & 128 / 64 \\
MIMO rank / chunk size & 4 / 16 & 4 / 16 \\
Rotary channel fraction & 0 versus 0.5 & 0 versus 0.5 \\
Peak learning rate & $3\times10^{-3}$ & $1.5\times10^{-3}$ \\
\bottomrule
\end{tabular}
\end{center}
\begin{itemize}
\item Each model: \textbf{100B FineWeb-Edu tokens}, Llama-3.1 tokenizer, 2048-token training context.
\item AdamW, cosine schedule, global batch 256; BF16 computation with FP32 parameters/states.
\item Evaluate PG-19, LongBench-E, standard tasks and four additional corpora.
\end{itemize}
\papersource{Paper, Sec. 5.2, PDF p. 7; App. D.3, Table 10, PDF pp. 24--25.}
\end{frame}

\begin{frame}[t]{Oscillation lowers PG-19 perplexity at both scales}
\begin{columns}[T,onlytextwidth]
\begin{column}{0.47\textwidth}
\centering\includegraphics[width=\linewidth,height=5.6cm,keepaspectratio]{figures/fig-03-pg19.pdf}
\end{column}
\begin{column}{0.50\textwidth}
\begin{itemize}
\item Blue: no oscillation. Orange: oscillation.
\item Train at 2K; evaluate up to 64K.
\item Reset state per window; score the \textbf{last 100 tokens}.
\item Lower PPL at both scales; larger long-context gap at 444M.
\item No-oscillation models remain stable.
\end{itemize}
\medskip
\small Compare variants at the same length: eligible books and targets vary with window length.
\end{column}
\end{columns}
\papersource{Paper, Fig. 3, PDF p. 7; Sec. 5.3, PDF p. 9; App. D.3, PDF pp. 24--25. PG-19 lengths use $K=1000$.}
\end{frame}

\begin{frame}[t]{LongBench-E gains appear at 444M, with task trade-offs}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{lrrr}
\toprule
Scale & No oscillation & Oscillation & Difference \\
\midrule
187M & 9.76 & 9.53 & $-0.23$ \\
444M & 10.61 & \textbf{13.22} & \alert{$+2.61$} \\
\bottomrule
\end{tabular}
\end{center}
\begin{itemize}
\item Scores average 13 tasks; this is not a single accuracy metric.
\item At 444M, major gains: SAMSum $+9.25$, code completion $+9.00$ points.
\item Passage Count $-2.39$ and Passage Retrieval $-1.24$ points.
\end{itemize}
\begin{block}{Interpretation}
Improved language-model perplexity at both scales does not imply uniform downstream gains.
\end{block}
\papersource{Paper, Sec. 5.2, Table 3, PDF p. 7. Differences calculated from reported scores. Full task names: Table 11, PDF p. 27.}
\end{frame}

\begin{frame}[t]{Standard benchmarks show modest aggregate changes}
\begin{center}
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{llrrr}
\toprule
Scale & Dynamics & FW-Edu PPL & LAMBADA PPL & Avg. acc. \\
\midrule
187M & No oscillation & 16.32 & 38.04 & 43.63 \\
187M & Oscillation & \textbf{16.18} & \textbf{32.97} & 43.43 \\
\midrule
444M & No oscillation & 12.68 & 18.38 & 50.18 \\
444M & Oscillation & \textbf{12.60} & \textbf{17.19} & \textbf{51.08} \\
\bottomrule
\end{tabular}
\end{center}
\begin{itemize}
\item PPL improves at both scales; lower is better.
\item Average accuracy changes by $-0.20$ pp at 187M and $+0.90$ pp at 444M.
\item The seven-task mean mixes the paper's task-specific \texttt{acc} / \texttt{acc\_n} metrics.
\end{itemize}
\papersource{Paper, Table 4, PDF p. 8. Seven tasks: LAMBADA, HellaSwag, PIQA, ARC-E, ARC-C, WinoGrande and OpenBookQA.}
\end{frame}

\begin{frame}[t]{Learned phase varies over multiple timescales}
\figfull{fig-04-phase.pdf}{4.7cm}
\par\smallskip\raggedright
\small Cosine/sine of accumulated phase; each panel uses its own token-offset range.\\
Selected near the 10th, 50th and 99th phase-travel percentiles among channels completing a cycle.
\papersource{Paper, Fig. 4, PDF p. 8; App. D.4, PDF p. 25. PG-19 validation, first 16,000 tokens.}
\end{frame}

\begin{frame}[t]{Interventions perturb distinct parts of a learned computation}
\begin{itemize}
\item \textbf{Phase shuffle:} reassign increments among rotary pairs within each layer, head and token, before accumulation.
\item \textbf{Phase removal:} set raw angle projections to zero.
\item \textbf{Phase reversal:} negate every phase increment.
\item \textbf{Decay permutation:} cyclically reassign decay coefficients $A$ across heads within a layer, keeping the receiving head's $\Delta$.
\end{itemize}
\[
\widetilde{(A\Delta)}_{\ell,h,t}
=A_{\ell,\pi_\ell(h),t}\Delta_{\ell,h,t}
\]
\alert{Protocol:} official 1.5B checkpoints; all layers and the whole window; matched windows and scored tokens within each context length.
\papersource{Paper, Sec. 5.3, Table 5, PDF p. 8; App. D.4, PDF pp. 25--26. Phase interventions preserve radial decay. Perturbation strengths are not matched.}
\end{frame}

\begin{frame}[t]{SISO becomes increasingly sensitive to phase disruption}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{lrrr}
\toprule
Intervention & 2K & 16K & 64K \\
\midrule
Native PPL & 21.09 & 21.41 & 20.60 \\
\midrule
Decay permutation & $5.43\times$ & $5.91\times$ & $6.08\times$ \\
Phase shuffle & $2.38\times$ & $4.49\times$ & $7.25\times$ \\
Phase removal & $3.22\times$ & $13.34\times$ & \textbf{$41.33\times$} \\
Phase reversal & $7.22\times$ & $27.93\times$ & \textbf{$74.10\times$} \\
\bottomrule
\end{tabular}
\end{center}
\medskip
Ratios are $\mathrm{PPL}_{\rm intervened}/\mathrm{PPL}_{\rm native}$; larger means more damage.
\par\medskip
At 64K, phase removal gives PPL \textbf{851.49}; reversal gives \textbf{1526.52}.
\papersource{Paper, Table 5, PDF p. 8. Frozen official 1.5B SISO checkpoint; this is an inference-time intervention.}
\end{frame}

\begin{frame}[t]{MIMO has a different pattern of phase sensitivity}
\begin{center}
\renewcommand{\arraystretch}{1.4}
\begin{tabular}{lrrr}
\toprule
Intervention & 2K & 16K & 64K \\
\midrule
Native PPL & 21.12 & 21.47 & 20.52 \\
\midrule
Decay permutation & $1.22\times$ & $1.21\times$ & $1.21\times$ \\
Phase shuffle & $3.05\times$ & $3.28\times$ & $3.20\times$ \\
Phase removal & \textbf{$9.34\times$} & \textbf{$14.92\times$} & \textbf{$15.08\times$} \\
Phase reversal & $3.52\times$ & $3.67\times$ & $3.60\times$ \\
\bottomrule
\end{tabular}
\end{center}
\begin{itemize}
\item Removal is most damaging; reversal and shuffle show weaker length dependence.
\item Phase interventions are more damaging than this decay-permutation control.
\end{itemize}
\papersource{Paper, Table 5, PDF p. 8. Ratios relative to native PPL at each length; official 1.5B MIMO checkpoint.}
\end{frame}

\begin{frame}[t]{Removing phase increases alignment and state magnitude}
\[
S_t=P_t+U_t,\qquad
\|S_t\|_F^2=\|P_t\|_F^2+\|U_t\|_F^2+2\langle P_t,U_t\rangle_F
\]
\begin{center}
\renewcommand{\arraystretch}{1.25}
\begin{tabular}{llcc}
\toprule
Model & Length & Cosine: native $\rightarrow$ remove & RMS: native $\rightarrow$ remove \\
\midrule
SISO & 16K & $0.293\rightarrow0.429$ & $7.635\rightarrow8.899$ \\
SISO & 64K & $0.294\rightarrow0.474$ & $7.621\rightarrow9.437$ \\
MIMO & 16K & $0.241\rightarrow0.488$ & $10.073\rightarrow12.022$ \\
MIMO & 64K & $0.242\rightarrow0.499$ & $10.078\rightarrow12.047$ \\
\bottomrule
\end{tabular}
\end{center}
\begin{itemize}
\item Larger positive alignment favors constructive accumulation.
\item Supports the authors' explanation that rotation regulates how updates combine.
\end{itemize}
\papersource{Paper, Table 6, PDF p. 9; App. D.4, PDF p. 26. Norm expansion is an explanatory identity. Statistics exclude the first 2,000 tokens; RMS is not dimension-normalized.}
\end{frame}

\begin{frame}[t]{Phase can control accumulation without changing retention}
\textbf{Fixed-coefficient toy system with constant input:}
\[
z_t=\rho e^{i\theta}z_{t-1}+u,\qquad 0\le\rho<1,\quad z_0=0
\]
\[
\boxed{\|z_\infty\|=\frac{\|u\|}{|1-\rho e^{i\theta}|}}
\qquad
|1-\rho e^{i\theta}|^2=(1-\rho)^2+2\rho(1-\cos\theta)
\]
\begin{block}{Illustrative calculation, not an experimental result}
For $\rho=0.99$ and $\|u\|=1$:
\[
\theta=0:\quad \|z_\infty\|=100
\qquad\qquad
\theta=\pi/2:\quad \|z_\infty\|\approx0.711
\]
\end{block}
Rotation changes accumulation while preserving the radial retention factor.
\papersource{Paper, App. C.3, Eqs. (115)--(120), PDF p. 23. Denominator expansion and numerical example added for explanation.}
\end{frame}

\begin{frame}[t]{Writing and readout need compatible phase coordinates}
\textbf{64K windows:} native history; intervene only during the last 100 tokens.
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{lrr}
\toprule
Tail intervention & SISO PPL & MIMO PPL \\
\midrule
Native & 20.60 & 20.52 \\
Remove rotation from $B$ & 186.28 ($9.04\times$) & 100.41 ($4.89\times$) \\
Remove rotation from $C$ & 250.90 ($12.18\times$) & 117.68 ($5.74\times$) \\
Remove rotation from both & 42.79 ($2.08\times$) & 59.46 ($2.90\times$) \\
\bottomrule
\end{tabular}
\end{center}
\begin{itemize}
\item Write/read vectors remain present; only their rotary transformations change.
\item Joint removal partially restores performance, supporting phase compatibility.
\item Earlier history still uses native coordinates, which may explain a residual gap.
\end{itemize}
\papersource{Paper, Table 7, PDF p. 9; App. D.4, PDF p. 26. Native state is carried into the intervention region without resetting.}
\end{frame}

\begin{frame}[t]{Phase shapes computation, accumulation and access}
\begin{enumerate}
\item \textbf{Compute:} input-dependent underdamping supports state tracking beyond two-coordinate coupling.
\medskip
\item \textbf{Combine:} rotation changes alignment between retained history and incoming writes; both scales improve long-context PPL.
\medskip
\item \textbf{Access:} disrupting phase or write/read compatibility damages pretrained prediction.
\end{enumerate}
\begin{block}{How to read the evidence}
Inference-time removal tests learned dependence. From-scratch training tests architectural benefit. Non-oscillatory models can still extrapolate stably.
\end{block}
\papersource{Liang and Song, Understanding Mamba-3 Through Oscillatory Dynamics, arXiv:2610.13341v1. Secs. 4--6, PDF pp. 5--9.}
\end{frame}

\appendix

\begin{frame}[t]{Backup: phase removal is not critical damping}
\begin{columns}[T,onlytextwidth]
\begin{column}{0.48\textwidth}
\begin{block}{Remove phase in rotary coordinates}
\[
\nu=0\quad\Rightarrow\quad A^{\rm rot}=-aI
\]
\[
e^{A^{\rm rot}\Delta}=e^{-a\Delta}I
\]
Two uncoupled equal-rate decays.
\end{block}
\end{column}
\begin{column}{0.49\textwidth}
\begin{block}{Critical second-order construction}
\[
A(0)=\begin{bmatrix}0&1\\-a^2&-2a\end{bmatrix}
\]
\[
e^{A(0)\Delta}=e^{-a\Delta}(I+\Delta N)
\]
$N\ne0$ and $N^2=0$: a nontrivial Jordan block.
\end{block}
\end{column}
\end{columns}
\medskip
\begin{itemize}
\item The coordinate map $P=-(\dot H+aH)/\nu$ becomes singular at $\nu=0$.
\item Fixed-frequency attenuation theory also differs from the learned spectral-family comparison.
\end{itemize}
\papersource{Reading analysis from Sec. 3, Eqs. (14)--(17), PDF p. 4, and Sec. 4.1--4.2, PDF p. 5.}
\end{frame}

\begin{frame}[t]{Backup: LongBench-E gains are concentrated at 444M}
\begin{columns}[T,onlytextwidth]
\begin{column}{0.49\textwidth}
\textbf{Largest gains (score points)}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{lr}
\toprule
Task & Change \\
\midrule
SAMSum & $+9.25$ \\
Long Code Completion & $+9.00$ \\
TREC & $+7.33$ \\
RepoBench-P & $+7.12$ \\
TriviaQA & $+2.27$ \\
\bottomrule
\end{tabular}
\end{center}
\end{column}
\begin{column}{0.48\textwidth}
\textbf{Regressions (score points)}
\begin{center}
\renewcommand{\arraystretch}{1.35}
\begin{tabular}{lr}
\toprule
Task & Change \\
\midrule
Passage Count & $-2.39$ \\
Passage Retrieval & $-1.24$ \\
MultiNews & $-0.29$ \\
Qasper & $-0.22$ \\
\bottomrule
\end{tabular}
\end{center}
\end{column}
\end{columns}
\medskip
\alert{Overall:} $10.61\rightarrow13.22$; synthetic state-tracking success does not imply gains on every long-context task.
\papersource{Paper, Table 3, PDF p. 7. Selected rows; changes calculated as oscillatory minus non-oscillatory scores.}
\end{frame}

\begin{frame}[t]{Backup: additional corpora show scale-dependent benefits}
\figfull{fig-06-other-datasets.pdf}{4.6cm}
\par\smallskip\raggedright
\small 187M loses on MATH-Hard; 444M improves on all four datasets.\\
The x-axis indexes consecutive 1024-token bins of 4096-token streams; state is carried across bins.
\papersource{Paper, App. E.2, Fig. 6, PDF pp. 27--28; evaluation protocol: App. D.3, PDF p. 25.}
\end{frame}

\begin{frame}[t]{Backup: validation PPL separates late in pretraining}
\figfull{fig-07-training.pdf}{4.7cm}
\par\medskip\raggedright
Both scales end with lower validation PPL for the oscillatory model. Insets show the final 20,000 training steps.
\papersource{Paper, App. E.3, Fig. 7, PDF p. 28. Horizontal axis: training steps; vertical axis: validation perplexity.}
\end{frame}

\begin{frame}[t]{Backup: 187M long-context gains develop during training}
\figfull{fig-08-training-187m.pdf}{5.6cm}
\papersource{Paper, App. E.3, Fig. 8, PDF p. 29. PG-19 PPL at training checkpoints; panels: 2K, 8K, 16K, 32K and 64K.}
\end{frame}

\begin{frame}[t]{Backup: 444M separates more at long evaluation lengths}
\figfull{fig-09-training-444m.pdf}{5.6cm}
\papersource{Paper, App. E.3, Fig. 9, PDF p. 29. Blue: no oscillation; orange: oscillation. Training uses a 2K context.}
\end{frame}

\begin{frame}[t]{Backup: frozen spectra connect recurrent architectures}
\[
S_t=T_tS_{t-1}+U_t,\quad
H=I-\beta kk^\top,\quad D=\operatorname{Diag}(\boldsymbol\alpha)
\]
\begin{center}
\renewcommand{\arraystretch}{1.12}
\begin{tabular}{lll}
\toprule
Model & Frozen transition & Modal structure \\
\midrule
Linear Attention & $I$ & Preserved modes \\
DeltaNet & $H$ & Decay plus preserved modes \\
Gated DeltaNet & $\alpha H$ & Two non-oscillatory decay rates \\
KDA & $HD$ & Positive real spectrum \\
Complex KDA & $HD$, signed gates & Can admit oscillatory modes \\
Momentum DeltaNet & Augmented $(S,M)$ & Can admit oscillatory modes \\
\bottomrule
\end{tabular}
\end{center}
\small Unit-norm keys; ordinary DeltaNet/Gated DeltaNet/KDA use $0<\beta<1$ and positive decay gates. These are frozen-interval statements.
\papersource{Paper, App. A, Table 8, PDF pp. 15--20. DeltaNet/Gated DeltaNet pairings share one key-direction mode; they are not independent state pairs.}
\end{frame}

\begin{frame}[t]{Backup: signed gates and momentum allow complex modes}
\textbf{Complex KDA construction}
\[
T_{\alpha,\theta}=R(\theta)\operatorname{Diag}(-\alpha,1),\quad -1<\alpha<0
\]
\[
\mathcal D=(1-\alpha)^2\cos^2\theta+4\alpha<0
\quad\Rightarrow\quad |\lambda_\pm|=\sqrt{-\alpha}<1
\]
\medskip
\textbf{Momentum DeltaNet: key-direction state--momentum block}
\[
B=\begin{bmatrix}\alpha(1-\beta\eta)&-\beta\mu\\\alpha\eta&\mu\end{bmatrix},
\quad \tau=\alpha(1-\beta\eta)+\mu
\]
\[
\tau^2<4\alpha\mu,\quad 0<\alpha,\mu<1
\quad\Rightarrow\quad |\lambda_\pm|=\sqrt{\alpha\mu}<1
\]
These conditions admit an underdamped realization in suitable real modal coordinates.
\papersource{Paper, App. A.6--A.7, PDF pp. 19--20; App. C.4, PDF pp. 23--24. Possibility conditions, not guarantees for every update.}
\end{frame}

\begin{frame}[t]{Backup: what the state measurements average}
\[
c_t=\frac{\langle P_t,U_t\rangle_F}{\|P_t\|_F\|U_t\|_F+10^{-12}},
\qquad P_t=\alpha_tS_{t-1}
\]
\begin{itemize}
\item PG-19 validation books with at least 64,000 tokens; one uninterrupted trajectory per condition, starting from zero state.
\item Cosine: average over positions $[2000,L)$, median over $24\times64$ layer--head pairs, then equal-weight mean across books.
\item RMS: average squared Frobenius norms over positions, median over layer--head pairs, square root, then mean across books.
\item All state coordinates are included; there is no dimension normalization.
\end{itemize}
\medskip
\alert{Mechanistic boundary:} altered alignment and norms support an accumulation account, while encoding and readout also change under phase removal.
\papersource{Paper, App. D.4, PDF p. 26; Table 6, PDF p. 9. $P_t$ here is propagated matrix state, not the scalar coordinate in Sec. 3.}
\end{frame}

\end{document}