Papers

Given a curve on a toric surface, a monomial projection induces a map from the normalization of the curve to the projective line. We determine the associated scrollar invariants for general integral curves of fixed geometric genus for a large class of toric surfaces. This generalizes a combinatorial formula to calculate such scrollar invariants for smooth curves in characteristic zero due to Castryck and Cools. We describe an expected behaviour for any toric surface, but provide examples where this fails at least on some irreducible component of the corresponding Severi variety.

We analyze the existence and stability of bright and dark pulses in a damped nonlinear Schrödinger (NLS) equation with periodic forcing. This Lugiato-Lefever equation is a canonical model in nonlinear optics describing pattern formation in a dissipative Kerr cavity driven by a bichromatic laser pump. Bifurcating from the bright and black NLS solitons, we rigorously construct bright and dark single-pulse solutions to the Lugiato-Lefever equation. Using a recently developed toolbox for concatenating and periodically extending pulse solutions in spatially periodic systems, we then obtain periodic multipulse solutions corresponding to experimentally observed broad-bandwidth frequency combs. While the existence and stability of the bright pulses follow from standard Lyapunov-Schmidt reduction and Krein index arguments, the construction of the dark pulses is substantially more delicate because the nonzero asymptotic states of the black NLS soliton yield neutral essential spectrum of the linearization. By carefully tracking the associated small spatial Floquet exponents along the bifurcation and employing exponential dichotomies, we establish the existence of dark pulse solutions. These consist of two domain walls connected by a long plateau with critical absolute spectrum. To our knowledge, this constitutes the first rigorous bifurcation result in the Lugiato-Lefever equation from the black NLS soliton.

In the first part we study excessive functions for a Markov process that are continuous semimartingales along the sample paths. The property then follows from the theory of local times of semimartingales. We next treat the case of Dirichlet spaces and show that, if u is a quasi-continuous version of a function u belonging to a regular Dirichlet space, then the image under u of the local energy measure of u is absolutely continuous with respect to Lebesgue measure. Finally, in a third part, we establish the occupation-time density property for certain Dirichlet processes.

We introduce cleft extensions for Hopf algebroids. We prove the equivalence between cleft extensions, $\sigma$-twisted crossed products, and Hopf-Galois extensions with the normal basis property, thereby generalizing the theory of cleft extensions for Hopf algebroids developed by Böhm and Brzeziński, and fitting in with the general theory of Galois and biGalois extensions over Hopf algebroids developed by the authors. We investigate the Ehresmann Hopf algebroid associated with a cleft extension and show that it is isomorphic to a generalized version of the Connes-Moscovici Hopf algebroid. A special case of the Connes-Moscovici Hopf algebroid, namely the case where the coinvariants of the cleft extension coincide with the base of the Hopf algebroid, is a Drinfeld twist of a Hopf algebroid by a two-cocycle, generalizing work of Böhm, Han and Majid.

Let $n\geq 2$ and let $q$ be an odd prime power. The first aim of this paper is to prove that, for every $E\subset \mathbb{H}_n(\mathbb{F}_q)$ and every $\lambda>0$, the following sharp rich-direction estimate holds \[ \left| \left\{ \vartheta\in D_n: M^{\mathrm{rd}}_{\mathbb{H}_n}\mathbf{1}_E(\vartheta)\geq\lambda \right\} \right| \lesssim_n q^{2n-1}|E|\lambda^{-2n}. \] The second aim is to determine, for every $1\leq u,v\leq\infty$, the sharp exponent of $q$ in the corresponding $\ell^u\to\ell^v$ estimate. More precisely, we prove that \[ A_n^{\mathrm{rd}}(u,v) = \max\left\{ \frac{2n-1}{v},\ 1-\frac1u,\ \frac{2n}{v}-\frac1u,\ 1+\frac{2n}{v}-\frac{2n+1}{u} \right\}. \] The proof combines the polynomial method with multiplicities and a probabilistic covering argument based on the action of the affine symplectic group.

The matrix Hilbert series of a locally finite elementary twisted Calabi–Yau algebra is the inverse of a matrix polynomial. The Smith normal form of this polynomial over the power series ring at $x=1$ produces a finite list of local exponents refining the Gelfand–Kirillov dimension, which records only the largest of them. We show that the Calabi–Yau symmetry makes this local data rigid: the algebra decomposes into ring factors according to the average Artin–Schelter index along Nakayama cycles, and after a local normalization the symmetry induces a nonsingular linking form on the Smith cokernel together with a finite-order Nakayama action on its layers, forcing reciprocal-eigenvalue and parity constraints on the multiplicities of the exponents. We compute the complete local data for cyclic skew-group algebras, derive a parity sieve for quiver classifications in dimension three, and realize, as an iterated smash product of a graded down-up algebra, a four-vertex type that a recent classification had left open.

We prove the $p$-Frobenius problems proposed as Conjectures 7.1 and 7.5 developed by T. Komatsu and R. Pandey (Bull. Korean Math. Soc. 2025;62:1397–1409.) for two families of consecutive odd integers. For integers $r,L,n\ge0$, the bounded restricted partition function $p_{\le r}^{(\le L)}(\le n)$ counts partitions of $n$ into at most $r$ parts, each at most $L$. Thus the bounded restricted partition functions $p_{\le 3}^{(\le a)}(\le s)$and $p_{\le 3}^{(\le a+1)}(\le s)$ play central roles in the proofs. Their generating functions are Gaussian polynomials, whose symmetry and unimodality provide a common tool for treating both families.

The existence of global weak solutions to a broad class of Navier-Stokes-Korteweg equations is established for large data in the three-dimensional torus, including the diffuse-interface and quantum Navier-Stokes systems as special cases. The model consists of the compressible Navier-Stokes equations with degenerate density-dependent viscosity and a general nonlinear third-order Korteweg term. The existence proof combines a priori estimates provided by the energy and Bresch-Desjardins (BD) entropy inequalities with a carefully designed approximation scheme. A crucial ingredient of the analysis is a new dissipation inequality associated with the Korteweg term, obtained via the systematic integration-by-parts method. To construct the solutions, artificial drag terms and a quantum Korteweg regularization are introduced, which are subsequently removed in the limit by establishing a renormalized formulation.

The aim of this article is two-fold. First, we obtain a normal form for automorphisms of the escaping sets of Hénon maps in a neighborhood of the attracting fixed point at infinity. Building on this local description, we characterize the global automorphisms of $\mathbb{C}^2$ that preserve the escaping sets. The analytic structure of the escaping sets, as established by Hubbard–Oberste-Vorth (\cite{HOV}), plays a crucial role in the proof of these results. In a similar spirit, we investigate the analytic structure of the super-level sets of the Green's functions associated with Hénon maps and give a description of the automorphisms of these domains.

We study nondegenerate $\mathbb Z_2$ harmonic functions on Euclidean spaces with branching loci given by ellipsoids and planar conics. We first prove that the parameter map associated with Yan's ellipsoidal family is injective. Together with Yan's surjectivity result, this identifies the ellipsoidal models, up to Euclidean motions and overall sign, with nondegenerate harmonic quadratic polynomials of index \(n-1\) and positive critical value. In \(\mathbb R^3\), we use modified ellipsoidal coordinates to construct a nondegenerate $\mathbb Z_2$ harmonic function in a neighborhood of any planar hyperbola. We then show that no global critical $\mathbb Z_2$ harmonic function with a planar hyperbola as branch locus can have finite Almgren frequency at infinity. Using modified paraboloidal coordinates, we construct a global nondegenerate $\mathbb Z_2$ harmonic function with any prescribed planar parabola as branch locus and obtain a precise asymptotic expansion and scale-down limit. Finally, we identify the parabolic model as the infinitesimal two-valued graph potential of a family of special Lagrangian three-folds constructed by Joyce.

For a closed connected Riemannian manifold $(M,g)$, the Gigli–Mantegazza construction pulls back the quadratic Wasserstein metric under the heat kernel embedding $x\mapsto p_t(x,\cdot)\,d\operatorname{vol}_g$. The resulting family $\widetilde{g}_t$ agrees with Ricci flow to first order in $t$, but in general not to second order. We prove that $$ \widetilde{g}_t =g-2t\operatorname{Ric}_g +t^2\left(-\Delta\operatorname{Ric}_g +2\operatorname{Ric}_g^2-\frac{2}{3}\mathcal{Q}_g\right) +O_{C^0}(t^3), $$ where $\mathcal{Q}_g$ is quadratic in the full curvature tensor. The term $-\Delta\operatorname{Ric}_g$ also occurs in the second-order expansion of Ricci flow, so the discrepancy depends pointwise and quadratically on the curvature. In particular, at a Ricci-flat metric that is not flat, Ricci flow is stationary while $\widetilde{g}_t$ is not. The Gromov–Hausdorff distance between the Gigli–Mantegazza and Ricci-flow metrics is $O(t^2)$, and round spheres show that this estimate is sharp.

In this paper, we prove that there is no complete two-sided stable immersed free boundary minimal hypersurface in the closed Euclidean unit ball $\mathbb{B}^{n+1}$ for $n\leq 5$.

We study Hausdorff limits of definable families over arbitrary models of o-minimal expansions of real closed fields. Over the real field, van den Dries proved that Hausdorff limits of definable families are definable, giving a geometric interpretation of the Marker–Steinhorn theorem. We prove a non-Archimedean analogue for definable families which are Lipschitz cells with a fixed cell presentation and a uniform Lipschitz bound. The proof replaces compactness of closed and bounded subsets of $\mathbb R^n$ by dense completions and long Cauchy sequences, and treats the Hausdorff distance as a metric valued in an ordered completion. We show that Hausdorff limits of such families are standard parts of external fibers over tame extensions, and use stable embeddedness of tame pairs to prove that every such limit is definable in the dense completion of the base model. We also prove a uniform version: the collection of these Hausdorff limits forms a definable family in the dense completion.

We present a demonstration showcasing the Autonomy Testbed for Multi-purpose Orbiting Systems (ATMOS), a planar spacecraft-analog robot designed for hardware-in-the-loop evaluation of guidance and control strategies in microgravity-like conditions. Using ATMOS as the physical test platform, we investigate the design, analysis, and performance evaluation of control architectures for remotely operated spacecraft under round-trip communication delays. In this work, we develop and experimentally validate a control strategy that combines state prediction and trajectory tracking control to perform a docking maneuver, accounting for time-varying random communication latency between ground operators and the ATMOS system. The demonstration includes a long-distance remote control experiment between Seoul and Stockholm, introducing realistic intercontinental delays and variability. The results highlight the capability of ATMOS to support rapid, reliable, and cost-effective testing of spacecraft teleoperation concepts, establishing a first step toward robust validation of on-orbit operations in microgravity-like environments.

We disprove Kusner's 1983 conjecture that every equilateral set in $\ell_p^n$ with $2<p<\infty$ has at most $n+1$ points: there exist $58$ points in $\mathbb{R}^{56}$ whose pairwise $\ell_5$ distances are all equal, so the maximum equilateral-set size satisfies $e(\ell_5^{56})\ge58>57$. This is the first equilateral set of more than $n+1$ points in $\ell_p^n$ for any finite $p\ge2$. The construction persists on an open interval of exponents around $5$; since Ge, Xu and Zhou recently proved the conjecture for $2\le p\le4$, the infimum of exponents at which it fails lies in $[4,5)$. The configuration is the unique solution of an explicit polynomial system with rational coefficients in a rational box, established in exact arithmetic.

In 1936, Zalcwasser proved the almost everywhere convergence of the arithmetic means of the square subsequence of trigonometric Fourier partial sums and asked whether this result extends to higher powers and to Cesàro means of fractional order. We give affirmative answers to both questions in a stronger pointwise form. Let $0<\alpha\leq 1$, and let $P$ be an integer-valued polynomial of degree with positive leading coefficient. We prove that the $(C,\alpha)$ means of the Fourier partial sums along any sequence whose $k$-th term equals $P(k)$ for all sufficiently large $k$ converges to $f(x)$ at every Lebesgue point $x$ of every $f\in L^{1}(\T)$.

We establish the limiting absorption principle for time-harmonic elastic scattering of plane waves by a periodic rigid diffraction grating. By perturbing the frequency with a small positive imaginary part, we regularize the ill-posed problem at propagative wavenumbers (that is, when uniqueness fails under the classical Rayleigh expansion condition) and characterize the limiting solution via a singular perturbation result from functional analysis. The limiting solution satisfies the original scattering problem together with an additional constraint that ensures uniqueness. Both incident pressure and shear waves are considered, and the same constraint condition is obtained in both cases. The results provide a rigorous selection mechanism for physically admissible solutions at resonance frequencies. Our framework extends naturally to the Neumann (cavity) boundary condition as well as other transmission conditions, in particular when guided waves exist in periodic structures.

High-order $h$-adaptive spectral element methods on tetrahedral meshes provide an effective framework for resolving localized singularities and multiscale structures in complex three-dimensional geometries. However, their development is often hindered by difficulties in maintaining $C^0$ continuity across refinement interfaces and efficiently transferring solutions between adaptive meshes. Such limitations are particularly relevant in demanding applications such as all-electron Kohn-Sham density functional theory, which place stringent requirements on the accurate resolution of both nuclear singularities and multiple physical scales. In this paper, we present an efficient $h$-adaptive tetrahedral spectral element framework. To address the continuity challenge, we develop an adaptive strategy that combines element orientation alignment with geometric red-green refinement, thereby eliminating the need for algebraic hanging-node constraints while preserving inter-element continuity. Furthermore, an efficient topology-based point-location algorithm is introduced to accelerate interpolation between adaptive meshes. Numerical experiments on Poisson and Laplacian eigenvalue problems confirm the spectral convergence of the proposed method. Applications to all-electron Kohn-Sham equations further demonstrate its capability to accurately resolve nuclear singularities. Moreover, parallel performance studies exhibit excellent scalability, with matrix assembly and adaptivity modules generally achieving speedups above 15 times and the proposed interpolation algorithm attaining speedups ranging from 25 to 35 on 64-core configurations compared to the single-core performance. These results indicate that the proposed framework provides an accurate, robust, and efficient solution for large-scale, high-resolution simulations.

Takens' time-delay embedding theorem provides conditions under which delay-coordinate maps, formed using uniformly-sampled time series of trajectories evolving on attractors of dynamical systems, can faithfully represent the dynamics of the original system. Nonlinear systems can be highly sensitive, and Takens' theorem does not provide guarantees about the stability of time-delay embeddings. In the linear setting, statements about the stability of time-delay embeddings are more tractable and have been proven for delay-coordinate maps with evenly-spaced delays. In many experimental applications, however, time series data may be non-uniformly-sampled, especially in systems with multiple timescales or when using event-based rather than time-based sampling techniques. In this paper, we extend the theorems for the stable linear Takens' embeddings to the setting where the delay-coordinate maps involve unevenly-spaced delays. We pose a conjecture about the rank of generalized Vandermonde matrices that capture the temporal structure of time-delay embeddings. We prove that, provided the conjecture holds, existing theorems about stable linear Takens' embeddings readily extend to unevenly-sampled settings, and the quality of the embedding converges to the same asymptotic bounds when using a large number of delays in the delay-coordinate map, a result which is supported by numerical simulations.

In this paper, we investigate the orbital stability of vortex patches in the two-dimensional incompressible Euler equations, extending the penalized energy variational framework pioneered by Abe and Choi \cite{abe2022stability} for Lamb dipoles. The recent work by Abe, Choi and Jeong \cite{Abe2025StabilityOL} (which removes $L^1$ constraint) and Dong and Luo \cite{Dong2026StabilityOV} (which treats domains lacking scaling or translation invariance) left open the challenge of a unified $L^p$ stability theory without any a priori $L^1$ or $L^p$ bounds on two-dimensional domains. We establish a unified $L^p$ stability theory on three typical two-dimensional domains. These domains are: the half-plane, strips of any width, and domains satisfying a weak finite volume condition. For each domain, we prove that the penalized energy functional admits a minimizer for suitable $p$, and that every such minimizer satisfies the elliptic equation $\omega^{p-1} = \lambda(\psi - W x_2)_+$. Furthermore, we demonstrate that the set of minimizers is orbitally stable under the Eulerian dynamics. The absence of spatial scaling and horizontal translation invariance necessitates novel strategies: on the strip, we refine a concentration-compactness argument to prove strict subadditivity; on weak finite volume domains, we bypass the need for subadditivity by exploiting the inherent decay rate $q$ of the domain to enforce compactness. This work synthesizes the approaches of \cite{abe2022stability}, \cite{Abe2025StabilityOL}, \cite{abe2025existence}, and \cite{Dong2026StabilityOV} into a comprehensive framework, significantly expanding the scope of provably stable vortex structures.

The Union Problem, which has its genesis in the classical Levi problem, asks for a classification of complex manifolds $M$ that can be exhausted by an increasing union of submanifolds $M_j \subset M$ which are all biholomorphic to a fixed domain in $\mathbb C^n$. We explore a quasiconformal variant of this question and seek to classify $n$-Riemannian manifolds $M$ such that each $M_j$ is quasiconformally equivalent to a bounded domain in $\mathbb R^n$. It turns out that this is possible when these quasiconformal equivalences have uniformly bounded dilatations. Using Kiernan's quasiconformal Schwarz lemma when $n=2$ and Ferrand's conformal capacity when $n \geq 3$, we classify a class of $n$-Riemannian manifolds $M$ such that each $M_j$ is $K_j$-quasiconformally equivalent to $\Omega \setminus A$, where $\sup K_j < \infty$ and $\Omega \subset \mathbb R^n$ is a $C^2$-smoothly bounded domain and $A \subset \Omega$ is at most finite. As a consequence, we obtain that Gehring's example of a bounded domain in $\mathbb R^n$ which has $C^1$-smooth boundary everywhere except at a point and is known to be quasiconformally inequivalent to the unit ball in $\mathbb R^n$, possesses the additional property that it cannot even be exhausted by quasiconformal images of the unit ball with uniformly bounded dilatations.

We develop a model of interbank networks with random liquidity shocks. Networks of dilutable debt—e.g., long-term, unsecured—facilitate efficient liquidity transfers: Shocked banks pledge interbank claims as collateral for new senior debt, diluting existing debt. Unlike with non-dilutable debt, indebtedness and connectedness are sources of stability, not fragility. Dilution is thus a “backdoor bail-in” that reallocates losses absent a resolution authority, trigger security, or ex post renegotiation. We uncover a class of networks, “exponential networks,” that implement optimal contingent transfers via plain debt. Yet exponential networks are not pairwise stable, whereas some core–periphery networks are, rationalizing observed interbank structures and their under-insurance against crises.

Agol in his 2014 ICM proceedings article \cite[Question 11]{Agol14} asks whether braid groups are (virtually) RFRS. We answer this positively by showing that pure braid groups are RFRS. As a consequence, several families of Artin groups are virtually RFRS, including those of type $A_n$ (the braid groups), $B_n=C_n$, $\widetilde A_n$, and $\widetilde C_n$. Our results also provide evidence toward the problem of whether braid groups, and more generally Artin groups, are virtually special; see \cite[Problem 9.4]{HagWi10}, \cite[Problem 13.4]{Wise14}.

A non-scale invariant BHP on any open set is obtained for a large class of discontinuous Hunt processes on metric measure spaces that are in weak duality with another Hunt process under suitable conditions in terms of estimates of exit distributions from open sets, and bounds on Green functions and the comparability of the jump density function. These conditions are easy to verify in concrete cases. Several examples are given to illustrate the main results and the new contributions of this paper. Our result in particular establishes the BHP on any open set for any Hunt process associated with a regular symmetric Dirichlet form on a metric measure space having both the strongly local term and the pure-jump term that admits a two-sided heat kernel estimates of the mixture of (sub-)Gaussian and stable-like form, as well as for a wide class of non-symmetric diffusion processes with jumps on $\mathbb R^d.$

We conduct a convergence analysis for the Schrödinger Bridge Time Series (SBTS) data generator. Starting from a regularized formulation in which the data ensemble is mixed with a standard multivariate Gaussian distribution with a prescribed probability, we prove that the Euler-Maruyama discretization converges to the mixed target distribution with half-order convergence rate, provided that the ensemble size and kernel bandwidth are chosen appropriately. We further show that the regularized distribution converges to the original target distribution as the mixing probability tends to zero. The analysis simultaneously accounts for the ensemble approximation error, kernel approximation error, and time-discretization error, and therefore provides a full distributional convergence result for the SBTS generator. Empirically, we further examine the flexibility of the method by replacing the Wiener reference measure with the path measure induced by a more general SDE. The numerical experiments show that the schemes based on both the original Wiener reference measure and the SDE-induced reference measure achieve comparable performance, demonstrating the robustness and stability of the SBTS framework.

We compare two natural Hardy quantizations carried by the unit cosphere bundle of the round two-sphere. Through $S^2\simeq CP^1$, the cosphere bundle is the unit circle bundle of the canonical bundle $O(-2)$, and its Hardy space assembles the section spaces $H^0(CP^1,O(2\ell))$. Through the real-analytic round metric, the imaginary-time exponential map identifies the same cosphere bundle with every Grauert-tube boundary carrying the adapted complex structure, whose Reeb flow is the geodesic flow. Both Hardy spaces are realized in $L^2(SO(3))$, are multiplicity-free under the left action, and select one line in each Peter-Weyl multiplicity space. We compute the normalized overlap of these lines in closed form as $\frac{\sqrt{\binom{2\ell}{\ell}}}{2^\ell}\frac{\sinh^\ell\tau}{\sqrt{P_\ell(\cosh 2\tau)}}$, with $P_\ell$ the Legendre polynomial. The normalized kernel vectors define an explicit equivariant unitary between the two Hardy spaces, and the squared overlaps are the eigenvalues of the positive trace-class operator $\Pi_h\Pi_\tau\Pi_h$. We derive four exact trace series with the Hardy projectors inserted and show that their sum differs from the full flat trace by a distribution whose Abel regularization has cubic growth at every geodesic period. The eigenvalues also give a genus-zero Fredholm determinant of order zero, and truncating the complete large-$\ell$ expansion at any fixed order gives a finite polylogarithmic expression. A Bargmann-Fock calculation identifies the $\ell$-independent prefactor $(2\sinh 2\tau)^{1/4}e^{-\tau/2}$ in the large-$\ell$ asymptotic of the overlap with the Gaussian matrix coefficient of a metaplectic operator comparing the two contact planes.

For a graph $G$, denote by $a(G)$ the number of vertices in the largest induced forest in $G$. The Albertson-Berman conjecture, which had been open since 1979, states that $a(G) \geq \frac{n}{2}$ for every simple planar graph $G$ on $n$ vertices. Although the Albertson-Berman conjecture was recently resolved in the negative by constructing a counterexample with the help of AI, we independently found a counterexample to the Albertson-Berman conjecture without AI and present it in this article. Our counterexample is on $39$ vertices with $a(G)=19$. Finally, we indicate, without a full proof, a variant of this construction with a larger number of vertices, but with a slightly smaller ratio $\frac{a(G)}{n}=\frac{37}{76}$.

A graph $G$ is called Hamiltonian if it possesses a Hamiltonian cycle; and $G$ is called Hamiltonian-connected if it contains a Hamiltonian path between any two distinct vertices. The toughness of a non-complete graph is the minimum ratio of $|S|$ to the number of components of $G-S$ for any cutset $S$. For a given graph $H$, a graph $G$ is called $H$-free if $G$ does not contain $H$ as an induced subgraph. In this paper, for an integer $k\ge 2$, we prove that every $2k$-connected, $1$-tough and $(P_{3}\cup kP_{1})$-free graph is Hamiltonian and every $(2k+1)$-connected $(P_{3}\cup kP_{1})$-free graph with toughness greater than $1$ is Hamiltonian-connected.

Dorbec et al. [SIAM J. Discrete Math., 27 (2013)] conjectured that, for all integers $k\geq1$ and $r\geq3$, every connected $r$-regular graph $G$ of order $n$, other than $K_{r,r}$, satisfies $\gamma_{P,k}(G)\leq n/(r+1)$. After disproving this conjecture, Chen et al.[Graphs Combin., 38 (2022)] proposed a corresponding conjecture for claw-free regular graphs. In this paper, we prove this conjecture: for integers $k\geq\ell\geq1$, every connected claw-free $(k+\ell+1)$-regular graph $G$ of order $n$ satisfies $\gamma_{P,k}(G)\leq n/(k+\ell+2)$, and this bound is tight. Moreover, without the claw-free assumption, we show that, for each fixed integer $k\geq1$, the supremum of $\gamma_{P,k}(G)/\lvert V(G)\rvert$ over all connected $r$-regular graphs $G$ is asymptotic to $(\ln r)/r$ as $r\to\infty$.

We study the Ricci curvature introduced by Lin, Lu, and Yau. A graph is called Ricci-flat if every edge has curvature zero. Lei and Bai classified $5$-regular symmetric Ricci-flat graphs by proving that every such graph is isomorphic to a particular $72$-vertex graph $\RF$, and conjectured that every $5$-regular Ricci-flat graph is either isomorphic to $\RF$ or admits a nontrivial Cartesian product decomposition. In this paper, we disprove this conjecture by constructing an infinite family of connected $5$-regular Ricci-flat graphs, none of which is isomorphic to $\RF$ or admits a nontrivial Cartesian product decomposition. This shows that the conjectured extension of the classification from the symmetric setting to general $5$-regular Ricci-flat graphs fails and that the class of such graphs is substantially richer than previously conjectured. To establish these results, we use an optimal-assignment formulation of Lin–Lu–Yau curvature to verify the Ricci-flatness of the constructed graphs.