\begin{filecontents*}{localrefs.bib}
@article{joyal1977remarques,
  author  = {Joyal, Andre},
  title   = {Remarques sur la theorie des jeux a deux personnes},
  journal = {Gazette des sciences mathematiques du Quebec},
  year    = {1977}
}
@article{laird2013constructing,
  author  = {Laird, James and Manzonetto, Giulio and McCusker, Guy},
  title   = {Constructing categories and models of game semantics},
  journal = {Mathematical Structures in Computer Science},
  year    = {2013}
}
@article{honsell2009conway,
  author  = {Honsell, Furio and Lenisa, Marina},
  title   = {Conway games and hypergames},
  journal = {Electronic Notes in Theoretical Computer Science},
  year    = {2009}
}
@article{honsell2011conway,
  author  = {Honsell, Furio and Lenisa, Marina},
  title   = {On the category of hypergames},
  journal = {Theoretical Computer Science},
  year    = {2011}
}
@article{bavsic2024categories,
  author  = {Bavsic, Boro and others},
  title   = {Categories of games and rulegraphs},
  journal = {Preprint},
  year    = {2024}
}
@article{blute2006differential,
  author  = {Blute, Richard and Cockett, Robin and Seely, Robert},
  title   = {Differential categories},
  journal = {Mathematical Structures in Computer Science},
  year    = {2006}
}
@article{cockett2019integral,
  author  = {Cockett, Robin and Lemay, Jean-Simon Pacaud},
  title   = {Integral categories and calculus categories},
  journal = {Logical Methods in Computer Science},
  year    = {2019}
}
@article{loregian2021differential,
  author  = {Loregian, Francesca and Trimble, Todd},
  title   = {Differential and integral 2-rigs},
  journal = {Theory and Applications of Categories},
  year    = {2021}
}
@article{bouton1901nim,
  author  = {Bouton, Charles L.},
  title   = {Nim, a game with a complete mathematical theory},
  journal = {Annals of Mathematics},
  year    = {1901}
}
@book{siegel2013combinatorial,
  author    = {Siegel, Aaron N.},
  title     = {Combinatorial Game Theory},
  publisher = {American Mathematical Society},
  year      = {2013}
}
@article{joyal1981theorie,
  author  = {Joyal, Andre},
  title   = {Une theorie combinatoire des series formelles},
  journal = {Advances in Mathematics},
  year    = {1981}
}
\end{filecontents*}

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\title[A Rota--Baxter equation for winning games]{A Rota--Baxter Equation for Winning Games}
\subtitle{Games as recursive coalgebras and differential invariants}
\author[R. Hora]{Ryuya Hora}
\institute{Graduate School of Mathematical Sciences, The University of Tokyo}
\date{Differentiation in category theory and program semantics\\Kyoto University, April 6, 2026}

\begin{document}

\begin{frame}
  \titlepage
  \vspace{-0.4em}
  \begin{center}
    \small Partially based on joint work in progress with Ryo Suzuki.
  \end{center}
\end{frame}

\begin{frame}{Where I come from}
\Large
I've been interested in toposes, \textbf{games, and coalgebras}.\\[0.7em]

\begin{figure}
\centering
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  \node[anchor=west, text=mygray] at (-3.82,0.25) {Google was not exactly helpful.};
  \node[anchor=west, text=myblue, font=\bfseries] at (-3.82,-0.42) {Did you mean: recursion};
  \node[anchor=west, text=mygray, font=\small] at (-3.82,-1.05) {So I decided to keep the joke in the slide.};
\end{tikzpicture}
\end{figure}

{\small When I tried to check the spelling of \dq{recursion}.}
\end{frame}

\begin{frame}{Where this talk comes from}
\small
\begin{columns}[T,totalwidth=\textwidth]
\begin{column}{0.56\textwidth}
\begin{itemize}
  \item My motivation comes from \textbf{Combinatorial Game Theory}.
  \item I am \textbf{not} a specialist in differential categories in the usual sense.
  \item So this talk is partly a proposal about how game recursion should be organized categorically.
\end{itemize}
\end{column}
\begin{column}{0.40\textwidth}
\begin{proposition}[What I hope to learn here]
\begin{itemize}
  \item the right map of prior work,
  \item links to differential / Cartesian differential categories,
  \item links to game semantics and linear logic,
  \item where the Rota--Baxter viewpoint fits.
\end{itemize}
\end{proposition}
\end{column}
\end{columns}

\vspace{0.3em}
\begin{center}
\emph{I am coming mainly from the game side, but trying to speak to this audience.}
\end{center}
\end{frame}

\begin{frame}{Short survey: games $\times$ category theory is not new}
\footnotesize
\begin{proposition}[Important positioning]
I know that \textbf{games + category theory} is already a rich story.
This talk isolates the narrower interface between
\[
\text{impartial combinatorial games}
\qquad\text{and}\qquad
\text{differential / Rota--Baxter ideas}.
\]
\end{proposition}

\begin{columns}[T,totalwidth=\textwidth]
\begin{column}{0.32\textwidth}
\begin{example}[Game semantics]
\begin{itemize}
  \item strategies as morphisms,
  \item linear logic / programming semantics.
\end{itemize}
\end{example}
\end{column}
\begin{column}{0.32\textwidth}
\begin{example}[Coalgebraic games]
\begin{itemize}
  \item Conway games / hypergames,
  \item recursive viewpoints on impartial games.
\end{itemize}
\end{example}
\end{column}
\begin{column}{0.32\textwidth}
\begin{example}[Differential side]
\begin{itemize}
  \item differential categories,
  \item integral / calculus categories.
\end{itemize}
\end{example}
\end{column}
\end{columns}

\vspace{0.15em}
{\scriptsize Representative references: game semantics \parencite{joyal1977remarques,laird2013constructing}; coalgebraic games \parencite{honsell2009conway,honsell2011conway,bavsic2024categories}; differential side \parencite{blute2006differential,cockett2019integral,loregian2021differential}.}
\end{frame}

\section{Games: Nim and Integration}

\begin{frame}{Rule of $n$-heap Nim}
\small
\begin{itemize}
  \item $n$ heaps of stones are given.
  \item Two players take turns choosing one heap and removing at least one stone from that heap.
  \item The player who is unable to take a stone loses.
\end{itemize}

\begin{figure}
\centering
\begin{tikzpicture}[scale=1]
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\end{figure}
\end{frame}

\begin{frame}[t]{Bouton's theorem}
\footnotesize
\begin{columns}[T,totalwidth=\textwidth]
\begin{column}{0.47\textwidth}
\begin{definition}[Nim-sum]
The \emph{Nim-sum} $\nimsum$ is bit-wise xor, i.e. the abelian group structure on $\N$ induced by the binary expansion
\[
\N \cong \bigoplus_{k=0}^{\infty} \Z/2\Z.
\]
\end{definition}
\begin{example}
$5\nimsum 7 = (101)_2 \nimsum (111)_2 = (010)_2 = 2$.
\end{example}
\end{column}
\begin{column}{0.49\textwidth}
\begin{theorem}[Bouton, Ann. of Math., 1901]
A state $(a_1,\dots,a_n)$ of $n$-heap Nim is winning for the previous player iff
\[
a_1\nimsum\cdots\nimsum a_n = 0.
\]
\end{theorem}
\begin{example}
$(1,2,3)$, $(0,1,1)$, and $(2,2,0)$ are $P$-positions of $3$-heap Nim.
\end{example}
\end{column}
\end{columns}
\end{frame}

\begin{frame}{Why xor in Nim?}
\small
\begin{itemize}
  \item Bouton's theorem says that an $n$-heap Nim position $(a_1,\dots,a_n)$ is a $P$-position iff
  \[
    a_1\nimsum \cdots \nimsum a_n = 0.
  \]
  \item Once the theorem is known, the proof is classical \parencite{bouton1901nim,siegel2013combinatorial}.
  \item But conceptually, why should \emph{xor} appear at all?
\end{itemize}

\begin{proposition}[My answer today]
First isolate the \emph{recursive} nature of games, then reinterpret the key Nim identity as a \emph{Rota--Baxter equation}:
\[
\mex(S)\nimsum \mex(T)
= \mex\bigl((\mex(S)\nimsum T)\cup (S\nimsum \mex(T))\bigr).
\]
\end{proposition}
\end{frame}

\section{Coalgebras: Games as recursive coalgebras}

\begin{frame}[t]{Games are recursive objects}
\footnotesize
\begin{columns}[T,totalwidth=\textwidth]
\begin{column}{0.54\textwidth}
A (short impartial) game is a directed graph $(X,\to)$ such that
\begin{itemize}
  \item each position has finitely many options;
  \item there is no infinite play.
\end{itemize}

Typical values are defined \emph{backwards from the options}:
\[
\oc(x)=
\begin{cases}
N & \text{if some }x\to x' \text{ has } \oc(x')=P,\\
P & \text{otherwise.}
\end{cases}
\]
\[
\G(x)=\mex\{\G(x')\mid x\to x'\}.
\]

Birthday, remoteness, and mis\`ere outcome fit the same recursive pattern.
\end{column}
\begin{column}{0.42\textwidth}
\begin{figure}
\centering
\begin{tikzpicture}[scale=0.82]
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  \draw[gameedge] (x1) -- (z2);
  \draw[gameedge] (x2) -- (z2);
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  \draw[gameedge] (x3) -- (z3);
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\end{figure}
\end{column}
\end{columns}
\end{frame}

\begin{frame}[t]{Coalgebra-algebra morphisms package the recursion}
\small
Write the neighborhood map as
\[
\thetaf\colon X\longrightarrow \Pf(X),\qquad x\longmapsto \{x'\mid x\to x'\}.
\]
A recursive game value with codomain $A$ is a map $f\colon X\to A$ determined by
\[
f(x)=\alpha\bigl(\{f(x')\mid x\to x'\}\bigr)
\]
for some $\Pf$-algebra $\alpha\colon \Pf(A)\to A$.

\begin{figure}
\centering
\begin{tikzpicture}[every node/.style={font=\small}]
  \node[bluepanel, minimum width=1.9cm] (pfx) at (-2,1.3) {$\Pf(X)$};
  \node[bluepanel, minimum width=1.9cm] (pfa) at ( 2,1.3) {$\Pf(A)$};
  \node[bluepanel, minimum width=1.9cm] (x)   at (-2,-1.1) {$X$};
  \node[bluepanel, minimum width=1.9cm] (a)   at ( 2,-1.1) {$A$};
  \draw[formulaarrow] (pfx) -- node[above] {$\Pf(f)$} (pfa);
  \draw[formulaarrow] (x) -- node[below] {$f$} (a);
  \draw[formulaarrow] (x) -- node[left] {$\thetaf$} (pfx);
  \draw[formulaarrow] (pfa) -- node[right] {$\alpha$} (a);
\end{tikzpicture}
\end{figure}

\begin{proposition}[Examples]
\begin{itemize}
  \item $\alpha=\np$ gives \textbf{outcome}.
  \item $\alpha=\mex$ gives \textbf{Grundy number}.
  \item $\alpha=\xem$ gives \textbf{birthday}.
\end{itemize}
The unique solution is the hylomorphism $\hylo_{\alpha,\thetaf}$.
\end{proposition}
\end{frame}

\begin{frame}[t]{Main statement of Part I}
\footnotesize
\begin{theorem}
The category of games is equivalent to the category of recursive $\Pf$-coalgebras:
\[
\Gs \simeq \RecCoalg{\Pf}.
\]
\end{theorem}

\begin{columns}[T,totalwidth=\textwidth]
\begin{column}{0.48\textwidth}
\begin{example}[One-heap Nim]
The coalgebra is
\[
\nu\colon \N\to \Pf(\N),\qquad \nu(n)=\{0,\dots,n-1\}.
\]
Hence $\G_{\mathrm{Nim}_1}=\mathrm{id}_{\N}$.
\end{example}
\end{column}
\begin{column}{0.48\textwidth}
\begin{example}[Terminal game]
The terminal recursive coalgebra is the game $\Hf$ of hereditarily finite sets, with
\[
A\to B \iff B\in A.
\]
Every game has a unique map $\mathrm{rd}_X\colon X\to\Hf$.
\end{example}
\end{column}
\end{columns}
\end{frame}

\begin{frame}{Game values are hylomorphisms}
\small
\begin{center}
\renewcommand{\arraystretch}{1.22}
\begin{tabular}{@{}ll@{}}
\toprule
\textbf{game value} & \textbf{$\Pf$-algebra} \\
\midrule
outcome & $\np\colon \Pf(\{N,P\})\to \{N,P\}$ \\
Grundy number & $\mex\colon \Pf(\N)\to \N$ \\
birthday & $\xem\colon \Pf(\N)\to \N$ \\
\bottomrule
\end{tabular}
\end{center}

\vspace{0.45em}
\begin{proposition}[Consequence]
A game morphism preserves every recursively defined invariant coming from a $\Pf$-algebra.
\end{proposition}

\vspace{0.25em}
\begin{center}
\emph{Next question: where is differentiation hiding in the game rules themselves?}
\end{center}
\end{frame}

\section{Differentiation: Differential structure and Rota--Baxter invariants}

\begin{frame}[t]{Move to pointed games and finite families}
\footnotesize
\begin{columns}[T,totalwidth=\textwidth]
\begin{column}{0.50\textwidth}
\begin{itemize}
  \item A pointed game is a pair $(X,x)$ with a chosen starting position.
  \item Let $\Gsp$ be the category of pointed games.
  \item Let $\Fam(\Gsp)$ be the free finite-coproduct completion.
\end{itemize}

Think of an object of $\Fam(\Gsp)$ as a finite family of \emph{local game situations}.

\vspace{0.3em}
\begin{definition}[Differential operator]
For a pointed game $(X,x)$, define
\[
\dd(X,x)\coloneqq \{(X,x')\}_{x\to x'}.
\]
\end{definition}
\end{column}
\begin{column}{0.46\textwidth}
\begin{figure}
\centering
\begin{tikzpicture}[scale=0.84]
  \node[bluepanel, minimum width=2.5cm, minimum height=0.75cm] (rootbox) at (0,2.05) {$ (X,x)$};
  \node[state] (x) at (0,1.15) {};
  \node[state] (x1) at (-1.2,0.15) {};
  \node[state] (x2) at (0,-0.05) {};
  \node[state] (x3) at (1.2,0.15) {};
  \draw[gameedge] (x) -- (x1);
  \draw[gameedge] (x) -- (x2);
  \draw[gameedge] (x) -- (x3);
  \node[greenpanel, fit=(x1) (x2) (x3), inner sep=7pt] (fam) {};
  \node[greenpanel, minimum width=3.1cm, minimum height=0.75cm] (d) at (0,-1.15) {$\dd(X,x)$};
  \draw[formulaarrow] (rootbox) -- (d);
  \draw[formulaarrow] (fam.south) -- (d.north);
\end{tikzpicture}
\end{figure}
\end{column}
\end{columns}
\end{frame}

\begin{frame}{Leibniz rule for Conway addition}
\small
For the Conway sum, every move changes \emph{either} the left component or the right component.

\vspace{0.2em}
\begin{theorem}
In $\Fam(\Gsp)$ we have a categorified Leibniz rule:
\[
\dd(X\otimes Y)\cong (\dd X)\otimes Y \;\sqcup\; X\otimes (\dd Y).
\]
\end{theorem}

\begin{figure}
\centering
\begin{tikzpicture}[scale=0.82]
  \node[bluepanel, minimum width=2.6cm, minimum height=0.8cm] (xy) at (0,0.05) {$X\otimes Y$};
  \node[bluepanel, minimum width=2.7cm, minimum height=0.8cm] (dx) at (-3.05,-1.7) {$(\dd X)\otimes Y$};
  \node[bluepanel, minimum width=2.7cm, minimum height=0.8cm] (dy) at (3.05,-1.7) {$X\otimes(\dd Y)$};
  \draw[formulaarrow] (xy) -- (dx) node[midway,left] {move in $X$};
  \draw[formulaarrow] (xy) -- (dy) node[midway,right] {move in $Y$};
\end{tikzpicture}
\end{figure}

{\scriptsize This is the differential structure emphasized here; compare with differential 2-rigs \parencite{joyal1981theorie,loregian2021differential}.}
\end{frame}

\begin{frame}[t]{Decategorification target: calculus / Rota--Baxter rigs}
\scriptsize
\begin{definition}[Differential rig]
A rig $(A,0,1,+,\times,\dd)$ with
\[
\dd(a+b)=\dd a+\dd b,
\qquad
\dd(ab)= (\dd a)b + a(\dd b).
\]
\end{definition}

\begin{definition}[Integral rig]
A rig with an operator $\Int$ satisfying
\[
1 = \Int 0,
\qquad
(\Int f)(\Int g)= \Int\bigl((\Int f)g + f(\Int g)\bigr).
\]
\end{definition}

\begin{definition}[Calculus rig]
A rig carrying both structures, with the fundamental theorem $\dd\,\Int f = f$.
\end{definition}

\begin{center}
{\footnotesize\emph{Game families can be sent to such algebraic differential / integral structures.}}
\end{center}
\end{frame}

\begin{frame}[t]{Rota--Baxter valued invariants of games}
\small
\begin{columns}[T,totalwidth=\textwidth]
\begin{column}{0.56\textwidth}
Let $A$ be an integral rig. For a pointed game $(X,x)$ define recursively
\[
F_{(X,x)}\coloneqq \Int\!\left(\sum_{x\to x'} F_{(X,x')}\right).
\]
For a finite family, define $F$ by finite sums.

\vspace{0.25em}
\begin{theorem}
This assignment preserves the rig operations:
\[
F_{\mathcal{X}\sqcup\mathcal{Y}} = F_{\mathcal{X}} + F_{\mathcal{Y}},
\qquad
F_{\mathcal{X}\otimes\mathcal{Y}} = F_{\mathcal{X}}\times F_{\mathcal{Y}}.
\]
\end{theorem}
\end{column}
\begin{column}{0.40\textwidth}
\begin{figure}
\centering
\begin{tikzpicture}[scale=0.86]
  \node[bluepanel, minimum width=2.7cm, minimum height=0.75cm] (fx) at (0,2.1) {$F_{(X,x)}$};
  \node[greenpanel, minimum width=1.25cm, minimum height=0.68cm] (int) at (0,1.05) {$\Int$};
  \node[bluepanel, minimum width=3.2cm, minimum height=0.78cm] (sum) at (0,-1.95) {$\sum_{x\to x'} F_{(X,x')}$};
  \node[state] (x) at (0,-0.02) {};
  \node[state] (x1) at (-1.15,-0.95) {};
  \node[state] (x2) at (0,-1.12) {};
  \node[state] (x3) at (1.15,-0.95) {};
  \draw[gameedge] (x) -- (x1);
  \draw[gameedge] (x) -- (x2);
  \draw[gameedge] (x) -- (x3);
  \draw[formulaarrow] (sum) -- (int);
  \draw[formulaarrow] (int) -- (fx);
\end{tikzpicture}
\end{figure}
\end{column}
\end{columns}

{\scriptsize Why multiplication works: apply the Rota--Baxter identity to the Leibniz rule for options of a Conway sum.}
\end{frame}

\begin{frame}[t]{The universal example: game families themselves}
\footnotesize
The family of all pointed positions in the terminal game carries a calculus-rig-like structure:

\begin{center}
\renewcommand{\arraystretch}{1.08}
\begin{tabular}{@{}ll@{}}
\toprule
\textbf{analysis} & \textbf{game families} \\
\midrule
$0$ & empty family \\
$1$ & the terminal pointed game \\
$+$ & disjoint union / union \\
$\times$ & gamewise Conway sum \\
$\dd$ & take all immediate options \\
$\Int$ & adjoin a new root / braces $A\mapsto \{A\}$ \\
\bottomrule
\end{tabular}
\end{center}

\begin{figure}
\centering
\begin{tikzpicture}[scale=0.68]
  \node[bluepanel, minimum width=1.2cm, minimum height=0.62cm] (A) at (-2.8,0) {$A$};
  \node[greenpanel, minimum width=1.4cm, minimum height=0.62cm] (IA) at (0,0) {$\{A\}$};
  \node[bluepanel, minimum width=1.7cm, minimum height=0.62cm] (dA) at (2.8,0) {$\dd A$};
  \draw[formulaarrow] (A) -- node[above] {$\Int$} (IA);
  \draw[formulaarrow] (A) -- node[above] {$\dd$} (dA);
  \node[state] (a1) at (2.45,-0.82) {};
  \node[state] (a2) at (3.12,-0.82) {};
  \draw[gameedge] (dA.south) -- (a1);
  \draw[gameedge] (dA.south) -- (a2);
\end{tikzpicture}
\end{figure}
\end{frame}

\begin{frame}[t]{Nim-sum gives a concrete Rota--Baxter rig}
\footnotesize
\begin{columns}[T,totalwidth=\textwidth]
\begin{column}{0.58\textwidth}
Start from the commutative monoid $(\N,\nimsum,0)$.
Its free idempotent rig is $\Pf(\N)$ with
\[
S+T \coloneqq S\cup T,
\qquad
S\times T \coloneqq \{s\nimsum t\mid s\in S,\ t\in T\}.
\]
Define the integral operator by
\[
\Int(S)\coloneqq \{\mex(S)\}.
\]

\begin{theorem}
This makes $\Pf(\N)$ into an integral (Rota--Baxter) rig.
\end{theorem}
\end{column}
\begin{column}{0.38\textwidth}
\begin{figure}
\centering
\begin{tikzpicture}[thick,>=Latex,scale=0.85]
  \node at (-1.0,0.65) {$3=$};
  \node at (-0.2,0.65) {$0$};
  \node at ( 0.4,0.65) {$1$};
  \node at ( 1.0,0.65) {$1$};

  \node at (-1.25,0.0) {$\nimsum$};

  \node at (-1.0,-0.65) {$5=$};
  \node at (-0.2,-0.65) {$1$};
  \node at ( 0.4,-0.65) {$0$};
  \node at ( 1.0,-0.65) {$1$};

  \draw[very thick] (-1.2,-1.05) -- (1.3,-1.05);

  \node at (-1.0,-1.7) {$6=$};
  \node at (-0.2,-1.7) {$1$};
  \node at ( 0.4,-1.7) {$1$};
  \node at ( 1.0,-1.7) {$0$};
\end{tikzpicture}
\end{figure}
\end{column}
\end{columns}
\end{frame}

\begin{frame}{The key identity is exactly a Rota--Baxter equation}
\small
In the rig $\Pf(\N)$,
\[
(\Int S)(\Int T)=\Int\bigl((\Int S)T + S(\Int T)\bigr)
\]
becomes
\[
\{\mex(S)\}\times \{\mex(T)\}
=
\Int\bigl((\{\mex(S)\}\times T)\cup (S\times \{\mex(T)\})\bigr).
\]
Unpacking the product gives exactly
\[
\mex(S)\nimsum\mex(T)
=\mex\bigl((\mex(S)\nimsum T)\cup (S\nimsum \mex(T))\bigr).
\]

\begin{proposition}[Interpretation]
The classical mex + xor identity is the integral shadow of the Leibniz rule for the option operator of Conway addition.
\end{proposition}
\end{frame}

\begin{frame}[t]{Recovering Bouton's theorem}
\scriptsize
\begin{columns}[T,totalwidth=\textwidth]
\begin{column}{0.56\textwidth}
Apply the construction to pointed Nim positions.
\begin{itemize}
  \itemsep0.2em
  \item One-heap Nim satisfies
  \[
  F_{(\mathrm{Nim}_1,n)} = \{n\}.
  \]
  \item Hence for the Conway sum of $n$ heaps,
  \[
  F_{(a_1,\dots,a_n)} = \{a_1\nimsum\cdots\nimsum a_n\}.
  \]
  \item Therefore $(a_1,\dots,a_n)$ is a $P$-position iff
  \[
  a_1\nimsum\cdots\nimsum a_n = 0.
  \]
\end{itemize}
\end{column}
\begin{column}{0.40\textwidth}
\begin{figure}
\centering
\begin{tikzpicture}[scale=0.75]
  \node[bluepanel, minimum width=2.65cm, minimum height=0.68cm] (n1) at (0,1.5) {$(\mathrm{Nim}_1,n)$};
  \node[greenpanel, minimum width=1.30cm, minimum height=0.64cm] (fn1) at (0,0.35) {$\{n\}$};
  \node[bluepanel, minimum width=2.85cm, minimum height=0.68cm] (nn) at (0,-1.05) {$(a_1,\dots,a_n)$};
  \node[greenpanel, minimum width=3.15cm, minimum height=0.68cm] (fnn) at (0,-2.25) {$\{a_1\nimsum\cdots\nimsum a_n\}$};
  \draw[formulaarrow] (n1) -- node[right] {$F$} (fn1);
  \draw[formulaarrow] (nn) -- node[right] {$F$} (fnn);
\end{tikzpicture}
\end{figure}
\end{column}
\end{columns}

\begin{proposition}[What changed conceptually?]
The xor criterion is now a Rota--Baxter shadow of the differential rule on games.
\end{proposition}
\end{frame}

\begin{frame}{Take-home messages}
\small
\begin{enumerate}
  \item \textbf{Games are recursive coalgebras.} The basic game values come from hylomorphisms for $\Pf$.
  \item \textbf{Conway addition has a differential flavor.} On pointed games / families, the option operator satisfies a Leibniz rule.
  \item \textbf{Nim-sum is a Rota--Baxter phenomenon.} The classical mex + xor identity is the algebraic image of that Leibniz rule.
\end{enumerate}

\vspace{0.65em}
\begin{proposition}[Outlook]
Partisan / probabilistic variants, a genuine chain rule, and a systematic interface with differential categories remain open.
\end{proposition}
\end{frame}

\begin{frame}[allowframebreaks]{References}
\printbibliography[heading=none]
\end{frame}

\appendix
\section{Appendix}

\begin{frame}{Rota--Baxter property on game values}
\small
\begin{proposition}
Suppose the Bouton monoid with respect to Conway addition is induced by a $\Pf$-algebra
\[
\beta\colon \Pf(B)\to B.
\]
Then its multiplication $\circledast$ satisfies the Rota--Baxter identity
\[
\beta(S)\circledast \beta(T)
=
\beta\bigl(S\circledast \beta(T)\cup \beta(S)\circledast T\bigr)
\]
for all finite $S,T\subset B$.
\end{proposition}

\vspace{0.3em}
This is the abstract version of the mex + xor identity we used for Nim.
\end{frame}

\begin{frame}{An open structural question}
\small
\begin{bluebox}{SMCC-style classification problem}
Classify symmetric monoidal closed structures on $\Gs$ whose underlying tensor on sets is cartesian product and for which the forgetful functor
\[
U\colon \Gs\to \mathbf{Set}
\]
is (strictly or at least lax) monoidal.
\end{bluebox}

\begin{figure}
\centering
\begin{tikzpicture}[scale=0.95]
  \node[bluepanel, minimum width=2.0cm, minimum height=0.7cm] (left) at (-2.6,0) {$\Gs\times \Gs$};
  \node[greenpanel, minimum width=1.4cm, minimum height=0.7cm] (mid) at (0,0) {$\ast$};
  \node[bluepanel, minimum width=1.6cm, minimum height=0.7cm] (right) at (2.6,0) {$\Gs$};
  \draw[formulaarrow] (left) -- (mid);
  \draw[formulaarrow] (mid) -- (right);
  \node[text=mygray] at (0,-1.2) {Which monoidal structures should count as natural game decompositions?};
\end{tikzpicture}
\end{figure}
\end{frame}

\end{document}