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\title[Effective recurrence]{What Makes Recurrence Effective\\in Looped Language Models?}
\subtitle{Architecture, state history, and inference budgets}
\author{\small Paper authors: Xinlin Zhuang, Siyuan Wang,\\Imran Razzak, and Weiyang Liu}
\institute{Independent research-paper discussion}
\date{arXiv:2609.36636v1\quad |\quad 15 October 2026}

\begin{document}

\begin{frame}
  \titlepage
  \vspace{-4mm}
  \begin{center}\color{AcademicTeal}
  Extra loops help when architecture and conditioning sustain useful computation.
  \end{center}
\end{frame}

\begin{frame}{A fixed training horizon leaves two deployment gaps}
  \[
  \underbrace{P}_{\text{input}}\ \longrightarrow\
  \underbrace{F\circ\cdots\circ F}_{r\text{ shared-core executions}}
  \ \longrightarrow\ \underbrace{C}_{\text{output}}
  \]
  \begin{center}
  \begin{tabular}{lll}
  \toprule
  Inference regime & Loop count & Question \\
  \midrule
  Under-unrolling & $r<K$ & Can an early state be read out? \\
  Training horizon & $r=K$ & Does the trained path work? \\
  Loop extrapolation & $r>K$ & Is extra computation still useful? \\
  \bottomrule
  \end{tabular}
  \end{center}
  \begin{block}{The empirical gap}
  Training supervises the final output at a fixed $K$. It does not directly optimize either shorter or longer execution.
  \end{block}
  \papersource{Paper, Secs. 1--2, PDF pp. 1--4. Diagram is a teaching schematic.}
\end{frame}

\begin{frame}{Physical depth and executed depth must be separated}
  \[
  L_{\mathrm{phys}}=p+s+c,\qquad D=L(r)=p+rs+c.
  \]
  \begin{itemize}
    \item $p$: Prelude layers; $s$: shared-core layers; $c$: Coda layers.
    \item $K$: training loops; $r$: inference loops.
    \item We consistently write configurations as $p+s\times K+c$.
  \end{itemize}
  \medskip
  \begin{center}
  \begin{tabular}{lcc}
    \toprule
    Configuration & Physical depth & Training depth \\
    \midrule
    BaseLoop $4\times5$ & 4 & 20 \\
    CoreLoop $0+2\times9+2$ & 4 & 20 \\
    NonLoop $20\times1$ & 20 & 20 \\
    \bottomrule
  \end{tabular}
  \end{center}
  \papersource{Paper, Sec. 2.1 and Table 2, PDF pp. 2--3, 18. Sec. 3 inconsistently swaps $s$ and $K$.}
\end{frame}

\begin{frame}{The study trains small models from scratch}
  \begin{center}
  \begin{tabular}{lcc}
  \toprule
  & Llama-style family & Qwen-style family \\
  \midrule
  Non-recurrent reference & 20 layers, 1.007B & 28 layers, 0.596B \\
  Hidden width $d$ & 1536 & 1024 \\
  Effective training depth & 20 & 28 \\
  Training tokens & 20.15B & 12.00B \\
  Representative BaseLoop & $4\times5$, 516.70M & $4\times7$, 218.51M \\
  \bottomrule
  \end{tabular}
  \end{center}
  \begin{itemize}
    \item FineWeb-Edu; packed 2048-token sequences; next-token loss.
    \item Fixed $K$, full recurrent backpropagation, final-output supervision.
    \item Muon + AdamW parameter groups; one random seed (42).
  \end{itemize}
  \papersource{Paper, App. C.1--C.2, Tables 1--3, PDF pp. 17--21. Family names identify architectures, not pretrained weights.}
\end{frame}

\begin{frame}{Separate capability groups before reading the average}
  \begin{columns}[T,onlytextwidth]
    \begin{column}{0.43\textwidth}
      \begin{block}{Knowledge: 3 tasks}
      SciQ, ARC-Easy, PIQA
      \end{block}
      \small Operational grouping based on the authors' view of computational demand.
    \end{column}
    \begin{column}{0.54\textwidth}
      \begin{block}{Reasoning: 8 tasks}
      ARC-Challenge, WinoGrande, OpenBookQA, HellaSwag, CommonsenseQA, ProofWriter, CLUTRR, BBH
      \end{block}
    \end{column}
  \end{columns}
  \medskip
  \[
    S_{\mathrm{Overall}}=\frac{3\Know+8\Reas}{11}
  \]
  \begin{itemize}
    \item Zero-shot accuracy or length-normalized accuracy; higher is better.
    \item BBH uses 14 selected subtasks, excluding near-chance tasks.
  \end{itemize}
  \papersource{Paper, App. C.3, PDF pp. 21--22. Overall weighting is inferred from and agrees with the result tables.}
\end{frame}

\begin{frame}{Extra loops improve reasoning while knowledge declines}
  \begin{columns}[c,onlytextwidth]
    \begin{column}{0.42\textwidth}
      \includegraphics[width=\linewidth,height=0.70\textheight,keepaspectratio]{figures/fig01a-task-tradeoff.pdf}
    \end{column}
    \begin{column}{0.55\textwidth}
      Llama BaseLoop $4\times5$
      \medskip
      \begin{tabular}{lrr}
      \toprule
      Score (\%) & $D=20$ & $D=40$ \\
      \midrule
      Knowledge & 62.80 & 52.11 \\
      Reasoning & 28.52 & 31.49 \\
      Overall & 37.87 & 37.11 \\
      \bottomrule
      \end{tabular}
      \begin{block}{The trade-off}
      Reasoning: $+2.97\pp$\\
      Knowledge: $-10.69\pp$
      \end{block}
      No test-time parameter updates.
    \end{column}
  \end{columns}
  \papersource{Paper, Fig. 1(a), PDF p. 3; Table 8, p. 29. The Sec. 3 prose mislabels this case as $2\times10$.}
\end{frame}

\begin{frame}{Harder instances do not consistently gain more}
  \begin{columns}[c,onlytextwidth]
    \begin{column}{0.62\textwidth}
      \includegraphics[width=\linewidth,height=0.68\textheight,keepaspectratio]{figures/fig01bc-difficulty.pdf}
    \end{column}
    \begin{column}{0.35\textwidth}
      Llama $2\times10$\\[2mm]
      $D=20\rightarrow24$:
      \begin{itemize}
        \item ProofWriter depths 3--5 gain 7.5--8.0 points.
        \item CLUTRR gains concentrate on short chains.
        \item Long-chain accuracy remains low and uneven.
      \end{itemize}
    \end{column}
  \end{columns}
  \papersource{Paper, Fig. 1(b,c), PDF p. 3; Sec. 3, p. 4. Cells show absolute accuracy (\%).}
\end{frame}

\begin{frame}{Equal training depth produces different scaling curves}
  \begin{columns}[c,onlytextwidth]
    \begin{column}{0.46\textwidth}
      \includegraphics[width=\linewidth,height=0.70\textheight,keepaspectratio]{figures/fig01d-allocation.pdf}
    \end{column}
    \begin{column}{0.51\textwidth}
      \begin{itemize}
        \item Shallow core: $2\times10$ peaks early.
        \item Deep core: $10\times2$ gains little from more loops.
        \item $5\times4$: 31.88 reasoning at $D=30$.
        \item NonLoop: 30.25 at its native $D=20$.
      \end{itemize}
      \begin{block}{Budget interpretation}
      Training depth is matched. The higher recurrent score uses 50\% more inference layer executions.
      \end{block}
    \end{column}
  \end{columns}
  \papersource{Paper, Fig. 1(d) and Sec. 3, PDF pp. 3--5. Physical depth and parameter count vary across these curves.}
\end{frame}

\begin{frame}{Boundary layers have different roles at different budgets}
  \begin{center}
  \begin{tabular}{lp{9.5cm}}
    \toprule
    Regime & Observation in the Llama comparisons \\
    \midrule
    Fewer loops & Coda often protects knowledge; e.g., $0+2\times9+2$. \\
    Training depth & Differences among boundary allocations become smaller. \\
    More loops & Prelude-heavy layouts can better sustain reasoning. \\
    \bottomrule
  \end{tabular}
  \end{center}
  \begin{block}{An informative exception}
  On Qwen at $D=56$, BaseLoop has higher Overall and Reasoning scores than all three tested CoreLoop layouts.
  \end{block}
  \papersource{Paper, Fig. 2 and Sec. 4, PDF pp. 5--6; App. D.2 and Fig. 9, pp. 26--27.}
\end{frame}

\begin{frame}{Small updates can coexist with continued state drift}
  \begin{columns}[T,onlytextwidth]
    \begin{column}{0.49\textwidth}
      \begin{block}{Local movement}
      Angular distance between loops decreases. Relative update norms also decrease:
      \[
      \frac{\|h^{t+1}-h^t\|_2}{\|h^t\|_2}.
      \]
      \end{block}
    \end{column}
    \begin{column}{0.48\textwidth}
      \begin{block}{Accumulated change}
      Normalized state variance continues growing:
      \[
      \widetilde V(t)=\frac{V(h^t)}{V(h^K)}.
      \]
      \end{block}
    \end{column}
  \end{columns}
  \medskip
  $V$ measures within-token dispersion across hidden channels, then averages over tokens.
  \begin{block}{Design implication}
  Update size alone is insufficient to decide whether another loop is useful.
  \end{block}
  \papersource{Paper, Fig. 3 and Sec. 4, PDF pp. 5--6; App. C.4, Eqs. (18)--(21), p. 23. Norm formula shown schematically.}
\end{frame}

\begin{frame}{Repeated static input injection can harm extrapolation}
  \[
  h_{\ell+1}=F_\theta\!\left(\mathcal I(h_\ell,h_0)\right)
  \]
  \begin{center}
  \begin{tabular}{llc}
  \toprule
  Injection & Map & Parameters \\
  \midrule
  Scalar & $h+\alpha h_0$ & 1 \\
  Channel-wise & $a\odot h+b\odot h_0$ & $2d$ \\
  Residual channel & $(\mathbf1+\delta_a)\odot h+b\odot h_0$ & $2d$ \\
  Dense & $W_hh+W_0h_0$ & $2d^2$ \\
  \bottomrule
  \end{tabular}
  \end{center}
  \begin{block}{Qwen $4\times7$, Dense injection}
  Knowledge falls from 58.77 at $D=28$ to 32.28 at $D=84$.\\
  BaseLoop retains 52.29 at $D=84$.
  \end{block}
  \papersource{Paper, Eq. (2), PDF p. 7; Table 10, p. 30. Every injection starts as the identity on the current state.}
\end{frame}

\begin{frame}{History injection conditions on recent state differences}
  \[
  h_{\ell+1}=F_\theta\!\left(h_\ell+
  \sum_{j=1}^{m_\ell}\mathcal B_j(h_{\ell-j}-h_\ell)\right),
  \quad m_\ell=\min\{w,\max(\ell-1,0)\}.
  \]
  \begin{itemize}
    \item $w$: history window; $j$: relative lag; $\mathcal B_j$: learned lag-specific map.
    \item Exclude $h_0$: only completed recurrent states enter the history.
    \item First history-dependent update:
    \[
    h_3=F_\theta\bigl(h_2+\mathcal B_1(h_1-h_2)\bigr).
    \]
    \item Zero-initialize history maps; retain gradients through buffered states.
  \end{itemize}
  \begin{block}{Teaching intuition, scalar $w=1$}
  $(1-\beta)h_\ell+\beta h_{\ell-1}$ can damp or extrapolate the recent trajectory; $\beta$ is unconstrained.
  \end{block}
  \papersource{Paper, Eq. (3), PDF p. 7; App. C.1, Eqs. (4)--(5), pp. 18--19.}
\end{frame}

\begin{frame}{The best history window depends on parameterization}
  Qwen $4\times7$, $D=84$; all scores in percent.
  \medskip
  \begin{center}
  \begin{tabular}{lrrr}
  \toprule
  Method & Overall & Knowledge & Reasoning \\
  \midrule
  BaseLoop & 35.60 & 52.29 & 29.34 \\
  Dense initial-state & 29.88 & 32.28 & 28.98 \\
  Dense history, $w=1$ & 37.15 & 54.16 & \textbf{30.77} \\
  Dense history, $w=4$ & 30.38 & 32.73 & 29.50 \\
  Scalar history, $w=4$ & \textbf{37.24} & \textbf{59.18} & 29.01 \\
  Channel history, $w=2$ & 36.14 & 53.86 & 29.49 \\
  \bottomrule
  \end{tabular}
  \end{center}
  \begin{itemize}
    \item One-step Dense history rescues static-injection collapse.
    \item Four-scalar history is a strong knowledge-preserving baseline.
  \end{itemize}
  \papersource{Paper, Table 10, PDF p. 30. Selected rows; bold identifies the best score among the displayed rows.}
\end{frame}

\begin{frame}{Loop gating adds a budget-aware update scale}
  \[
  t_\ell=\ell/T,\quad \Delta t=1/T,\quad \psi_\ell\in\mathbb R^8,
  \qquad g_\ell=1+q^\top\psi_\ell
  \]
  \[
  \boxed{h_{\ell+1}=h_\ell+g_\ell\bigl(F_\theta(h_\ell)-h_\ell\bigr)}
  \]
  \begin{itemize}
    \item Train with $T=K$; rescale to $T=r$ at inference.
    \item Time features include $t$, $\Delta t$, polynomial and sine terms.
    \item Only 8 learned parameters; initialize $q=0$ to recover BaseLoop.
  \end{itemize}
  \begin{block}{What the formula does}
  The step size is a conditioning feature. There is no additional explicit $\Delta t$ multiplier in the update.
  \end{block}
  \papersource{Paper, App. C.1, Eqs. (7)--(8), PDF pp. 19--20. Gates are unconstrained.}
\end{frame}

\begin{frame}{Finer time modulation trades capacity for control}
  \[
  u^+=u+g_{\ell,k}^b\odot\mathcal B_k^b\!\left(
  (\mathbf1+s_{\ell,k}^b)\odot N_k^b(u)\right)
  \]
  \begin{center}
  \begin{tabular}{lp{6.7cm}r}
  \toprule
  Method & Modulation granularity & Parameters \\
  \midrule
  LG & One scalar for the entire core update & $8$ \\
  BG & One scalar per attention/MLP branch & $16s$ \\
  AdaLN & Channel gates and norm-output scales & $32sd$ \\
  \bottomrule
  \end{tabular}
  \end{center}
  \begin{itemize}
    \item $k$ indexes a core layer; $b$ indexes its attention or MLP branch.
    \item BG sets the additional normalization scale to zero.
    \item All start from the unconditioned computation.
  \end{itemize}
  \papersource{Paper, App. C.1, Eqs. (9)--(11), PDF pp. 19--20. Here $s$ consistently denotes shared-core layer count.}
\end{frame}

\begin{frame}{Time conditioning helps some architectures and capabilities}
  Qwen $7\times4$, $D=56$; all scores in percent.
  \medskip
  \begin{center}
  \begin{tabular}{lrrr}
  \toprule
  Method & Overall & Knowledge & Reasoning \\
  \midrule
  BaseLoop & 36.59 & 60.33 & 27.68 \\
  LG & 36.91 & 59.24 & 28.54 \\
  BG & \textbf{38.46} & 58.67 & \textbf{30.88} \\
  AdaLN & 38.01 & \textbf{61.72} & 29.12 \\
  \bottomrule
  \end{tabular}
  \end{center}
  \begin{itemize}
    \item BG gains $1.87\pp$ Overall, driven by reasoning.
    \item AdaLN better preserves knowledge in this comparison.
    \item Qwen $2\times14$: none improves Overall at $D=28$ or 56.
  \end{itemize}
  \papersource{Paper, Table 12, PDF p. 31; Fig. 6, p. 8. The effect depends on core depth and loop count.}
\end{frame}

\begin{frame}{History plus loop gating is the strongest matched pair}
  I: initial state; H: history; T: time conditioning.
  \medskip
  \begin{columns}[c,onlytextwidth]
    \begin{column}{0.55\textwidth}
      \centering
      Qwen $4\times7$, $D=84$
      \par\medskip
      \begin{tabular}{lr}
      \toprule
      Method & Overall (\%) \\
      \midrule
      BaseLoop & 35.60 \\
      H only & 36.14 \\
      I+LG & 36.33 \\
      I+H & 36.91 \\
      \textbf{H+LG} & \textbf{37.30} \\
      \bottomrule
      \end{tabular}
    \end{column}
    \begin{column}{0.42\textwidth}
      \begin{block}{Matched setting}
      Channel-wise states, $w=2$.
      \end{block}
      \begin{block}{Small adapter}
      $wd+8=2056$ parameters.\\[2mm]
      $+1.16\pp$ over H alone;\\
      $+1.70\pp$ over BaseLoop.
      \end{block}
      Knowledge: 56.32 vs. 52.29 for BaseLoop.
    \end{column}
  \end{columns}
  \papersource{Paper, Sec. 6 and Fig. 7, PDF p. 9; Table 10 and App. D.6, pp. 30--31. Parameter total is calculated from the stated maps.}
\end{frame}

\begin{frame}{Adding another conditioning signal need not help}
  \begin{itemize}
    \item \textbf{I+H can help at a particular budget.} At $D=56$, channel-wise $w=2$ reaches 37.74, vs. 36.67 (I) and 36.77 (H).
    \item \textbf{I+T has little complementarity.} At training depth, I+LG reaches 35.99, below LG alone at 36.68.
    \item \textbf{The triple is weaker at $D=56$.} Four tested I+H+T variants score 35.69--36.60; H+LG reaches 37.49.
  \end{itemize}
  \begin{block}{Keep a strong lightweight comparator}
  Scalar H with $w=4$ alone reaches 37.24 at $D=84$, only 0.06 points below the highlighted H+LG result. The study uses one seed.
  \end{block}
  \papersource{Paper, App. D.6 and Table 10, PDF pp. 29--31. The triple variants and highlighted pair have their stated distinct parameterizations.}
\end{frame}

\begin{frame}{A skip intervention measures computational dependence}
  Skip one block occurrence $i$ by replacing its output with its input. Measure how a later block's \emph{update} changes:
  \[
  u_{j,p}=h^{\mathrm{out}}_{j,p}-h^{\mathrm{in}}_{j,p},\qquad
  C_{i\to j}=\frac{\mathbb E_p\|u^{(-i)}_{j,p}-u_{j,p}\|_2}
  {\mathbb E_p\|u_{j,p}\|_2}.
  \]
  \begin{itemize}
    \item Only one execution is skipped; other uses of the shared weights remain.
    \item $C=0.1$: average update change equals 10\% of the baseline update norm.
    \item Held-out validation text: 49 sequences, 100,000 token positions.
    \item Aggregate pairs beyond the training horizon by loop lag.
  \end{itemize}
  \papersource{Paper, App. C.4, Eqs. (25)--(32), PDF pp. 22--25. $C$ is a ratio of mean norms.}
\end{frame}

\begin{frame}{H+LG shows stronger interactions across loops}
  \begin{columns}[c,onlytextwidth]
    \begin{column}{0.51\textwidth}
      \includegraphics[width=\linewidth,height=0.65\textheight,keepaspectratio]{figures/fig08c-pairwise.pdf}
    \end{column}
    \begin{column}{0.46\textwidth}
      At $D=84$, weighted mean over lags 1--6:
      \begin{itemize}
        \item H+LG vs. I+H: $+18.6\%$ in $C$.
        \item H+LG vs. I+LG: $+10.4\%$ in $C$.
        \item Raw responses also rise: $+6.4\%$ and $+10.2\%$.
      \end{itemize}
      \medskip
      Consistent with later loops building on earlier computation; the link to task accuracy remains interpretive.
    \end{column}
  \end{columns}
  \papersource{Paper, Fig. 8(c) and Sec. 6, PDF p. 9; Fig. 23 and App. E.2, p. 36. Dashed: $D=56$; solid: $D=84$.}
\end{frame}

\begin{frame}{Design for the intended inference budget}
  \begin{enumerate}
    \item Track Knowledge and Reasoning separately as loops increase.
    \item Tune physical depth and boundary allocation jointly with loop count.
    \item Compare channel-wise H+LG against strong scalar-history baselines.
  \end{enumerate}
  \begin{block}{Scope of the evidence}
  Reference models up to about 1B; fixed-budget pretraining; one seed. History combinations mainly use Qwen $4\times7$, tested up to $3\times$ training depth. No end-to-end latency measurements or adaptive halting.
  \end{block}
  \small
  Xinlin Zhuang, Siyuan Wang, Imran Razzak, Weiyang Liu.\\
  \emph{What Makes Recurrence Effective in Looped Language Models?}\\
  \href{https://arxiv.org/abs/2609.36636v1}{arXiv:2609.36636v1}, 29 September 2026.
  \papersource{Paper, App. A, C.2, and D.4--D.6, PDF pp. 15, 21, 27--35.}
\end{frame}

\appendix

\begin{frame}{Backup: exact time features and adapter sizes}
  \[
  \psi(t,\Delta t)=\left(
  \begin{array}{c}
  t,\ \Delta t,\ t^2,\ (\Delta t)^2,\ t\Delta t,\\
  \sin(\pi t),\ \cos(\pi t)-1,\ \sin(2\pi t)
  \end{array}\right)^{\!\top}
  \]
  \begin{center}
  \begin{tabular}{lr}
    \toprule
    Qwen $4\times7$: $d=1024$, $s=4$, $w=2$ & Added parameters \\
    \midrule
    Channel history & 2,048 \\
    Dense history & 2,097,152 \\
    LG / BG / AdaLN & 8 / 64 / 131,072 \\
    Channel history + LG & 2,056 \\
    \bottomrule
  \end{tabular}
  \end{center}
  \small Direct history buffering requires $O(wnd)$ state elements. Parameter efficiency alone does not determine serving cost.
  \papersource{Paper, App. C.1, PDF pp. 18--20. Counts are calculated from the stated formulas; buffer cost is an implementation inference.}
\end{frame}

\begin{frame}{Backup: the same function class can train differently}
  \begin{columns}[T,onlytextwidth]
    \begin{column}{0.48\textwidth}
      \begin{block}{Channel-wise}
      \[
      a\odot h+b\odot h_0
      \]
      Initialize $a=\mathbf1$, $b=0$.\\[2mm]
      Weight decay on $a$ favors a current-state coefficient of zero.
      \end{block}
    \end{column}
    \begin{column}{0.49\textwidth}
      \begin{block}{Residual channel-wise}
      \[
      (\mathbf1+\delta_a)\odot h+b\odot h_0
      \]
      Initialize $\delta_a=b=0$.\\[2mm]
      Weight decay on $\delta_a$ favors a current-state coefficient of one.
      \end{block}
    \end{column}
  \end{columns}
  \medskip
  Both add $2d$ parameters and can represent the same maps. Their regularization is different.
  \papersource{Paper, App. C.1, PDF p. 18.}
\end{frame}

\begin{frame}{Backup: the numerical comparisons need three corrections}
  \begin{enumerate}
    \item \textbf{Configuration convention.} Sec. 3 swaps the meanings of $s$ and $K$. Use Table 2's core-depth $\times$ loop-count convention.
    \item \textbf{Task divergence example.} The 28.52 $\to$ 31.49 reasoning result belongs to $4\times5$, as shown by Fig. 1(a) and Table 8, not the $2\times10$ label in the prose.
    \item \textbf{Qwen $14\times2$, $D=56$.} Sec. 5.2 reports LG/AdaLN reasoning as 29.61/28.78. Table 12 reports 29.22/29.36; Fig. 6(d) supports the table's ordering.
  \end{enumerate}
  \medskip
  These slides use the consistent figure/table evidence for those comparisons.
  \papersource{Paper, Sec. 3, PDF p. 4; Sec. 5.2, p. 8; Tables 2, 8, 12, pp. 18, 29, 31.}
\end{frame}

\begin{frame}{Backup: optimizer and noise ablations support the recipe}
  \begin{center}
  \begin{tabular}{lrr}
    \toprule
    Pilot model & AdamW mean & Muon mean \\
    \midrule
    NonLoop $12\times1$ & 41.95 & 42.04 \\
    BaseLoop $3\times4$ & 39.98 & 40.74 \\
    LoopFormer $3\times4$ & 39.96 & 40.05 \\
    Huginn $1+2\times5+1$ & Failed to converge & 41.72 \\
    \bottomrule
  \end{tabular}
  \end{center}
  \begin{itemize}
    \item The pilot averages ten tasks, distinct from the main eleven-task metric.
    \item Initialization noise 0.03 gives no consistent benefit over no noise for the tested initial-state injections.
  \end{itemize}
  \papersource{Paper, Table 7 and App. D.1, PDF p. 26; Fig. 22 and App. E.1, pp. 33, 35--36. Selected pilot rows.}
\end{frame}

\begin{frame}{Backup: how the interaction statistic is aggregated}
  \begin{itemize}
    \item For Qwen $4\times7$, include only source and target occurrences beyond training loop 7.
    \item For loop lag $d$, average $C_{i\to j}$ over occurrence pairs $\mathcal P_d$.
    \item Cross-lag summaries weight pairs equally:
  \end{itemize}
  \[
  C_{\mathcal D}=\frac{\sum_{d\in\mathcal D}|\mathcal P_d|\,C(d)}
  {\sum_{d\in\mathcal D}|\mathcal P_d|}.
  \]
  \begin{itemize}
    \item At $D=84$: 21 loops and lags 0--13. Main pair comparison uses lags 1--6.
    \item Adding I to H+LG lowers mean $C$ over lags 2--6 by 4.2\%, and the raw numerator by 6.0\%.
    \item Only 9/16 shared-block pairings show lower $C$ in that triple comparison.
  \end{itemize}
  \papersource{Paper, App. C.4, Eqs. (30)--(33), PDF pp. 24--25; App. E.2 and Fig. 23, p. 36.}
\end{frame}

\end{document}