Papers

In this paper, we investigate various forms of proximal relations for non-autonomous dynamical systems. In particular, we introduce the notion of pointwise asymptotic commutativity and asymptotic commutativity, and show that these are strictly weaker than classical commutativity. We establish conditions under which different proximal relations, including proximality, syndetic proximality, and regional proximality, exhibit invariance and structural properties. In particular, we relate proximality with equicontinuity and weak mixing, and characterize conditions where proximal relations become trivial or maximal. Further, for systems generated by uniformly and strongly convergent sequences, we derive connections between proximal pairs of the system, its product systems, and the corresponding limiting system. Several examples are provided to demonstrate the necessity of the assumptions and the sharpness of the results.

For $x\ge1$, let $G_3(x)$ be the maximum sum of the side lengths of three pairwise interior-disjoint, arbitrarily rotated squares contained in a $1\times x$ rectangle. We determine this function exactly: $G_3(x)=x+\tfrac12$ for $1\le x\le\tfrac32$, $G_3(x)=2$ for $\tfrac32\le x\le2$, $G_3(x)=x$ for $2\le x\le3$, and $G_3(x)=3$ for $x\ge3$. This completes the $n=3$ case of the rectangular square-packing question posed by Richard Stanley in a 2021 MathOverflow comment. The values for $\tfrac32\le x\le3$ follow from a strip theorem stating that three squares in $[0,1]\times[0,H]$, $H\ge2$, have total side length at most $H$. For $x\ge3$, the formula is immediate because each square has side length at most $1$. The range $1\le x\le\tfrac32$ is handled by combining the two-square theorem with an additional semi-perimeter estimate for a triangle whose two nonhorizontal sides have opposite slopes. In particular, the special case $x=2$ answers the question asked in the MathOverflow post. In connection with Erdős problem #106, we also prove that every five-square packing in the unit square with an axis-parallel guillotine cut has total side length at most $2$.

We study the one-dimensional reaction-diffusion equation \[ u_t=[A(u)]_{xx}+f(x,u). \] The diffusion operator belongs to a broad class of nonlinear degenerate diffusion operators that includes the porous medium operator as a special case. We develop a systematic regularity theory for the solutions and their free boundaries. First, we establish the $C^1$ regularity of the pressure variable $v$, together with a lower bound for its second spatial derivative. Next, we prove Darcy's law and that, after the waiting time, a right (resp.\ left) free boundary moves with strictly positive (resp.\ negative) velocity (Theorem 3.1), thereby strengthening the previously known results which only gave nonnegativity (resp.\ nonpositivity). Finally, under additional structural assumptions on the diffusion and reaction terms, we obtain higher regularity for both the solution and its free boundaries (Theorems 4.5 and 4.6). These results extend several classical regularity properties of the porous medium equation to a much broader class of degenerate diffusion equations with reactions.

We study first-order sensitivity of a scalar backward stochastic differential equation with a deterministic rough driver. The driver belongs to the nonlinear space of step-two weakly geometric $p$-rough paths, $2<p<3$, so an ordinary Banach-space difference quotient is not available. At a fixed rough path $x$, we use a weakly geometric, finite-$(p,p/2)$-variation tensor realization of the Qian-Tudor tangent structure, represented intrinsically by a first-level direction $h$ and a compatible second-level direction $\kappa$. Admissible rough-path secants are required to converge in a strong levelwise variation topology. Under bounded smooth rough vector fields, a bounded terminal condition, and a globally Lipschitz generator, we construct a continuous linear map $A_x:T_x^p\to S^\infty\times H^2_{\mathrm{BMO}}$. The proof first establishes a uniform four-jet expansion for reset rough flows along bounded realizations of full rough tangents. A Doss-Sussmann transformation transfers this expansion to quadratic generators. Uniform BMO and reverse-Holder estimates then yield a local difference-quotient theorem, which is propagated over a fixed deterministic partition and reconstructed in the original coordinates. The resulting derivative is independent of the joint lift, central decomposition, and radial realization. Consequently, solution difference quotients converge to $A_x(h,\kappa)$ for varying directions and arbitrary admissible secants with strong levelwise variation contact. This is intrinsic tangential Hadamard differentiability on that tensor-coordinate tangent class. Locally bounded ray-homogeneous selections give Frechet-differentiable chart pullbacks at the parameter origin. The latter statement is chartwise; it is not Frechet differentiability of the solution map in the homogeneous rough-path metric.

Combinatorial Mesh Calculus (CMC) formulates conservation laws directly on cell complexes using combinatorial differential forms and their cochain representations. This note develops an incidence-based geometric extension of that framework for directional quantities and local geometric structure on explicit cellular organisation. The central construction is a family of local fibres generated intrinsically by the incidence structure of the complex. Each vertex carries a vector space spanned by its incident edge directions, providing a combinatorial analogue of a tangent space whose dimension reflects local topology. These fibres define local coefficient spaces for incidence-based vector-, covector- and endomorphism-valued cochains. Directed transport maps compare states attached to neighbouring fibres, while a canonical solder form relates fibre directions to the underlying cell-complex structure. Together they give rise to finite combinatorial analogues of transport, torsion, curvature and metric structure. Evaluation cup products provide pairings between kinematic and force-like quantities, while bundle Hodge operators induced by the fibre metric relate vector- and covector-valued cochains, and weighted covariant incidence operators define degree-raising transport-corrected operations on incidence cochains. The framework does not assume a smooth manifold, fixed-rank bundle, cellular sheaf or local system. Instead, geometric structure is generated directly from the organisation represented by the cell complex. The resulting theory establishes a combinatorial scaffold for geometry on explicit cellular organisation and identifies the principal mathematical questions required for its further development, including admissible transport classes, metric compatibility, Cartan-type structure equations and locality-dependent algebraic structures.

Let k be the algebraic closure of F_2. We study the polynomial endomorphism F=(P,Q) of the affine plane, where P=x+x^2 y+x^4+x^6 y^2 and Q=y+x^5+x^6 y+x^7 y^2+x^8 y^3. Its Jacobian determinant is 1, while the three distinct points (0,1), (1,0), and (1,1) have the same image. We prove that [k(x,y):k(P,Q)]=3 and that the extension is separable. Thus the generic degree is prime to the characteristic although F is not an automorphism, giving a dimension-two counterexample to the separable Jacobian conjecture in characteristic two. The proof uses explicit recovery from a hidden cubic, irreducibility over the actual target field, and a bridge between function-field embeddings and the geometric generic fiber. We also give an explicit graph presentation proving that F is etale and derive the map from a coordinate-permuted form of a three-variable map of Irit Huq-Kuruvilla. An appendix records the precise scope and evidence boundaries of a Lean formalization and an independent Harmonic Aristotle replay.

pyMOR is a free and open-source software library of model order reduction algorithms for the Python programming language. Designed with classical model-based reduction methods for large-scale parametric partial differential equation problems in mind, algorithms in pyMOR are implemented in terms of operations on abstract VectorArray, Operator and Model interfaces, allowing for a seamless integration with external solver codes implementing the full-order model. For cases where a tight integration with the full-order model code is not feasible, data-driven model order reduction algorithms, which only require simulation or measurement data of the full-order model, are an attractive alternative. In this work we discuss the data-driven methods that have been recently added to pyMOR, show practical examples of their application using pyMOR and compare their performance with classical model-based methods. We show that pyMOR serves as a unified framework for combining model-based and data-driven methods, enabling the construction of flexible and efficient hierarchical model reduction pipelines.

Consider read-once polynomial functions over a finite field of order $k$, i.e., functions defined by expressions built from field addition, multiplication, and constants, in which every variable occurs at most once. Let $p=(p_1,\dots,p_k)$ be the distribution of the values of a read-once function on independent uniform inputs. We prove that for every $k\ge4$ all such distributions belong to a body $\mathcal{B}_k$, defined by the following relation on the sorted atoms $p_1^\downarrow\geq\cdots\geq p_k^\downarrow$ of the distribution: \[ \mathcal{B}_k=\left\{p:p_k^\downarrow\ge \frac{1-p_2^\downarrow-(1-p_2^\downarrow)^k}{k-1}\right\}, \] or equivalently, $\mathcal{B}_k = \{ p \colon 1-p_2^\downarrow-(k-1)p_k^\downarrow\le(1-p_2^\downarrow)^k\}$. Our main theorem is that this body is stable under the convolutions corresponding to both field operations. More generally, convolution for any quasigroup operation on $k$ points preserves $\mathcal{B}_k$; the multiplicative conclusion needs only an absorbing zero and a quasigroup operation on the nonzero elements. The body is full-dimensional and contains the uniform law and every point mass; its normalized volume is given by an exact one-dimensional integral, and the $k$th root of that volume tends to $0.2183305369\ldots$. The complete development, including the two stability theorems, the exact volume formula, and its sharp exponential rate, has been formalized in Lean 4.

This paper studies asymptotic expansions of the Gaussian and Archimedean compounds of two arbitrary bivariate means that need not be symmetric. Recursive algorithms for the coefficients of such expansions were previously obtained only in the symmetric case. Here, this restriction is removed and recursions are derived for the general nonsymmetric setting. All coefficient calculations are carried out in the algebra of formal power series. This introduces an additional case in the recursion, as well as singular configurations in which the formal invariance equation does not determine the coefficients recursively. The formal problem is kept separate from the existence and convergence of the iterative procedures. We also connect the first formal coefficients with Farhi's metric, recall its known global convergence criterion, and derive the corresponding local rates. The algorithms are illustrated using weighted power means and several known identities. Finally, the method is extended to expansions with sign-dependent coefficients and is applied to the neo-Pythagorean means and to a recently introduced two-parameter family.

Zsolt Páles asked whether there exists a function $f: \mathbb{R}\to \mathbb{R}$ such that $$ f\left(\frac{x+2y}{3}\right) \le \frac{f(x)+2f(y)}{3} \quad \text{ for all }x\le y $$ which is not midconvex. We show that the answer is negative.

In this paper, we study the sample complexity of the empirical plug-in estimator for the $2$-Gromov-Wasserstein distance $D_2$ between compactly supported probability measures on Euclidean spaces. Let $\mu$ and $\nu$ be supported on compact subsets of $\mathbb{R}^{d_x}$ and $\mathbb{R}^{d_y}$, respectively, and let $\widehat\mu_n$ and $\widehat\nu_n$ be their empirical measures based on independent samples of size $n$. We prove that \[ \mathbb{E}\left|D_2^2(\widehat\mu_n,\widehat\nu_n)-D_2^2(\mu,\nu)\right| \lesssim n^{-2/((d_x\wedge d_y)\vee 4)} (\log n)^{\mathbf 1_{\{d_x\wedge d_y=4\}}}. \] This rate is sharp up to the logarithmic factor in the critical dimension. The proof is based on a geometric representation of the Euclidean distance as a squared $L^2$-distance between half-space feature maps. This yields a variational dual formulation of the Gromov-Wasserstein functional in terms of a family of classical optimal transport problems indexed by an infinite-dimensional auxiliary parameter. Although the resulting cost functions need not be semiconcave in either argument, we introduce a marginal recentering of the costs that restores the concavity structure needed for sharp metric-entropy bounds. Combining this representation with empirical-process estimates gives a rate governed by the smaller of the two ambient dimensions.

Let $$ A_q(z)=\sum_{k=0}^{\infty}\frac{q^{k^2}}{(q;q)_k}(-z)^k $$ be Ramanujan's entire function. We study it in the turning-point scaling $$ q=e^{-\varepsilon},\qquad z=\frac{\sqrt q}{4}e^{-\varepsilon^{2/3}\zeta}, $$ with $\zeta$ in a compact subset of $\mathbb{C}$. After an explicit exponential normalization, a direct coalescing-saddle analysis gives, uniformly on compact subsets, the expansion $$ \operatorname{Ai}(\zeta) +\frac{\varepsilon^{2/3}}{30} \left(4\zeta\operatorname{Ai}(\zeta) +\zeta^2\operatorname{Ai}'(\zeta)\right) +O_K(\varepsilon). $$ Thus the first correction is explicit and comes with a quantitative remainder. For every fixed $n$, the same analysis locates the positive zero associated with the $n$-th Airy zero and proves that it is globally the $n$-th positive zero of $A_q$. Expanding the normalized $q$-difference equation recovers the Airy differential equation and confirms the scaling. An appendix records Morita's antisymmetric companion, whose normalized limit is $\operatorname{Bi}$, together with a single-valued meromorphic descent of it. Numerical tables illustrate the normalization, the correction, and the zero formulas.

Let $\chi$ be a primitive odd Dirichlet character of conductor $f$. For the universal projector polynomials $P_m$ introduced in arXiv:2607.23177, defined by $\sum_m P_m(X)Y^m = -\log(1-X(1-e^{-Y}))$, we evaluate the character Fourier spectrum at $h_t = \zeta_f^t/(\zeta_f^t-1)$: for every odd $m$, $\sum_t \bar\chi(t)P_m(h_t) = \tau(\bar\chi)B_{m,\chi}/(m\,m!)$. After reduction at any prime above $p \nmid f$, the case $m = p-j$ identifies this spectrum, including its exact nonzero scalar, with the divided generalized Bernoulli value attached to $\chi\omega^{-j}$; the spectral-zero and Bernoulli-zero criteria therefore agree over every residue field, with no splitting hypothesis on the coefficient field. We connect the identity with the local Kummer spectrum of the circular unit $1-\zeta_f\zeta_p$ and carry the programme through in the first non-real case: for the two primitive quartic characters modulo 5 and primes $p < 500$, $p \equiv 1 \pmod{20}$, exactly eleven zero lines occur. On each line an integral character projection of $1-\zeta_5\zeta_p$ is everywhere locally unramified, a finite split-prime Artin computation proves it is not a global $p$-th power, and the character-wise Main Conjecture shows the radical generates the complete order-$p$ Hilbert class component. Six of the eleven components occur at classically regular primes. At $p = 61$ the two conjugate characters contribute on different indices. A deterministic integer-arithmetic program (ancillary file) verifies the enumeration, the divided digits, and every certificate.

The purpose of this paper is to show that, under mild inductive assumptions, if a group $G$ contains a component that is a simple group of Lie type in characteristic $p$ and $O_p(G) = 1$, then the Quillen poset of $G$ at $p$ has nonzero rational homology. In particular, this shows that such components cannot arise in a minimal counterexample to Quillen's conjecture. This result is of particular interest at the prime $p=2$, where the conjecture is still open.

Climate change reshapes food webs by altering species distributions and interactions, making it essential to identify structural backbones of interactions that persist across ecosystems. Deterministic alignment methods are computationally slow and restricted to one-to-one correspondences. We introduce a scalable, non-deterministic alignment framework inspired by optimal transport that captures overlapping species roles via many-to-many mappings. Framed via motif-role profiles as a Gromov-Wasserstein transport problem, our method is both efficient and interpretable. We apply the proposed method to a large continental-scale data set of 129 mammal food webs in Sub-Saharan Africa. Pairwise alignments are identified, and we uncover robust backbones with greater connectivity and transitivity than those expected under null models. The proposed approach provides a formal, reproducible tool for forecasting ecosystem reorganization and conservation efforts.

We study the variational properties of the spectrum of the Dirichlet-to-Robin map $\mathcal{D}_g$ on connected compact manifolds with boundary of dimension at least three. For the first eigenvalue, we show that Type II Yamabe metrics extremize the first normalized eigenvalue functional, and we characterize all extremals. If $[g]$ is a conformal class for which $\mathcal{D}_g$ has at least two negative eigenvalues, then we show the existence of a generalized metric that maximizes the second normalized eigenvalue of $\mathcal{D}_g$ in the conformal class. Moreover, we show that each such metric either defines a solution to an Escobar–Yamabe type equation on manifolds with boundary that changes sign along the boundary, or a weakly free-boundary harmonic map into the unit Euclidean ball.

We extend the vertex polynomial to graphs of arbitrary degree and prove local relations that hold when a graph contains a digon, triangle, quadrilateral or pentagon.

A plank of width $w$ in a Euclidean space is the set of points lying between two parallel hyperplanes at distance $w$ from each other. Bang's theorem says that if a family of planks covers a convex body $K$, then their total width is at least the width of $K$, that is, the width of the thinnest plank containing $K$. We study a quantitative variant of this problem in the case where the total width of the planks is fixed. How should the planks be placed so as to cover as much of the volume of the body as possible? For the central case where $K$ is a Euclidean ball, Károly Bezdek asked whether the optimal arrangement consists of a single plank centered at the origin. We give an affirmative answer to this question. We also show that for every planar convex body an optimal partial covering is attained by a single plank.

We consider the limit of small Mach number and small vertical-to-horizontal aspect ratio for the isothermal compressible Navier-Stokes system. In addition, we consider the scale in which the stratification is weak. Owing to the anisotropic nature of the problem, the dynamics exhibit a three-wave separation phenomenon, consistent of a slow wave, a fast horizontal acoustic wave, and an even faster vertical acoustic wave. These three waves, unfortunately, are not mutually orthogonal, and the corresponding projections are parametrized by the small parameter, which significantly complicates the nonlinear analysis. Without any restriction on the size of the initial waves, we establish the uniform existence and uniqueness of solutions to the compressible Navier-Stokes system for any fixed small parameter, by carefully analyzing the evolutions of both the energy and the acoustic waves. Moreover, we prove that, as the small parameter tends to zero, the solutions converge to that of the incompressible primitive equations governing atmospheric and oceanic flows.

We prove an $\Omega(L/T)$ lower bound for the convergence rate of minimization in the class of functions that are convex and $L$-smooth relative to negative entropy on the standard $d$-simplex, valid for every first-order method when $d = \Omega(T^2)$. In particular, this shows that mirror descent is optimal up to a logarithmic factor in this class. This may be surprising due to the fact that accelerated methods are readily available under the assumption of smoothness in $\ell_1$-norm. While Dragomir et al. (Mathematical Programming, 2022) have already showed that acceleration might be impossible under relative smoothness, their prox-function is pathological and constructed together with the hard instance. In contrast, we show non-acceleration for a specific prox-function with particularly favorable structure. We also extend the result to the quantum setting, proving the same lower bound in the class of functions $L$-smooth relative to negative von Neumann entropy on the spectrahedron of $d \times d$ Hermitian positive-semidefinite matrices with unit trace.

We study stochastic, parabolic-parabolic Keller–Segel equations on the $d$-dimensional torus in scaling critical Besov spaces, for $d \geq 3$. Using stochastic maximal regularity estimates, we prove local well-posedness of the equation, i.e., that there exists a unique solution up to a maximal time of existence. We show that the time of existence can be made arbitrarily large with arbitrarily high probability, provided that the initial data is sufficiently small in these critical spaces. Contribution to the Oberwolfach Report for the Seminar 'Stochastic Partial Differential Equations in Critical Spaces' organized by Antonio Agresti and Mark Veraar.

We introduce a residual-driven procedure for training nonlinear-manifold reduced-order models for parametrized linear partial differential equations that, given prescribed latent and lifting spaces, identifies the nonlinear lifting without high-fidelity solution snapshots. The approximation is represented by a low-dimensional latent coordinate together with a nonlinear lifting into a richer reduced space. Rather than fitting the lifting to snapshot data, we determine it by minimizing a computable residual-based upper bound for the state error. For affinely parametrized operators, the resulting training objective admits an efficient offline–online decomposition, and the lifting update reduces to a sequence of low-dimensional weighted least-squares problems. Numerical evaluations on an advection–diffusion problem and a plane-strain elasticity benchmark show that the proposed approach substantially improves accuracy over linear subspaces. The resulting nonlinear models achieve accuracy comparable to snapshot-driven nonlinear-manifold training while avoiding high-fidelity snapshots in the lifting-identification stage.

We study planar continuous piecewise-smooth slow–fast return circuits in which an invariant-line entry–exit passage is followed by a separated quadratic grazing and the reference cycle has unit multiplier. We first prove that the physical entry–exit map for positive slow parameter extends, uniformly to any prescribed finite order, to the singular parameter, including nonvertical endpoint fibers. In the exact moving penetration coordinate, composition with the grazing passage gives the extended Poincare displacement $\Delta(q,p)=S(q,p)+q_+^{3/2}K(\sqrt{q_+},q,p)$. Under a rank-two unfolding, we use the exact constant and linear coefficients of $\Delta$ as parameters and construct, for every sufficiently small fixed positive slow parameter, a unique nonpenetrating fold, a unique penetrating fold, and the grazing-incidence stratum. We give the complete marked chamber decomposition by nonpenetrating, grazing, and penetrating cycles and determine their stability from the Poincare multiplier. The open chambers contain zero or two cycles when the smooth and grazing coefficients have the same sign, and one or three when their signs are opposite; the corresponding cyclicity bounds are sharp corollaries. A compact polynomial family realizes both sign classes. In a cutoff-Gause family, interval certificates isolate one singular balanced-neutral point in each of two parameter boxes and verify the required signs and rank; an analytic local pullback transfers the diagram to the response parameters. Thus one local classification records cycle number, itinerary, and stability across the grazing and fold boundaries.

WOW-284 asserts that the minimum dual degree of every connected graph of order at least three and girth at least five does not exceed the negative of its least distance eigenvalue. We refute it with exact counterexamples of orders $38,39,40,42$, and $50$, and develop a structural theory of the failure. For a connected $k$-regular graph of girth at least five and diameter three, we prove $\delta^*(G)+\lambda_{\min}(D(G))=2k-2-\max_{\theta\ne k}(\theta+1)^2$. Here $\theta$ ranges over the nonprincipal adjacency eigenvalues. We further prove that every regular strict counterexample has degree at least six and diameter at most four, while diameter four forces degree at least ten. We solve the associated one-variable nonbacktracking linear program exactly, including optimizer rigidity. For regular strict counterexamples of diameter three, the optimizer yields a positive-semidefinite slack matrix whose integral excess gives the stronger bound $|V(G)|\le\left\lfloor 3(k+2)^2(k^2+3)/(18k+41)\right\rfloor$; this follows from a three-to-one quantization theorem for the integral excess. The slack matrix's principal minors also recover local cycle constraints. In particular, regular degree-six counterexamples have order at most $50$, and at the degree-six, order-$50$ boundary the associated signed complement is necessarily disconnected. We determine the distance spectra of one- and two-vertex punctures of Moore graphs and establish a uniform deletion-stability bound: every deletion of at most five vertices from the Hoffman–Singleton graph remains a strict counterexample, whereas an explicit six-vertex deletion does not. All theorem-level computations use exact arithmetic. Lean 4.31 kernel-checks the explicit $50$-vertex Hoffman–Singleton counterexample at graph level, finite spectral certificates at orders $38,39,40,42$, and the analytic LP optimum and rigidity for every integer $k\ge4$.

This note proves existence and uniqueness of strong solutions to a broad class of fourth-order quasi-linear parabolic equations in the class of Wiener spaces. It improves upon previous results in the literature, devoted to some special cases falling within the general framework developed here, in which uniqueness held true in a smaller class than the space where existence was proven. Gain of analyticity, as well as generalisations to higher-order equations and to propagation of higher regularities, are also discussed.

We prove that a complete, connected, embedded, minimal hypersurface in $\mathbb{R}^{n+1}$ with finite total curvature and Morse index one is the higher-dimensional catenoid.

We study the equivariant CR Yamabe problem for a fixed-point-free torus action on a co-oriented compact contact manifold. Such examples arise naturally in the study of Sasakian manifolds with constant scalar curvature. In the toric case, we show that this problem is equivalent to a kind of boundary value problem for an elliptic PDE on a pair of functions defined on a convex, polyhedral domain. We provide examples of solutions and prove that many contact toric manifolds admit compatible CR structures with distinct signs of CR Yamabe invariants.

Computational costs often make smoothing procedures prohibitive for high-dimensional data assimilation problems. To address this challenge, we propose a dynamical low-rank approximation (DLRA) methodology for smoothing concerning frameworks based on stochastic differential equations. We extend the previously developed joint mean-and-covariance optimization (JMCO) filtering setting to derive a reduced-order smoother via the Rauch–Tung–Striebel recursion and establish the corresponding Kalman–Bucy smoothing for affine drift dynamics. The resulting algorithms retain the adaptive nature of DLRA while significantly reducing the computational time and storage of the whole smoothing procedure.

We introduce and study continued fractions defined by Schneider-like maps over polynomial rings, where the maps are associated with a fixed polynomial of arbitrary degree. In particular, we prove the existence and uniqueness of the continued fraction expansion for every element of the field of Laurent series. We then establish precise Diophantine approximation properties of the corresponding convergents. We also study the dynamical aspects of the underlying map, proving that the Haar measure is invariant and ergodic. As an arithmetic consequence, we determine the asymptotic sum of the digits of the expansion for almost every element. Finally, we identify the set of elements that are worst approximable in this framework and compute its Hausdorff dimension, showing that it is a fractal set of positive dimension.

The capacity region of the $K$-user discrete memoryless broadcast channel is fully characterized when non-signaling (NS) assistance is available to the transmitter and all $K$ receivers. The NS-assisted capacity region is shown to coincide with Sato's region, i.e., the region defined by sum-rate bounds over all subsets of messages, where each bound corresponds to full cooperation among that subset of receivers under a worst-case joint channel law consistent with the marginal channels.