Papers

We develop a functorial framework for ample C$^*$-diagonal pairs and their Weyl groupoids. The main contribution is a partial functoriality theorem in the untwisted setting: from a diagonal-preserving $*$-homomorphism whose restriction to the diagonal is an isomorphism, we construct an open subgroupoid $\mathcal H_\Phi\subseteq\mathcal G_B$ and a continuous, open, injective groupoid homomorphism $\rho_\Phi: \mathcal H_\Phi\to\mathcal G_A$, together with the induced homeomorphism on unit spaces. We prove that this construction is well-defined and extends to a contravariant functor into a category of partial groupoid morphisms, thereby giving a concrete partial Weyl-groupoid functor. We record consequences for expectation-compatible ideals, faithful conditional expectations, and quotient reductions, and formulate questions concerning the interaction with dynamical comparison. We also record tensor-product permanence results, including the product description of Weyl groupoids and the relevant diagonal-dimension estimate. Finally, in the untwisted principal case, we prove a reconstruction/rigidity theorem showing that the underlying Weyl groupoid determines the diagonal pair up to isomorphism. Results imported from Bönicke–Li, Li–Liao–Winter, Renault, Kumjian, and Kopsacheilis–Winter are used explicitly as background or supporting permanence results rather than claimed as new contributions.

Adopting the powerful methods introduced in \cite{li2021carnotcaratheodory, LZ2025}, we investigate the asymptotic behaviour at infinity for the heat kernel associated with the Grushin operator $\Delta_G = \Delta_x + |x|^2 \Delta_u$ on $ \mathbb{R}_x ^n \times \mathbb{R}_u ^{n'} $ for all $ n , \, n' \ge 1 $. We further establish sharp bounds for its spatial derivatives. As a by-product, precise estimates and small-time asymptotic behaviour for the Grushin heat kernel will be provided, as explicit as one can possibly hope for.

This paper studies the strong quasiconvexity of norm and distance functions in finite-dimensional normed spaces. Although the Euclidean norm is known to be strongly quasiconvex on bounded convex sets, a complete characterization of this property for general norms remains open. We establish necessary and sufficient conditions for a norm function to be strongly quasiconvex on a convex set. We also initiate the study of the strong quasiconvexity of distance functions. Our results provide new insights into the geometric properties of norm and distance functions and extend several existing results in the literature.

This paper presents a Jacobi-type iteration for computing a given specified eigenpair of a symmetric matrix. For a certain class of diagonally dominant matrices, the procedure is shown to converge at a linear rate depending on how the matrix is significantly dominated. The cost per iteration is generally quadratic. Therefore, the proposed procedure can compute an approximation of the desired eigenpair in quadratic time.

Classification of matches played in the last rounds of sports competitions is a well-established tool for evaluating tournament designs. Both deterministic and probabilistic approaches are available for this purpose. Our paper offers the first comparison of them by analysing the most prominent example of four-team round-robin competitions, the group stage of the FIFA World Cup. We show that both methods are highly relevant in practice: all (four) deterministic and (six) probabilistic match types occurred in the 2014 and 2018 FIFA World Cups, respectively. The probabilistic model, which accounts for the relative benefits of attacking and defending, provides deeper insights; for instance, the competitive matches from the deterministic approach can be of any of the six probabilistic types. Finally, the probabilistic framework is used to quantify and decompose the impact of the main reforms introduced for the 2026 FIFA World Cup: the expansion to 48 teams, as well as the modified qualification and tie-breaking rules.

Motivated by nonclassical Weyl laws arising in various contexts (including Connes' approach to the Riemann Hypothesis), we develop a systematic theory of Dixmier traces and Connes' noncommutative integration for weak Lorentz ideals associated with regularly varying functions. A key ingredient is an asymptotic additivity property for eigenvalue partial sums, obtained by combining Karamata's theorem with results of Kalton and Lord-Sukochev-Zanin. This yields a direct construction of Dixmier traces in terms of eigenvalue sequences and a complete spectral characterization of measurable operators, answering a question of Connes in this general setting. We also extend to weak Lorentz ideals the Birman-Solomyak perturbation theory for eigenvalue and singular-value asymptotics. Weyl operators (those admitting precise asymptotic limits for their rescaled eigenvalue sequences) are shown to form a closed subset of the ideal, stable under compact perturbations, extending classical results of Weyl and Birman-Solomyak. We further study strong measurability (measurability with respect to all positive normalized traces). We prove that every Weyl operator is strongly measurable, so spectral measurability implies strong measurability. The converse does not hold in general; a spectral characterization via Pietsch's correspondence is obtained in the forthcoming companion pape. Finally, as an application, we establish spectral measurability for operators arising from nonclassical Weyl laws: operators associated with the Riemann Hypothesis (assuming RH); Schrödinger operators with anisotropic potentials and Dirichlet Laplacians on infinite-volume domains; Dirac operators on open spin manifolds with conformally cusp metrics; and the operator formed by the Dirac operator of the Podles̀ quantum sphere and the Laplacian on the 2-torus.

A difference operator on an associative algebra is an algebraic abstraction of the forward and backward difference operators. In this paper, we first introduce difference operators on associative $2$-algebras and consider the category of difference associative $2$-algebras. Subsequently, we also introduce difference operators on a given $A_\infty$-algebra in terms of their Maurer-Cartan characterization. We prove that the category of difference associative $2$-algebras and the category of $2$-term difference $A_\infty$-algebras are equivalent. We characterize skeletal and strict $2$-term difference $A_\infty$-algebras by respectively third cocycles and crossed modules of difference algebras. Finally, we define the notion of a $2$-term bimodule up to homotopy over a difference algebra, which in turn yields a construction of a $2$-term difference $A_\infty$-algebra.

We establish local variational characterizations of non-degenerate free-boundary minimal hypersurfaces in compact Riemannian manifolds with boundary. For a two-sided strictly stable hypersurface, we show that it is the unique mass minimizer in its relative homology class, both in a small tubular and flat-neighborhood. As an application, for generic metrics in dimensions $3\le n+1\le 7$, we show that the first free-boundary width is realized by a multiplicity-one hypersurface of Morse index one. For an orientable non-degenerate free-boundary hypersurface of Morse index $k>0$, we construct a canonical local $k$-parameter family and obtain a local min–max characterization.

This paper develops a trace-regular variational framework for time-harmonic Maxwell scattering problems involving pointwise nonlinear boundary and interface responses. We investigate three canonical classes of models: nonlinear impedance, nonlinear perfect electric conductor, and nonlinear transmission conditions. Since the standard Maxwell tangential trace belongs to a space of negative order, the nonlinearities are formulated in refined functional spaces where the tangential electric field admits an $L^2(\Gamma)$-trace. Under the assumption of a sufficiently small Lipschitz constant for the nonlinear response, we establish the well-posedness of the direct problems via fixed-point arguments leveraging the mapping properties of the associated linear Maxwell operators. Within this framework, we perform a rigorous sensitivity analysis of the electromagnetic fields with respect to perturbations of the scattering interface. By employing the covariant Piola transform, we prove the continuity and Fréchet differentiability of the pulled-back solutions with respect to domain variations. The material derivative is characterized as the unique solution to a corresponding $\mathbb{R}$-linearized Maxwell system, and the shape derivative is shown to satisfy explicit boundary or interface conditions for each of the three nonlinear models. We further demonstrate that the resulting sensitivity expressions possess the Hadamard structure, depending exclusively on the normal component of the boundary deformation. The resulting derivative characterizations provide a mathematical basis for subsequent adjoint-based sensitivity analysis, shape optimization, and gradient-driven inverse reconstruction.

A Lie-Yamaguti algebra is a non-associative algebraic structure that generalizes both Lie algebras and Lie triple systems. We first consider the factorization problem for Lie-Yamaguti algebras that essentially related to the bicrossed product of Lie-Yamaguti algebras. Next, given an inclusion $\mathfrak{g} \subset E$ of Lie-Yamaguti algebras and a strong $\mathfrak{g}$-complement $\mathfrak{h}$, we describe and classify all $\mathfrak{g}$-complements in $E$. In particular, we show that any other $\mathfrak{g}$-complement in $E$ is isomorphic to $\mathfrak{h}$ by some deformation map $r: \mathfrak{h} \rightarrow \mathfrak{g}$. Despite this importance, it turns out that a deformation map generalizes homomorphisms, derivations, crossed homomorphisms and relative Rota-Baxter operators on Lie-Yamaguti algebras. We define the cohomology of a deformation map unifying the cohomologies of all the operators mentioned above. Finally, we provide a Maurer-Cartan characterization and construct the governing $L_\infty$-algebra of a deformation map $r$ that controls the linear deformations of $r$.

It is known that invariant subspaces of classical Jordan blocks of the Hardy space over the open unit disc are described by factorizations of inner functions. In the polydisc setting, Jordan blocks are tensor products of one-variable Jordan blocks. We provide representations of their doubly commuting invariant subspaces.

We propose Consistent CutPINN, a framework for partial differential equations posed on bounded curved domains defined implicitly by a $\C^2$ level-set function, $\Omega = \{\varphi < 0\}$. In this paper we develop the framework for second-order elliptic problems in two dimensions. The standard PINN loss penalises the boundary mismatch in $L^2(\partial\Omega)$, but $L^2(\partial\Omega)$ does not control the $H^{1/2}(\partial\Omega)$ trace norm that appears in the $H^1(\Omega)$ energy estimate. The consistent PINN framework of Bonito et al.~\cite{bonito2025} fixes this on the unit cube $(0,1)^d$ via a Kuhn–Tucker simplicial decomposition of the flat boundary faces, but the construction relies on the affine structure of the faces and does not carry over to smooth curved boundaries. We address this gap. Specifically, (i) we introduce a discrete $H^{1/2}(\partial\Omega)$ surrogate built directly from collocation points on a $\C^2$ curve, (ii) we prove a Chord-arc norm equivalence between this surrogate and the continuous trace norm, (iii) we establish an a priori $H^1$ error bound on cut domains, and (iv) we derive convergence rates under Besov regularity using optimal recovery theory. Numerical experiments on a disk and a non-convex flower domain confirm that the consistent loss is much more accurate than the standard PINN loss and far more robust to cut-cell configurations.

In this paper we combine a flexible covariant formulation of the shallow water equations with the semi-implicit numerical scheme developed over the years by Casulli and collaborators. After adopting an orthogonal, but non-orthonormal, coordinate basis on two dimensional manifolds, and by writing the divergence of symmetric tensors in a way that avoids the introduction of Christoffel symbols, the shallow water equations preserve a very close resemblance to the usual one expressed in Cartesian coordinates. In this way, a stable semi-implicit scheme can be derived by using an implicit discretization for the gradient of surface elevation in the momentum equations and for the velocity in the continuity equation, with stability properties that are independent of the celerity. We have tested the new method over a variety of challenging benchmarks, including, among the others, the smooth wave propagation over a water globe and the deformation of an artery branch. Two appealing additional features make the method particularly powerful with respect to oceanographic applications: firstly, thanks to the wetting and drying ability of our semi-implicit approach, no pathological behaviors occur at the poles; secondly, the scheme is naturally well-balanced, and it is able to preserve perfect stationarity, up to machined precision, of the entire ocean configuration of the earth.

The greatest integer that does not belong to a numerical semigroup $S$ is called the Frobenius number of $S$, and finding the Frobenius number is called the Frobenius problem. In this paper, we resolve the conjecture of Frobenius problem for shifted square sequences suggested by Liu and Xin.

Motivated by structural biology applications, we study the projected multi-reference alignment (MRA) model, in which an unknown signal is observed through noisy samples, each generated by applying a random cyclic shift followed by a fixed projection. The projection merges reflection-symmetric index pairs, thereby discarding orientation information. The goal is to recover the dihedral orbit of the signal. We prove that in the high-noise regime, the first three moments of the projected observations determine a generic dihedral orbit. The main mechanism is a reduction, at the moment level, from projected MRA to the reflection-invariant phase-coupling structure of dihedral MRA. In Fourier-cosine coordinates adapted to the projection, the first moment determines the mean component, the second moment determines the Fourier magnitudes, and selected third moments yield the cosine phase-coupling relations appearing in the dihedral bispectrum. These relations lead to a constructive recovery scheme from moments up to order three. We complement the population theory with finite-sample experiments comparing expectation–maximization (EM), direct moment optimization, and direct Fourier-cosine moment optimization. The results show that, in the high-noise regime, both EM and direct moment optimization are consistent with the predicted third-moment sample-complexity scaling $n \gtrsim \sigma^6$, where $n$ is the number of observations and $\sigma^2$ is the noise variance.

We prove $\ell^2(\mathbb{Z}^n)-$estimate of long $r$-variational seminorm for the family of discrete averages associated to simplices.

Korsky, Saffat and Aiylam introduced a growth constant $c(G)$ for integer-valued $h$-Lipschitz functions on a finite graph $G$ and proved that, for $G=G(n,d/n)$, \[ \frac{1}{2d}+O(d^{-2})\le \log c(G)\le \frac{4\log^2 d}{d}+O(d^{-1}) \] with high probability. We sharpen the random-graph part of their result; as $n\to\infty$ and then $d\to\infty$, we prove \[ \log c(G)=\frac{\pi^2}{6d}+o(d^{-1}) \] with high probability. Additionally, we derive bounds on $\log c(Q_d)$ where $Q_d$ is the $d$-dimensional hypercube graph: \[ \frac{\pi^2}{6d}+o(d^{-1}) \le \log{c(Q_d)}\le \left(\frac{3}{4} + o(1)\right)\frac{\log d}{d}. \]

We establish sharp higher-order heat estimates with complete bound on the noncommutative tori \(\mathbb{T}_{\theta}^{n}\) and show the optimality in the small-time order. As an application in polynomial semilinear heat equations on \(\mathbb{T}_{\theta}^{n}\), we give local well-posedness, the blow-up alternative, persistence of higher regularity, and instantaneous smoothing in the Sobolev algebra scale \(H^k_\theta\), \(k>n/2\).

Based on the work of Guillarmou, Kupiainen, and Rhodes, we construct compactified imaginary Toda theory on closed Riemann surfaces, extending the rank-one construction to the higher-rank setting. This theory is expected to describe critical higher-rank models with extended symmetries, such as web models. We construct the correlation functions and prove that they satisfy the axioms of conformal field theory, including Segal's gluing axioms. On the Riemann sphere, we express the correlation functions as Dotsenko–Fateev type integrals. In the case $\mathfrak g=\mathfrak{sl}_n$, under a semidegenerate condition, we obtain a closed formula for the three-point structure constant.

Firmly nonexpansive operators arise naturally as resolvents of monotone operators and as generalizations of projections and proximal mappings in convex optimization and fixed point theory. While their iterates are known to converge weakly to a fixed point, strong convergence is not guaranteed (Genel and Lindenstrauss, 1975). Strong convergence of Cesàro means of iterates is also known to fail for general nonlinear nonexpansive mappings (Krengel and Lin, 1987). In this paper, we show that this failure persists in the much smaller class of firmly nonexpansive mappings. Using suitable meshes, we construct a new explicit family of counterexamples in infinite-dimensional Hilbert spaces with the origin as the unique fixed point. In the harmonic case, the Cesàro means of the iterates remain bounded away from the origin. Another variant yields Cesàro means that converge strongly to the origin. A third variant presents Cesàro means whose norms oscillate in the sense that their liminf is zero while their limsup is positive. Thus the strong convergence conclusion in von Neumann's linear mean ergodic theorem does not extend to Baillon's nonlinear mean ergodic theorem, even for firmly nonexpansive mappings.

This paper deals with nonsmooth convex optimization problems in Euclidean spaces. We identify special elements of the subdifferential of a convex function, called specular gradients. Based on this observation, we propose three numerical methods that use specular gradients in subgradient methods. We prove the convergence of the proposed methods under suitable step sizes. Numerical experiments demonstrate that the proposed methods are capable of minimizing non-differentiable functions that classical methods fail to minimize.

Derived brackets provide a mechanism for generating algebraic structures from graded Lie superalgebras, with applications in Poisson geometry, mathematical physics, and the theory of algebroids. In this paper, we present a complete structural and isomorphism classification of a family of Lie algebras constructed from the general linear Lie superalgebra $\mathfrak{gl}_{m|n}$ over a field $\mathbb{K}$ of characteristic zero via the derived bracket generated by an odd element $B$ satisfying $B^2 = 0$, which endows $\mathfrak{g}_{-1}$ with a Lie algebra structure denoted $\mathfrak{g}_{-1}^{B}$. We prove that for fixed dimensions $m$ and $n$, the isomorphism type of $\mathfrak{g}_{-1}^{B}$ is entirely determined by $r=\operatorname{rank}(B)$. In arbitrary dimensions, two such algebras are isomorphic if and only if they share the same rank $r$ and satisfy $\{m,n\}=\{p,q\}$. We explicitly compute the Levi-Malcev decomposition, proving the semisimple Levi factor is isomorphic to $\mathfrak{sl}(r)$, and provide exact formulas for the solvable radical and center.

Classical hyperbolic uniformization identifies every hyperbolic log-orbi curve with a compactified quotient of the upper half-plane by a cofinite Fuchsian lattice. The lattice is unique up to conjugacy. We reconstruct it intrinsically. For each hyperbolic log-orbi curve C we construct a canonical maximal principal PSL2-Higgs object. Etale-locally it comes from the standard square-root SL2-model. The central mu2 ambiguity disappears after passage to PSL2. Using vector tame non-abelian Hodge theory and regular-singular Riemann–Hilbert as input we assemble the required principal realizations Tannakianly. Parahoric structures encode the orbifold and cusp data on the coarse curve. After choosing a base point and conjugating the Betti realization is represented by a discrete faithful finite-covolume representation whose image is the uniformizing lattice. Compatibility with finite etale pullback makes the lattice construction a quasi-inverse to the compactified quotient functor. Thus classical uniformization is recast as an intrinsic Tannakian reconstruction theorem. We also identify finite etale covers with finite continuous sets for the profinite completion of the reconstructed lattice. After fixing a separable closure and the resulting geometric generic point we recover the absolute Galois group of the function field of C as the inverse limit of the based etale fundamental groups of orbifold models over C.

In this paper, we construct Poincaré-Einstein 4-manifolds with various kinds of cusps. In particular, we construct: (1) Infinite families of Einstein metrics on $(0,\infty)\times \mathscr{N}$, where $\mathscr{N}\to T^2$ is a principal $\mathbb{S}^1$-bundle over $T^2$, with one Poincaré-Einstein end and one end asymptotic to a real or complex hyperbolic cusp. (2) Infinite families of Einstein metrics on $(0,\infty)\times P$, where $P\to \Sigma_{\mathtt{g}}$ is a principal $\mathbb{S}^1$-bundle over a closed Riemann surface $\Sigma_{\mathtt{g}}$ of genus $\mathtt{g}\geq 2$, with one Poincaré-Einstein end and one end asymptotic to a bundle of two-dimensional hyperbolic cusps over hyperbolic $\Sigma_{\mathtt{g}}$. Universal covers of (1) and (2) provide new complete negative Einstein metrics on $\mathbb{R}^4$. These Einstein metrics also exhibit interesting degeneration phenomena. With this construction, we give a negative answer to a question of Anderson concerning cusp formation for Poincaré-Einstein 4-manifolds.

The Fefferman–Szegő metric \(g_{\operatorname{FS}}^\Omega\) on a \(C^\infty\)-smooth bounded strongly pseudoconvex domain \(\Omega\subset\mathbb C^n\) is an invariant metric defined via the Fefferman surface measure. For this metric, we first establish the vanishing of its \(L^2\)-Dolbeault cohomology outside the middle degree: \(\dim H^{p,q}_2(\Omega)=0\) if \(p+q\ne n\), while \(\dim H^{p,q}_2(\Omega)=\infty\) if \(p+q=n\). We also prove that the metric has \(C^\infty\)-bounded geometry. Using this analytic property, we obtain several rigidity results. In particular, if the Fefferman–Szegő metric is a gradient Kahler–Ricci soliton, then \(\Omega\) is biholomorphic to the unit ball \(\mathbb B^n\). Moreover, if the metric has constant scalar curvature, then it is Einstein, and again \(\Omega\) is biholomorphic to \(\mathbb B^n\). We also give a Ramadanov-type criterion in terms of the Fefferman–Szegő invariant function. Finally, in dimension \(n=2\), assuming the existence of a Kahler immersion into a finite-dimensional ball that maps boundary to boundary transversally, we show that the logarithmic term of the Fefferman–Szegő kernel vanishes to infinite order. Consequently, the boundary is locally spherical; if, in addition, \(\Omega\) is simply connected, then \(\Omega\) is biholomorphic to \(\mathbb B^2\).

The sliced Wasserstein distance $SW_2(\mu,\nu)$ compares high-dimensional probability measures by averaging one-dimensional optimal transport distances over linear projections. Although sliced Wasserstein distances are now standard computational tools in statistics, imaging, and machine learning, the rigidity behind the elementary comparison \[ SW_2^2(\mu,\nu)\leq \frac1d W_2^2(\mu,\nu) \] has not been systematically studied. Let $\mu,\nu\in\mathcal P_2(\mathbb R^d)$, $d\ge2$, with $\mu\ll\mathcal L^d$, and define the sliced Wasserstein deficit by \[ {\mathrm D}(\mu,\nu):=\frac1d W_2^2(\mu,\nu)-SW_2^2(\mu,\nu)\geq 0. \] We prove that ${\mathrm D}(\mu,\nu)=0$ if and only if the Brenier map $T=\nabla\varphi$ from $\mu$ to $\nu$ is homothetic affine, \[ T(x)=\lambda x+b \qquad \mu\text{-a.e.}, \] for some $\lambda\ge0$ and $b\in \mathbb R^d$. For quantitative stability, we introduce the sliced Poincaré–Korn (SPK) constant $\kappa_{\mathrm{SPK}}(\mu)$, defined as an new spectral gap of an averaged ridge-projection quadratic form on gradient fields modulo the family $\{\lambda x+b\}$. Whenever this constant is positive, we prove a stability estimate for the sliced Wasserstein deficit, up to a one-dimensional Lipschitz scale for the projected monotone transports. We obtain the sharp SPK constant for the Gaussian measures as the most important example, and establish positive SPK bounds for bounded perturbations of the Gaussian and compact classes of gradient fields for fixed source measures. Finally, we show that anisotropic Gaussians give a sharp obstruction: neither a Bakry–Émery lower curvature bound nor a usual Poincaré inequality alone can imply a global sliced Poincaré–Korn inequality.

We prove quantitative stability estimates for Wasserstein barycenters on Alexandrov spaces with curvature bounded from below. The proof combines the variational strategy of Carlier–Delalande–Mérigot with heat-kernel regularization, which supplies the regularity needed for dual convexity arguments in this non-smooth curved setting. The main result is an explicit strong-convexity modulus for the barycentric variance functional. As a consequence, barycenters depend Hölder-continuously on the underlying distributions with respect to the $1$-Wasserstein distance on the space of probability measures. We derive empirical-barycenter consistency and entropy-based sample-complexity bounds. Our proof does not rely on linear structure; in particular, the resulting estimates appear to be new even on smooth compact Riemannian manifolds.

We investigate the local topological structure of non-metrizable topological groups through the lens of Tukey order and cofinal types. Motivated by recent advances in topological groups admitting an $\omega^\omega$-base, we introduce the fineness index, denoted $\f(P)$, for arbitrary directed partially ordered sets. This cardinal invariant fundamentally generalizes the bounding number $\mathfrak{b}$ by capturing the exact threshold where a poset evades domination by its countable subsets, thereby establishing a universal lower bound for the character of topological groups with a $P$-base: $\chi(G) \in \{1, \omega\} \cup [fi(P), \text{cof}(P)]$. Furthermore, we resolve a structural problem regarding the exact cofinal types of free topological groups over uniform spaces. While classical results by Nickolas, Tkachenko, and others successfully computed the character of these groups via cardinal equalities (e.g., $\chi(F(X, \mathcal{U})) = \text{cof}(\mathcal{U}^\omega)$), lifting these equalities to strict Tukey equivalences has remained a persistent combinatorial challenge. By developing the novel machinery of neat trees to refine uniform covering trees, we overcome the structural obstructions and prove the Tukey equivalence $\Ne_e(F(X, \U))=_T \U^\omega$ for any compact uniform space $(X, \U)$.

Given a signing $\sigma\colon E(K_{n,n})\to\{-1,+1\}$ of the complete bipartite graph, when does $K_{n,n}$ admit a $1$-factorization in which every perfect matching has discrepancy bounded below by a positive absolute constant? Unlike the complete-graph case resolved by Ai, He, Im, and Lee, the bipartite setting carries an unavoidable obstruction: any balanced one-sided signing – one whose edge signs depend on a single bipartition class, with the two labels split as evenly as possible – forces every perfect matching to have discrepancy at most $1/n$. We prove that this is essentially the only obstruction: For every $\varepsilon>0$ there exists $c=c(\varepsilon)>0$ such that, for all sufficiently large $n$, every signing of $K_{n,n}$ either (i) admits a $1$-factorization in which every perfect matching has discrepancy at least $c$, or (ii) is $\varepsilon$-close, in normalized Hamming distance, to a balanced one-sided signing. A key ingredient is a spectral stability argument forcing the sign matrix to be close to a balanced one-sided pattern when both the overall discrepancy and the density of local switching patterns are small.

We study freely infinitely divisible $R$-diagonal elements in the unbounded setting and Brown measures for free additive perturbations by such elements. This class includes circular elements, circular Cauchy elements, and other previously studied $R$-diagonal models. We construct examples and prove stability under several algebraic operations, including homogeneous noncommutative polynomials in bounded, freely independent elements from this class. Using results for general $R$-diagonal perturbations, together with several analytic estimates specific to freely infinitely divisible $R$-diagonal elements, we prove that, in the bounded case, the support of the Brown measure coincides with the spectrum, and we obtain a criterion for property (H) in this non-normal setting. Finally, we study the free convolution semigroup associated with the symmetrized law of the modulus and derive a Hamilton–Jacobi equation for the regularized logarithmic potential.