\label{thm_main}(Unique determination)
Let $\Omega\subset\mathbb{R}^n$ be a bounded open set with smooth
boundary $\partial\Omega$. Let $s\in \mathbb{R}^+\setminus \mathbb{Z}$, $\lfloor s \rfloor>\max\{m,\,n/2\}$.
Suppose that $q_j\in L^\infty(\Omega)$ satisfies $q_j\geq 0$ in $\Omega$ for $j=1,\,2$.
Let $\mathbf{P}_1$ and $\mathbf{P}_2$ be nonlinearities of the form \eqref{DEF:nonlinear P} with coefficients $a_{\sigma,k}^{(1)},\, a_{\sigma,k}^{(2)}(x) \in C (\overline\Omega)$. Let $W_1,\, W_2\subset\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\Lambda_{\mathbf{P}_j}: \mathcal{X}_{\varepsilon_0}(W_1)\rightarrow (H^{-s}(\Omega_e))^*$ satisfy
\[
\Lambda_{\mathbf{P}_1}f|_{W_2} = \Lambda_{\mathbf{P}_2}f|_{W_2} \quad \mbox{ for all } f \in \mathcal{X}_{\varepsilon_0}(W_1),
\]
then for all multi-indices $\sigma$ such that $0\le |\sigma|\le m$ and for all $k=1,\ldots, K-1$ it holds
$$
q_1=q_2\quad\hbox{and}\quad	a_{\sigma,k}^{(1)} = a_{\sigma,k}^{(2)} \quad \hbox{ in }\Omega,
$$
and thus the nonlinearities $\mathbf P_1$ and $\mathbf P_2$ coincide.

Let $\Omega$ be a bounded open set in $\mathbb{R}^n$, $n\geq 1$, with smooth boundary $\partial\Omega$, and
$\Omega_e := \mathbb{R}^n\setminus \overline\Omega$ be the exterior of $\Omega$.
Let $m\geq 0$ be an integer and $s\in \mathbb{R}^+\setminus \mathbb{Z}$, where $\mathbb{R}^+:=(0,\infty)$ and $\mathbb{Z}$ denotes the set of all integers in $\mathbb{R}$.
We consider inverse problems for the nonlinear fractional Schr\"odinger equation (NLFSE):
\begin{equation}\label{eq_1_IBVP}
\begin{cases}
(-\Delta)^su + q(x)u +\mathbf{P}(u)= 0 &\hbox{ in } \Omega,\\
u = f &\hbox{ in } \Omega_e,
\end{cases}
\end{equation}
where the fractional Laplacian $(-\Delta)^s$ is the nonlocal pseudo-differential operator defined by $(-\Delta)^su: = \mathcal{F}^{-1}(|\cdot|^{2s}\mathcal{F}(u))$. Here $\mathcal{F} (u)$ represents the Fourier transform of $u$.
The local nonlinear perturbation $\mathbf{P} (u)$ is defined as
\begin{align}\label{DEF:nonlinear P}
\mathbf{P} (u) := uP_1(x,D)u + u^2 P_2(x,D)u+\ldots+u^{K-1}P_{K-1}(x,D)u,
\end{align}
where $K\geq 2$ is an integer, and the linear differential operators $P_k(x,D)$ of order $m$ are of the form
$$
P_k(x,D):=\sum_{\sigma: \,|\sigma|\leq m}a_{\sigma,k}(x)D^\sigma,\quad k=1,\ldots,K-1,
$$
with scalar coefficients $a_{\sigma,k}$ depending only on $x$. Here $\sigma=(\sigma_1,\ldots,\sigma_n)$ is multi-index of length $|\sigma|=\sigma_1+\ldots+\sigma_n=\ell$ for $\ell\leq m$, and therefore $D^\sigma u := \frac{\partial^{\sigma_1}}{\partial x_1^{\sigma_1}}\ldots\frac{\partial^{\sigma_n}}{\partial x_n^{\sigma_n}} u$. \\

We show in Theorem \ref{thm_wellposdness} that \eqref{eq_1_IBVP} is well-posed in $H^s(\mathbb{R}^n)$ for any data $f$ in a small ball $\mathcal{X}_{\varepsilon_0}(\Omega_e)$ in $C^\infty_c(\Omega_e)$, where for any open set $O\subset \mathbb R^n$ we define
\[
\mathcal{X}_{\varepsilon_0}(O):=\{f\in C^\infty_c(O):\,\|f\|_{C^\infty_c(O)}\le\varepsilon_0\}.
\]
This well-posedness result for \eqref{eq_1_IBVP} allows us to define the corresponding Dirichlet-to-Neumann (DN) map
$$
\Lambda_{\mathbf{P}}: H^{s}(\Omega_e)\rightarrow (H^{-s}(\Omega_e))^*,\quad f\mapsto (-\Delta)^s u_f|_{\Omega_e}.
$$

In this framework, we ask the following inverse problem: Does the map $\Lambda_{\mathbf{P}}$ uniquely determine the nonlinearity $\mathbf P$? The answer is affirmative, and we give a full description of the results in the following Theorem~\ref{thm_main}:

\begin{theorem}\label{thm_main}(Unique determination)
Let $\Omega\subset\R^n$ be a bounded open set with smooth
boundary $\p\Omega$. Let $s\in \R^+\setminus \mathbb{Z}$, $\floor{s}>\max\{m,\,n/2\}$.
Suppose that $q_j\in L^\infty(\Omega)$ satisfies $q_j\geq 0$ in $\Omega$ for $j=1,\,2$.
Let $\mathbf{P}_1$ and $\mathbf{P}_2$ be nonlinearities of the form \eqref{DEF:nonlinear P} with coefficients $a_{\sigma,k}^{(1)},\, a_{\sigma,k}^{(2)}(x) \in C (\overline\Omega)$. Let $W_1,\, W_2\subset\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\Lambda_{\mathbf{P}_j}: \mathcal{X}_{\varepsilon_0}(W_1)\rightarrow (H^{-s}(\Omega_e))^*$ satisfy
\[
\Lambda_{\mathbf{P}_1}f|_{W_2} = \Lambda_{\mathbf{P}_2}f|_{W_2} \quad \mbox{ for all } f \in \mathcal{X}_{\varepsilon_0}(W_1),
\]
then for all multi-indices $\sigma$ such that $0\le |\sigma|\le m$ and for all $k=1,\ldots, K-1$ it holds
$$
q_1=q_2\quad\hbox{and}\quad	a_{\sigma,k}^{(1)} = a_{\sigma,k}^{(2)} \quad \hbox{ in }\Omega,
$$
and thus the nonlinearities $\mathbf P_1$ and $\mathbf P_2$ coincide.
\end{theorem}

\begin{theorem}[Well-posedness for the nonlinear equation]
\label{thm_wellposdness}
Let $\Omega\subset\R^n$ be a bounded open domain with smooth
boundary $\p\Omega$. Let $s\in \R^+\setminus \mathbb{Z}$ and $m\in \mathbb{N}$ satisfy $\floor{s}>\max\{m,\,n/2\}$.
Suppose that $q\in L^\infty(\Omega)$ satisfies $q\geq 0$ in $\Omega$, and the coefficients in $\mathbf{P}(u)$ satisfy $a_{\sigma,k}\in C (\overline\Omega)$.
Then there exists a sufficiently small $\varepsilon_0>0$ such that for any $f\in \mathcal{X}_{\varepsilon_0}(\Omega_e)$,
the Dirichlet problem \eqref{eq_1_IBVP} has a unique solution $u\in H^s(\R^n)\cap C^s(\R^n)$. Moreover, there exists a constant $C>0$, independent of $u$ and $f$, such that
\[
\|u\|_{C^{s}(\R^n)}+ \|u\|_{H^{s}(\R^n)}\le C\|f\|_{C^\infty_c(\Omega_e)}.
\]
\end{theorem}

\subsection{Motivation and connection to the literature}
This work addresses a natural generalization of the inverse problems for linear fractional elliptic equations, such as the fractional Schr\"odinger equation $(-\Delta)^s+q$ for $s\in (0,1)$. Our unique determination result of Theorem \ref{thm_main} particularly finds its place within the context of the very active field of inverse problems of fractional Calder\'on problem, known as the fractional analogue of Calder\'on problem. First introduced in the seminal work \cite{GSU20}, the fractional Calder\'on problem asks to uniquely determine a scalar potential $q$ in the fractional Schr\"odinger equation from the related DN-map. It corresponds to our problem \eqref{eq_1_IBVP} in the case $\mathbf P \equiv 0$ for $s\in (0,1)$. Already in the first work \cite{GSU20}, the authors were able to provide a uniqueness result from partial data for $L^\infty$ potentials.

Because the DN-map acts on a quotient space, we have to make sure that it is well-defined as indicated above. We also need to show that it is indeed a bounded linear operator. In what follows, we will use $f$ rather than $[f]$ in order to simplify the notation.
\begin{prop}
Let $\Omega\subset\R^n$ be a bounded open domain with smooth
boundary $\p\Omega$. Let $s\in \R^+\setminus \mathbb{Z}$ and $m\in \mathbb{N}$ satisfy $\floor{s}>\max\{m,\,n/2\}$.
Suppose that $q\in L^\infty(\Omega)$ satisfies $q\geq 0$ in $\Omega$, and the coefficients in $\mathbf{P}(u)$ satisfy $a_{\sigma,k}\in C (\overline\Omega)$.
Then the DN-map $\Lambda_\mathbf{P}$ is well-defined and bounded.
\end{prop}
\begin{proof}
We first show that $\Lambda_\mathbf{P}$ only depends on the equivalence classes. Let $\phi,\,\psi\in \widetilde{H}^s(\Omega)$. Since $u_f$ and $u_{f+\phi}$ both solve problem \eqref{eq_1_IBVP} with the same exterior data, the well-posedness of \eqref{eq_1_IBVP} implies $u_f = u_{f+\phi}$.
By the linearity of $B_\mathbf{P}$ in the second component, we have
\[
B_\mathbf{P}(u_{f+\phi},v+\psi) = B_\mathbf{P}(u_f, v+\psi) = B_\mathbf{P}(u_f, v)+ B_\mathbf{P}(u_f, \psi).
\]
Using the fact that $\psi = 0$ in $\Omega_e$ and $u_f$ solves \eqref{eq_1_IBVP}, we get $B_\mathbf{P}(u_f,\psi) = 0$. This proves that $\LA \Lambda_\mathbf{P}(f+\phi), v+\psi \RA =\LA \Lambda_\mathbf{P}f,v\RA $, and thus $\Lambda_\mathbf{P}$ is well-defined.

\subsection{Linearization}\label{sec_linearization}
For $f=(f_1,\ldots,f_K)$ with $f_\ell\in C^\infty_c(\Omega_e)$, $\|f_\ell\|_{C_c^\infty(\Omega_e)}\le 1/K$, Theorem \ref{thm_wellposdness} yields that, for sufficiently small $\varepsilon = (\varepsilon_1,\ldots,\varepsilon_K)$ with $|\varepsilon|<\varepsilon_0$, there exists a unique solution $u_\varepsilon \in H^s(\R^n)\cap C^s(\R^n)$ of the following problem
\begin{equation}\label{eq_3_linearization}
\begin{cases}
[(-\Delta)^s + q(x)]u_\varepsilon =- \mathbf{P}(u_\varepsilon) &\hbox{ in } \Omega,\\
u_\varepsilon = \sum_{\ell=1}^K \varepsilon_\ell f_\ell=:\varepsilon\cdot f &\hbox{ in } \Omega_e.
\end{cases}
\end{equation}
Moreover, we have
\begin{align}\label{eq_3_linear_u_est}
\|u_\varepsilon\|_{H^{s}(\R^n)}+
\|u_\varepsilon\|_{C^{s}(\R^n)}\le C\|\varepsilon\cdot f\|_{C_c^\infty(\Omega_e)}\le C|\varepsilon|\sum_{\ell=1}^K\|f_\ell\|_{C_c^\infty(\Omega_e)}\le C|\varepsilon|,
\end{align}
where the constant $C > 0$ is independent of $u_\varepsilon$ and $f_\ell$. We will expand the solution $u_\varepsilon$ in terms of the small parameter $\varepsilon$. Let $\alpha=(\ell_1,\ldots,\ell_K)\in \mathbb{N}^K_0 $ be a $K$-dimensional multi-index of nonnegative integers. Combining Propositions \ref{prop_wellposedness_linear} and \ref{prop_Holder_linear}, we know that there exists a unique solution $w_{\alpha}\in H^s(\R^n)\cap C^{s}(\R^n)$ to the linear fractional Schr\"odinger equation
\begin{equation}
\label{eq_3_linear_w_3}
\begin{cases}
[(-\Delta)^s + q(x)]w_{\alpha} =- \mathcal{T}_{\alpha} &\hbox{ in } \Omega,\\
w_{\alpha} = D^\alpha_\varepsilon (\varepsilon\cdot f)|_{\varepsilon=0} &\hbox{ in } \Omega_e,
\end{cases}
\end{equation}
where the inhomogeneous term is
\begin{equation}\label{eq_3_def_T}
\begin{aligned}
\mathcal{T}_{\alpha} :=
\sum^{|\alpha|-1}_{\ell=1} \sum_{\substack{ \beta_1,\ldots, \beta_\ell\in \mathbb{N}^K_0\setminus \{0,\cdots,0\} : \\ \beta:= \sum_{j=1}^\ell\beta_j < \alpha }}\binom{\alpha}{\beta} \binom{\beta}{\beta_1,\cdots, \beta_\ell}\bigg( \prod_{j=1}^\ell w_{\beta_j } \bigg) P_\ell(x,D)w_{\alpha-\beta}.
\end{aligned}
\end{equation}
Note here $\mathcal{T}_{\alpha} = 0$ when $|\alpha| = 0,\, 1$.
Furthermore, the solution satisfies the stability estimate
\begin{equation}\label{eq_3_linear_w3_est}
\begin{aligned}
\|w_\alpha\|_{C^{s}(\R^n)}+ \|w_\alpha\|_{H^{s}(\R^n)}
\le
\begin{cases}
C\|D^\alpha_\varepsilon (\varepsilon\cdot f)|_{\varepsilon=0}\|_{C^\infty_c(\Omega_e)}&|\alpha|=1,\\
C\|\mathcal{T}_\alpha\|_{L^\infty(\Omega)},&|\alpha|\geq 2.
\end{cases}
\end{aligned}
\end{equation}
Here we denote
$$
\binom{\alpha}{\beta} = \frac{\alpha !}{\beta !(\alpha-\beta)!} \quad \hbox{and} \quad
\binom{\beta}{\beta_1, \cdots, \beta_k}= \frac{\beta!}{\beta_1! \cdots\beta_k!}.
$$
Observe that the definition of $\mathcal T_\alpha$ only requires knowledge of those $w_\beta$ with $\beta<\alpha$, and that in turn $w_\alpha$ can be computed from $\mathcal T_\alpha$ as the solution of \eqref{eq_3_linear_w_3}. Thus the definitions of $\mathcal T_\alpha$ and $w_\alpha$ are not circular.

By using the derivatives given above, and the fact $u_\varepsilon=0$ when $\varepsilon=0$, we obtain the following lemma through direct computation.
\begin{lemma}\label{lamma_DN_higher}
Let $f=(f_1,\ldots,f_K)$, where $f_\ell\in C^\infty_c(\Omega_e)$ satisfies $\|f_\ell\|_{C_c^\infty(\Omega_e)}\le 1/K$. For sufficiently small $\varepsilon = (\varepsilon_1,\ldots,\varepsilon_K)$ such that $|\varepsilon|<\varepsilon_0$, we have for $|\alpha|\ge 1$
that
\begin{align*}
\LA \frac{\p^{|\alpha|}}{\p \varepsilon^\alpha} \Big|_{\varepsilon=0}\Lambda_\mathbf{P}(\varepsilon\cdot f),g \RA = \int_{\R^n} (-\Delta)^{\frac{s}{2}}w_\alpha (-\Delta)^{\frac{s}{2}}g \dx + \int_\Omega q(x)w_\alpha\, g \dx+
\int_{\Omega}\mathcal{T}_{\alpha}g \dx, \quad\hbox{for }g\in H^s(\R^n).
\end{align*}
Notice that $\mathcal T_\alpha \equiv 0$ when $|\alpha|=1$.
\end{lemma}
\begin{proof}
We first show the first-order derivative of the DN-map.
For any $g\in H^s(\R^n)$, by the definition of $\Lambda_\mathbf{P}$, taking the derivatives on the bilinear form yields that
\begin{align*}
\LA \p _{\varepsilon_\ell}\big|_{\varepsilon=0}\Lambda_\mathbf{P}(\varepsilon\cdot f),g\RA =\p_{\varepsilon_{\ell}} \big|_{\varepsilon=0}B_\mathbf{P}(u_\varepsilon,g) = \int_{\R^n}(-\Delta)^{\frac{s}{2}}v_\ell (-\Delta)^{\frac{s}{2}}g \dx + \int_\Omega q(x)v_\ell\, g \dx.
\end{align*}
Note that all nonlinear terms vanish as $u_\varepsilon=0$ when $\varepsilon=0$.
Similarly, let $\alpha=e_{\ell_1}+e_{\ell_2}$, for second-order partial derivatives of $\Lambda_\mathbf{P}$, since only linear terms and the quadratic nonlinearity in $B_\mathbf{P}$ remain after taking derivatives, we get
\begin{align*}
\LA \p_{\varepsilon_{\ell_1}} \p_{\varepsilon_{\ell_2}} \big|_{\varepsilon=0}\Lambda_\mathbf{P}(\varepsilon\cdot f),g \RA &=\p_{\varepsilon_{\ell_1}} \p_{\varepsilon_{\ell_2}} \big|_{\varepsilon=0}B_\mathbf{P}(u_\varepsilon,g)\\
&=\int_{\R^n} (-\Delta)^{\frac{s}{2}}w_{\alpha} (-\Delta)^{\frac{s}{2}}g \dx + \int_\Omega q(x) w_{\alpha}\, g \dx\\
&\quad
+\int_{\Omega} \LC v_{\ell_1}P_1(x,D)v_{\ell_2} + v_{\ell_2}P_1(x,D)v_{\ell_1} \RC g \dx.
\end{align*}
Following a similar argument, we can also derive the integral representation of
$\frac{\p^{|\alpha|}}{\p \varepsilon^\alpha}
\Big|_{\varepsilon=0}\Lambda_\mathbf{P}(\varepsilon\cdot f)$ for $|\alpha|\geq 3$.
\end{proof}

\subsection{Motivation and connection to the literature}
This work addresses a natural generalization of the inverse problems for linear fractional elliptic equations, such as the fractional Schr\"odinger equation $(-\Delta)^s+q$ for $s\in (0,1)$. Our unique determination result of Theorem \ref{thm_main} particularly finds its place within the context of the very active field of inverse problems of fractional Calder\'on problem,  known as the fractional analogue of Calder\'on problem. First introduced in the seminal work \cite{GSU20}, the fractional Calder\'on problem asks to uniquely determine a scalar potential $q$ in the fractional Schr\"odinger equation from the related DN-map. It corresponds to our problem \eqref{eq_1_IBVP} in the case $\mathbf P \equiv 0$ for $s\in (0,1)$. Already in the first work \cite{GSU20}, the authors were able to provide a uniqueness result from partial data for $L^\infty$ potentials.

Besides the mathematical interest, the fractional Calder\'on problem arises naturally in the sciences, due to the close connection between the fractional Laplace operator and anomalous diffusion \cite{CL24,V09}. A novel application is that of the recent paper \cite{LNOS25}, which considers an inverse problem of fractional Calder\'on type related to quantum field theory. Many aspects of the fractional Calder\'on problem have been studied in recent years: among these, we recall uniqueness for low regularity partial data \cite{RS20a} and anisotropic background metrics \cite{GLX17}, reconstruction for a single measurement \cite{GRSU20} and by monotonicity methods \cite{ HL20, HL19}, the stability and instability results of \cite{ BCR25,RS18, RS20b}, general anisotropic metrics \cite{Fei24, FGKRSU25, FGKU21},
and boundary reconstruction \cite{CR25}. Moreover, the entanglement principles are studied for fractional poly-elliptic \cite{FKU24, FL24} and poly-parabolic operators \cite{LLYan25} to decouple the entangled effect of nonlocal perturbations. We refer to the survey \cite{S17} and to the recent book \cite{LL25book} for many more related results on the topic of nonlocal inverse problems. Incidentally, the fractional Calder\'on problem has been studied also in its conductivity formulation, as shown in \cite{Cov20a} and the subsequent works \cite{Cov20b, Cov22, CMR21, CMRU22, CRZ22}. We refer to \cite{Cov25} for a recent survey on the topic. Other equations of fractional and, more generally, nonlocal nature have also been the object of intensive studies. Among these, we recall inverse problems for fractional elasticity \cite{CdHS22}, the fractional Helmholtz equation via geometrical optics solutions \cite{CdHS25}, and the fractional wave equation \cite{KLW22, LZ25}, among many others \cite{DGM25, Li23, LW25, Lin23}. The relations between the fractional and classical Calder\'on problems have also attracted substantial attention \cite{BCR25, BR25, CGRU23, R23}.

The two distinctive features of our equation \eqref{eq_1_IBVP}  include the \textit{higher-order nonlocal leading operator} and \textit{general local nonlinear perturbations} of the fractional Schr\"odinger equation. It is a local nonlinear perturbation of the higher-order fractional Schr\"odinger equation. As such, it differs substantially from the perturbations considered in \cite{BGU21, CLR20, Cov21,Li21}, which are linear (and sometimes nonlocal), and from those of \cite{CMRU22}, where the most general local linear perturbations of the fractional Schr\"odinger equation were studied.

The consideration of the higher-order fractional Laplace operator $(-\Delta)^s$ with $s\in \mathbb R^+\setminus\mathbb Z$ in this paper is inspired by the work \cite{CMRU22}, which investigated the unique determination of a general linear term.
Indeed, the fractional Laplace operator with power $s>1$ has been object of study in several recent publications: we recall \cite{CMRU22, DHP23} for higher-order inverse problems of fractional Schr\"odinger type, and \cite{AJ24, CSZ25, DPS25, FF20, GR19, RS15} for theoretical results concerning the unique continuation property and existence of solutions for the higher-order fractional Laplacian.

Meanwhile, the latter feature falls into a rapidly developing research direction in inverse problems, dealing with reconstruction of nonlinear terms in partial differential equations (PDEs), which are highly motivated by natural phenomena, such as nonlinear optics. To study inverse problems for nonlinear PDEs, a classical linearization technique first introduced in \cite{Isakov93} is to perform first-order linearization of the DN-map.
Lately, the higher-order linearization method \cite{KLU18} was launched to treat inverse problems for various kind of PDEs. In particular, it was observed that the presence of nonlinearity in an equation turns out to be a great benefit to the study of inverse problems, despite the potential difficulty nonlinearity introduced to solve the forward problems. Specifically, since the nonlinear interactions generate more fruitful information in the reconstruction process, many inverse problems for the nonlinear equations can be resolved, while the underlying inverse problems in the linear setting are still open.
The interested reader is referred to the following relevant literature and the reference therein. The nonlinear Schr\"odinger equation has been studied in \cite{lai_partial_2023, LUY24, Lai_inverse_2021,lassas_inverse_2024}, while a nonlinear version of the wave equation and related acoustic problems were the object of \cite{HTT25, Kal25b, Kal25,LLPT25}. Many more nonlinear-type inverse problems, including for the Boltzmann equation, were considered in \cite{KK25, lai_reconstruction_2021, lai_stable_2023, LZ24,LL25, NS25}. We also recall a recent result about corrosion detection by identification of a nonlinear Robin boundary condition \cite{J25}. The fractional Schr\"odinger \cite{KMS23, KMSS25, lai_global_2018}, wave \cite{LTZ24} and several other related equations \cite{JNS23, JNS25, lai_inverse_2022, lai_recovery_2022} were studied also in the semilinear case.

There is a major difference between the cases $s\in (0,1)$ and $s\in \mathbb{R}^+\setminus\mathbb{Z}$ in $(-\Delta)^s$. For the nonlinear Schr\"odinger equation with $s\in (0,1)$, the availability of the maximum principle helps to ensure the $L^\infty$ bounded solution to the linear Schr\"odinger equation. This can be used to deduce the $C^s$ regularity solution for the nonlinear one, see \cite{lai_global_2018}, which studied the unique determination of the nonlinear potential in $(-\Delta)^s u +q(x,u)=0$. For our case, however, it is necessary to require the condition $\lfloor s \rfloor>\max\{m,\,n/2\}$ to guarantee the existence of such bounded solution so that we can further derive the $C^s$ regularity, see Section~\ref{sec:pre} for details. With this, the contraction mapping principle is applied to establish the well-posedness for \eqref{eq_1_IBVP}.

\subsection{Outline of the article}

The remaining part of the paper is organized as follows. In Section~\ref{sec:pre} we survey and extend some existing results for the linear Schr\"odinger equation to suit our current setting, and we establish the well-posedness Theorem \ref{thm_wellposdness}. In Section~\ref{sec_linearization} we show that the solution to the nonlinear Schr\"odinger equation \eqref{eq_1_IBVP} can be decomposed with respect to the order of a small parameter. This enables us to translate the  inverse problem for \eqref{eq_1_IBVP} into the study of recovery of the nonhomogeneous term in the linearized equation in Section~\ref{sec:proof of theorem}, where we prove our main result, Theorem \ref{thm_main}. After the acknowledgments of Section~\ref{sec:ack}, we included an Appendix dedicated to computations.

To prove Theorem~\ref{thm_main}, the paper proceeds by first establishing well-posedness: in Section~\ref{sec:pre} the authors “survey and extend some existing results for the linear Schr\"odinger equation to suit our current setting” and “establish the well-posedness Theorem \ref{thm_wellposdness}.” A key point for the forward problem is that, since for $s\in\mathbb{R}^+\setminus\mathbb{Z}$ one cannot rely on the maximum principle as in the case $s\in(0,1)$, they “require the condition $\lfloor s \rfloor>\max\{m,\,n/2\}$ to guarantee the existence of such bounded solution so that we can further derive the $C^s$ regularity,” and “with this, the contraction mapping principle is applied to establish the well-posedness for \eqref{eq_1_IBVP}.”

For the inverse problem, in Section~\ref{sec_linearization} they “show that the solution to the nonlinear Schr\"odinger equation \eqref{eq_1_IBVP} can be decomposed with respect to the order of a small parameter,” which “enables us to translate the inverse problem for \eqref{eq_1_IBVP} into the study of recovery of the nonhomogeneous term in the linearized equation.” Finally, in Section~\ref{sec:proof of theorem} they “prove our main result, Theorem~\ref{thm_main},” i.e. uniqueness of the coefficients via this linearization-based reduction.

To prove the main theorem, the paper proceeds by first establishing well-posedness: next the authors survey and extend some existing results for the linear Schr\"odinger equation to suit the current setting, and establish the following theorem.

\begin{theorem}[Well-posedness for the nonlinear equation]
\label{thm_wellposdness}
Let $\Omega\subset\R^n$ be a bounded open domain with smooth
boundary $\p\Omega$. Let $s\in \R^+\setminus \mathbb{Z}$ and $m\in \mathbb{N}$ satisfy $\floor{s}>\max\{m,\,n/2\}$.
Suppose that $q\in L^\infty(\Omega)$ satisfies $q\geq 0$ in $\Omega$, and the coefficients in $\mathbf{P}(u)$ satisfy $a_{\sigma,k}\in C (\overline\Omega)$.
Then there exists a sufficiently small $\varepsilon_0>0$ such that for any $f\in \mathcal{X}_{\varepsilon_0}(\Omega_e)$,
the Dirichlet problem \eqref{eq_1_IBVP} has a unique solution $u\in H^s(\R^n)\cap C^s(\R^n)$. Moreover, there exists a constant $C>0$, independent of $u$ and $f$, such that
\[
\|u\|_{C^{s}(\R^n)}+ \|u\|_{H^{s}(\R^n)}\le C\|f\|_{C^\infty_c(\Omega_e)}.
\]
\end{theorem}

A key point for the forward problem is that, since for $s\in\mathbb{R}^+\setminus\mathbb{Z}$ one cannot rely on the maximum principle as in the case $s\in(0,1)$, they require the condition $\lfloor s \rfloor>\max\{m,\,n/2\}$ to guarantee the existence of such bounded solution so that one can further derive the $C^s$ regularity; with this, the contraction mapping principle is applied to establish well-posedness for
\begin{equation}\label{eq_1_IBVP}
\begin{cases}
(-\Delta)^su + q(x)u +\mathbf{P}(u)= 0 &\hbox{ in } \Omega,\\\nu = f &\hbox{ in } \Omega_e,
\end{cases}
\end{equation}

For the inverse problem, they then show that the solution to the nonlinear Schr\"odinger equation given by the equation above can be decomposed with respect to the order of a small parameter, which enables one to translate the inverse problem for the same equation into the study of recovery of the nonhomogeneous term in the linearized equation. Finally, they complete the argument by proving the main theorem, i.e. uniqueness of the coefficients via this linearization-based reduction.

\label{thm_main}(Unique determination)
Let $\Omega\subset\mathbb{R}^n$ be a bounded open set with smooth
boundary $\partial\Omega$. Let $s\in \mathbb{R}^+\setminus \mathbb{Z}$, $\lfloor s \rfloor>\max\{m,\,n/2\}$.
Suppose that $q_j\in L^\infty(\Omega)$ satisfies $q_j\geq 0$ in $\Omega$ for $j=1,\,2$.
Let $\mathbf{P}_1$ and $\mathbf{P}_2$ be nonlinearities of the form
\begin{align}\label{DEF:nonlinear P}
\mathbf{P} (u) := uP_1(x,D)u + u^2 P_2(x,D)u+\ldots+u^{K-1}P_{K-1}(x,D)u,
\end{align}
with coefficients $a_{\sigma,k}^{(1)},\, a_{\sigma,k}^{(2)}(x) \in C (\overline\Omega)$. Let $W_1,\, W_2\subset\Omega_e$ be two arbitrary nonempty bounded open sets. If the DN-maps $\Lambda_{\mathbf{P}_j}: \mathcal{X}_{\varepsilon_0}(W_1)\rightarrow (H^{-s}(\Omega_e))^*$ satisfy
\[
\Lambda_{\mathbf{P}_1}f|_{W_2} = \Lambda_{\mathbf{P}_2}f|_{W_2} \quad \mbox{ for all } f \in \mathcal{X}_{\varepsilon_0}(W_1),
\]
then for all multi-indices $\sigma$ such that $0\le |\sigma|\le m$ and for all $k=1,\ldots, K-1$ it holds
$$
q_1=q_2\quad\hbox{and}\quad	a_{\sigma,k}^{(1)} = a_{\sigma,k}^{(2)} \quad \hbox{ in }\Omega,
$$
and thus the nonlinearities $\mathbf P_1$ and $\mathbf P_2$ coincide.,

Question:
Let $\Omega\subset \mathbb{R}^n$ be a bounded open set with smooth boundary, and let $\Omega_e:=\mathbb{R}^n\setminus \overline{\Omega}$. Fix an integer $m\ge 0$, an integer $K\ge 2$, and $s\in \mathbb{R}^+\setminus \mathbb{Z}$ such that $\lfloor s\rfloor>\max\{m,n/2\}$. For $j=1,2$, assume $q_j\in L^\infty(\Omega)$ with $q_j\ge 0$ in $\Omega$, and let $\mathbf P_j$ be nonlinearities of the form
\[
\mathbf P_j(u)=uP^{(j)}_1(x,D)u+u^2P^{(j)}_2(x,D)u+\cdots+u^{K-1}P^{(j)}_{K-1}(x,D)u,
\]
where
\[
P^{(j)}_k(x,D)=\sum_{|\sigma|\le m} a^{(j)}_{\sigma,k}(x)D^\sigma,
\qquad a^{(j)}_{\sigma,k}\in C(\overline\Omega),
\]
for every $k=1,\dots,K-1$. Let $W_1,W_2\subset \Omega_e$ be arbitrary nonempty bounded open sets, and define
\[
\mathcal X_{\varepsilon_0}(W_1):=\{f\in C_c^\infty(W_1):\ \|f\|_{C_c^\infty(W_1)}\le \varepsilon_0\}.
\]
For each $j$, let $\Lambda_{\mathbf P_j}$ be the Dirichlet-to-Neumann map for the exterior value problem
\[
(-\Delta)^s u+q_j(x)u+\mathbf P_j(u)=0\ \text{in }\Omega,\qquad u=f\ \text{in }\Omega_e,
\]
namely $\Lambda_{\mathbf P_j}(f)=(-\Delta)^s u_f|_{\Omega_e}$. Suppose that the partial DN data agree in the sense that
\[
\Lambda_{\mathbf P_1}f|_{W_2}=\Lambda_{\mathbf P_2}f|_{W_2}\qquad \text{for all }f\in \mathcal X_{\varepsilon_0}(W_1).
\]
Which statement is guaranteed for every such pair of nonlinear equations?

One must have $q_1=q_2$ in $\Omega$, and for every multi-index $\sigma$ with $0\le |\sigma|\le m$ and every $k=1,\dots,K-1$, the coefficients satisfy $a^{(1)}_{\sigma,k}=a^{(2)}_{\sigma,k}$ in $\Omega$; equivalently, the nonlinearities $\mathbf P_1$ and $\mathbf P_2$ coincide in $\Omega$.

Choices:
B. One must have $q_1=q_2$ in $\Omega$, and for every multi-index $\sigma$ with $0\le |\sigma|< m$ and every $k=1,\dots,K-1$, the coefficients satisfy $a^{(1)}_{\sigma,k}=a^{(2)}_{\sigma,k}$ in $\Omega$; hence the lower-order parts of the operators $P_k^{(j)}(x,D)$ agree, though the coefficients of order exactly $m$ need not be determined.
C. One must have $q_1=q_2$ in $\Omega$.
D. One must have $q_1=q_2$ in $\Omega$, and for every multi-index $\sigma$ with $0\le |\sigma|\le m$ there exists an index $k=k(\sigma)\in\{1,\dots,K-1\}$ such that $a^{(1)}_{\sigma,k}=a^{(2)}_{\sigma,k}$ in $\Omega$; in particular, each differential order is represented by some matching coefficient, although not necessarily for every $k$.
E. There exists a sufficiently small $\varepsilon_*>0$, depending only on $\Omega$, $n$, $m$, $s$, and the a priori bounds for the coefficients, such that if the partial DN maps agree on $W_2$ for all $f\in \mathcal X_{\varepsilon_*}(W_1)$, then one must have $q_1=q_2$ in $\Omega$ and $a^{(1)}_{\sigma,k}=a^{(2)}_{\sigma,k}$ in $\Omega$ for all $0\le |\sigma|\le m$ and all $k=1,\dots,K-1$.

full recovery up to top differential order $m$

dropped identification of all nonlinear coefficients $a_{\sigma,k}$

quantifier structure over $(\sigma,k)$ in coefficient recovery

dependence on the given small-data radius $\varepsilon_0$ versus a newly asserted uniform threshold


\label{bc}

We note that
\begin{align}\label{bn}
\sum_{n=1}^\infty B(n)q^{n+1}&=q\sum_{n=1}^\infty\frac{q^{2n-1}}{(q;q^2)_n} \notag \\
&=\sum_{n=1}^\infty\frac{q^{2n}}{(q;q^2)_n},
\end{align}
here, an

\begin{theorem}\label{main theorem}
For $n>0$, we have
\begin{align}\label{main theorem eqn}
A(n)=B(n)=C(n+1)=\frac{1}{2}D(n+1),
\end{align}
where $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.
\end{theorem}

The grandfather of all partition identities is Euler's theorem \cite{andrews book}, namely,
\begin{align*}
A(n)=B(n),
\end{align*}
where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.

This theorem contains all the elements that would suggest generalizations, and, over the centuries, generalizations have been found in profusion. The Rogers-Ramanujan identities \cite[p.~104]{andrews book} and Schur's 1926 theorem \cite[p.~116]{andrews book} kicked off the twentieth century's contributions, consult Henry Alder's survey \cite{alder} for an account of some of the results in the late 20th century.

However, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.

\subjclass[2020]{Primary 11P84, 05A17}
 \keywords{Euler's theorem, Partitions, Glaisher's bijection}
\maketitle
\pagenumbering{arabic}
\pagestyle{headings}
\begin{abstract}
Euler's theorem asserts that $A(n)=B(n)$ where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts. In this paper, it is proved that for $n>0$,
\begin{align*}
A(n)=B(n)=C(n+1)=\frac{1}{2}D(n+1),
\end{align*}
where $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.

The grandfather of all partition identities is Euler's theorem \cite{andrews book}, namely,
\begin{align*}
A(n)=B(n),
\end{align*}
where $A(n)$ is the number of partitions of $n$ into distinct parts and $B(n)$ is the number of partitions of $n$ into odd parts.

However, little has ever been added to Euler's theorem itself. In this paper, we shall add two further partition functions.

For example, if $n=6$, the four sets of partitions are following:
\begin{center}
\begin{tabular}{ p{2cm} p{3.5cm} p{3cm} p{4.1cm} p{4.1cm}}

On the other hand, we have
\begin{align}
\sum_{n=1}^\infty C(n)q^{n}&=\sum_{n=1}^\infty\frac{(-q;q)_nq^{2n}}{(q^{n+1};q)_n}\nonumber\\
&=\sum_{n=1}^\infty\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}} \label{c1n}\\
&=\sum_{n=1}^\infty\frac{q^{2n}}{(q;q^2)_n}.\label{cn}
\end{align}

Let us fix $C(0)=1$. From \eqref{c1n}, we have
\begin{align}\label{before p19}
\sum_{n=0}^\infty C(n)q^{n}&=\sum_{n=0}^\infty\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}}\nonumber\\
&=(q^2;q^2)_\infty\sum_{n=0}^\infty\frac{q^{2n}}{(q;q)_{2n}(q^{2n+2};q^2)_\infty}.
\end{align}
We now employ the following Euler's identity \cite[p.~19, (2.2.5)]{andrews book}
\begin{align}\label{euler identity}
\frac{1}{(t;q)_\infty}=\sum_{m=0}^\infty\frac{t^m}{(q,q)_m},
\end{align}
(with replacing $q$ by $q^2$ and then letting $t=q^{2n+2}$) in \eqref{before p19} so as to obtain
\begin{align}\label{series}
\sum_{n=0}^\infty C(n)q^{n}&=(q^2;q^2)_\infty\sum_{n=0}^\infty\frac{q^{2n}}{(q;q)_{2n}}\sum_{m=0}^\infty\frac{q^{2nm+2m}}{(q^2;q^2)_m}\nonumber\\
&=(q^2;q^2)_\infty\sum_{m,n=0}^\infty\frac{q^{2n+2nm+2m}}{(q;q)_{2n}(q^2;q^2)_m}\nonumber\\
&=(q^2;q^2)_\infty\sum_{m,n=0}^\infty\frac{1}{2}\left(1+(-1)^n\right)\frac{q^{n+nm+2m}}{(q;q)_{n}(q^2;q^2)_m}\nonumber\\
&=\frac{1}{2}(q^2;q^2)_\infty\sum_{m=0}^\infty\frac{q^{2m}}{(q^2;q^2)_m}\left\{\sum_{n=0}^\infty\frac{q^{n(m+1)}}{(q;q)_n}+\sum_{n=0}^\infty\frac{(-1)^nq^{n(m+1)}}{(q;q)_n}\right\}.
\end{align}
Upon invoking \eqref{euler identity} twice, once with letting $t=q^{m+1}$ and once with letting $t=-q^{m+1}$, and then substituting both resulting expressions in \eqref{series}, we conclude that
\begin{align}\label{3.4}
\sum_{n=0}^\infty C(n)q^{n}&=\frac{1}{2}(q^2;q^2)_\infty\sum_{m=0}^\infty\frac{q^{2m}}{(q^2;q^2)_m}\left\{\frac{1}{(q^{m+1};q)_\infty}+\frac{1}{(-q^{m+1};q)_\infty}\right\}\nonumber\\
&=\frac{1}{2}(-q;q)_\infty\sum_{m=0}^\infty\frac{q^{2m}}{(-q;q)_m}+\frac{1}{2}(q;q)_\infty\sum_{m=0}^\infty\frac{q^{2m}}{(q;q)_m}\nonumber\\
&=\frac{1}{2}\sum_{m=0}^\infty q^{2m}(-q^{m+1};q)_\infty+\frac{1}{2}(1-q),
\end{align}
where the last step follows upon again using \eqref{euler identity} with letting $t=q^2$.

\subsection{A Bijective proof of $A(n-1)=\frac{1}{2}D(n)$}
We can first remove the non-negative part condition for $D(n)$ as follows:
\begin{quote}
For $n\ge 2$, $D(n)$ counts the number of partitions of $n$ where only the smallest part can repeat at most twice and all other parts are distinct.
\end{quote}
In other words, a partition $\lambda=(\lambda_1,\ldots, \lambda_{\ell})$ counted by $D(n)$ satisfies
$$
\lambda_1>\lambda_2> \cdots > \lambda_{\ell-1}\ge \lambda_{\ell} \text{ if $\ell>1$}
$$
or
$\lambda=(\lambda_1)$ has only one part.

Let $M \le 2N$ be an even number, which can be written as
$$
M=2^k a \text{ for some odd integer $a$ and some $ k\ge 1$}.
$$
We apply Glaisher's bijection $\phi$ to $M$ and obtain $2^k$ parts of size $a$. Note that since
$
2^k a \le 2N,
$
$$
a \le N.
$$
Also, if $M\le N$, then $M$ can appear only once, so we get all distinct positive powers $2^k$ such that
\begin{equation}
2^k\le N/a. \label{power1}
\end{equation}
On the other hand, if $N< M\le 2N$, then
\begin{equation}
2^k> N/a, \label{power2}
\end{equation}
and $M$ can repeat.
Suppose $M$ appears $f$ many times. Upon applying Glaisher's bijection $\phi$ to $f$ copies of $M$, we obtain
$$
 f 2^{k} \text{ copies of $a$}.
$$
By writing $f$ as a binary expansion,
$$
f 2^k =(f_0 \,2^{0}+f_1 \, 2^{1} +\cdots ) 2^k,
$$
where $f_j$ is either $0$ or $1$ for $j\ge 0$. By \eqref{power2}, we see that each summand in the above expression represents a distinct power of $2$ greater than $N/a$, i.e.,
\begin{equation}
f_j \, 2^{j+k} >N/a. \label{power3}
\end{equation}
It follows from \eqref{power1} and \eqref{power3} that $a$ can appear with any multiplicity greater than $1$. This proves that the resulting partition is counted by $B(n)$.

For example, if $n=6$, the four sets of partitions are following:
\begin{center}
\begin{tabular}{ p{2cm} p{3.5cm} p{3cm} p{4.1cm} p{4.1cm}}

 \vspace{2mm}$A(6)$\newline  &  \vspace{2mm} $B(6)$ & \vspace{2mm} $C(7)$& \vspace{2mm} $D(7)$ \\

 $6$ \newline 5+1 \newline 4+2\newline 3+2+1   & 5+1 \newline 3+3 \newline 3+1+1+1 \newline 1+1+1+1+1+1  & 6+1\newline 4+3 \newline 4+2+1 \newline 2+2+2+1  & 0+0+7\newline 0+0+6+1\newline 0+0+5+2\newline 0+0+4+3\newline 0+0+4+2+1\newline 1+1+5\newline 1+1+2+3\newline 2+2+3
\end{tabular}
\end{center}

We conclude the introduction with the following remark.
\begin{remark}
In their recent paper \cite{ba}, M. El Bachraoui and the first author considered partitions with multiple appearances by the first part.  All parts were assumed to be positive. It would be a simple matter to extend the results of that paper to the case of non-negative parts in that this would add $(-q;q)_\infty$ to the generating functions in question.
\end{remark}

This paper is organised as follows. In section \ref{bc}, we provide the brief proof that $B(n)=C(n+1)$. In Section \ref{cd}, we prove that $C(n)=\frac{1}{2}D(n)$. We also provide bijective proofs our assertions in Theorem \ref{main theorem} in Section \ref{bijective proofs}.

The post-theorem introduction does not give a proof sketch beyond an outline of the paper: in Section~\ref{bc} they "provide the brief proof that $B(n)=C(n+1)$"; in Section~\ref{cd} they "prove that $C(n)=\frac{1}{2}D(n)$"; and in Section~\ref{bijective proofs} they "provide bijective proofs" of the assertions in Theorem~\ref{main theorem}.

The post-theorem introduction does not give a proof sketch beyond an outline of the paper: next they note that
\begin{align}\label{bn}
\sum_{n=1}^\infty B(n)q^{n+1}&=q\sum_{n=1}^\infty\frac{q^{2n-1}}{(q;q^2)_n} \notag \\
&=\sum_{n=1}^\infty\frac{q^{2n}}{(q;q^2)_n},
\end{align}
here, an
and also that
\begin{align}
\sum_{n=1}^\infty C(n)q^{n}&=\sum_{n=1}^\infty\frac{(-q;q)_nq^{2n}}{(q^{n+1};q)_n}\nonumber\\
&=\sum_{n=1}^\infty\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}} \label{c1n}\\
&=\sum_{n=1}^\infty\frac{q^{2n}}{(q;q^2)_n}.\label{cn}
\end{align}

Comparing \eqref{bn} and \eqref{cn}, we see that $B(n)=C(n+1)$.

\section{Proof that $C(n)=\frac{1}{2}D(n)$}\label{cd}

Let us fix $C(0)=1$. From \eqref{c1n}, we have
\begin{align}\label{before p19}
\sum_{n=0}^\infty C(n)q^{n}&=\sum_{n=0}^\infty\frac{(q^2;q^2)_nq^{2n}}{(q;q)_{2n}}\nonumber\\
&=(q^2;q^2)_\infty\sum_{n=0}^\infty\frac{q^{2n}}{(q;q)_{2n}(q^{2n+2};q^2)_\infty}.
\end{align}
They then prove that $C(n)=\frac{1}{2}D(n)$, and finally they provide bijective proofs as follows:
\label{bijective proofs}

\subsection{A Bijective proof of $A(n-1)=\frac{1}{2}D(n)$}
We can first remove the non-negative part condition for $D(n)$ as follows:
\begin{quote}
For $n\ge 2$, $D(n)$ count

\label{main theorem}
For $n>0$, we have
\begin{align}\label{main theorem eqn}
A(n)=B(n)=C(n+1)=\frac{1}{2}D(n+1),
\end{align}
where $C(n)$ is the number of partitions of $n$ with largest part even and parts not exceeding half of the largest part are distinct, and $D(n)$ is the number of partitions of $n$ into non-negative parts wherein the smallest part appear exactly twice and no other parts are repeated.,

Question: For each positive integer n, let A(n) denote the number of partitions of n into distinct parts, let B(n) denote the number of partitions of n into odd parts, let C(n) denote the number of partitions of n whose largest part is even and for which every part not exceeding half of the largest part occurs at most once, and let D(n) denote the number of partitions of n into nonnegative parts in which the smallest part appears exactly twice and no other part is repeated. Which statement holds for every positive integer n?

Choices:
B. A(n)=B(n)=C(n)=\frac{1}{2}D(n).
C. A(n)=B(n)=C(n+1).
D. A(n)=B(n)=C(n+1)=D(n+1).
E. A(n)=B(n)=C(n)+1=\frac{1}{2}D(n+1).


\begin{theorem}\label{thm:main}
    The properad $\mathscr{Y}^{(n)}_{prop}$ is Koszul contractible, and therefore it is a Koszul properad.
\end{theorem}

In this note we work over a field $\kk$ of characteristic zero, and everything is in $\kk$-vector spaces. A \emph{properad} is an object that encodes a type of algebra whose operations have multiple inputs and outputs; one allows composition along any connected graph, and imposes relations that involve compositions given by graphs of any genera. In a \emph{dioperad}, one only records the data of compositions along trees, and is constrained to impose relations that only involve compositions along trees. A \emph{planar dioperad} is an even more restrictive notion, where one requires these trees to be embedded in the plane in a coherent manner. There are functors
\[ \PDiop \xrightarrow{\Phi} \Diop \xrightarrow{F} \mathsf{Properads}\]
that, starting from a planar dioperad, freely adjoin compositions along any connected graphs, quotienting out by planar genus zero relations.

If $\mathscr{Q}$ is a quadratic dioperad that is Koszul (as a dioperad), it is not true in general that $F\mathscr{Q}$ will be Koszul as a properad; this is notably the case with non-commutative Frobenius algebras \cite{merkulov2009deformation} and $\scrV^{(n)}$-algebras \cite{poirier2019koszuality}. There is, however, a class of dioperads for which dioperadic Koszulity implies properadic Koszulity; this happens for a quadratic dioperad $\mathscr{Q}$ when the Koszul dual coproperad of $F\mathscr{Q}$ is in fact a codioperad, that is, its decomposition map does not generate any higher genus terms; in this case, one says that the properad is \emph{contractible}.

The properad $\scrY^{(n)}$ was defined in \cite{emprin2025properadic}, and one of its uses is to encode the data of an orientation on a space $X$ into the chains on the based loop space. This properad is defined by planar genus zero relations, that is, it satisfies
\[ \mathscr{Y}^{(n)}_{prop} = F \mathscr{Y}^{(n)} = F\Phi(\mathscr{Y}^{(n)}_{pl})\]
for a certain planar dioperad $\mathscr{Y}^{(n)}_{pl}$. An algebra $A$ over this planar dioperad has
\begin{itemize}
    \item A product $\mu \colon A \otimes A \to A$ of degree zero,
    \item A coproduct $\alpha \colon A \to A \otimes A$ of homological degree $(1-n)$,
    \item A cyclically symmetric rank three tensor $\beta \in A \otimes A \otimes A$ of homological degree $(2-2n)$,
\end{itemize}
satisfying the following relations.

\small\begin{align*}
    &\tikzfig{
        \node [vertex] (bot) at (0,-0.2) {$\mu$};
        \node [vertex] (top) at (-0.5,0.5) {$\mu$};
        \draw [->-] (-1,1) to (top);
        \draw [->-] (0,1) to (top);
        \draw [->-] (top) to (bot);
        \draw [->-] (1,1) to (bot);
        \draw [->-] (bot) to (0,-1);
    } ~ - ~ \tikzfig{
        \node [vertex] (bot) at (0,-0.2) {$\mu$};
        \node [vertex] (top) at (0.5,0.5) {$\mu$};
        \draw [->-] (-1,1) to (bot);
        \draw [->-] (0,1) to (top);
        \draw [->-] (top) to (bot);
        \draw [->-] (1,1) to (top);
        \draw [->-] (bot) to (0,-1);
    } ~ \quad ; \quad \tikzfig{
        \node [vertex] (top) at (0,0.5) {$\mu$};
        \node [vertex] (bot) at (0,-0.5) {$\alpha$};
        \draw [->-] (-1,1) to (top);
        \draw [->-] (top) to (bot);
        \draw [->-] (1,1) to (top);
        \draw [->-] (bot) to (-1,-1);
        \draw [->-] (bot) to (1,-1);
    } ~ - ~ \tikzfig{
        \node [vertex] (top) at (0.5,0) {$\alpha$};
        \node [vertex] (bot) at (-0.5,0) {$\mu$};
        \draw [->-] (-1,1) to (bot);
        \draw [->-] (top) to (bot);
        \draw [->-] (1,1) to (top);
        \draw [->-] (bot) to (-1,-1);
        \draw [->-] (top) to (1,-1);
    } ~ - ~ \tikzfig{
        \node [vertex] (top) at (-0.5,0) {$\alpha$};
        \node [vertex] (bot) at (0.5,0) {$\mu$};
        \draw [->-] (-1,1) to (top);
        \draw [->-] (top) to (bot);
        \draw [->-] (1,1) to (bot);
        \draw [->-] (top) to (-1,-1);
        \draw [->-] (bot) to (1,-1);
    } ~;\\
    &\tikzfig{
        \node [vertex] (left) at (-0.5,0) {$\alpha$};
        \node [vertex] (right) at (0.5,0) {$\mu$};
        \draw [->-] (0,1) to (left);
        \draw [->-] (0,-1) to (right);
        \draw [->-] (left) to (-1.2,0);
        \draw [->-] (left) to (right);
        \draw [->-] (right) to (1.2,0);
    } ~ - ~ \tikzfig{
        \node [vertex] (left) at (-0.5,0) {$\mu$};
        \node [vertex] (right) at (0.5,0) {$\alpha$};
        \draw [->-] (0,1) to (right);
        \draw [->-] (0,-1) to (left);
        \draw [->-] (left) to (-1.2,0);
        \draw [->-] (right) to (left);
        \draw [->-] (right) to (1.2,0);
    } ~ + (-1)^n \left(\tikzfig{
        \node [vertex] (left) at (-0.5,0) {$\alpha$};
        \node [vertex] (right) at (0.5,0) {$\mu$};
        \draw [->-] (0,1) to (right);
        \draw [->-] (0,-1) to (left);
        \draw [->-] (left) to (-1.2,0);
        \draw [->-] (left) to (right);
        \draw [->-] (right) to (1.2,0);
    } ~ - ~ \tikzfig{
        \node [vertex] (left) at (-0.5,0) {$\mu$};
        \node [vertex] (right) at (0.5,0) {$\alpha$};
        \draw [->-] (0,1) to (left);
        \draw [->-] (0,-1) to (right);
        \draw [->-] (left) to (-1.2,0);
        \draw [->-] (right) to (left);
        \draw [->-] (right) to (1.2,0);
    }\right) ~;
\end{align*}
\begin{align*}
    &\tikzfig{
        \node [vertex] (top) at (0,0.4) {$\mu$};
        \node [vertex] (bot) at (0,-0.6) {$\beta$};
        \draw [->-] (bot) to (top);
        \draw [->-] (bot) to (-1.2,-1);
        \draw [->-] (bot) to (1.2,-1);
        \draw [->-] (-1.2,0.4) to (top);
        \draw [->-] (top) to (0,1);
    } ~ - \qquad \tikzfig{
        \node [vertex] (top) at (0,0.1) {$\beta$};
        \node [vertex] (bot) at (-0.7,-0.5) {$\mu$};
        \draw [->-] (top) to (0,1);
        \draw [->-] (bot) to (-1.2,-1);
        \draw [->-] (top) to (1.2,-1);
        \draw [->-] (-1.2,0.4) to (bot);
        \draw [->-] (top) to (bot);
    } ~ + ~ \tikzfig{
        \node [vertex] (top) at (0,0.4) {$\alpha$};
        \node [vertex] (bot) at (0,-0.6) {$\alpha$};
        \draw [->-] (top) to (bot);
        \draw [->-] (bot) to (-1.2,-1);
        \draw [->-] (bot) to (1.2,-1);
        \draw [->-] (-1.2,0.4) to (top);
        \draw [->-] (top) to (0,1);
    } ~ - (-1)^{n-1} ~ \tikzfig{
        \node [vertex] (top) at (0,0.1) {$\alpha$};
        \node [vertex] (bot) at (-0.7,-0.5) {$\alpha$};
        \draw [->-] (top) to (0,1);
        \draw [->-] (bot) to (-1.2,-1);
        \draw [->-] (top) to (1.2,-1);
        \draw [->-] (-1.2,0.4) to (bot);
        \draw [->-] (bot) to (top);
    } ~; \\ \\
    &\tikzfig{
        \node [vertex] (top) at (0,0.5) {$\alpha$};
        \node [vertex] (bot) at (0,-0.5) {$\beta$};
        \draw [->-] (bot) to (top);
        \draw [->-] (top) to (-1,1);
        \draw [->-] (top) to (1,1);
        \draw [->-] (bot) to (-1,-1);
        \draw [->-] (bot) to (1,-1);
    } ~ - ~ \tikzfig{
        \node [vertex] (left) at (-0.5,0) {$\alpha$};
        \node [vertex] (right) at (0.5,0) {$\beta$};
        \draw [->-] (right) to (left);
        \draw [->-] (left) to (-1,1);
        \draw [->-] (left) to (-1,-1);
        \draw [->-] (right) to (1,-1);
        \draw [->-] (right) to (1,1);
    } ~ + (-1)^n \left( ~ \tikzfig{
        \node [vertex] (top) at (0,0.5) {$\beta$};
        \node [vertex] (bot) at (0,-0.5) {$\alpha$};
        \draw [->-] (top) to (bot);
        \draw [->-] (top) to (-1,1);
        \draw [->-] (top) to (1,1);
        \draw [->-] (bot) to (-1,-1);
        \draw [->-] (bot) to (1,-1);
    } ~ - ~ \tikzfig{
        \node [vertex] (left) at (-0.5,0) {$\beta$};
        \node [vertex] (right) at (0.5,0) {$\alpha$};
        \draw [->-] (right) to (left);
        \draw [->-] (left) to (-1,1);
        \draw [->-] (left) to (-1,-1);
        \draw [->-] (right) to (1,-1);
        \draw [->-] (right) to (1,1);
    } ~ \right)
\end{align*}\normalsize

Note that all of these relations, and the cyclicity condition on $\beta$, can be given by linear combinations of diagrams embedded in a disc with a consistent labeling of inputs and outputs, using only the cyclic action, which comes from the rotational symmetry of the disc; this is why this dioperad is planar, see \cref{sec:planar}. In this note we establish the following theorem.

The properad $\scrY^{(n)}$ was defined in \cite{emprin2025properadic}, and one of its uses is to encode the data of an orientation on a space $X$ into the chains on the based loop space. This properad is defined by planar genus zero relations, that is, it satisfies
\[ \mathscr{Y}^{(n)}_{prop} = F \mathscr{Y}^{(n)} = F\Phi(\mathscr{Y}^{(n)}_{pl})\]
for a certain planar dioperad $\mathscr{Y}^{(n)}_{pl}$. An algebra $A$ over this planar dioperad has
\begin{itemize}
 \item A product $\mu \colon A \otimes A \to A$ of degree zero,
 \item A coproduct $\alpha \colon A \to A \otimes A$ of homological degree $(1-n)$,
 \item A cyclically symmetric rank three tensor $\beta \in A \otimes A \otimes A$ of homological degree $(2-2n)$,
\end{itemize}
satisfying the following relations.

\small\begin{align*}
 &\tikzfig{
 \node [vertex] (bot) at (0,-0.2) {$\mu$};
 \node [vertex] (top) at (-0.5,0.5) {$\mu$};
 \draw [->-] (-1,1) to (top);
 \draw [->-] (0,1) to (top);
 \draw [->-] (top) to (bot);
 \draw [->-] (1,1) to (bot);
 \draw [->-] (bot) to (0,-1);
 } ~ - ~ \tikzfig{
 \node [vertex] (bot) at (0,-0.2) {$\mu$};
 \node [vertex] (top) at (0.5,0.5) {$\mu$};
 \draw [->-] (-1,1) to (bot);
 \draw [->-] (0,1) to (top);
 \draw [->-] (top) to (bot);
 \draw [->-] (1,1) to (top);
 \draw [->-] (bot) to (0,-1);
 } ~ \quad ; \quad \tikzfig{
 \node [vertex] (top) at (0,0.5) {$\mu$};
 \node [vertex] (bot) at (0,-0.5) {$\alpha$};
 \draw [->-] (-1,1) to (top);
 \draw [->-] (top) to (bot);
 \draw [->-] (1,1) to (top);
 \draw [->-] (bot) to (-1,-1);
 \draw [->-] (bot) to (1,-1);
 } ~ - ~ \tikzfig{
 \node [vertex] (top) at (0.5,0) {$\alpha$};
 \node [vertex] (bot) at (-0.5,0) {$\mu$};
 \draw [->-] (-1,1) to (bot);
 \draw [->-] (top) to (bot);
 \draw [->-] (1,1) to (top);
 \draw [->-] (bot) to (-1,-1);
 \draw [->-] (top) to (1,-1);
 } ~ - ~ \tikzfig{
 \node [vertex] (top) at (-0.5,0) {$\alpha$};
 \node [vertex] (bot) at (0.5,0) {$\mu$};
 \draw [->-] (-1,1) to (top);
 \draw [->-] (top) to (bot);
 \draw [->-] (1,1) to (bot);
 \draw [->-] (top) to (-1,-1);
 \draw [->-] (bot) to (1,-1);
 } ~;\\
 &\tikzfig{
 \node [vertex] (left) at (-0.5,0) {$\alpha$};
 \node [vertex] (right) at (0.5,0) {$\mu$};
 \draw [->-] (0,1) to (left);
 \draw [->-] (0,-1) to (right);
 \draw [->-] (left) to (-1.2,0);
 \draw [->-] (left) to (right);
 \draw [->-] (right) to (1.2,0);
 } ~ - ~ \tikzfig{
 \node [vertex] (left) at (-0.5,0) {$\mu$};
 \node [vertex] (right) at (0.5,0) {$\alpha$};
 \draw [->-] (0,1) to (right);
 \draw [->-] (0,-1) to (left);
 \draw [->-] (left) to (-1.2,0);
 \draw [->-] (right) to (left);
 \draw [->-] (right) to (1.2,0);
 } ~ + (-1)^n \left(\tikzfig{
 \node [vertex] (left) at (-0.5,0) {$\alpha$};
 \node [vertex] (right) at (0.5,0) {$\mu$};
 \draw [->-] (0,1) to (right);
 \draw [->-] (0,-1) to (left);
 \draw [->-] (left) to (-1.2,0);
 \draw [->-] (left) to (right);
 \draw [->-] (right) to (1.2,0);
 } ~ - ~ \tikzfig{
 \node [vertex] (left) at (-0.5,0) {$\mu$};
 \node [vertex] (right) at (0.5,0) {$\alpha$};
 \draw [->-] (0,1) to (left);
 \draw [->-] (0,-1) to (right);
 \draw [->-] (left) to (-1.2,0);
 \draw [->-] (right) to (left);
 \draw [->-] (right) to (1.2,0);
 }\right) ~;
\end{align*}
\begin{align*}
 &\tikzfig{
 \node [vertex] (top) at (0,0.4) {$\mu$};
 \node [vertex] (bot) at (0,-0.6) {$\beta$};
 \draw [->-] (bot) to (top);
 \draw [->-] (bot) to (-1.2,-1);
 \draw [->-] (bot) to (1.2,-1);
 \draw [->-] (-1.2,0.4) to (top);
 \draw [->-] (top) to (0,1);
 } ~ - \qquad \tikzfig{
 \node [vertex] (top) at (0,0.1) {$\beta$};
 \node [vertex] (bot) at (-0.7,-0.5) {$\mu$};
 \draw [->-] (top) to (0,1);
 \draw [->-] (bot) to (-1.2,-1);
 \draw [->-] (top) to (1.2,-1);
 \draw [->-] (-1.2,0.4) to (bot);
 \draw [->-] (top) to (bot);
 } ~ + ~ \tikzfig{
 \node [vertex] (top) at (0,0.4) {$\alpha$};
 \node [vertex] (bot) at (0,-0.6) {$\alpha$};
 \draw [->-] (top) to (bot);
 \draw [->-] (bot) to (-1.2,-1);
 \draw [->-] (bot) to (1.2,-1);
 \draw [->-] (-1.2,0.4) to (top);
 \draw [->-] (top) to (0,1);
 } ~ - (-1)^{n-1} ~ \tikzfig{
 \node [vertex] (top) at (0,0.1) {$\alpha$};
 \node [vertex] (bot) at (-0.7,-0.5) {$\alpha$};
 \draw [->-] (top) to (0,1);
 \draw [->-] (bot) to (-1.2,-1);
 \draw [->-] (top) to (1.2,-1);
 \draw [->-] (-1.2,0.4) to (bot);
 \draw [->-] (bot) to (top);
 } ~; \\ \\
 &\tikzfig{
 \node [vertex] (top) at (0,0.5) {$\alpha$};
 \node [vertex] (bot) at (0,-0.5) {$\beta$};
 \draw [->-] (bot) to (top);
 \draw [->-] (top) to (-1,1);
 \draw [->-] (top) to (1,1);
 \draw [->-] (bot) to (-1,-1);
 \draw [->-] (bot) to (1,-1);
 } ~ - ~ \tikzfig{
 \node [vertex] (left) at (-0.5,0) {$\alpha$};
 \node [vertex] (right) at (0.5,0) {$\beta$};
 \draw [->-] (right) to (left);
 \draw [->-] (left) to (-1,1);
 \draw [->-] (left) to (-1,-1);
 \draw [->-] (right) to (1,-1);
 \draw [->-] (right) to (1,1);
 } ~ + (-1)^n \left( ~ \tikzfig{
 \node [vertex] (top) at (0,0.5) {$\beta$};
 \node [vertex] (bot) at (0,-0.5) {$\alpha$};
 \draw [->-] (top) to (bot);
 \draw [->-] (top) to (-1,1);
 \draw [->-] (top) to (1,1);
 \draw [->-] (bot) to (-1,-1);
 \draw [->-] (bot) to (1,-1);
 } ~ - ~ \tikzfig{
 \node [vertex] (left) at (-0.5,0) {$\beta$};
 \node [vertex] (right) at (0.5,0) {$\alpha$};
 \draw [->-] (right) to (left);
 \draw [->-] (left) to (-1,1);
 \draw [->-] (left) to (-1,-1);
 \draw [->-] (right) to (1,-1);
 \draw [->-] (right) to (1,1);
 } ~ \right)
\end{align*}\normalsize

Note that all of these relations, and the cyclicity condition on $\beta$, can be given by linear combinations of diagrams embedded in a disc with a consistent labeling of inputs and outputs, using only the cyclic action, which comes from the rotational symmetry of the disc; this is why this dioperad is planar, see \cref{sec:planar}. In this note we establish the following theorem.

If $\mathscr{Q}$ is a quadratic dioperad that is Koszul (as a dioperad), it is not true in general that $F\mathscr{Q}$ will be Koszul as a properad; this is notably the case with non-commutative Frobenius algebras \cite{merkulov2009deformation} and $\scrV^{(n)}$-algebras \cite{poirier2019koszuality}. There is, however, a class of dioperads for which dioperadic Koszulity implies properadic Koszulity; this happens for a quadratic dioperad $\mathscr{Q}$ when the Koszul dual coproperad of $F\mathscr{Q}$ is in fact a codioperad, that is, its decomposition map does not generate any higher genus terms; in this case, one says that the properad is \emph{contractible}.

It was proven in \cite{emprin2025properadic} that the dioperad $\mathscr{Y}^{(n)}$ is contractible; the other component that allows us to establish Koszulity of the properad $\scrY^{(n)}_{prop}$ is a geometric interpretation of the bar complexes associated to (the Koszul dual of) the planar dioperad $\scrY^{(n)}_{pl}$. This interpretation is in the same spirit as the description of the ``PROP of open-closed marked surfaces'' of \cite{kontsevich2025pre} using moduli spaces of meromorphic Strebel differentials. In \cref{sec:planar} we will recall this theory, in the specific case of differentials on $\mathbb{CP}^1$ with a higher-order pole at infinity, or equivalently, polynomial quadratic differentials on $\CC$. The moduli spaces of such objects has a natural non-compact cell complex structure that is dual to the cell complex structure of the \emph{assocoipahedra} of \cite{poirier2018combinatorics}, and gives an alternative proof of their compatibility, established in that reference; by the results of \cite{poirier2019koszuality} this implies Koszulity of the dioperad $\mathscr{V}^{(n)}$.

As a consequence of \cref{thm:main}, the map
\[ \scrY^{(n)}_\infty = \Omega (\scrY^{(n)}_{prop})^{\antish} \to \scrY^{(n)}_{prop} \]
is a quasi-isomorphism of dg properads and therefore gives a cofibrant resolution of the properad $\scrY^{(n)}_{prop}$; the results on formality in \cite{emprin2025properadic} can then be seen as completely analogous to the formality of $A_\infty$-algebras, but in the properadic setting of $\scrY^{(n)}_{prop}$-algebras. Together with the explicit formulas for homotopy transfer of \cite{hoffbeck2021properadic}, we have as a consequence of our main theorem.
\begin{corollary}
 The homotopy transferred structures for $\scrY^{(n)}_{prop}$-algebras can be computed by formulas only involving sums of planar trees.
\end{corollary}

Therefore, for any $\mathbf{C}$ as above, the subspace
\[ \mathrm{ClovQ}_\mathbf{C} \subset Q^\mathrm{Str}_\mathrm{reg}(k;i_1,\dots,i_k) \]
of regularized Strebel differentials that are cloven by $\mathbf{C}$ is a union of cells, each of which labeled by a directed tree that has at least $r$ bivalent vertices. This union of cells is coclosed, in the sense that the if $a$ is in $\mathrm{ClovQ}_\mathbf{C}$ and is on the boundary of $b$, then so is $b$. So we can write
\[ \mathrm{Clov}_\mathbf{C} \subset Z_{(k;i_1,\dots,i_k)} \]
for the corresponding union of cells in the assocoipahedron, which will be a cell subcomplex. We now give the main proposition of this section.
\begin{proposition}\label{prop:contractible}
 For any collection of pairwise distinct equivalence classes
 \[ \mathbf{C} = \{C_1,\dots,C_r\}, \]
 the space $\mathrm{Clov}_\mathbf{C}$ is contractible if and only if there are $r$ cuts $\gamma_1,\dots,\gamma_r$, one in each equivalence class, that are pairwise disjoint and divide the complex plane into regions such that each contains at least one of the $k$ output directions. In every other case, $\mathrm{Clov}_\mathbf{C}$ is empty.
\end{proposition}
\begin{proof}
 Let us start by proving the contrapositive of the last statement. If $\mathrm{Clov}_\mathbf{C}$ is not empty, $\mathrm{ClovQ}_\mathbf{C}$ must contain at least one point; taking the $r$ saddle cuts gives us the desired representatives $\gamma_1,\dots,\gamma_r$.

\begin{proposition}\label{prop:barcomplexKoszul}
 The complex $(\mathbf{B}^*_{pl} (\scrY^{(n)}_{pl})^!)_{(k;i_1,\dots,i_k)}$ has cohomology concentrated in degree zero.
\end{proposition}
\begin{proof}
 For ease of notation, let us write
 \[ Y^* = (\mathbf{B}^*_{pl} (\mathscr{Y}^{(n)}_{pl})^!)_{(k;i_1,\dots,i_k)}, \quad Z^* = (\mathbf{B}^*_{pl} (\mathscr{V}^{(n)}_{pl})^!)_{(k;i_1,\dots,i_k)} \cong C^*(Z_{(k;i_1,\dots,i_k)},\ZZ) \]
 By definition, we have an exact sequence
 \[ Y^{*-k+1} \hookrightarrow Z^* \twoheadrightarrow R^* \]
 and the result then follows from the induced long exact sequence in cohomology, together with the vanishing results above, and contractibility of $Z^*$.
\end{proof}

\subsection{Dioperadic and properadic Koszulity}
The contractibility of these planar bar complexes immediately implies dioperadic Koszulity.
\begin{corollary}\label{cor:diopKoszul}
 The dioperad $\mathscr{Y}^{(n)} = \Phi(\scrY^{(n)}_{pl})$ is a Koszul dioperad.
\end{corollary}
\begin{proof}
 From \cref{prop:KoszulDiop,prop:barcomplexKoszul} it follows that $(\scrY^{(n)})^! \cong \Phi((\scrY^{(n)}_{pl})^!)$ is a Koszul dioperad; since $\scrY^{(n)}$ is finitely generated, we conclude that $\scrY^{(n)} \cong (\scrY^{(n)})^{!!}$ is also a Koszul dioperad.
\end{proof}

The functor $F$ \emph{does not} intertwine dioperadic and properadic Koszul duality. That is, for an arbitrary finitely-generated quadratic dioperad $\mathscr{Q}$, the properads
\[ F(\mathscr{Q}^!) \quad \text{and} \quad (F \mathscr{Q})^! \]
need not be isomorphic, even if $\mathscr{Q}$ is Koszul; in other words, the Koszul dual properad $(F \mathscr{Q})^!$ might have nontrivial higher-genus relations, or dually, the decomposition maps of the Koszul dual coproperad $(F \mathscr{Q})^{\antish}$ is not a codioperad, that is, if its decomposition map generates higher-genus graphs from genus-zero ones.
\begin{definition}
 A quadratic dioperad $\mathscr{Q}$ is called \emph{contractible} when $(F \mathscr{Q})^{\antish}$ is a codioperad.
\end{definition}
If $\mathscr{Q}$ is a finitely-generated quadratic dioperad, then it is contractible if and only if the canonical map $F(\mathscr{Q}^!) \to (F \mathscr{Q})^!$ is an isomorphism of properads. Contractibility means that one can deduce Koszulity of the properad from Koszulity of the dioperad.
\begin{proposition}
 If a quadratic dioperad $\mathscr{Q}$ is Koszul (as a dioperad) and contractible, then the properad $F\mathscr{Q}$ is Koszul (as a properad).
\end{proposition}
In the case the proposition above holds, one says that the properad $F\mathscr{Q}$ is \emph{Koszul contractible}. Together with \cite[Thm.1.23]{emprin2025properadic}, which shows that $\scrY^{(n)}$ is a contractible dioperad, and \cref{cor:diopKoszul}, we deduce the main theorem of this paper.
\begin{theorem*}(\cref{thm:main})
 The properad $\scrY^{(n)}_{\mathrm{prop}} = F(\scrY^{(n)})$ is Koszul contractible, and therefore a Koszul properad.
\end{theorem*}

It was proven in \cite{emprin2025properadic} that the dioperad $\mathscr{Y}^{(n)}$ is contractible; the other component that allows us to establish Koszulity of the properad $\scrY^{(n)}_{prop}$ is a geometric interpretation of the bar complexes associated to (the Koszul dual of) the planar dioperad $\scrY^{(n)}_{pl}$. This interpretation is in the same spirit as the description of the ``PROP of open-closed marked surfaces'' of \cite{kontsevich2025pre} using moduli spaces of meromorphic Strebel differentials. In \cref{sec:planar} we will recall this theory, in the specific case of differentials on $\mathbb{CP}^1$ with a higher-order pole at infinity, or equivalently, polynomial quadratic differentials on $\CC$. The moduli spaces of such objects has a natural non-compact cell complex structure that is dual to the cell complex structure of the \emph{assocoipahedra} of \cite{poirier2018combinatorics}, and gives an alternative proof of their compatibility, established in that reference; by the results of \cite{poirier2019koszuality} this implies Koszulity of the dioperad $\mathscr{V}^{(n)}$.

In the $\mathscr{Y}^{(n)}$ case, one needs to remove a certain subcomplex of \emph{cloven Strebel differentials}; these are quadratic differentials that split the complex plane into regions in a specific way. Using geometric arguments, we can control the topology of this subcomplex, concluding that it is always homologous to a bouquet of $(k-2)$-spheres, where $k$ is the number of outputs we are looking at. Together with a comparison between the bar complexes associated to $\mathscr{V}^{(n)}$ and $\mathscr{Y}^{(n)}$, vanishing of this homology outside of that specific degree implies vanishing of higher cohomology of the relevant bar complex, establishing our main theorem.

As a consequence of \cref{thm:main}, the map
\[ \scrY^{(n)}_\infty = \Omega (\scrY^{(n)}_{prop})^{\antish} \to \scrY^{(n)}_{prop} \]
is a quasi-isomorphism of dg properads and therefore gives a cofibrant resolution of the properad $\scrY^{(n)}_{prop}$; the results on formality in \cite{emprin2025properadic} can then be seen as completely analogous to the formality of $A_\infty$-algebras, but in the properadic setting of $\scrY^{(n)}_{prop}$-algebras. Together with the explicit formulas for homotopy transfer of \cite{hoffbeck2021properadic}, we have as a consequence of our main theorem.
\begin{corollary}
    The homotopy transferred structures for $\scrY^{(n)}_{prop}$-algebras can be computed by formulas only involving sums of planar trees.
\end{corollary}

\subsection{Relation to other known results}
We would like to point out that there are three other closely related dioperads for which Koszulity has been studied. The first one is the dioperad $\scrV^{(n)}$ of Poirier and Tradler, following earlier work of Tradler and Zeinalian \cite{tradler2007infinity}; algebras over this dioperad are pre-Calabi--Yau algebras, in the formulation of \cite{kontsevich2025pre}. We cite results of \cite{poirier2019koszuality}.
\begin{theorem*}
    The dioperad $\scrV^{(n)}$ is Koszul (as a dioperad) but not contractible.
\end{theorem*}

Another one is the dioperad of balanced infinitesimal bialgebras $\mathrm{BIB}^\lambda$ used by Quesney in \cite{quesney2024balanced}; the algebras that this dioperad encodes are associative analogues of Lie bialgebras, and have a coproduct of degree $\lambda$. It was proved in that paper that this dioperad is contractible, but using deformation theory techniques, Merkulov showed in \cite{merkulov2025complex} the following result.
\begin{theorem*}
    The properad $F(\mathrm{BIB}^\lambda)$ is not Koszul, so as a consequence the dioperad $\mathrm{BIB}^\lambda$ cannot be Koszul.
\end{theorem*}

Finally, the third dioperad that I would like to mention is the dioperad $\mathrm{DPois}$ encoding the double Poisson algebras of van den Bergh; this dioperad was studied by Leray in \cite{leray2019protoperadsI,leray2020protoperadsII}, who showed the following result, see also \cite{leray2025precalabiyau} for a more recent discussion.
\begin{theorem*}
    The dioperad $\mathrm{DPois}$ is Koszul contractible.
\end{theorem*}

The relationship between $\scrY^(n)$ and these three other dioperads was established in \cite{leray2025precalabiyau,quesney2024balanced,emprin2025properadic}. Their Koszul dual dioperads are related by inclusions
\[ \mathrm{DPois}^! \overset{n=2}\hookrightarrow (\mathrm{BIB}^{1-n})^! \hookrightarrow (\scrY^{(n)})^! \hookrightarrow (\scrV^{(n)})^! \]
where the superscript on the first arrow means that this map only exists for that value of $n$.\footnote{There are shifted versions of double Poisson algebras, but the results we cite are only phrased for the unshifted case.} We note that the first and third maps do not respect weight-grading, and do not come from taking Koszul dual maps.

Dually, we have quotients of codioperads
\[ \mathrm{DPois}^{\antish} \overset{n=2}\twoheadleftarrow (\mathrm{BIB}^{1-n})^{\antish} \twoheadleftarrow (\scrY^{(n)})^{\antish} \twoheadleftarrow (\scrV^{(n)})^{\antish} \]
and taking cobar, quotients of dg codioperads
\[ \mathrm{DPois}_\infty \overset{n=2}\twoheadleftarrow \Omega (\mathrm{BIB}^{1-n})^{\antish} \twoheadleftarrow \scrY^{(n)}_\infty \twoheadleftarrow \scrV^{(n)}_\infty = \mathrm{pCY}, \]
so the notion of a  $\scrY^{(n)}_\infty$-algebra is extremely close to the notion of a $\scrV^{(n)}_\infty$ or pre-CY algebra; in fact, it is just a pre-CY algebra with vanishing copairing. When $n \ge 1$, every pre-CY algebra that is concentrated in non-negative homological degree must have vanishing copairing by degree reasons, and will be a $\scrV^{(n)}_\infty$-algebra. We summarize the results cited above in a table for convenience.
\begin{center}
    \begin{tabular}{|c||c|c|c|c|}
    \hline
                            & $\mathrm{DPois}$ & $\mathrm{BIB}^{1-n}$  & $\scrY^{(n)}$ & $\scrV^{(n)}$ \\
    \hline\hline
    Dioperad is Koszul?     & Yes              & No                    & Yes           & Yes \\
    \hline
    Properad is Koszul?     & Yes              & No                    & Yes           & Unknown \\
    \hline
    \end{tabular}
\end{center}

\noindent \textbf{Acknowledgments:} I would like to thank C.~Emprin, S.~Merkulov and B.~Vallette for helpful discussions, and Uppsala University for the great working environment. This work was supported by the Knut and Alice Wallenberg foundation.

It is recalled that \cite{emprin2025properadic} proves the dioperad $\mathscr{Y}^{(n)}$ is contractible; to establish Koszulity (and hence Koszul contractibility) for the properad $\scrY^{(n)}_{prop}$, the other ingredient is “a geometric interpretation of the bar complexes associated to (the Koszul dual of) the planar dioperad $\scrY^{(n)}_{pl}$.” The introduction outlines that one studies moduli spaces of meromorphic Strebel differentials (here specialized to $\mathbb{CP}^1$ with a higher-order pole at infinity / polynomial quadratic differentials on $\CC$), which carry “a natural non-compact cell complex structure” related to assocoipahedra; by \cite{poirier2019koszuality} this type of input yields Koszulity in the $\mathscr{V}^{(n)}$ setting.

For $\mathscr{Y}^{(n)}$, the sketch says one must “remove a certain subcomplex of \emph{cloven Strebel differentials}.” One then uses “geometric arguments” to control its topology, concluding this subcomplex “is always homologous to a bouquet of $(k-2)$-spheres, where $k$ is the number of outputs.” Finally, “together with a comparison between the bar complexes associated to $\mathscr{V}^{(n)}$ and $\mathscr{Y}^{(n)}$,” the fact that this homology vanishes outside that degree implies “vanishing of higher cohomology of the relevant bar complex,” which is said to “establish[] our main theorem” (i.e. Theorem~\ref{thm:main}).

It is recalled that \cite{emprin2025properadic} proves the dioperad $\mathscr{Y}^{(n)}$ is contractible; to establish Koszulity (and hence Koszul contractibility) for the properad $\scrY^{(n)}_{prop}$, the other ingredient is “a geometric interpretation of the bar complexes associated to (the Koszul dual of) the planar dioperad $\scrY^{(n)}_{pl}$.” The introduction outlines that one studies moduli spaces of meromorphic Strebel differentials (here specialized to $\mathbb{CP}^1$ with a higher-order pole at infinity / polynomial quadratic differentials on $\CC$), which carry “a natural non-compact cell complex structure” related to assocoipahedra; by \cite{poirier2019koszuality} this type of input yields Koszulity in the $\mathscr{V}^{(n)}$ setting.

For $\mathscr{Y}^{(n)}$, the sketch says one must “remove a certain subcomplex of \emph{cloven Strebel differentials}.” One then uses “geometric arguments” to control its topology, concluding this subcomplex “is always homologous to a bouquet of $(k-2)$-spheres, where $k$ is the number of outputs.” Finally, “together with a comparison between the bar complexes associated to $\mathscr{V}^{(n)}$ and $\mathscr{Y}^{(n)}$,” the fact that this homology vanishes outside that degree implies “vanishing of higher cohomology of the relevant bar complex,” which is said to establish the main theorem.

Question: Let \(\mathscr{Y}^{(n)}\) be the quadratic dioperad associated to the planar dioperad \(\mathscr{Y}^{(n)}_{pl}\), and let \(\mathscr{Y}^{(n)}_{\mathrm{prop}} = F(\mathscr{Y}^{(n)})\) be the properad obtained from it by the functor \(F\). Here, a properad of the form \(F\mathscr{Q}\) is called \emph{Koszul contractible} when the quadratic dioperad \(\mathscr{Q}\) is both Koszul and contractible; for finitely generated quadratic dioperads, contractibility is equivalent to the canonical map \(F(\mathscr{Q}^!) \to (F\mathscr{Q})^!\) being an isomorphism of properads. Which statement holds for \(\mathscr{Y}^{(n)}_{\mathrm{prop}}\)?

The properad \(\mathscr{Y}^{(n)}_{\mathrm{prop}}\) is Koszul contractible, and therefore it is a Koszul properad.

Choices:
B. The properad \(\mathscr{Y}^{(n)}_{\mathrm{prop}}\) is a Koszul properad, but it is not Koszul contractible because the canonical map \(F((\mathscr{Y}^{(n)})^!) \to (F\mathscr{Y}^{(n)})^!\) need not be an isomorphism of properads.
C. The properad \(\mathscr{Y}^{(n)}_{\mathrm{prop}}\) is a Koszul properad.
D. The properad \(\mathscr{Y}^{(n)}_{\mathrm{prop}}\) is contractible in the sense that the canonical map \(F((\mathscr{Y}^{(n)})^!) \to (F\mathscr{Y}^{(n)})^!\) is an isomorphism of properads, but it is not Koszul contractible because Koszulity of \(\mathscr{Y}^{(n)}\) as a dioperad does not imply Koszulity of \(F(\mathscr{Y}^{(n)})\) as a properad.
E. The properad \(\mathscr{Y}^{(n)}_{\mathrm{prop}}\) is Koszul contractible only after restricting to the genus-zero part; equivalently, the canonical map \(F((\mathscr{Y}^{(n)})^!) \to (F\mathscr{Y}^{(n)})^!\) is an isomorphism merely as a dioperad, not as a properad.

comparison-plus-vanishing needed to upgrade contractibility to properadic Koszul contractibility

drops the contractibility/Koszul-contractible assertion while keeping the properadic Koszul conclusion

proposition that Koszul + contractible dioperad implies \(F\mathscr Q\) is Koszul as a properad

full properadic isomorphism versus genus-zero/dioperadic-only control