Papers

Let $F$ be a rank-two symmetrizable Nahm sum, with any symmetrizer $\operatorname{diag}(d_1,d_2)$, a positive-definite rational matrix and rational shifts, and let $K$ be the field generated by its Nahm point; write $m=d_2/d_1$. We prove that if $q^cF$ is modular, then $[K:\mathbb{Q}]\le 24$. Modularity forces the higher coefficients of the logarithmic radial asymptotic expansion to vanish; we study the resulting polynomial equations through their Newton polygons and a pole-order analysis along the determinant conic, and for $m=1$ show that their common components occur only on one explicit family, whose Nahm point is rational. For equal row sums, Bloch-group torsion gives $[K:\mathbb{Q}]\le 2$. For complementary Nahm coordinates, the bound is $2$ when $m\ne 1$ and $4$ when $m=1$, using the first two asymptotic corrections in the latter case. Several steps of the proofs rely on exact computation.

In this paper, we study favorite sites of one-dimensional transient random walk on $\mathbb{Z}$ beyond the simple case. Denote by $\mathcal{K}(n)$ the set of favorite sites at time $n$. The goal of this paper is twofold. First, we study $\# \mathcal{K}(n)$, the cardinality of $\mathcal{K}(n)$, and show that, under some mild conditions, for any positive integer $k$, $\mathbf{P}(\#\mathcal{K}(n)=k, {\rm i.o.})=1$, and $\limsup_{n\to\infty}\#\mathcal{K}(n)/\log\log n=-1/\log \gamma$, where $\gamma$ is the probability that the random walk never returns to the starting point. Second, we study the escape rate of favorite sites, and give the limsup and liminf behaviors, where the liminf behavior is characterized via an integral test. To the best of our knowledge, this integral test is the first result of this nature for random walks (in non-random environments).

Strong well-filteredness is a refinement of well-filteredness arising naturally in non-Hausdorff topology and domain theory. In this paper, we construct a countable locally compact strong R-space X such that neither its Smyth power space nor its Scott power space is a strong d-space. This example gives negative answers to the questions concerning the preservation of strong well-filteredness and the strong R-space property under power space constructions. Furthermore, for the two-point discrete space \two, the function space (\two,X) is not a strong d-space under any of the pointwise convergence, compact-open, Isbell, core-open or Scott topologies, where the Scott topology is induced by the pointwise order.

Rates of convergence in normal approximation are fundamental to probability and statistics. The theory has evolved from normalized sums to Studentized statistics, smooth functions of sample means, and $U$-statistics, and more generally to symmetric statistics. A central question throughout this development has been to identify the weakest assumptions under which optimal rates of convergence can be established. First, in this paper we address this question by establishing optimal bounds for adjusted normal approximations of symmetric statistics based on independent and identically distributed observations. More precisely, we bound the Kolmogorov error by a very neat sum of the classical third-moment term and the variance of the remainder term. This resolves a long-standing conjecture posed by Bentkus, Jing and Zhou (Ann. Probab. 37 (2009), 2174–2199). Secondly, we establish the optimal Berry-Esseen rate for jackknife Studentized \(U\)-statistics of order two under minimal conditions, i.e., a finite third absolute moment of the first projection and a finite \(5/3\)-absolute moment of the canonical kernel. This resolves a long-standing conjecture in classical probability theory. Our proofs combine signed Prawitz smoothing with a conditional-variance submartingale, polynomial approximation, and a bounded change of measure.

A flag complex $L$ determines a locally CAT(0) cube complex $P_L$ with the links of all vertices isomorphic to $L$. The fundamental group of $P_L$ is the commutator subgroup of the right-angled Coxeter group $W_L$. We observe that an edge contraction $L \rightarrow L/e$ determines a $2$-fold branched cover $P_L \rightarrow P_{L/e}$ (up to a homotopy equivalence). We use this branched cover to give an inequality between the $\mathbb{F}_2$-$L^2$-Betti numbers of $W_L$ and $W_{L/e}$. This reduces an $\mathbb{F}_2$-version of the Singer Conjecture for right-angled Coxeter groups to minimal flag triangulations of even dimensional spheres. We also use this inequality to compute the $L^2$-Betti numbers of $W_L$ when $L$ is a flag triangulation of the projective plane or the torus.

We study large-data continuation below the critical energy for the two-dimensional sphere-valued Schrödinger map \[ \partial_t u=u\times\Delta u, \qquad u:[0,T_+)\times\mathbb{R}^2\to\mathbb{S}^2, \qquad E(u_0)<4\pi. \] Working directly with the original equation through the complex tangential fields $W_m=\partial_m u+\mathrm{i} u\times\partial_m u$ and the ambient connections $\Gamma_\alpha$, we prove frequency-envelope bounds governed by the critical norm $\|\nabla u\|_{\ell^2L^4_{t,x}}$. As consequences, $T_+=\infty$ if $\|\nabla u\|_{\ell^2L^4([0,T_+)\times\mathbb{R}^2)}<\infty$, and also if a terminal one-window high-frequency tail $\mathcal{A}_{\mathrm{win}}(t)$ satisfies $\liminf_{t\uparrow T_+}\mathcal{A}_{\mathrm{win}}(t)<\varepsilon_*$. Thus any finite-time failure must force persistent high-frequency concentration on all sufficiently late critical parabolic windows.

This paper studies finite-terminal continuation and subthreshold global existence for the two-dimensional Landau–Lifshitz–Gilbert equation in the $H_Q^2$ strong-solution class. The proof shows that if a maximal strong solution terminates at a finite time and the full-frequency $L^4_{t,x}$ norm of $\nabla u$ is finite on a terminal time window, then the strong solution can be continued. Furthermore, the solution can still be continued as long as the ultraviolet tail above one fixed dyadic cutoff belongs to $L^4_{t,x}$. As applications, we obtain global strong solutions when $E(u_0)<4\pi$; in the degree-zero sector, this energy threshold can be raised to $8\pi$.

In this paper, we obtain extreme values of Dirichlet $L$-functions on the critical line with a growing prime modulus. Our result yields a larger lower bound for extreme values than that from Soundararajan's classical resonance method. Consequently, we show that one branch of the dichotomy established by Bondarenko, Darbar, Hagen, Heap, and Seip (2023) can be realized for this family of Dirichlet $L$-functions. The proof relies on the resonance method together with precise estimates for GCD sums.

We prove that the Füredi-Hajnal limit and the Stanley-Wilf limit of a uniformly random permutation matrix of order $k$ are at most $\exp\bigl(O(\sqrt{k}(\log k)^{5/2})\bigr)$ with probability tending to one as $k\to\infty$. This improves the bound $\exp\bigl(O(k^{2/3}(\log k)^{7/3}/(\log\log k)^{1/3})\bigr)$ of Cibulka and Kynčl. Together with the lower bound due to Fox, these bounds show that the logarithms of both limits are $k^{1/2+o(1)}$ for almost all permutations.

This article proposes and articulates structure-driven methodology as an emerging cross-domain paradigm. In contrast to the objective-driven paradigm that has dominated inverse-problem solving for over two centuries, structure-driven methodology advocates identifying and exploiting the intrinsic structure of a problem to construct the solution method directly, rather than approaching the solution through iterative search. Drawing on examples from the Liouville-variant Goldbach conjecture, the three-dimensional Kakeya conjecture, Noether's theorem, and the author's nearly four decades of geophysical inversion practice, the article illustrates the common logic of structure-driven thinking in number theory, pure mathematics, physics, and seismic imaging. The author also systematically proposes the Structure-Driven Inversion (SDI) paradigm, comprising Mathematical Structure-Driven Inversion (MSDI) and Physical Structure-Driven Inversion (PSDI). Structure-driven methodology does not negate objective-driven methodology but complements it: objective-driven is a general search strategy, whereas structure-driven is a custom direct strategy. The article concludes that the significance of this shift lies not only in efficiency gains but also in cognitive deepening—when facing complex problems, the first question is no longer “how to solve the problem faster” but “what is the structure of the problem.”

Given a group $G$, the prime graph of $G$, denoted $\Gamma(G)$, is the graph with vertex set the set of primes $p$ such that there exists an element of order $p$ in $G$, and such that there is an edge between primes $p$ and $q$ if and only if there exists an element of order $pq$ in $G$. Let $T$ be nonabelian finite simple group. A $T$-solvable group $G$ is a group such that there exists a finite subnormal series of $G$ such that each factor is either abelian or isomorphic to $T$. In this paper, we continue the classification of the prime graphs of $T$-solvable locally torsion groups and classify prime graphs of locally torsion metanilpotent groups using tools from model theory. We also classify the prime graphs of ultragroups with certain first-order properties and prove results on the number and type of certain classes of groups that can realize a given graph as a prime graph.

Let $\eta_{1},\cdots,\eta_{d}$ be nonconstant one-variable inner functions satisfying $\eta_{i}(0)=0$ for all $1\leq i\leq d$. Denote by $M_{\eta}$ the submodule of the Hardy space $H^{2}(\mathbb{D}^{d})$ generated by the function family $\{\eta_{i}(z_{i})-\eta_{i+1}(z_{i+1})\mid 1\leq i\leq d-1\}$. For the corresponding quotient module $N_{\eta}=H^{2}(\mathbb{D}^{d})\ominus M_{\eta}$, we define the compressed multiplication operators $S_{z_{i}}=P_{N_{\eta}}M_{z_{i}}\vert_{N_{\eta}}$, where $P_{N_{\eta}}$ is the orthogonal projection from $H^{2}(\mathbb{D}^{d})$ onto $N_{\eta}$. In this paper, we characterize the Hilbert-Schmidt property of the self-commutators $[S_{z_{i}}^{\ast},S_{z_{i}}]$ and cross-commutators $[S_{z_{i}}^{\ast},S_{z_{j}}]$ with $i\neq j$. Moreover, explicit formulas for their Hilbert-Schmidt norms are established whenever these commutators are Hilbert-Schmidt.

Assuming a weak form of resolution of singularities in positive characteristic, we show several at-the-characteristic numerical invariants, including Hodge numbers, are given by homomorphisms out of the Grothendieck ring of varieties. Under the same assumption, we obtain two consequences: (i) $\widehat K$-equivalent varieties have the same Hodge numbers, answering a question of de Fernex–Mere; and (ii) universally homeomorphic varieties need not have the same class in the Grothendieck ring of varieties, answering a question of Nicaise–Sebag. In the appendix, we provide unconditional evidence for (i) by proving that birational Calabi–Yau varieties of dimension at most four have the same Hodge numbers.

For a smooth projective $3$-fold $V$ of intermediate Kodaira dimension, we establish Noether-type inequalities relating the Iitaka volume $\operatorname{Ivol}(V)$ and the geometric genus $p_g(V)$. Specifically, for a smooth projective $3$-fold $V$ of Kodaira dimension $2$, we show that $\operatorname{Ivol}(V)\geq \frac{3}{4}p_g(V)-\frac{5}{4};$ while if the Kodaira dimension of $V$ is $1$, then $\operatorname{Ivol}(V)\geq p_g(V)-1.$ Both inequalities are sharp. In the Kodaira dimension $2$ case, we prove that the equality holds only if the bicanonical system of $V$ induces the Iitaka fibration and the canonical system of $V$ is composed with a pencil of surfaces whose general member has Iitaka volume $\frac12$ provided that $p_g(V)\geq 4$.

We develop a rigorous justification for a parameter estimation algorithm which couples continuous data assimilation to generic optimization. For finite dimensional systems, we provide a rigorous justification of an asymptotic approximation of the sensitivity for the underlying modeled dynamical system, prove that the $L^2$ loss function satisfies an approximate Polyak-Lojasiewicz inequality, and use that result to justify convergence of gradient descent for the proposed algorithm. Numerical examples are provided that demonstrate the precision of the rigorous results.

A bounded time-dependent drift together with an arbitrarily small real potential can produce double exponential decay of the full spatial $L^2$ norm. We prove matching quantitative two-time lower bounds for energy solutions of $\partial_tu-\Delta u=vu+w\cdot\nabla u$ on $\mathbb{R}^n$, for $n\geq3$, when $w\in L^\infty_{t,x}$ and either $v$ is sufficiently small in $L^\infty_tL^{n/2}_x$ or $v\in L^\infty_tL^p_x$ for some $n/2<p\leq\infty$, without a smallness assumption. In particular, a nonzero global solution satisfies a double exponential lower bound at large times. This decay scale is optimal for these coefficient classes: In every dimension $n\geq3$, we construct a real bounded drift, a real potential arbitrarily small in the critical space and in any prescribed subcritical $L^p$ space, and a nonzero real energy solution with double exponential decay. The construction localizes Rowan's torus example by absorbing the localization error.

Boolean factor models, including prominent subfamilies such as Boolean matrix decompositions and cognitive diagnosis models, find broad applications ranging from social sciences to engineering. Despite their flexibility, a key challenge lies in establishing the identifiability of their graphical structures, which specify how latent variables influence observed variables. Existing identifiability conditions typically rely on the strong assumption of pure nodes, which may be unrealistic in many applications. We develop a novel approach leveraging the Hasse diagram to represent the distribution of observed variables and transform identifiability into a graph isomorphism challenge. Based on this, we establish sufficient and necessary graphical identifiability conditions that do not require pure nodes. We further derive equivalent algebraic conditions and develop an efficient Boolean satisfiability (SAT)-based verification algorithm. We extend the analysis to probabilistic Boolean factor models, establishing conditions for jointly identifying the graphical structure and additional model parameters without requiring pure nodes. Our results substantially broaden the class of identifiable and interpretable Boolean factor models by removing the pure-node requirement, yielding new theoretical insights, while also providing practitioners with a concrete and easily implementable tool to assess model identifiability.

We prove that, for every $\eps>0$, any $n$-vertex graph that needs at least $\eps n^2$ edge changes to become a Turán graph has positive discrepancy at least $c_\eps n^{5/4}$. Consequently, every such regular graph has second eigenvalue at least $c'_\eps n^{1/4}$. These results prove two conjectures of Räty, Sudakov and Tomon.

Multiparametric optimal control in continuous time partitions the space of initial states into critical regions, each with its own control law involving a set of time-varying active constraints. The boundaries between two neighbouring regions are either hyperplanes, which are given in closed form by one linear equation, or curved boundaries, which must be computed. The sequence of time-varying active constraints plays a key role in defining the linear or nonlinear shape of a critical-region boundary. In this paper we show how to determine that shape prior to any computation. In particular, we show that the boundary is a hyperplane under two conditions. First, the new constraint becomes active at the same instant at every point of the boundary. Second, the durations for which the other constraints stay active are the same at every point of the boundary. Otherwise, the boundary is curved. Both conditions can be read from the sequences of active constraints of the two neighbouring regions, which gives five rules, each yielding a hyperplane or a curved boundary. We illustrate the rules on the Newell-Lee evaporator, where they classify the eight boundaries of a seven-region partition into four hyperplanes, given in closed form, and four curved boundaries, computed exactly.

Inferring an Allen-Cahn constitutive force from images requires more than an accurate optimizer: nonlinear observation noise alters the mean of force features, shared noise couples instruments to the response, and a temporal balance borrowed from another solver can remain biased even on clean data. We develop observation-aware, noise-adjusted moment equations that address these effects before inversion. The surface-tension-calibrated constitutive transformer (SCCT) couples the resulting moments to a positive Bernstein inverse and can adapt its prior and precision from images and moments. A conditional stability bound separates moment residuals from prior error. Synthetic tests show that correcting the estimating equation matters more than increasing regularizer flexibility; the benefit of learned adaptation is smaller and depends on the training comparison. Forces inferred from planar movies predict unseen planar and three-dimensional evolutions. A ring-cage breakup shows why diffuse-interface accuracy distinguishes forces even when their topological predictions agree, while external surface tension sets the energy scale separately.

In this article, we construct, for a broad class of one-dimensional stochastic reaction-diffusion equations, a sequence of spatially-structured individual-based models whose rescaled empirical processes converge to the solution of the equation. The purpose of this article is to show that, whenever the rescaled one-step mean and variance of a reasonable individual-based model converge to those of an SDE with drift $b(x)x(1-x)+\theta^+(1-x)-\theta^-x$ and noise coefficient $\sqrt{\sigma(x)x(1-x)}$, under certain moment conditions there also exists a corresponding multi-island model, composed of independent and identically distributed copies of the individual-based model on each island, with selection and mutation scaled to the diffusive time scale, coupled through migration, which converges to an SPDE with a Laplacian plus the same drift and noise coefficients, provided this SPDE is unique in law.

Despite the remarkable empirical success of flow-matching models, their statistical generalization guarantees remain underdeveloped. Existing analyses often impose restrictive assumptions on the estimated velocity field and yield convergence rates that fail to reflect the intrinsic low-dimensional structure common in real data, such as natural images and molecular geometries. In this work, we study the statistical generalization of flow-matching models for learning an unknown distribution $P_{\mathrm{data}}$ from finitely many samples. We derive finite-sample error bounds on the learned generative distribution, measured in the Wasserstein-$p$ distance, for all $p\geq 1$. Specifically, given $n$ i.i.d. samples from $P_{\mathrm{data}}$, we show that, for every $d>d_p^\ast(P_{\mathrm{data}})$ and appropriately chosen network architectures and hyperparameters, the learned distribution $\widehat{P}^{\mathrm{FM}}$ satisfies $ \mathbb{W}_p(\widehat{P}^{\mathrm{FM}},P_{\mathrm{data}}) \lesssim n^{-1/d}+n^{-1/(2p)}\bigl(\log(1/\xi)\bigr)^{1/(2p)}$ with probability at least $1-\xi$, where $d_p^\ast(P_{\mathrm{data}})$ denotes the Wasserstein-$p$ dimension of the target measure. Our results demonstrate that flow matching naturally adapts to the intrinsic geometry of data and mitigates the curse of dimensionality, as the convergence exponent depends on the intrinsic rather than ambient dimension. These guarantees remain meaningful in high-dimensional regimes and provide a theoretical explanation for the empirical success of flow matching on structured data distributions under substantially more relaxed assumptions than those in existing analyses.

Modern state space models (SSMs), such as Mamba2, provide a compelling alternative to transformers by combining linear-time sequence modeling with recurrent state-space dynamics. However, the behavior of SSMs in distributed learning settings remains poorly understood. In particular, the existing standard federated learning methods are largely architecture-agnostic, and do not account for the stability, selectivity, and state-space parameterization that characterize modern selective SSMs. To address this, we derive architecture-aware gradient and smoothness bounds for single- and multi-layer selective SSMs, and convergence bounds for FedAvg and FedProx, characterizing how recurrent stability, input-dependent discretization, and state projection norms affect federated optimization. We then numerically validate the single-layer bounds on sequences generated by a teacher SSM, using a learner that follows the analyzed recurrence. We use this analysis to formulate expectations about the effects of local training and client heterogeneity, and examine these expectations by comparing nine federated learning algorithms on Mamba2 language modeling across six text domains. These experiments illustrate how SSM-specific bounds can provide a basis for interpreting the behavior of practical federated learning algorithms.

In the French pedagogic tradition of Gaspard Monge, geometry was learned by drawing and handling models: drawing was knowing. The formalism of the early twentieth century placed illustration on the side of intuition, perhaps useful for discovery but kept apart from proof. Yet illustration stayed active in mathematics and provides a model for mathematical knowledge as proofs can be generated by automated systems and the Leiden Declaration asks again what mathematical knowledge is. We trace a persistent tradition of illustration through Hadamard, Coxeter, Conway, Thurston and the Geometry Center, and draw on the philosophy of mathematical practice, including Manders' distinction between exact and coexact claims, De Toffoli's account of diagrams with justificatory force, and Giardino's representational affordances. Using examples from Schwartz's minimal flat torus, early images of the Mandelbrot set, and shader renderings of algebraic starscapes, we argue for the development of rigorous illustration, to help inspire and certify the insights it affords. We close by asking the mathematical community to recognise the role illustration plays in helping develop mathematical research and understanding.

We study a family of aperiodic metric graphs generated by one-dimensional tilings. We prove that, generically, the spectrum of the associated Kirchhoff Laplacian has Lebesgue measure zero and, apart from a discrete set of isolated eigenvalues, is a generalized Cantor set. For a large subclass of these graphs, we further show that isolated eigenvalues are generically absent. This extends several well-known results from the setting of discrete ergodic Schrödinger operators on $\mathbb{Z}$ to a large family of metric graphs. Earlier results of this kind for metric graphs typically assume equilateral edges and exploit symmetries that reduce the analysis to a Sturm-Liouville operator on the real line. By contrast, the graph families considered here need not have equilateral edges or the symmetries needed for such a reduction and therefore require a different analytical approach.

We study compactness of the product of two weighted Hardy primitives into $L^q(u)$, $0<q<\infty$, on the input boundary $1\le p_1,p_2\le\infty$, where at least one input exponent is $1$ or $\infty$. For $L^1\times L^p$, $1<p<\infty$, boundedness implies compactness when $q<p/(p+1)$. For $L^1\times L^1$, the corresponding range is $q<1/2$. At and above these boundaries compactness is characterized by additional active-tail decay, and critical power weights give bounded noncompact examples. The endpoint reductions use cumulative Stieltjes measures with their initial atoms and strict Hardy-Copson brackets. An output-mass allocation argument proves local finite-rank density, including quasi-Banach targets. Applying the localization principles developed for the interior-input regime identifies the essential distance, finite-rank distance and Hausdorff noncompactness radius with the coupled active tail, with the appropriate metric power when $q<1$. For a finite left endpoint, ordinary $L^\infty$ inputs admit an exact scalar transfer completed by linear nodal approximation.

We show that for any $R,\epsilon >0$ and any prime $p\geq 2$, there exists an infinite family of $p$-ary quantum low-density parity-check (QLDPC) codes, rate $R$, checks of weight $O_\epsilon(1)$, and normalized distance at least $\delta_{\mathrm{GV}}(p,R)-\epsilon$. Here, $\delta_{\mathrm{GV}}(p,R)$ denotes the quantum Gilbert-Varshamov (GV) bound for $p$-ary stabilizer codes. In fact, we construct an ensemble of such QLDPC codes, such that a random code from this ensemble is close to the GV bound with high probability. Moreover, this ensemble matches the performance of random stabilizer codes on several quantum channels. Specifically, it approaches the quantum capacity of the erasure channel and the hashing bound for memoryless Pauli channels, including the depolarizing channel. A significant challenge in working with QLDPC codes is that they are necessarily degenerate, i.e., contain many low-weight stabilizers. A key contribution of our work is a construction of QLDPC codes with quantitative control on their degeneracy. These codes are obtained by combining known constructions of asymptotically good QLDPC codes with the expander-based distance amplification procedure of Alon, Edmonds, and Luby [FOCS'95]. Our random ensemble is constructed by starting with these low-degeneracy QLDPC codes near the quantum Singleton bound and concatenating each coordinate with a random inner code. This can be viewed as a quantum analogue of Thommesen's construction, and as an LDPC version of a result of Ouyang, with an appropriately designed outer code.

We identify a common action–density structure for the Hartree and Vlasov–Riesz equations and their coupling. Along free rays, the feedback between nonlinear actions and rescaled densities governs wave phase modulations and kinetic momentum translations. This structure persists under the exchange of sources, providing convergent modified profiles and an explicit recursive construction of the asymptotic corrections. We thereby establish small-data global existence and modified scattering in three spatial dimensions for the Hartree equation, the Vlasov–Riesz equation, and the coupled Vlasov–Hartree system, with interaction potential $\lambda|x|^{-\alpha}$, $\lambda\in\mathbb R$ and $0<\alpha\leq 1/2$. In this strongly long-range regime, the asymptotic action contains finitely many successive corrections beyond its leading term. To our knowledge, these are the first forward modified-scattering results for all three models throughout this range.

We study Scott-product compatibility and sobriety for countable meet-continuous directed-complete partial orders (dcpos). We construct a countable meet-continuous dcpo $\mathcal R$ whose Scott space is well-filtered but not sober, although the Scott topology on every finite power is the product of the factor Scott topologies. The ideals of $\mathcal R$ admit a unique finite-parameter description. We also construct a countable meet-continuous dcpo $\mathbb{H}$ with a greatest element whose Scott space is coherent and well-filtered, but whose Scott square has a topology strictly finer than the ordinary product topology. These constructions answer negatively two questions about countable meet-continuous dcpos. On the positive side, we prove that a meet-continuous $L$-dcpo, meaning a dcpo whose principal ideals are complete lattices, is Scott sober whenever its Scott square carries the ordinary product topology. This answers Jia's core-compactness question affirmatively for $L$-dcpos without any countability assumption. Finally, we characterize well-filteredness of $L$-dcpos by boundedness of closed Rudin sets and formulate related open questions.

The ability to derive the governing equations of a dynamical system from data is essential for prediction and control across many scientific fields. Two challenges, however, are the expansion of candidate terms when reducing higher-order equations to first-order form and noise from estimating derivatives. We propose Higher-Order-KANDy, which addresses both problems by using delay and derivative ("jet") embeddings of the original data with derivative-ordered Kolmogorov-Arnold Networks for Dynamics (KANDy) layers stacked into a machine learning architecture. Compared with SINDy and WSINDy on noisy data, Higher-Order-KANDy learns governing equations without a candidate library (e.g., recovering the full-period $-\sin(u)$ with nothing named in advance), natively for higher-order systems, and holds the correct term structure as noise grows, selecting the exact active set on every seed at noise levels up to 7.5% where an oracle weak-form regression admits a spurious term, though the oracle remains the more accurate coefficient estimator. We leverage the relationship between delay and differential embeddings, along with the Kolmogorov-Arnold representation theorem, to construct a delay-to-jet map and learn it as a Kolmogorov-Arnold network to preserve derivative structure and provide meaningful terms that appear in the governing equation. In our synthetic benchmarks, this construction recovers equations in the presence of coarse sampling and noise.