\label{thm:intro-main}
For every $r\ge 1$ there is a natural equivalence of groupoids
\begin{equation}\label{eq:intro-main-equiv}
\Rep_r(\G_U)\times^{(2)}_{\Rep_r(\G_{\times})} \operatorname{Sec}(\pi)
\ \simeq\ \Stokes_r(X^{\log}_n;D,\Phi),
\end{equation}
compatible with restriction to $\Rep_r(\G_U)$ (forgetting the wild layer) and with restriction to $\operatorname{Sec}(\pi)$
(forgetting the interior).

\begin{proof}
The congruence generated by $R$ forces the subgroupoid structures of $\GU$ and $\GC$ and imposes the strict identifications prescribed by $j_U,j_C$ on $\GX$.
This is exactly the generators-and-relations model of the explicit $2$-pushout.
\end{proof}

\subsection{A completely explicit curve model: the explicit $2$-pushout groupoid $\G_{\Phi,n}$ and the forgetful map $\pi:\GC\to\CB$}\label{subsec:curve-model-clean}

This subsection is expository and is not used later.
Its goal is to make the explicit $2$-pushout
\begin{equation}\label{eq:curve-pushout-clean}
\G_{\Phi,n}\ :=\ \GU\sqcup_{\GX}\GC
\end{equation}
completely concrete in complex dimension~$1$, and to clarify the comparison with the groupoids
in \cite[\S3,\S5]{GualtieriLiPym} and the Kato--Nakayama viewpoint.

\medskip

\medskip

\providecommand{\Cech}{\check{\mathrm C}}

\providecommand{\CB}{\Cech(\mathcal B)}

\subsubsection*{(A) Geometry on a curve and sector boxes on $X^{\log}_n$}

Let $X$ be a connected Riemann surface and let $D=\{p_1,\dots,p_m\}\subset X$ be a reduced finite divisor.
Write $U:=X\setminus D$.
Fix a Kummer level $n\ge 1$ and an irregular type $\Phi$ along $D$ (at level~$n$).
Let $\tau_n:X^{\log}_n\to X$ be the level-$n$ Kato--Nakayama/Kummer map.
Over $U$, $\tau_n$ is a homeomorphism; over each $p_i$ the fiber is the level-$n$ circle of directions
$S^1_{p_i,n}:=\tau_n^{-1}(p_i)$.
Choose pairwise disjoint closed discs $\Delta_i$ around $p_i$ and set $\Delta_i^\times:=\Delta_i\setminus\{p_i\}$.
Let $N_i^\times:=\tau_n^{-1}(\Delta_i^\times)\subset X^{\log}_n$ be the punctured collar near $p_i$.
The Stokes walls at $p_i$ cut $S^1_{p_i,n}$ into finitely many open arcs
\begin{equation}\label{eq:arcs-clean}
S^1_{p_i,n}\setminus\Sigma_{p_i}\ =\ \bigsqcup_{a\in\ZZ/\ell_i\ZZ} I_{i,a},
\end{equation}
and on each arc the Stokes preorder is constant.
Choose \emph{sector boxes}
\begin{equation}\label{eq:boxes-clean}
B_{i,a}\ \subset\ N_i^\times,
\qquad
B_{i,a}\ \simeq\ \Delta_i^\times\times I_{i,a},
\qquad
B_{i,a}\cap B_{i,a+1}\neq\varnothing,
\end{equation}
each contractible, and with overlaps also contractible.
\begin{figure}[t]
\centering
\begin{tikzpicture}[
  >=Stealth,
  font=\scriptsize,
  lab/.style={fill=white,inner sep=1pt},
  vtx/.style={circle,draw,inner sep=1.2pt},
  box/.style={draw,rounded corners,inner sep=3pt,align=center}
]

\node  (Xbox) at (-6.0,2.35) {$X$ with $D=\{p_1,\dots,p_m\}$,\quad $U=X\setminus D$};

\draw[thick] (-7.6,0.55) .. controls (-7.4,1.75) and (-4.6,1.75) .. (-4.4,0.55)
             .. controls (-4.6,-0.65) and (-7.4,-0.65) .. (-7.6,0.55);

\node[vtx] (pi) at (-6.85,0.95) {$p_i$};
\draw[dashed,thick] (-6.85,0.95) circle (0.55);
\node[lab] at (-6.85,1.62) {$\Delta_i$};

\draw[->,thick,bend left=12] (-4.4,0.55)
  to node[lab,above,yshift=4pt] {zoom at $p_i$} (-1.70,0.9);

\node (Fbox) at (0.0,2.6) {$\tau_n^{-1}(p_i)=S^1_{p_i,n}$};

\def\R{1.15}
\coordinate (O) at (0,0.75);
\draw[thick] (O) circle (\R);

\foreach \ang in {25,115,205,295}{
  \draw[thick] (O) -- ({\R*cos(\ang)},{0.75+\R*sin(\ang)});
}
\node[lab,anchor=west] at (1.32,0.75) {$\Sigma_{p_i}$};

\node[lab] at ({1.48*cos(70)},{0.75+1.48*sin(70)}) {$I_{i,a}$};
\node[lab,xshift=-6pt] at ({1.48*cos(150)},{0.75+1.48*sin(150)}) {$I_{i,a+1}$};
\node[lab] at ({1.48*cos(250)},{0.75+1.48*sin(250)}) {$I_{i,a+2}$};

\node[lab,align=center] at (0,-1.05)
{$S^1_{p_i,n}\setminus\Sigma_{p_i}=\bigsqcup_{a\in\ZZ/\ell_i\ZZ} I_{i,a}$};

\draw[->,thick,bend left=10] (1.70,0.85) to node[lab,above] {thicken} (3.55,0.95);

\node (Nbox) at (5.6,2.35) {$N_i^\times=\tau_n^{-1}(\Delta_i^\times)$,\quad $\Delta_i^\times=\Delta_i\setminus\{p_i\}$};

\draw[thick] (3.90,-0.10) rectangle (7.30,1.65);

\draw[->,thin] (3.95,-0.55) -- (7.25,-0.55)
  node[lab,midway,below] {radial in $\Delta_i^\times$};
\draw[->,thin] (7.65,-0.10) -- (7.65,1.65)
  node[lab,midway,right] {angular};

\draw[dashed,thick] (4.55,-0.10) -- (4.55,1.65);
\draw[dashed,thick] (5.25,-0.10) -- (5.25,1.65);
\draw[dashed,thick] (5.95,-0.10) -- (5.95,1.65);
\draw[dashed,thick] (6.65,-0.10) -- (6.65,1.65);

\node[lab] at (4.90,0.85) {$B_{i,a}$};
\node[lab] at (5.60,0.85) {$B_{i,a+1}$};

\node[lab,align=center] at (5.60,-1.30)
{$B_{i,a}\simeq \Delta_i^\times\times I_{i,a}$,\quad
$B_{i,a}\cap B_{i,a+1}\neq\varnothing$\\
boxes and overlaps chosen contractible};

\end{tikzpicture}
\caption{At a puncture $p_i$, the level-$n$ circle of directions $S^1_{p_i,n}$ is cut by Stokes walls $\Sigma_{p_i}$ into finitely many arcs $I_{i,a}$.
A punctured collar $N_i^\times$ over $\Delta_i^\times$ is then covered by sector boxes $B_{i,a}\simeq \Delta_i^\times\times I_{i,a}$ with adjacent overlaps.}
\label{fig:curve-collar-sector-boxes}
\end{figure}

Let $\St_\Phi$ be the Stokes sheaf on the sector cover (as in Definition~\ref{def:stokes-sheaf}).
Since $B_{i,a}\cap B_{i,a+1}$ is contractible, the group
\begin{equation}\label{eq:stokes-groups-clean}
\St_{i;a,a+1}\ :=\ \St_\Phi(B_{i,a}\cap B_{i,a+1})
\end{equation}
is a constant (typically unipotent) subgroup of the relevant $\Aut(\Gr)$.

\subsubsection*{(B) The three groupoids $\GU,\GX,\GC$ and the \v{C}ech collar groupoid $\CB$}

Fix once and for all the following covers.

\begin{itemize}
\item \textbf{Interior cover on $U$.} Choose a good cover $\mathcal U=\{U_\alpha\}$ of $U$.
Set $\GU:=\Cech(\mathcal U)$.

\item \textbf{Collar cover on $N^\times:=\bigsqcup_i N_i^\times$.}
Set $\mathcal B:=\{B_{i,a}\}_{i,a}$ and define the \emph{plain collar \v{C}ech groupoid}
\begin{equation}\label{eq:CB-def-clean}
\CB\ :=\ \Cech(\mathcal B).
\end{equation}

\item \textbf{Overlap cover.} Consider the overlaps $U_\alpha\cap B_{i,a}\subset U\cap N^\times$ and let
$\mathcal W:=\{W_{\alpha;i,a}\}$ be any refinement by contractible opens.
Set $\GX:=\Cech(\mathcal W)$.
\end{itemize}

Now define the \emph{collar \v{C}ech--Stokes groupoid} $\GC$ as the Stokes decoration of $\CB$:
\begin{equation}\label{eq:GC-objects-clean}
(\GC)_0\ :=\ \bigsqcup_{i,a} B_{i,a},
\end{equation}
and arrows
\begin{equation}\label{eq:GC-arrows-clean}
(\GC)_1\ :=\ \bigsqcup_{i,a,a'}\Bigl((B_{i,a}\cap B_{i,a'})\times \St_\Phi(B_{i,a}\cap B_{i,a'})\Bigr),
\end{equation}
where $(x,u)$ is an arrow from the object $B_{i,a}$ to the object $B_{i,a'}$ (at the point $x$),
and composition is
\begin{equation}\label{eq:GC-comp-clean}
(x,u')\circ(x,u)\ :=\ (x,u'u).
\end{equation}

\medskip

\noindent\textbf{The forgetful projection.}
There is a \emph{strict} functor
\begin{equation}\label{eq:pi-forgetful-clean}
\pi:\GC\longrightarrow \CB
\end{equation}
defined by \emph{forgetting the Stokes decoration}:
on objects it is the identity (same sector boxes), and on arrows
\begin{equation}\label{eq:pi-on-arrows-clean}
\pi(x,u)\ :=\ x \in (B_{i,a}\cap B_{i,a'})\ \subset\ (\CB)_1.
\end{equation}
So here $\CB$ is the \v{C}ech groupoid of the cover $\mathcal B$, \emph{not} $\CC$.

\subsubsection*{(C) The explicit $2$-pushout and its ``generators-and-relations'' meaning}

There are canonical strict functors $j_U:\GX\to\GU$ and $j_C:\GX\to\GC$ induced by inclusions:
on objects, $W_{\alpha;i,a}\subset U_\alpha$ and $W_{\alpha;i,a}\subset B_{i,a}$;
on arrows, overlaps map to the corresponding \v{C}ech arrows, and in $\GC$ we take the
\emph{trivial} Stokes label $u=1$.

\begin{definition} \label{def:curve-pushout-clean}
Define
\begin{equation}\label{eq:pushout-clean}
\G_{\Phi,n}\ :=\ \GU\sqcup_{\GX}\GC.
\end{equation}
\end{definition}

Concretely, $\G_{\Phi,n}$ is obtained from the disjoint union groupoid $\GU\bigsqcup\GC$
by imposing precisely the identifications dictated by $\GX$ (gluing the overlap charts),
while keeping the Stokes-decorated arrows as genuinely new arrows.
Equivalently, $\G_{\Phi,n}$ admits the standard coequalizer presentation
\begin{equation}\label{eq:coeq-clean}
\G_{\Phi,n}\ \cong\ \mathrm{coeq}\Bigl(F(R)\rightrightarrows F(E)\Bigr),
\end{equation}
where $E$ consists of the generating arrows coming from $\GU$ and $\GC$,
and $R$ encodes only the overlap relations from $\GX$.

\subsubsection*{(D) A finite skeleton (graph-of-groupoids picture) on curves}

The \v{C}ech presentation is explicit but indexed by covers.
On a curve one can replace it by a finite skeleton without changing representation theory (Morita reduction).

Fix a basepoint $x_0\in U$.
For each sector box $B_{i,a}$ pick a point $b_{i,a}\in B_{i,a}$ and choose a path class
$\gamma_{i,a}:x_0\to b_{i,a}$ inside $U\cup N^\times$ entering the collar through the sector $I_{i,a}$.

Define a small groupoid $\G^{\mathrm{sk}}_{\Phi,n}$ by:
\begin{itemize}
\item objects: $\{x_0\}\cup\{b_{i,a}\}_{i,a}$;
\item arrows generated by:
  \begin{enumerate}
  \item $\pi_1(U,x_0)$ as loops at $x_0$ (ordinary monodromy),
  \item the spokes $\gamma_{i,a}:x_0\to b_{i,a}$ and formal inverses,
  \item for each fixed $i$ and adjacent sectors $a\to a+1$, arrows
  $\sigma_{i,a}(u):b_{i,a}\to b_{i,a+1}$ for each $u\in \St_{i;a,a+1}$,
  \end{enumerate}
\item relations:
  \begin{enumerate}
  \item $\sigma_{i,a+1}(v)\circ \sigma_{i,a}(u)=\sigma_{i,a}(vu)$ whenever the compositions correspond to the same overlap,
  \item the overlap/gluing relations expressing compatibility between the restriction of monodromy on $U$
  and the boundary Stokes jumps (the same relations encoded by $\GX$),
  \item the usual groupoid relations (identities/inverses).
  \end{enumerate}
\end{itemize}

\begin{proposition}\label{prop:curve-skeleton-clean}
There is a canonical chain of Morita equivalences connecting the cover-based presenter $\G_{\Phi,n}$ and the finite skeletal presenter
$\G^{\mathrm{sk}}_{\Phi,n}$. In particular, they present the same classifying stack, and for every $r\ge 1$ there is a natural equivalence
of groupoids
\begin{equation}\label{eq:repr-clean}
\Rep_r(\G_{\Phi,n})\ \simeq\ \Rep_r(\G^{\mathrm{sk}}_{\Phi,n}).
\end{equation}
\end{proposition}

\begin{proof}
Fix the good cover $\mathcal U=\{U_\alpha\}$ of $U$, the sector-box cover $\mathcal B=\{B_{i,a}\}$ of $N^\times$, and a contractible refinement
$\mathcal W=\{W_{\alpha;i,a}\}$ of the overlap $U\cap N^\times$ as in \Cref{subsec:curve-model-clean}. The global presenter is the explicit $2$-pushout
\begin{equation}\label{eq:curve-pushout-cech}
\G_{\Phi,n}\ :=\ \GU\ \sqcup_{\GX}\ \GC,
\qquad
\GU=\Cech(\mathcal U),\ \GX=\Cech(\mathcal W),\ \GC=\Cech\text{--Stokes}(\mathcal B).
\end{equation}

Choose basepoints $x_\alpha\in U_\alpha$ and $b_{i,a}\in B_{i,a}$, and choose basepoints in each $W_{\alpha;i,a}$ compatibly with the inclusions
$W_{\alpha;i,a}\subset U_\alpha$ and $W_{\alpha;i,a}\subset B_{i,a}$.
Since all charts and all nonempty finite intersections are contractible, each connected overlap carries a unique homotopy class of paths between the
chosen basepoints. Therefore each \v{C}ech presenter admits the standard Morita reduction to a small groupoid with one object per chart and one arrow
per nonempty overlap:
\begin{equation}\label{eq:morita-reductions}
\GU\ \simeq_{\mathrm{Morita}}\ \GU^{\mathrm{bp}},
\qquad
\GX\ \simeq_{\mathrm{Morita}}\ \GX^{\mathrm{bp}},
\qquad
\GC\ \simeq_{\mathrm{Morita}}\ \GC^{\mathrm{bp}}.
\end{equation}
Moreover, the refinement functors $j_U:\GX\to\GU$ and $j_C:\GX\to\GC$ induce functors
$j_U^{\mathrm{bp}}:\GX^{\mathrm{bp}}\to\GU^{\mathrm{bp}}$ and $j_C^{\mathrm{bp}}:\GX^{\mathrm{bp}}\to\GC^{\mathrm{bp}}$, compatible with
\eqref{eq:morita-reductions}. Taking explicit $2$-pushouts (realized by Lemma~\ref{lem:strict-pushout-is-bicategorical}) and using the $2$-pushout universal property yields a comparison functor between the two pushouts, unique up to unique isomorphism. This comparison is a Morita equivalence, hence a canonical chain
\begin{equation}\label{eq:morita-chain-curve}
\G_{\Phi,n}=\GU\sqcup_{\GX}\GC\ \simeq_{\mathrm{Morita}}\ \GU^{\mathrm{bp}}\sqcup_{\GX^{\mathrm{bp}}}\GC^{\mathrm{bp}}
=: \G^{\mathrm{bp}}_{\Phi,n}.
\end{equation}
This passage is summarized by Diagram~\eqref{eq:curve-morita-cube} below.

Finally, by construction the reduced pushout $\G^{\mathrm{bp}}_{\Phi,n}$ is the explicit finite skeleton
$\G^{\mathrm{sk}}_{\Phi,n}$ introduced in \Cref{subsec:curve-model-clean}: it has objects $x_0$ and the sector basepoints $b_{i,a}$, and it is
generated by ordinary monodromy in $U$ together with the Stokes-labelled arrows between adjacent sectors, modulo the overlap relations coming from
$\GX$. Thus $\G^{\mathrm{bp}}_{\Phi,n}\cong \G^{\mathrm{sk}}_{\Phi,n}$. Applying $\Rep_r(-)$, which is invariant under Morita equivalence of
presenters, yields \eqref{eq:repr-clean}.

\begin{equation}\label{eq:curve-morita-cube}
\begin{tikzcd}[row sep=large, column sep=large]
\GU
\arrow[d, "\simeq_{\mathrm{Morita}}" description]
\arrow[rrr, bend left=18, "i_U"]
&
\GX
\arrow[l, "j_U"']
\arrow[r, "j_C"]
\arrow[d, "\simeq_{\mathrm{Morita}}" description]
&
\GC
\arrow[d, "\simeq_{\mathrm{Morita}}" description]
\arrow[r, "i_C"]
&
\GU \sqcup_{\GX} \GC
\arrow[d, "\simeq_{\mathrm{Morita}}" description]
\\
\GU^{\mathrm{bp}}
\arrow[rrr, bend right=18, "i_U^{\mathrm{bp}}"']
&
\GX^{\mathrm{bp}}
\arrow[l, "j_U^{\mathrm{bp}}"']
\arrow[r, "j_C^{\mathrm{bp}}"]
&
\GC^{\mathrm{bp}}
\arrow[r, "i_C^{\mathrm{bp}}"]
&
\GU^{\mathrm{bp}} \sqcup_{\GX^{\mathrm{bp}}} \GC^{\mathrm{bp}}
\arrow[d, "\cong"]
\\
&&&
\G^{\mathrm{sk}}_{\Phi,n}.
\end{tikzcd}
\end{equation}
\end{proof}

Irregular singular connections and Stokes phenomena may be viewed, at their core, as a problem of
\emph{topological encoding of analytic continuation near a divisor}.
Let $X$ be a complex manifold (or a smooth complex algebraic variety), and let $D\subset X$ be a simple normal crossings divisor.
Write $U:=X\setminus D$.
A meromorphic flat connection on $(X,D)$ restricts to $U$ as an ordinary local system, but its behavior near $D$
involves additional, genuinely nonabelian gluing data: the \emph{Stokes jumps}.
In dimension one this is classical (see e.g.\ \cite{BV89,Sabbah,Boalch2014}); in higher dimension, the same principle persists,
with one essential geometric change:
the relevant ``directions'' around $D$ vary along the strata of $D$ and must be handled functorially across charts.
A Betti-side presenter that is useful for geometry should therefore keep track simultaneously of
(i) the directional boundary geometry, (ii) tame/Kummer branching along all strata, and
(iii) the wild Stokes gluing across sectorial overlaps.

A standard form of the irregular Riemann--Hilbert correspondence asserts that meromorphic connections
with prescribed formal/irregular type are classified on the Betti side by \emph{Stokes-filtered}
(or Stokes-graded) local systems on a real boundary space surrounding $D$.
Analytically, this boundary is often realized by the real oriented blow-up along $D$,
while modern formulations use enhanced or perverse-sheaf avatars of irregularity (cf.\ \cite{DAgnoloKashiwara,Mochizuki}).
In the Stokes-filtered description, the Stokes jumps are encoded by a Stokes sheaf on the boundary,
and the classification problem becomes a torsor problem.
While this framework is well-established, the Betti objects are frequently presented either analytically
or as stacks defined by descent; in particular, the underlying \emph{presenting groupoid} is often left implicit.
For applications where one wants strict gluing along covers (for instance as input to Betti-stack or Tannakian formalisms),
it is useful to have a small and explicit strict groupoid model adapted to normal crossings strata and compatible with tame/Kummer ramification.

A complementary viewpoint, especially effective on curves, is to encode wild monodromy by groupoids:
Stokes data can be organized into ``wild'' fundamental groupoids, and moduli can be described as representation groupoids
of these presenters (see e.g.\ \cite{GualtieriLiPym} and the references therein).
This has two structural advantages: it makes cutting-and-pasting arguments transparent, and it interfaces naturally with
geometric structures on moduli (Poisson/symplectic forms, generators-and-relations descriptions, and related constructions).
In higher dimension, however, one faces an additional bookkeeping issue: the boundary geometry is naturally \emph{logarithmic},
and the tame/Kummer layer interacts with the Stokes layer along strata of arbitrary depth.
The aim of this paper is to give a single, strictly $1$-categorical, cover-based model in which the directional, tame, and wild layers
appear together through explicit maps of small groupoids governed by universal properties (pullbacks and pushouts),
and which admits finite skeletal models well-suited to computation (compare the \textit{skeleton} perspective in \cite{TeyssierSkeletons}).

We work with the logarithmic Betti realization of the pair $(X,D)$.
Let $X^{\log}$ denote the Kato--Nakayama space of $(X,D)$, equipped with the canonical map $\tau:X^{\log}\to X$ \cite{KatoNakayama}.
Locally, near a point where $k$ components of $D$ meet, the fiber of $\tau$ is a real torus $(S^1)^k$,
canonically recording the angular directions around each local branch.
To incorporate tame ramification uniformly we pass to the Kummer level $n$ and use the Kummer log cover
$X^{\log}_n\to X^{\log}$ (cf.\ \cite{NakayamaOgus,Ogus}); along a depth-$k$ stratum the natural deck group is $(\mu_n)^k$.
This is the bookkeeping forced by multi-branch behavior once one insists on a boundary model compatible with descent.

A recurring point of terminology concerns the word \textit{representation}.
If $G$ is a small (topological) groupoid, we write $\Rep_r(G)$ for the groupoid of \emph{torsorial} data on $G$:
principal $\GL_r(\mathbb C)^\delta$-bundles on $G_0$ equipped with descent isomorphisms along $G_1$
(here $\GL_r(\mathbb C)^\delta$ denotes the discrete group).
This unpointed notion is stable under refinement and Morita equivalence, and it is the natural target for strict gluing arguments.
When a basepoint and a trivialization are fixed on a connected space, one recovers the familiar $\mathrm{Hom}(\pi_1,-)$ description,
but we do not impose such auxiliary choices.

Fix an irregular type $\Phi$ along $D$, and let $N^\times\subset X^{\log}_n$ be a punctured collar neighborhood of the logarithmic boundary.
We consider the groupoid $\LocSys_r(U)$ of rank-$r$ local systems on $U$ and the groupoid
$\StokesLocal_r(N^\circ;\Phi)$ of rank-$r$ Stokes local systems on the collar of type $\Phi$,
together with their restrictions to the overlap $N^\circ\subset U$.
We define the global Stokes object intrinsically as the $2$-fiber product
\begin{equation}\label{eq:intro-global-stokes}
\Stokes_r(X^{\log}_n;D,\Phi)
:=
\LocSys_r(U)\times^{(2)}_{\LocSys_r(N^\circ)}
\StokesLocal_r(N^\circ;\Phi).
\end{equation}
This formalizes the basic gluing principle: an irregular object is obtained by gluing an ordinary local system on $U$
with a Stokes enhancement on a collar, compatibly on the overlap.

Our main technical step is to present \eqref{eq:intro-global-stokes} by an explicit \emph{small strict groupoid}
built from \v{C}ech data.
Choose a good cover of $U$ and a sectorial/Stokes cover of $N^\times$ compatible on $U\cap N^\times$.
Let $\G_U$ be the \v{C}ech groupoid of the interior cover, let $\G_C$ be the \v{C}ech--Stokes collar groupoid on $N^\times$,
and let $\G_{\times}$ be the induced overlap groupoid.
The collar groupoid comes equipped with a canonical projection
$\pi:\G_C\to \check{\mathrm C}(\mathcal B)$ to the underlying sector \v{C}ech groupoid (forgetting the Stokes labels);
the correct collar moduli is the section groupoid $\operatorname{Sec}(\pi)$, rather than a representation groupoid of $\G_C$.
We then form the explicit $2$-pushout in small groupoids
\begin{equation}\label{eq:pushout-intro}
\G_{\Phi,n}\ :=\ \G_U \sqcup_{\G_{\times}} \G_C.
\end{equation}
The pushout is the groupoid-theoretic expression of \textit{glue interior monodromy and boundary Stokes torsors
along their common restriction.}

Irregular singular connections and Stokes phenomena may be viewed, at their core, as a problem of
\emph{topological encoding of analytic continuation near a divisor}.
Let $X$ be a complex manifold (or a smooth complex algebraic variety), and let $D\subset X$ be a simple normal crossings divisor.
Write $U:=X\setminus D$.
A meromorphic flat connection on $(X,D)$ restricts to $U$ as an ordinary local system, but its behavior near $D$
involves additional, genuinely nonabelian gluing data: the \emph{Stokes jumps}.
In dimension one this is classical (see e.g.\ \cite{BV89,Sabbah,Boalch2014}); in higher dimension, the same principle persists,
with one essential geometric change:
the relevant ``directions'' around $D$ vary along the strata of $D$ and must be handled functorially across charts.
A Betti-side presenter that is useful for geometry should therefore keep track simultaneously of
(i) the directional boundary geometry, (ii) tame/Kummer branching along all strata, and
(iii) the wild Stokes gluing across sectorial overlaps.

A complementary viewpoint, especially effective on curves, is to encode wild monodromy by groupoids:
Stokes data can be organized into ``wild'' fundamental groupoids, and moduli can be described as representation groupoids
of these presenters (see e.g.\ \cite{GualtieriLiPym} and the references therein).
This has two structural advantages: it makes cutting-and-pasting arguments transparent, and it interfaces naturally with
geometric structures on moduli (Poisson/symplectic forms, generators-and-relations descriptions, and related constructions).
In higher dimension, however, one faces an additional bookkeeping issue: the boundary geometry is naturally \emph{logarithmic},
and the tame/Kummer layer interacts with the Stokes layer along strata of arbitrary depth.
The aim of this paper is to give a single, strictly $1$-categorical, cover-based model in which the directional, tame, and wild layers
appear together through explicit maps of small groupoids governed by universal properties (pullbacks and pushouts),
and which admits finite skeletal models well-suited to computation (compare the \textit{skeleton} perspective in \cite{TeyssierSkeletons}).

We work with the logarithmic Betti realization of the pair $(X,D)$.
Let $X^{\log}$ denote the Kato--Nakayama space of $(X,D)$, equipped with the canonical map $\tau:X^{\log}\to X$ \cite{KatoNakayama}.
Locally, near a point where $k$ components of $D$ meet, the fiber of $\tau$ is a real torus $(S^1)^k$,
canonically recording the angular directions around each local branch.
To incorporate tame ramification uniformly we pass to the Kummer level $n$ and use the Kummer log cover
$X^{\log}_n\to X^{\log}$ (cf.\ \cite{NakayamaOgus,Ogus}); along a depth-$k$ stratum the natural deck group is $(\mu_n)^k$.
This is the bookkeeping forced by multi-branch behavior once one insists on a boundary model compatible with descent.

A recurring point of terminology concerns the word \textit{representation}.
If $G$ is a small (topological) groupoid, we write $\Rep_r(G)$ for the groupoid of \emph{torsorial} data on $G$:
principal $\GL_r(\mathbb C)^\delta$-bundles on $G_0$ equipped with descent isomorphisms along $G_1$
(here $\GL_r(\mathbb C)^\delta$ denotes the discrete group).
This unpointed notion is stable under refinement and Morita equivalence, and it is the natural target for strict gluing arguments.
When a basepoint and a trivialization are fixed on a connected space, one recovers the familiar $\mathrm{Hom}(\pi_1,-)$ description,
but we do not impose such auxiliary choices.

Fix an irregular type $\Phi$ along $D$, and let $N^\times\subset X^{\log}_n$ be a punctured collar neighborhood of the logarithmic boundary.
We consider the groupoid $\LocSys_r(U)$ of rank-$r$ local systems on $U$ and the groupoid
$\StokesLocal_r(N^\circ;\Phi)$ of rank-$r$ Stokes local systems on the collar of type $\Phi$,
together with their restrictions to the overlap $N^\circ\subset U$.
We define the global Stokes object intrinsically as the $2$-fiber product
\begin{equation}\label{eq:intro-global-stokes}
\Stokes_r(X^{\log}_n;D,\Phi)
:=
\LocSys_r(U)\times^{(2)}_{\LocSys_r(N^\circ)}
\StokesLocal_r(N^\circ;\Phi).
\end{equation}
This formalizes the basic gluing principle: an irregular object is obtained by gluing an ordinary local system on $U$
with a Stokes enhancement on a collar, compatibly on the overlap.

Our main technical step is to present \eqref{eq:intro-global-stokes} by an explicit \emph{small strict groupoid}
built from \v{C}ech data.
Choose a good cover of $U$ and a sectorial/Stokes cover of $N^\times$ compatible on $U\cap N^\times$.
Let $\G_U$ be the \v{C}ech groupoid of the interior cover, let $\G_C$ be the \v{C}ech--Stokes collar groupoid on $N^\times$,
and let $\G_{\times}$ be the induced overlap groupoid.
The collar groupoid comes equipped with a canonical projection
$\pi:\G_C\to \check{\mathrm C}(\mathcal B)$ to the underlying sector \v{C}ech groupoid (forgetting the Stokes labels);
the correct collar moduli is the section groupoid $\operatorname{Sec}(\pi)$, rather than a representation groupoid of $\G_C$.
We then form the explicit $2$-pushout in small groupoids
\begin{equation}\label{eq:pushout-intro}
\G_{\Phi,n}\ :=\ \G_U \sqcup_{\G_{\times}} \G_C.
\end{equation}
The pushout is the groupoid-theoretic expression of \textit{glue interior monodromy and boundary Stokes torsors
along their common restriction.}

\begin{equation}\label{eq:intro-global-stokes}
\Stokes_r(\Xlogn;D,\Phi)
:=
\LocSys_r(U)\times^{(2)}_{\LocSys_r(N^\circ)}
\StokesLocal_r(N^\circ;\Phi).
\end{equation}

Our main technical step is to present \eqref{eq:intro-global-stokes} by an explicit \emph{small strict groupoid}
built from \v{C}ech data.
Choose a good cover of $U$ and a sectorial/Stokes cover of $N^\times$ compatible on $U\cap N^\times$.
Let $\GU$ be the \v{C}ech groupoid of the interior cover, let $\GC$ be the \v{C}ech--Stokes collar groupoid on $N^\times$,
and let $\GX$ be the induced overlap groupoid.
The collar groupoid comes equipped with a canonical projection
$\pi:\GC\to \Cech(\mathcal B)$ to the underlying sector \v{C}ech groupoid (forgetting the Stokes labels);
the correct collar moduli is the section groupoid $\Sec(\pi)$, rather than a representation groupoid of $\GC$.
We then form the explicit $2$-pushout in small groupoids
\begin{equation}\label{eq:pushout-intro}
\G_{\Phi,n}\ :=\ \GU \sqcup_{\GX} \GC.
\end{equation}
The pushout is the groupoid-theoretic expression of \textit{glue interior monodromy and boundary Stokes torsors
along their common restriction.}

Although the global Stokes groupoid \eqref{eq:intro-global-stokes} is defined intrinsically as a $2$-fiber product,
the main point of this paper is to provide an explicit small strict presenter for it, adapted to SNC depth and Kummer descent.
Concretely, we:
\begin{itemize}
\item construct a strict \v{C}ech--Stokes collar groupoid $\GC$ and identify the correct collar moduli as $\Sec(\pi)$
(rather than $\Rep_r(\GC)$), making the Stokes layer strictly functorial and cover-based;
\item build the explicit $2$-pushout presenter $\G_{\Phi,n}$ and prove that it computes global Stokes objects via the torsorial gluing problem
in \Cref{thm:intro-main};
\item make the tame/Kummer layer along depth strata functorial at the level of presenters, enabling explicit descent constraints
and, in particular, a hands-on description in terms of equivariant cocycles;
\item provide finite skeletal models (invariant up to Morita equivalence, with contractible ambiguity) and an explicit
generators-and-relations/chamber-complex description in SNC corners, solving the local presentation problem highlighted in
Problem~12.3.
\end{itemize}

\begin{theorem}\label{thm:main}
For every $r\ge 1$ there is a natural equivalence of groupoids
\begin{equation}\label{eq:main-equiv}
\Rep_r(\GU)\times^{(2)}_{\Rep_r(\GX)}\Rep^{\mathrm{St}}_r(\GC)
\ \simeq\
\mathrm{Stokes}_r(X^{\log}_n;D,\Phi).
\end{equation}
Under this equivalence, the projection to $\Rep_r(\GU)$ corresponds to the forgetful functor
$$\mathrm{Stokes}_r(X^{\log}_n;D,\Phi)\to\LocSys_r(U)$$
\end{theorem}

\begin{proof}
By \Cref{prop:rep-cech-locsys} there are natural equivalences
\begin{equation}\label{eq:rep-locsys-identifications}
\Rep_r(\GU)\ \simeq\ \LocSys_r(U),
\qquad
\Rep_r(\GX)\ \simeq\ \LocSys_r(N^\circ),
\end{equation}
and by \Cref{lem:cech-naturality} these identifications intertwine the restriction functor $$j_U^*:\Rep_r(\GU)\to\Rep_r(\GX)$$ with the usual
restriction $\LocSys_r(U)\to\LocSys_r(N^\circ)$.
On the collar side, \Cref{prop:RepStokesGC-eq-StokesLocal} gives a natural equivalence
\begin{equation}\label{eq:collar-identification}
\Rep^{\mathrm{St}}_r(\GC)=\Sec(\pi)\ \simeq\ \mathrm{StokesLocal}_r(N^\circ;\Phi),
\end{equation}
and the forgetful functor $\Rep^{\mathrm{St}}_r(\GC)\to \Rep_r(\GX)$ corresponds to the functor
$\mathrm{StokesLocal}_r(N^\circ;\Phi)\to \LocSys_r(N^\circ)$ sending a Stokes-local object to its underlying local system.
Consequently, replacing each term in the $2$-fiber product
$\Rep_r(\GU)\times^{(2)}_{\Rep_r(\GX)}\Rep^{\mathrm{St}}_r(\GC)$ using \eqref{eq:rep-locsys-identifications} and \eqref{eq:collar-identification}
identifies it canonically with
\[
\LocSys_r(U)\times^{(2)}_{\LocSys_r(N^\circ)}\mathrm{StokesLocal}_r(N^\circ;\Phi),
\]
which equals $\mathrm{Stokes}_r(X^{\log}_n;D,\Phi)$ by Definition~\ref{def:global-stokes}. This proves \eqref{eq:main-equiv}.
The final statement follows from the fact that both sides are defined as $2$-fiber products over the restriction to $N^\circ$, hence the projection to
$\Rep_r(\GU)\simeq \LocSys_r(U)$ is exactly the forgetful functor.
\end{proof}
\begin{equation}\label{eq:main-diagram}
\begin{tikzcd}[row sep=large, column sep=huge]
\Rep_r(\GU)\times^{(2)}_{\Rep_r(\GX)}\Rep^{\mathrm{St}}_r(\GC)
\arrow[r,"\sim"] \arrow[d]
&
\LocSys_r(U)\times^{(2)}_{\LocSys_r(N^\circ)}\mathrm{StokesLocal}_r(N^\circ;\Phi)
\arrow[d]
\\
\Rep_r(\GU) \arrow[r,"\sim"]
&
\LocSys_r(U)
\end{tikzcd}
\end{equation}

\begin{proposition}\label{prop:van-kampen-presenters}
There is a canonical zig-zag of Morita equivalences exhibiting the global presenter $\G_{\Phi,n}$ as a bicategorical pushout of
$\G_{\Phi,n}^{(1)}$ and $\G_{\Phi,n}^{(2)}$ over $\G_{\Phi,n}^{(12)}$ in $\mathbf{Grpd}$.
Consequently, for every $r\ge 1$ there is a natural equivalence
\begin{equation}\label{eq:vk-rep-pullback}
\Rep^{\mathrm{St}}_r(\G_{\Phi,n})
\ \simeq\
\Rep^{\mathrm{St}}_r(\G_{\Phi,n}^{(1)})\times^{(2)}_{\Rep^{\mathrm{St}}_r(\G_{\Phi,n}^{(12)})}\Rep^{\mathrm{St}}_r(\G_{\Phi,n}^{(2)}).
\end{equation}
\end{proposition}

\begin{proposition}\label{prop:curve-skeleton-clean}
There is a canonical chain of Morita equivalences connecting the cover-based presenter $\G_{\Phi,n}$ and the finite skeletal presenter
$\G^{\mathrm{sk}}_{\Phi,n}$. In particular, they present the same classifying stack, and for every $r\ge 1$ there is a natural equivalence
of groupoids
\begin{equation}\label{eq:repr-clean}
\Rep_r(\G_{\Phi,n})\ \simeq\ \Rep_r(\G^{\mathrm{sk}}_{\Phi,n}).
\end{equation}
\end{proposition}

\subsubsection*{Global Stokes objects from the skeletal presenter}
\begin{theorem} \label{thm:global-skeletal-pushout}
For every $r\ge 1$ there is a natural equivalence of groupoids
\begin{equation}\label{eq:global-skeletal-pushout}
\Stokes_r(X^{\log}_n;D,\Phi)\ \simeq\
\Rep_r(\GU^{\mathrm{sk}})\times^{(2)}_{\Rep_r(\GX^{\mathrm{sk}})}\Sec(\pi^{\mathrm{sk}}),
\end{equation}
where $\Stokes_r(X^{\log}_n;D,\Phi)$ is the intrinsic global Stokes groupoid from Definition~\ref{def:global-stokes}.
Equivalently, global Stokes objects are classified by Stokes-corrected torsorial representations of the small presenter
$\G_{\Phi,n}^{\mathrm{sk}}$.
\end{theorem}

\begin{theorem} \label{thm:kummer-descent-skeletal}
Fix a depth-$k$ chart and let $G_k=(\mu_n)^k$ be the local deck group of $q_n$.
Then pullback along $q_n$ identifies global Stokes objects downstairs with homotopy fixed points upstairs:
\begin{equation}\label{eq:kummer-descent-skeletal}
\Stokes_r(X^{\log};D,\Phi)\ \simeq\ \Stokes_r(X^{\log}_n;D,q_n^*\Phi)^{hG_k},
\end{equation}
where $(-)^{hG_k}$ denotes the homotopy fixed point groupoid (Definition~\ref{def:hfp-finite}).
Moreover, after choosing equivariant triangulations as in Proposition~\ref{prop:equivariant-triangulation} on the collar (and similarly on
$U$ and on the overlap), the descent equivalence is visible on the skeletal presenter: the $G_k$-action on $\G_{\Phi,n}^{\mathrm{sk}}$
induces a strict action on
$$\Rep_r(\GU^{\mathrm{sk}})\times^{(2)}_{\Rep_r(\GX^{\mathrm{sk}})}\Sec(\pi^{\mathrm{sk}}),$$ and its homotopy fixed points compute the left-hand side of
\eqref{eq:kummer-descent-skeletal} under the identification of Theorem~\ref{thm:global-skeletal-pushout}.
\end{theorem}

\begin{equation}\label{eq:intro-global-stokes}
\Stokes_r(\Xlogn;D,\Phi)
:=
\LocSys_r(U)\times^{(2)}_{\LocSys_r(N^\circ)}
\StokesLocal_r(N^\circ;\Phi).
\end{equation}

\begin{theorem} \label{thm:intro-main}
For every $r\ge 1$ there is a natural equivalence of groupoids
\begin{equation}\label{eq:intro-main-equiv}
\Rep_r(\GU)\times^{(2)}_{\Rep_r(\GX)} \Sec(\pi)
\ \simeq\ \Stokes_r(\Xlogn;D,\Phi),
\end{equation}
compatible with restriction to $\Rep_r(\GU)$ (forgetting the wild layer) and with restriction to $\Sec(\pi)$
(forgetting the interior).
\end{theorem}

Although the global Stokes groupoid \eqref{eq:intro-global-stokes} is defined intrinsically as a $2$-fiber product,
the main point of this paper is to provide an explicit small strict presenter for it, adapted to SNC depth and Kummer descent.
Concretely, we:
\begin{itemize}
\item construct a strict \v{C}ech--Stokes collar groupoid $\G_C$ and identify the correct collar moduli as $\operatorname{Sec}(\pi)$
(rather than $\Rep_r(\G_C)$), making the Stokes layer strictly functorial and cover-based;
\item build the explicit $2$-pushout presenter $\G_{\Phi,n}$ and prove that it computes global Stokes objects via the torsorial gluing problem
in \Cref{thm:intro-main};
\item make the tame/Kummer layer along depth strata functorial at the level of presenters, enabling explicit descent constraints
and, in particular, a hands-on description in terms of equivariant cocycles;
\item provide finite skeletal models (invariant up to Morita equivalence, with contractible ambiguity) and an explicit
generators-and-relations/chamber-complex description in SNC corners, solving the local presentation problem highlighted in
Problem~12.3.
\end{itemize}

Theorem~\Cref{thm:intro-main} makes it possible to treat global Stokes objects using a concrete presenter:
questions about existence, comparison, and descent can be translated into torsors and nonabelian \v{C}ech cocycles
on an explicit skeleton underlying $\G_{\Phi,n}$.
We record three consequences that will be used later; each is proved in the indicated section.

\begin{corollary} \label{cor:intro-morita}
If one replaces the interior and collar covers by refinements, the resulting pushout groupoid $G'_{\Phi,n}$
is Morita equivalent to $\G_{\Phi,n}$; in particular the associated torsorial representation groupoids are equivalent:
$\Rep_r(G'_{\Phi,n})\simeq \Rep_r(\G_{\Phi,n})$.
\end{corollary}

\begin{corollary} \label{cor:intro-kummer}
Along a depth-$k$ stratum of $D$, the Kummer cover $X^{\log}_n\to X^{\log}$ has deck group $(\mu_n)^k$,
and the groupoid $\G_{\Phi,n}$ records this tame layer functorially through the collar groupoid $\G_C$.
Consequently, Kummer descent constraints for Stokes objects can be expressed as strict descent conditions
on the collar data (equivalently, as equivariant cocycle constraints on a finite skeleton).
\end{corollary}

\begin{corollary} \label{cor:intro-van-kampen}
Let $X= X^{(1)}\cup X^{(2)}$ be an open cover compatible with $D$, and assume the sectorial/Stokes data are chosen compatibly on overlaps.
Then the presenting groupoids admit a pushout description reflecting this decomposition, so that questions about
$\Stokes_r(X^{\log}_n;D,\Phi)$ may be reduced to representations of an explicit groupoid obtained by gluing the local presenters.

\end{corollary}

For a fully explicit local model (including a finite Morita skeleton), the reader may jump directly to
\Cref{subsec:curve-model-clean}.
There we compute the presenting groupoid near a simple normal crossings corner and spell out the resulting
generators-and-relations (equivalently, nonabelian cocycle) description of the Stokes data.
This example is included as a concrete reference point: it displays, in a single diagram, the directional layer
($X^{\log}$), the tame/Kummer layer (the $(\mu_n)^k$-actions along strata), and the wild Stokes layer
(sector overlaps and Stokes jumps), and it provides a template for later computations.

After recalling the basic geometry of $X^{\log}_n$ in the SNC case and fixing conventions, we define the intrinsic groupoid
$\Stokes_r(X^{\log}_n;D,\Phi)$ and establish its torsor description.
We then construct the interior \v{C}ech groupoid $\G_U$, the collar \v{C}ech--Stokes groupoid $\G_C$, and the overlap groupoid $\G_{\times}$,
form the pushout \eqref{eq:pushout-intro}, and prove \Cref{thm:intro-main} in \Cref{sec:pushout}.
Finally, we discuss Morita invariance, Kummer descent along depth strata, and an explicit coequalizer
(generators-and-relations) model for $\G_{\Phi,n}$, including the SNC computation announced above.

Although the global Stokes groupoid \eqref{eq:intro-global-stokes} is defined intrinsically as a $2$-fiber product, the paper’s proof strategy for Theorem~\Cref{thm:intro-main} is to “provide an explicit small strict presenter for it,” adapted to SNC depth and Kummer descent. Concretely, the outline given is:
\begin{itemize}
\item “construct a strict \v{C}ech--Stokes collar groupoid $\G_C$ and identify the correct collar moduli as $\operatorname{Sec}(\pi)$ (rather than $\Rep_r(\G_C)$), making the Stokes layer strictly functorial and cover-based;”
\item “build the explicit $2$-pushout presenter $\G_{\Phi,n}$ and prove that it computes global Stokes objects via the torsorial gluing problem in \Cref{thm:intro-main};”
\item “make the tame/Kummer layer along depth strata functorial at the level of presenters, enabling explicit descent constraints and, in particular, a hands-on description in terms of equivariant cocycles;”
\item “provide finite skeletal models (invariant up to Morita equivalence, with contractible ambiguity) and an explicit generators-and-relations/chamber-complex description in SNC corners.”
\end{itemize}
More procedurally, the introduction says: “After recalling the basic geometry of $X^{\log}_n$… we define the intrinsic groupoid $\Stokes_r(X^{\log}_n;D,\Phi)$ and establish its torsor description. We then construct the interior \v{C}ech groupoid $\G_U$, the collar \v{C}ech--Stokes groupoid $\G_C$, and the overlap groupoid $\G_{\times}$, form the pushout \eqref{eq:pushout-intro}, and prove \Cref{thm:intro-main} in \Cref{sec:pushout}.”

Although the global Stokes groupoid
\begin{equation}\label{eq:intro-global-stokes}
\Stokes_r(\Xlogn;D,\Phi)
:=
\LocSys_r(U)\times^{(2)}_{\LocSys_r(N^\circ)}
\StokesLocal_r(N^\circ;\Phi).
\end{equation}
is defined intrinsically as a $2$-fiber product, the paper’s proof strategy for the main theorem is to “provide an explicit small strict presenter for it,” adapted to SNC depth and Kummer descent. Concretely, the outline given is:
\begin{itemize}
\item “construct a strict \v{C}ech--Stokes collar groupoid $\G_C$ and identify the correct collar moduli as $\operatorname{Sec}(\pi)$ (rather than $\Rep_r(\G_C)$), making the Stokes layer strictly functorial and cover-based;”
\item “build the explicit $2$-pushout presenter $\G_{\Phi,n}$ and prove that it computes global Stokes objects via the torsorial gluing problem in the main theorem;”
\item “make the tame/Kummer layer along depth strata functorial at the level of presenters, enabling explicit descent constraints and, in particular, a hands-on description in terms of equivariant cocycles;”
\item “provide finite skeletal models (invariant up to Morita equivalence, with contractible ambiguity) and an explicit generators-and-relations/chamber-complex description in SNC corners.”
\end{itemize}
More procedurally, the introduction says: “After recalling the basic geometry of $X^{\log}_n$… we define the intrinsic groupoid $\Stokes_r(X^{\log}_n;D,\Phi)$ and establish its torsor description. We then construct the interior \v{C}ech groupoid $\G_U$, the collar \v{C}ech--Stokes groupoid $\G_C$, and the overlap groupoid $\G_{\times}$, form the pushout
\begin{equation}\label{eq:pushout-intro}
\G_{\Phi,n}\ :=\ \GU \sqcup_{\GX} \GC.
\end{equation}
and prove the main theorem later.”

Question:
Let $X$ be a complex manifold (or a smooth complex algebraic variety), let $D\subset X$ be a simple normal crossings divisor, set $U:=X\setminus D$, and fix an irregular type $\Phi$ along $D$. Let $X^{\log}_n$ be the level-$n$ Kummer log cover of the Kato--Nakayama space $X^{\log}$. The intrinsic global Stokes groupoid is
\[
\Stokes_r(X^{\log}_n;D,\Phi)
:=
\LocSys_r(U)\times^{(2)}_{\LocSys_r(N^\circ)}
\StokesLocal_r(N^\circ;\Phi),
\]
where $N^\circ$ is the overlap with a collar neighborhood of the logarithmic boundary. Choose a good cover of $U$ and a compatible sectorial/Stokes cover of a punctured collar; let $\G_U$ be the resulting interior \v{C}ech groupoid, let $\G_{\times}$ be the induced overlap groupoid, and let $\pi$ be the canonical projection from the collar \v{C}ech--Stokes groupoid to the underlying overlap \v{C}ech groupoid. For a small groupoid $G$, write $\Rep_r(G)$ for the groupoid of rank-$r$ torsorial representations of $G$ (principal $\mathrm{GL}_r(\mathbb C)^\delta$-bundles on objects with descent isomorphisms along arrows), and write $\operatorname{Sec}(\pi)$ for the paper's groupoid of sections associated with $\pi$, equipped with its natural restriction functor to $\Rep_r(\G_{\times})$. Which statement holds for every integer $r\ge 1$?

There is a natural equivalence of groupoids
\[
\Rep_r(\G_U)\times^{(2)}_{\Rep_r(\G_{\times})} \operatorname{Sec}(\pi)
\ \simeq\ \Stokes_r(X^{\log}_n;D,\Phi),
\]
compatible with the restriction to $\Rep_r(\G_U)$ (forgetting the wild layer) and with the restriction to $\operatorname{Sec}(\pi)$ (forgetting the interior).

Choices:
B.
There is a natural equivalence of groupoids
\[
\Rep_r(\G_U)\times^{(2)}_{\Rep_r(\G_{\times})} \Rep_r(\G_C)
\ \simeq\ \Stokes_r(X^{\log}_n;D,\Phi),
\]
compatible with the restriction to $\Rep_r(\G_U)$ and with the restriction to $\Rep_r(\G_C)$.
C.
There is a natural functor of groupoids
\[
\Rep_r(\G_U)\times^{(2)}_{\Rep_r(\G_{\times})} \operatorname{Sec}(\pi)
\longrightarrow \Stokes_r(X^{\log}_n;D,\Phi),
\]
compatible with the restriction to $\Rep_r(\G_U)$ and with the restriction to $\operatorname{Sec}(\pi)$.
D.
There is a natural equivalence of groupoids
\[
\Rep_r(\G_U)\times_{\Rep_r(\G_{\times})} \operatorname{Sec}(\pi)
\ \simeq\ \Stokes_r(X^{\log}_n;D,\Phi),
\]
compatible with the restriction to $\Rep_r(\G_U)$ (forgetting the wild layer) and with the restriction to $\operatorname{Sec}(\pi)$ (forgetting the interior).
E.
For every $r\ge 1$ there is a natural equivalence of groupoids
\[
\Rep_r(\G_U)\times^{(2)}_{\Rep_r(\G_{\times})} \operatorname{Sec}(\pi)
\ \simeq\ \Stokes_r(X^{\log};D,\Phi),
\]
compatible with the restriction to $\Rep_r(\G_U)$ (forgetting the wild layer) and with the restriction to $\operatorname{Sec}(\pi)$ (forgetting the interior).

correct collar moduli as Sec(pi) rather than Rep_r(G_C)

equivalence dropped to mere natural functor

homotopy-coherent 2-fiber product replaced by ordinary fiber product

Kummer level upstairs X^{log}_n replaced by downstairs X^{log}

The stem does not state the equivalence itself and does not explicitly reveal the correct option. It sets up the notation used in the theorem, but the key choice points—equivalence vs functor, 2-fiber product vs ordinary fiber product, the correct collar moduli, and the correct log level—are not given away.

This is essentially a theorem-recall item: the correct choice is the precise theorem statement, with distractors formed by small perturbations. It does not substantially move beyond restating a known result.

Some reasoning is needed to distinguish subtle categorical variants and reject plausible weakenings or nearby false statements. However, the item mainly tests exact recall/recognition of the theorem rather than open-ended mathematical generation.

The distractors are technically plausible and target realistic failure modes: replacing Sec(pi) by ordinary collar representations, weakening equivalence to a functor, confusing 2-pullback with ordinary pullback, and using the wrong log/Kummer level.

Technically well-crafted with strong distractors and little direct answer leakage, but it is mostly a precise theorem-recognition question rather than a generative reasoning task.


For given graphs $H$ and $G$, and a positive integer $r$, we say that $G$ is \emph{Ramsey for} $H$, denoted by $G \longrightarrow (H)_r$, if for every $r$-edge coloring of $G$ with $r$ colors, there exists a monochromatic subgraph of $G$ isomorphic to $H$.
Ramsey's pioneering work~\cite{Ramsey} established that for any graph
$H$ and any positive integer $r$, there exists $N\in \mathbb{N}$ such
that $K_N \longrightarrow (H)_r$. The smallest such~$N$ is called the
\emph{$r$-multicolor Ramsey number} of $H$ and is denoted by $R_r(H)$.

While this perspective focuses on minimizing the number of vertices,
an alternative viewpoint—initiated by Erd\H{o}s, Faudree, Rousseau,
and Schelp~\cite{S.R.Erd}—seeks to minimize the number of edges. This
leads to the notion of \emph{size-Ramsey numbers}. For a graph $H$ and
integer $r \geq 2$, the \emph{$r$-multicolor size-Ramsey number}
of~$H$ is
\[
\widehat{R}_r(H) = \min \{ |E(G)| : G \longrightarrow (H)_r \}.
\]

One of the central problems in this area is to determine for which
families of graphs the size-Ramsey number grows linearly with the
number of vertices. Erd\H{o}s~\cite{Erd_1} asked this question in the
case of paths and this question was addressed by Beck~\cite{Beck_1},
who proved that $\widehat{R}_2(P_n) < 900n$. The linearity of size-Ramsey
numbers has since been established for various graph families:
cycles~\cite{Sudakov-cycle, Haxell-cycle, JKhOP, JM}, bounded-degree
trees~\cite{Friedman, Haxell-tree, Xin}, powers of paths and
cycles~\cite{Power_path, jie}, bounded powers of bounded-degree trees
(equivalently graphs of bounded treewidth and maximum
degree)~\cite{Power_tree, treewidth, jiang13:_degree_ramsey}, and logarithmic subdivisions of bounded-degree graphs~\cite{Rolling}. On the negative side, R\"{o}dl and Szemer\'{e}di~\cite{Rodl} constructed graphs on $n$ vertices with maximum degree $3$ and size-Ramsey number at least $n \log^cn$, for a small constant $c>0$. This bound has been
improved to $ cne^{c\sqrt{\log n}} $ for some $ c > 0 $ by Tikhomirov~\cite{Tikh}. Thus
the linearity of size-Ramsey numbers does not extend universally to bounded degree graphs.

Given a graph $H$ and a function $\sigma:E(H)\to \mathbb{N}$, the \emph{subdivision} $H^\sigma$ is obtained by replacing each edge $e \in E(H)$ with a path of length $\sigma(e)$.
Answering a conjecture of
Pak~\cite{Pak}, Dragani\'c, Krivelevich, and Nenadov~\cite{Rolling} proved the following theorem.

\begin{abstract}
 For a positive integer $r$, the $r$-color size-Ramsey
 number~$\widehat{R}_r(H)$ of a graph $H$ is the minimum number of
 edges in a graph $G$ such that every $r$-edge coloring of $G$
 contains a monochromatic copy of $H$. For a graph~$H$ and a
 function $\sigma:E(H)\to \mathbb{N}$, the \emph{subdivision}
 $H^\sigma$ is obtained by replacing every $e \in E(H)$ with a path
 of length $\sigma(e)$. In~\cite{javadi25:_induced_long} it is shown
 that for all integers $r,\, D\geq 2 $, there exists a constant
 $c=c(r, D)$ such that for every graph $ H $ with maximum degree $D$
 if $H^{\sigma}$ is a subdivision of~$H$ in which
 $\sigma(e) > c \log n $ for every $e \in E(H)$, where
 $n=|V(H^\sigma)|$, then
 $ \widehat{R}_r(H^\sigma) = O\big(2^{34r}
 r^6 \log^5(r) D^5\log D\big)n. $ We improve upon this result in the case that~$H^{\sigma}$
 is a bipartite graph and the number of colors~$r$ is large using a
 significantly different argument, obtaining the bound
 $ \widehat{R}_r(H^{\sigma}) \leq r^{400D \log D} \, n $.
\end{abstract}

Given a graph $H$ and a function $\sigma:E(H)\to \mathbb{N}$, the \emph{subdivision} $H^\sigma$ is obtained by replacing each edge $e \in E(H)$ with a path of length $\sigma(e)$.
Answering a conjecture of
Pak~\cite{Pak}, Dragani\'c, Krivelevich, and Nenadov~\cite{Rolling} proved the following theorem.

The proof of Theorem~\ref{conj:pak} is based on Szemer\'edi's regularity lemma and makes no attempt to optimize the constant~$C$.

\begin{theorem}{\rm~\cite{javadi25:_induced_long}}
 \label{thm:odd_subd}
 Let $r,\, D \geq 2$ be integers, and let $H^\sigma$ be a subdivision
 of a graph~$H$ with maximum degree~$D$ with
 $ \sigma(e) > {2\log_2^2r\log_{D}n+Cr^4 2^{2r}\log^2 D}$
 for every $e \in E(H)$, where $n=|V(H^\sigma)|$ and~$C$ is a
 suitably large absolute constant. Then
 \begin{equation}
 \label{eq:1}
 \widehat{R}_r(H^\sigma) = O\big(2^{34r}
 r^6 \log^5(r) D^5\log D\big)n.
 \end{equation}
\end{theorem}

\begin{theorem}{\rm~\cite{javadi25:_induced_long}}
 \label{thm:even_subd}
 Let $r, \, D \geq 2$ be integers, and let $H^\sigma$ be a
 subdivision of a graph~$H$ with maximum degree~$D$ with $\sigma(e)$
 even and larger than ${2\log_2^2r\log_{D}n+Cr^4 2^{2r}\log^2 D}$ for
 every $e \in E(H)$, where $n=|V(H^\sigma)|$ and~$C$ is a suitably
 large absolute constant. Then
 \begin{equation}
 \label{eq:2}
 \widehat{R}_r(H^\sigma) = O\big(r^{34}
 D^5\log D\big)n.
 \end{equation}
\end{theorem}

\begin{theorem}
 \label{thm:main}
 Let $r, \, D\geq 2$ be integers, and let~$H$ be a graph with maximum
 degree~$D$. Suppose~$H^\sigma$ is a subdivision of~$H$
 with~$\sigma(e)\geq 2\log_{D-1} n$ for every $e \in E(H)$, where
 $n=|V(H^\sigma)|$. Furthermore, suppose~$H^\sigma$ is bipartite.
 Then
 \begin{equation}
 \label{eq:3}
 \widehat{R}_r(H^\sigma) \leq r^{\,400 D\log D} \, n.
 \end{equation}
\end{theorem}

It was shown in~\cite{JM} that there exist explicit bipartite graphs whose local edge densities mimic those expected in random bipartite graphs. The following lemma that is Lemma 2.2 in~\cite{JM}, provides a precise quantitative formulation of this phenomenon.
\begin{lemma}{\rm~\cite{JM}}\label{quasi}
 Let $c_1$ be a positive integer and $c_2$, $c_3$, $\varepsilon$, $\delta$ be positive numbers such that $c_3\leq c_1$, $0<\epsilon\leq 1/2$ and $3/2\geq \delta >\sqrt{6{\log({c_1e}/{c_3})/(c_2c_3)}} $. Then there exists $n_0=n_0(c_1,c_2,c_3,\epsilon, \delta)$ for which the following holds for every $n\geq n_0$.

In a complementary direction, the following lemma shows that the goodness property is also preserved when successively removing degree (at most) one vertices from the embedded graph. This is the bipartite analogue of~\cite[Lemma~2.4]{Rolling} and we include a proof in Appendix~\ref{app2} for completeness.
\begin{lemma}\label{lem:good_removing}
Let $n$ and $D$ be positive integers. Consider an $(n,D)$-bipartite graph $F$ and suppose that $\phi: F \hookrightarrow G$ is an $(n,D)$-good embedding. Assume that $F'$ is obtained from $F$ by successively deleting vertices of degree at most one. Then the restriction of $\phi$ to $F'$ is also $(n,D)$-good.
\end{lemma}
We next recall another useful structural concept introduced in~\cite{JM}, which allows us to pass from certain dense bipartite configurations to bipartite graphs with strong expansion.
\begin{definition}{\rm~\cite{JM}}
Let $G=G(V_1,V_2)$ be a bipartite graph with $|V_1|=|V_2|=N$ and let $\alpha\in(0,1)$ be a real number. We say that $G$ is \emph{$\alpha$-joined} if for every pair of subsets $A\subseteq V_1$ and $B\subseteq V_2$ with $|A|, \,|B|\geq \alpha N$, we have $e(A,B)\neq0$.
\end{definition}
The next lemma shows that every $\alpha$-joined bipartite graph contains a large induced subgraph with strong expansion properties. Moreover, this subgraph also supports a good embedding of a suitable null graph.
\begin{lemma} \label{lem:expender_in_join}
Let $G=G(V_1,V_2)$ be an $\alpha$-joined bipartite graph with $0<\alpha<1/5$ and $|V_1|=|V_2|=N$. Then there exists an induced subgraph $G'=G'(V'_1,V'_2)$ of $G$ such that for each $i\in \{1,2\}$,
 \begin{itemize}
 \item $|V'_i|\geq (1-\alpha)N$,
 \item for every $X\subset V'_i$, with $|X|\leq \alpha N$, we have $|N_{G'}(X)|> (1-4\alpha)/{(2\alpha)} |X|$, and
 \item for every $X\subset V'_i$, with $|X|> \alpha N$, we have $|N_{G'}(X)|> (1-2\alpha)N$.
 \end{itemize}
 In particular, $G'$ is $(6\alpha N, (1-2\alpha)/(6\alpha))$-expanding.
 Moreover, there exists an embedding $\phi: F\hookrightarrow G'$, where $F$ is a null bipartite graph with $\alpha N$ vertices in each part, such that $\phi$ is a
 $(6\alpha N, {(1-4\alpha)}/{(6\alpha)})$-good embedding.
\end{lemma}
\begin{proof}

\begin{theorem}
  {\rm~\cite{Rolling}}
  \label{conj:pak}
  For every $ r,\, D\in \mathbb{N}  $ there exist  $ C,\,L > 0 $ such that if $ H $
  is a graph with $ \Delta(H) \leq D $ and $ \sigma(e)\ge L\log(|V(H^{\sigma})|) $ for all $ e \in E(H) $, then
  $\widehat{R}_r(H^{\sigma})\leq C|V(H^{\sigma})| $.
\end{theorem}

The proof of Theorem~\ref{conj:pak} is based on Szemer\'edi's regularity lemma and makes no attempt to optimize the constant~$C$.

In a recent paper~\cite{javadi25:_induced_long} we reprove
Theorem~\ref{conj:pak} with the goal of obtaining reasonable
explicit bounds for the constants, in particular avoiding the use of
regularity methods. Specifically, our main result for general
subdivisions in~\cite{javadi25:_induced_long} is as follows.

\begin{theorem}{\rm~\cite{javadi25:_induced_long}}
  \label{thm:odd_subd}
  Let $r,\, D \geq 2$ be integers, and let $H^\sigma$ be a subdivision
  of a graph~$H$ with maximum degree~$D$ with
  $ \sigma(e) > {2\log_2^2r\log_{D}n+Cr^4 2^{2r}\log^2 D}$
  for every $e \in E(H)$, where $n=|V(H^\sigma)|$ and~$C$ is a
  suitably large absolute constant. Then
  \begin{equation}
    \label{eq:1}
    \widehat{R}_r(H^\sigma) = O\big(2^{34r}
        r^6 \log^5(r) D^5\log D\big)n.
  \end{equation}
\end{theorem}

In~\cite{javadi25:_induced_long} we also prove that if we further
restrict to the case that $\sigma(e)$ is even for every $e \in E(H)$,
then Theorem~\ref{thm:odd_subd} can be significantly improved.

\begin{theorem}{\rm~\cite{javadi25:_induced_long}}
  \label{thm:even_subd}
  Let $r, \, D \geq 2$ be integers, and let $H^\sigma$ be a
  subdivision of a graph~$H$ with maximum degree~$D$ with $\sigma(e)$
  even and larger than ${2\log_2^2r\log_{D}n+Cr^4 2^{2r}\log^2 D}$ for
  every $e \in E(H)$, where $n=|V(H^\sigma)|$ and~$C$ is a suitably
  large absolute constant.  Then
  \begin{equation}
    \label{eq:2}
    \widehat{R}_r(H^\sigma) = O\big(r^{34}
        D^5\log D\big)n.
  \end{equation}
\end{theorem}

Clearly, the subdivisions~$H^\sigma$ considered in
Theorem~\ref{thm:even_subd} are bipartite graphs.  In this paper we
prove that an additional improvement of Theorem~\ref{thm:odd_subd} is
possible when the subdivided graph is bipartite, \textit{without}
assuming that every~$\sigma(e)$ is even.

\begin{theorem}
  \label{thm:main}
  Let $r, \, D\geq 2$ be integers, and let~$H$ be a graph with maximum
  degree~$D$.  Suppose~$H^\sigma$ is a subdivision of~$H$
  with~$\sigma(e)\geq 2\log_{D-1} n$ for every $e \in E(H)$, where
  $n=|V(H^\sigma)|$.  Furthermore, suppose~$H^\sigma$ is bipartite.
  Then
  \begin{equation}
    \label{eq:3}
    \widehat{R}_r(H^\sigma) \leq r^{\,400 D\log D} \, n.
  \end{equation}
\end{theorem}

In general, the bound in~\eqref{eq:3} is worse than the bound
in~\eqref{eq:2}, but of course~\eqref{eq:3} applies to more general
subdivisions~$H^\sigma$.  Furthermore, the method of proof employed
here is entirely different from the one
in~\cite{javadi25:_induced_long}.

\subsection*{Conventions and notation}

For a graph $ G $, we write $ V(G) $, $ E(G) $ and $ e(G) $ for the vertex set, the edge set and the number of edges of $ G $, respectively.
For $ v \in V(G) $, by $ N_{G}(v) $ we mean the set of all neighbors of $ v $ in $G$.
For a subset $ X\subseteq V(G) $, we define the neighborhood of $X$ as~$ N_G(X)=\bigcup_{x\in X}N_G(x) $.
In a rooted tree $T$, the maximum distance of a vertex from the root is the height of $ T $. If a tree has only one vertex (the root), its height is zero.
Let $ A,\, B \subset V(G) $ with $A\cap B=\emptyset$. Then $ E_G(A,B)= \{ xy\in E(G) : x\in A, \, y\in B\} $
is the set of edges connecting a vertex of $ A $ to a vertex of $ B $. Also, $e_G(A,B)=|E_G(A,B)|$. A bipartite graph $ G $ with a bipartition $ (V_1, V_2) $ is denoted by $ G=G(V_1, V_2) $.

Given functions $f(n), \, g(n)$ and $h(n)$, we write $\Omega(h(n))=f(n)=O(g(n))$ if there exist  absolute constants $ C_1,\,  C_2>0 $ such that $C_1h(n)\leq f(n)\leq C_2g(n)$ for  all sufficiently large~$n$.

All logarithms are taken to base
$ e $, unless stated otherwise. Throughout the paper, we omit floor and ceiling symbols whenever they are not essential.

The post-theorem introduction only states that "The proof of Theorem~\ref{conj:pak} is based on Szemer\'edi's regularity lemma and makes no attempt to optimize the constant~$C$." No further outline, steps, or proof structure for Theorem~\ref{conj:pak} is given there.

{\rm~\cite{Rolling}}
  \label{conj:pak}
  For every $ r,\, D\in \mathbb{N}  $ there exist  $ C,\,L > 0 $ such that if $ H $
  is a graph with $ \Delta(H) \leq D $ and $ \sigma(e)\ge L\log(|V(H^{\sigma})|) $ for all $ e \in E(H) $, then
  $\widehat{R}_r(H^{\sigma})\leq C|V(H^{\sigma})| $ .

Question:
For a graph $J$ and a positive integer $r$, write $G\to (J)_r$ if every $r$-edge-coloring of a graph $G$ contains a monochromatic copy of $J$, and define the $r$-color size-Ramsey number by
\[
\widehat R_r(J)=\min\{ |E(G)|: G\to (J)_r\}.
\]
If $H$ is a graph and $\sigma:E(H)\to \mathbb N$, let the subdivision $H^{\sigma}$ be the graph obtained by replacing each edge $e\in E(H)$ with a path of length $\sigma(e)$. For positive integers $r$ and $D$, which statement about subdivisions of graphs $H$ with maximum degree $\Delta(H)\le D$ is valid?

For every $r,D\in \mathbb N$, there exist constants $C,L>0$ such that whenever $H$ is a graph with $\Delta(H)\le D$ and $\sigma:E(H)\to\mathbb N$ satisfies $\sigma(e)\ge L\log\bigl(|V(H^{\sigma})|\bigr)$ for every $e\in E(H)$, the subdivision $H^{\sigma}$ has linear $r$-color size-Ramsey number:
\[
\widehat R_r(H^{\sigma})\le C\,|V(H^{\sigma})|.
\]

Choices:
B.
For every $r,D\in \mathbb N$, there exist constants $C,L>0$ such that whenever $H$ is a graph with $\Delta(H)\le D$ and $\sigma:E(H)\to\mathbb N$ satisfies $\sigma(e)\le L\log\bigl(|V(H^{\sigma})|\bigr)$ for every $e\in E(H)$, the subdivision $H^{\sigma}$ has linear $r$-color size-Ramsey number:
\[
\widehat R_r(H^{\sigma})\le C\,|V(H^{\sigma})|.
\]
C.
For every $r,D\in \mathbb N$, there exists a constant $C>0$ such that for every graph $H$ with $\Delta(H)\le D$, there is some choice of subdivision lengths $\sigma:E(H)\to\mathbb N$ for which
\[
\widehat R_r(H^{\sigma})\le C\,|V(H^{\sigma})|.
\]
D.
For every $r,D\in \mathbb N$, there exist absolute constants $C,L>0$ such that whenever $H$ is a graph with $\Delta(H)\le D$ and $\sigma:E(H)\to\mathbb N$ satisfies $\sigma(e)\ge L\log\bigl(|V(H^{\sigma})|\bigr)$ for every $e\in E(H)$, one has
\[
\widehat R_r(H^{\sigma})\le C\,|V(H^{\sigma})|.
\]
E.
For every $r,D\in \mathbb N$, there exist constants $C,L>0$ such that whenever $H$ is a graph with $\Delta(H)\le D$ and $\sigma:E(H)\to\mathbb N$ satisfies $\sigma(e)\ge L\log\bigl(|V(H^{\sigma})|\bigr)$ for every $e\in E(H)$, the original graph $H$ has linear $r$-color size-Ramsey number:
\[
\widehat R_r(H)\le C\,|V(H)|.
\]

The stem gives only definitions and the setup; it does not explicitly reveal the correct option. The correct answer is not signposted by wording in the stem.

The item is essentially asking for the exact theorem statement among nearby variants. This is more than a verbatim restatement, but it is still primarily theorem recognition rather than a genuinely new conclusion derived from the setup.

Some reasoning is needed to compare quantifier strength, dependence of constants, the logarithm argument, and upper bound versus equality. However, success depends mostly on recalling or matching the theorem statement, not on substantial generative mathematical reasoning.

The distractors are mostly plausible and target meaningful failure modes: uniform vs parameter-dependent constants, weaker hypotheses, boundary changes, and replacing an inequality by equality. But the presence of a weaker-true option makes the distractor set problematic for a single-correct MCQ.

A reasonably well-disguised theorem-recognition item with no clear answer leakage, but it does not strongly test generative reasoning and is weakened by a distractor that appears to be true in a weaker form.


Question:
For a graph $J$ and a positive integer $r$, write $G\to (J)_r$ if every $r$-edge-coloring of a graph $G$ contains a monochromatic copy of $J$, and define the $r$-color size-Ramsey number by
\[
\widehat R_r(J)=\min\{ |E(G)|: G\to (J)_r\}.
\]
If $H$ is a graph and $\sigma:E(H)\to \mathbb N$, let the subdivision $H^{\sigma}$ be the graph obtained by replacing each edge $e\in E(H)$ with a path of length $\sigma(e)$. For positive integers $r$ and $D$, which statement about subdivisions of graphs $H$ with maximum degree $\Delta(H)\le D$ is valid?

Choices:
B.
For every $r,D\in \mathbb N$, there exist constants $C,L>0$ such that whenever $H$ is a graph with $\Delta(H)\le D$ and $\sigma:E(H)\to\mathbb N$ satisfies $\sigma(e)\le L\log\bigl(|V(H^{\sigma})|\bigr)$ for every $e\in E(H)$, the subdivision $H^{\sigma}$ has linear $r$-color size-Ramsey number:
\[
\widehat R_r(H^{\sigma})\le C\,|V(H^{\sigma})|.
\]
C.
For every $r,D\in \mathbb N$, there exists a constant $C>0$ such that for every graph $H$ with $\Delta(H)\le D$, there is some choice of subdivision lengths $\sigma:E(H)\to\mathbb N$ for which
\[
\widehat R_r(H^{\sigma})\le C\,|V(H^{\sigma})|.
\]
D.
For every $r,D\in \mathbb N$, there exist absolute constants $C,L>0$ such that whenever $H$ is a graph with $\Delta(H)\le D$ and $\sigma:E(H)\to\mathbb N$ satisfies $\sigma(e)\ge L\log\bigl(|V(H^{\sigma})|\bigr)$ for every $e\in E(H)$, one has
\[
\widehat R_r(H^{\sigma})\le C\,|V(H^{\sigma})|.
\]
E.
For every $r,D\in \mathbb N$, there exist constants $C,L>0$ such that whenever $H$ is a graph with $\Delta(H)\le D$ and $\sigma:E(H)\to\mathbb N$ satisfies $\sigma(e)\ge L\log\bigl(|V(H^{\sigma})|\bigr)$ for every $e\in E(H)$, the original graph $H$ has linear $r$-color size-Ramsey number:
\[
\widehat R_r(H)\le C\,|V(H)|.
\]

The stem only gives definitions and asks which statement is valid; it does not reveal the key logarithmic lower bound, the linear conclusion, or the dependence on parameters in a way that singles out choice A.

This is not a direct restatement of a theorem in the stem. The task is to distinguish among closely related variants with different quantifiers, parameter dependencies, and target graphs.

There is some reasoning pressure because the solver must track subtle differences such as  vs  bounds, dependence of constants on r and D, and whether the conclusion concerns H or H^sigma. However, it mainly tests theorem recognition/quantifier parsing rather than deeper generative mathematical reasoning.

Although several distractors are plausible, choice C is a weaker statement that is also true if A is true, so the item does not have a uniquely correct option. That is a serious distractor-design flaw.

Good at avoiding answer leakage and not tautological, but flawed as an MCQ because one distractor is also true. It tests careful theorem-form discrimination more than genuine generative reasoning.


\subsection{Background and Motivation}
Low-dimensional topology quantifies knot complexity through numerical invariants. While classical invariants, such as the standard crossing number, rely on planar diagrams, these two-dimensional projections compress spatial information which does not fully reflect the global topology of the knot exterior within the ambient space $S^3$.

To capture this three-dimensional structure, we extend the notion of diagrammatic complexity to arbitrary closed surfaces. For a knot $K$ and a closed surface $F \subset S^3$, we define the \emph{surface crossing number} $c(K;F)$ as the minimal number of crossings among all regular diagrams of $K$ obtained by isotoping the knot into a regular neighborhood of $F$.

In this paper, we establish an inequality demonstrating that $c(K;F)$ is bounded below by the tunnel number of $K$ and the Heegaard deficiency of $F$.

\subsection{Main Results}
The objective of this paper is to establish a quantitative relationship between a knot's three-dimensional complexity, measured via its tunnel number, and its diagrammatic complexity on an arbitrary fixed closed surface.

\documentclass[11pt]{amsart}
\usepackage{amsmath,amssymb,amsthm}
\usepackage{hyperref}
\usepackage{tikz}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}[theorem]{Definition}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{example}[theorem]{Example}
\newtheorem{question}[theorem]{Question}
\title[Crossing Numbers of Knots on Closed Surfaces]
{Crossing Numbers of Knots on Closed Surfaces}
\author{Makoto Ozawa}
\address{Department of Natural Sciences, Faculty of Arts and Sciences, Komazawa University, 1-23-1 Komazawa, Setagaya-ku, Tokyo, 154-8525, Japan}
\email{w3c@komazawa-u.ac.jp}
\date{\today}
\dedicatory{Dedicated to Professor Kanji Morimoto on the occasion of his retirement.}
\keywords{knot, surface crossing number, tunnel number, Heegaard deficiency, Heegaard splitting, surface bridge number, surface ascending number}
\begin{document}
\begin{abstract}
Let $c(K;F)$ denote the surface crossing number of a knot $K$ with respect
to a closed surface $F \subset S^{3}$. We establish the lower bound
\[
c(K;F) \ge 2\bigl(t(K)-\delta(F)\bigr)+1,
\]
where $t(K)$ is the tunnel number of $K$ and $\delta(F)$ is the Heegaard
deficiency of $F$. In particular, for any fixed closed surface $F$, the
surface crossing number $c(K;F)$ is unbounded over all knots $K$.
Furthermore, we construct a family of knots $K_m$ demonstrating that $c(K_m;F) = \Theta(t(K_m))$, which shows that this lower bound is asymptotically sharp.
\end{abstract}
\maketitle

\subsection{Main Results}
The objective of this paper is to establish a quantitative relationship between a knot's three-dimensional complexity, measured via its tunnel number, and its diagrammatic complexity on an arbitrary fixed closed surface.

The proof relies on a chain of inequalities relating the surface crossing number $c(K;F)$, the surface ascending number $a(K;F)$, and the surface bridge number $b(K;F)$:
$$\frac{c(K;F)-1}{2} \ge a(K;F) \ge b(K;F)-1 \ge t(K)-\delta(F).$$
This sequence demonstrates that the geometric obstruction to simplifying a knot diagram is bounded below by a combination of the knot exterior's complexity, $t(K)$, and the ambient surface's Heegaard deficiency, $\delta(F)$. Unlike the classical planar crossing number, $c(K;F)$ directly reflects the Heegaard-theoretic complexity of the knot.

Furthermore, we prove that the linear lower bound in Theorem \ref{thm:main_intro} is asymptotically optimal. For iterated connected sums $K_m$ of a prime knot (such as the trefoil), we obtain the linear growth estimate:
$$c(K_m;F) \ge 2m - 2\delta(F) + 1.$$
Pairing this with a general upper bound derived from the subadditivity of the planar crossing number yields $c(K_m;F) = \Theta(t(K_m))$. This confirms that no general lower bound of higher order exists, and that the surface crossing number grows linearly with respect to the tunnel number.

\begin{theorem}[Fundamental inequality]\label{thm:fundamental}
Let $F\subset S^{3}$ be a closed separating surface, and let $K\subset S^{3}$ be a knot. Then
\[
c(K;F) \ge 2\bigl(t(K)-\delta(F)\bigr)+1.
\]
\end{theorem}

Applying the fundamental inequality to $K_m$ yields:
\begin{proposition}
For every closed surface $F\subset S^{3}$,
\[
c(K_m;F) \ge 2\bigl(m-\delta(F)\bigr)+1 \to \infty \quad \text{as } m \to \infty.
\]
\end{proposition}

\section{Proof of the Main Inequality}
In this section, we prove Theorem \ref{thm:fundamental} by establishing the chain
$$\frac{c(K;F)-1}{2} \ge a(K;F) \ge b(K;F)-1 \ge t(K)-\delta(F).$$
Throughout the proof, let $F\subset S^{3}$ be a closed separating surface with decomposition
$S^{3}=M_{1}\cup_{F}M_{2}$.

\subsection{Generalization to spatial graphs}
The framework extends to spatial graphs. A spatial graph $G$ is a 1-dimensional CW complex embedded in $S^{3}$. The key arguments rely on the 1-dimensionality of the object: in the proof of Lemma \ref{lem:amalgamation}, a general position argument shows that the 2-dimensional attaching disks of the 1-handles can be made disjoint from $G$. The tunnel number $t(G)$, bridge position, and ascending number extend to this setting, and one expects a lower bound of the form
\[
c(G;F) \ge 2\bigl(t(G)-\delta(F)\bigr) + C_G,
\]
where $C_G$ is a constant depending on the combinatorial structure of $G$ (such as its Euler characteristic or vertex degrees).

\begin{theorem}[Fundamental inequality]\label{thm:main_intro}
Let $F\subset S^3$ be a closed surface. Then every knot $K\subset S^3$ satisfies:
\[
c(K;F) \ge 2\bigl(t(K)-\delta(F)\bigr)+1,
\]
where $t(K)$ is the tunnel number of $K$, and $\delta(F)=g(M_1)+g(M_2)-g(F)$ is the Heegaard deficiency of $F$.
\end{theorem}

The proof relies on a chain of inequalities relating the surface crossing number $c(K;F)$, the surface ascending number $a(K;F)$, and the surface bridge number $b(K;F)$:
$$\frac{c(K;F)-1}{2} \ge a(K;F) \ge b(K;F)-1 \ge t(K)-\delta(F).$$
This sequence demonstrates that the geometric obstruction to simplifying a knot diagram is bounded below by a combination of the knot exterior's complexity, $t(K)$, and the ambient surface's Heegaard deficiency, $\delta(F)$. Unlike the classical planar crossing number, $c(K;F)$ directly reflects the Heegaard-theoretic complexity of the knot.

Furthermore, we prove that the linear lower bound in Theorem \ref{thm:main_intro} is asymptotically optimal. For iterated connected sums $K_m$ of a prime knot (such as the trefoil), we obtain the linear growth estimate:
$$c(K_m;F) \ge 2m - 2\delta(F) + 1.$$
Pairing this with a general upper bound derived from the subadditivity of the planar crossing number yields $c(K_m;F) = \Theta(t(K_m))$. This confirms that no general lower bound of higher order exists, and that the surface crossing number grows linearly with respect to the tunnel number.

\subsection{Organization of the Paper}
The paper is organized as follows. Section 2 recalls basic definitions and establishes the monotonicity of $c(K;F)$ under surface compression. Section 3 states the main results. Section 4 proves the fundamental inequality using the ascending number, bridge position, and the $(g,b)$-decomposition framework of \cite{MSY}. Section 5 provides the explicit construction of the knot family $K_m$ to demonstrate the asymptotic sharpness of our lower bound. Finally, Section 6 discusses further implications and open questions. These include the hierarchy of surface invariants, their behavior under connected sums, and a comparison with the surface trunk. We outline generalizations to surfaces with boundary and spatial graphs, and remark on the necessity of the genuine manifold condition compared to singular spaces \cite{D,G}. The section concludes with questions regarding the asymptotic behavior for prime knots, lower bounds by the Haken number, and the minimality of alternating projections on closed surfaces.

To prove Theorem~\ref{thm:main_intro}, the argument “relies on a chain of inequalities relating the surface crossing number $c(K;F)$, the surface ascending number $a(K;F)$, and the surface bridge number $b(K;F)$”:
\[
\frac{c(K;F)-1}{2} \ge a(K;F) \ge b(K;F)-1 \ge t(K)-\delta(F).
\]
This sequence yields the stated lower bound for $c(K;F)$ in terms of “a combination of the knot exterior's complexity, $t(K)$, and the ambient surface's Heegaard deficiency, $\delta(F)$.”

For optimality, the introduction says they “prove that the linear lower bound in Theorem~\ref{thm:main_intro} is asymptotically optimal” by taking iterated connected sums $K_m$ of a prime knot and obtaining
\[
c(K_m;F) \ge 2m - 2\delta(F) + 1.
\]
“Pairing this with a general upper bound derived from the subadditivity of the planar crossing number yields $c(K_m;F)=\Theta(t(K_m))$,” so “no general lower bound of higher order exists,” and $c(K;F)$ grows linearly with tunnel number.

To prove the main theorem, the argument “relies on a chain of inequalities relating the surface crossing number $c(K;F)$, the surface ascending number $a(K;F)$, and the surface bridge number $b(K;F)$”:
\[
\frac{c(K;F)-1}{2} \ge a(K;F) \ge b(K;F)-1 \ge t(K)-\delta(F).
\]
This sequence yields the stated lower bound for $c(K;F)$ in terms of “a combination of the knot exterior's complexity, $t(K)$, and the ambient surface's Heegaard deficiency, $\delta(F)$.”

For optimality, the introduction says they “prove that the linear lower bound in the main theorem is asymptotically optimal” by taking iterated connected sums $K_m$ of a prime knot and obtaining
\[
c(K_m;F) \ge 2m - 2\delta(F) + 1.
\]
“Pairing this with a general upper bound derived from the subadditivity of the planar crossing number yields $c(K_m;F)=\Theta(t(K_m))$,” so “no general lower bound of higher order exists,” and $c(K;F)$ grows linearly with tunnel number.

Question: Let $F\subset S^3$ be any closed surface, and write $S^3=M_1\cup_F M_2$. Define the Heegaard deficiency of $F$ by $\delta(F)=g(M_1)+g(M_2)-g(F)$, where $g(\cdot)$ denotes genus. For a knot $K\subset S^3$, let $t(K)$ be its tunnel number, and let $c(K;F)$ denote the surface crossing number of $K$ with respect to $F$, meaning the minimum number of crossings among all regular diagrams obtained by isotoping $K$ into a regular neighborhood of $F$. Which statement holds for every such closed surface $F$ and every knot $K\subset S^3$?

$c(K;F) \ge 2\bigl(t(K)-\delta(F)\bigr)+1$.

Choices:
B. $c(K;F) \ge 2\bigl(t(K)-\delta(F)\bigr)$.
C. $c(K;F) \ge t(K)-\delta(F)+1$.
D. $c(K;F) \ge 2\bigl(t(K)+\delta(F)\bigr)+1$.
E. $c(K;F) \ge 2\bigl(t(K)-\delta(F)\bigr)+3$.

dropped leading factor 2 while keeping the same dependence on $t(K)$ and $\delta(F)$

sharp additive constant in the final bound

The stem gives definitions and asks for the correct universal estimate, but it does not explicitly or implicitly reveal the exact bound. The correct additive constant and the use of (F) must be inferred from the options.

This is largely a theorem-recall item: the task is to identify the exact published inequality among close variants. It is not a pure tautology because the choices compete on sharpness, constants, and the correct invariant, but it still mainly tests recognition of the theorem statement.

Some reasoning is required to compare a sharp bound, a weaker true bound, and false strengthenings or parameter substitutions. However, the item does not demand substantial derivation; it primarily rewards remembering or recognizing the exact estimate.

The distractors are strong: one is a stronger-but-false variant, one is a weaker true statement, one swaps in the wrong invariant, and one suggests an unjustified uniform constant. These are plausible and mathematically meaningful failure modes.

A solid MCQ with no answer leakage and high-quality distractors, but it mainly tests recognition of a theorem statement rather than deeper generative reasoning.


Question: Let $F\subset S^3$ be any closed surface, and write $S^3=M_1\cup_F M_2$. Define the Heegaard deficiency of $F$ by $\delta(F)=g(M_1)+g(M_2)-g(F)$, where $g(\cdot)$ denotes genus. For a knot $K\subset S^3$, let $t(K)$ be its tunnel number, and let $c(K;F)$ denote the surface crossing number of $K$ with respect to $F$, meaning the minimum number of crossings among all regular diagrams obtained by isotoping $K$ into a regular neighborhood of $F$. Which statement holds for every such closed surface $F$ and every knot $K\subset S^3$?

Choices:
B. $c(K;F) \ge 2\bigl(t(K)-\delta(F)\bigr)$.
C. $c(K;F) \ge t(K)-\delta(F)+1$.
D. $c(K;F) \ge 2\bigl(t(K)+\delta(F)\bigr)+1$.
E. $c(K;F) \ge 2\bigl(t(K)-\delta(F)\bigr)+3$.

The stem only defines the quantities and asks for a universal inequality; it does not explicitly reveal the correct bound or give strong hints about the exact coefficients or constant term.

This is close to a theorem-statement recognition item: the task is essentially to identify the correct universal inequality. However, the nearby alternative bounds mean it is not a completely verbatim restatement.

Some reasoning is needed to distinguish the sharp bound from weaker true or false variants, especially because the options differ by subtle coefficient/sign/constant changes. Still, the item mainly tests precise recall or recognition rather than substantial derivation.

The distractors are mathematically plausible and target realistic errors: dropping the factor of 2, changing the sign of the deficiency term, or altering the additive constant. Including a weaker true statement also improves quality.

A solid MCQ with no answer leakage and strong distractors, but it remains primarily a theorem-recognition question rather than one that strongly tests generative mathematical reasoning.