Papers

Let $A$ be a module-finite algebra over a commutative Noetherian ring. We give a local criterion for the $n$-Gorenstein condition. For such algebras satisfying the Auslander condition, we prove that finite self-injective dimension is symmetric. Under a one-sided local finiteness hypothesis, we further characterize the Auslander condition by a fiberwise form of Iyama's grade bijection and obtain a criterion for the Auslander–Gorenstein property.

For an undirected graph $G$, its Sylvester–Gallai dimension $\text{SGdim}(G)$ is the largest affine dimension of a configuration of distinct real points indexed by $V(G)$ in which every line determined by an edge of $G$ is a special line (contains at least three points). Hence, the classical Sylvester–Gallai theorem can be stated as $\text{SGdim}(K_n)=1$ for the complete graph $K_n$. We initiate the systematic study of this new graph parameter and prove lower bounds for certain graph families (bounded degree, sparse, minor-free) as well as an upper bound for random graphs.

We study stochastic approximation of fixed points of a non-expansive operator $T$ when the oracle samples originate from a continuing Markovian trajectory. A direct block-minibatch implementation of Halpern iteration attains an expected last-iterate residual of order $O(\log N/N)$, but accrues a substantive complexity of $\tilde O(\epsilon^{-5})$ Markovian samples. We therefore introduce a variance-reduced Markovian PAGE-Halpern method whose refresh and same-state difference blocks are analyzed through the Poisson equation. In Hilbert spaces, the cocoercivity of $I-T$ results in an $O(\epsilon^{-3})$ sample complexity. Our main result extends this construction to a general finite-dimensional Banach space. A displacement-level Halpern bound replaces the Hilbert-space potential and yields $\tilde O(\epsilon^{-3})$ sample complexity. We also establish a high-probability guarantee with the same leading accuracy dependence by measuring the estimator in an auxiliary smooth norm.

For Legendrian surfaces in 1-jet spaces, we consider augmentation number invariants that together generalize the ruling polynomial of 1-dimensional Legendrian knots and establish skein relations for these invariants. The main skein relation involves four ways of resolving a Legendrian with a double point: perturbing the double point to a contractible Reeb chord in two different ways, and removing the double point via the two Lagrangian surgeries. We apply the skein relations in examples, and use them to provide an alternate proof of the relation between augmentations of Legendrian 2-weaves and face colorings due to Casals, Murphy, and Sackel.

We prove that the global minimizer of the Ginzburg-Landau energy in the disk of radius $R$ with boundary value $ u(x)=\frac{x}{|x|}$ is the degree-one radial solution of the planar Ginzburg–Landau equation. This gives an affirmative answer to Open Problem~2.2 in Brezis' open-problem list. This is achieved by comparing the radial solution $f$ in the disk with the degree-one radial solution $F$ in the whole plane. Multiplying a disk competitor by $F/f$ enables us to use the known minimality of the whole-plane vortex without changing the boundary trace. The difference of the two energies can be decomposed into Fourier modes. Every nonzero mode is nonnegative, and the zero mode is then handled by a Picone type identity.

We classify nearly Gorenstein monomial ideals over an arbitrary field. In dimension zero, we compute the canonical trace ideal using a graph on the socle monomials. In positive dimension, we prove a dimension bound under unmixedness and a connectedness condition, with weaker assumptions on the canonical trace ideal.

We investigate the residual Eisenstein cohomology of an arbitrary unitary group $U(V)$ attached to an arbitrary quadratic extension of number fields $E/F$. Our focus lies on the contribution of the maximal parabolic $F$-subgroups of $U(V)$, for which we identify the cohomologically relevant poles of Eisenstein series and prove that the resulting residues all survive as non-trivial classes in automorphic cohomology in an explicit degree. To illustrate the range of phenomena involved, we study in detail the case of a unitary group over $F=\mathbb{Q}(\sqrt[3]{2})$ of $F$-rank $3$, for which we explicitly construct cuspidal automorphic representations, which satisfy all the assumptions of our main theorem and hence explicitly construct non-zero residual Eisenstein cohomology classes for this unitary group. The methods used in this construction are paradigmatic however, i.e., generalize to other unitary groups over other ground fields $F$ by the use of base change.

Let $d\ge 1$ be an integer. We study $d$-polynomial convexity of smooth Jordan arcs in terms of their total absolute curvature $\T(K)$. We prove that every $\cC^2$-smooth Jordan arc $K\subset\C$ satisfying $\T(K)\le \frac{d-1}{d}\pi$ is $d$-polynomially convex. This bound is sharp: for every $\tau>\frac{d-1}{d}\pi$, there exists a smooth Jordan arc $K\subset\C$ such that $\T(K)<\tau$ and $K$ is not $d$-polynomially convex. We also show that, for $0<\alpha<\pi$, the circular arc $A_\alpha=\{e^{it}:|t|\le \alpha\}$ is $d$-polynomially convex if and only if $\alpha\le \frac{d-1}{d}\pi$.

In the theory of error correcting codes, list-decoding refers to the following problem. Given a code $C \subseteq \Sigma^N$ and a received word $y \in \Sigma^N$, find all codewords $c \in C$ so that $\delta(c,y) \leq \rho$, where $\delta$ is relative Hamming distance and $\rho \in (0,1)$. Codes that approach the optimal trade-off between the rate $R := \log_{|\Sigma|}(|C|) / N$ and the list-decoding radius $\rho$ are said to achieve capacity.By now, there are constructions of capacity-achieving list-decodable codes with fast near-linear-time list-decoding algorithms, but most existing work has not considered space complexity. In a recent line of work, Cook and Moshkovitz (2024, 2025, 2026) initiated the study of low-space deterministic algorithms for error correcting codes. In particular, in their 2026 paper, they gave a construction of list-decodable codes with deterministic near-linear-time and sublinear space list-decoding algorithms. However, these codes were far from achieving capacity. In this paper, we present list-decodable codes approaching capacity with deterministic time- and space-efficient list-decoding algorithms. More precisely, for any $R \in (0,1)$ and any arbitrarily small constant $\tau > 0$, we present a family of codes $C\subseteq \Sigma^N$ with rate $R$ that are deterministically list-decodable up to radius $\rho = 1 - R - \tau$, in time $N^{1 + \tau}$ and space $N^{\tau}$ with constant output list size and constant alphabet size. Our results can be extended to capacity-achieving list-recoverable codes.

We study nondegenerate equilateral triangles whose vertices are centroids of the $4^n$ order-$n$ floretion tiles. Scaled integer triangular coordinates give exact equilateral completions and, by an obstruction modulo $3$, show that all three vertices have the same tile orientation. In particular, the complete triangle family and its synchronized local cyclic subfamily have respective sizes $|E_n|=\frac{4^n(4^n-1)}{12},\quad |L_n|=\frac{7^n-4^n}{3},$ so the local proportion tends to zero. Our principal new result answers the radial-homogeneity question left open in the preceding revision: for every order, the number of triangles containing a given tile is given by an exact affine formula in its squared scaled centroid radius. Although the two tile orientations have different formulas, their attainable squared radii belong to disjoint residue classes modulo $3$, yielding a single radial-incidence law on the full centroid set. We also determine local and nonlocal vertex incidences and obtain circular-shell and squared-radius moment identities. The unsigned vertex product defines an algebraic product point. By a rigidity theorem of Ma and Zhang, the multiplication-generated triangles are precisely the nontrivial global cyclic orbits; combining this with our geometric shell classification, we show that the product point is the origin exactly when the Euclidean triangle center is the origin. Further results enumerate central-vertex, no-$e$, and main-axis families, derive Fibonacci and finite-state recurrences for product-centered and reflection-restricted classes, and characterize pairwise anticommutation.

We prove a universality theorem for a broad class of two-dimensional gauge theories on compact surfaces. Each admissible conjugation-invariant Lévy process on a compact connected Lie group determines a universality class of lattice gauge theories whose continuum limit is the associated Markovian holonomy process. Our result can be seen as a gauge-theoretic analogue of invariance principles for random walks and Lévy processes. This framework includes the Yang–Mills holonomy process and the standard heat-kernel (Villain), Wilson, and Manton lattice actions. The proof uses state-sum formulas of independent interest, separating action-dependent spectral coefficients from action-independent topological coefficients determined by the surface and the marked ribbon type of the loop configuration.

We study the differential, statistical, and geometrical consequences generated by the geodesic quantum $f$-divergences introduced in [14], which are constructed by interpolating the relative modular operator and the commutant Radon-Nikodym derivative using a geodesic with parameter $t\in [0,1]$. For an invertible state $\rho$ and an operator convex function $f$, we compute the Hessian and obtain an explicit formula for the induced monotone quantum information metric $g_{\rho,t}^{(f)}$. Furthermore, we also compare these metrics with the Petz-Hasegawa metric and find the meaning of the interpolation parameter $t$ in this new geometry. We next show that the interpolation of relative modular operators $\Gamma_t$ in the reference purification of a state $\sigma$ defines a canonical finite binary experiment $(p_t,q_t)$, and introduce a log-likelihood cumulant function $\Psi_{\rho,\sigma}(t,s)$, recovering the Nussbaum-Szkoła distributions at $t=0$ and the Matsumoto construction at $t=1$. Finally, using Busemann functions, we endow this statistical framework with a geometric meaning in the cone of positive operators.

We study Gaussian random dynamical systems with coordinatewise $\tanh$ nonlinearity, where finite precision is modeled by nearest-grid rounding after each step. Gaussian symmetry reduces the dynamics to an exact Markov chain for the normalized squared radius. Rounding makes the origin absorbing, and the total variation distance to the absorbing equilibrium equals the survival probability of the absorption time. At fixed width, we identify the critical gain and prove an absorption cutoff with Gaussian profile as the mesh tends to zero. At fixed precision, global contraction yields a large-dimension absorption cutoff, while positive drift produces metastability. In the supercritical regime, we prove a large-dimension cutoff to a nonzero invariant law and show that, at fixed dimension, its mass near the repelling origin has a power-law asymptotic. All six main results are formalized in Lean 4 on top of Mathlib and independently checked against restatements that import only Mathlib.

Let $d\geq1$, $\Lambda>0$, and $\rho\in\mathbb{R}$, let $\mu=\mathcal{N}(0,\Lambda^{-1}\mathrm{Id})$ on $\mathbb{R}^{d}$, where $\mathrm{Id}$ is the identity matrix, and let $\nu$ be a probability measure whose support is a convex body $K$ of diameter $D=\operatorname{diam}(K)$ and whose density is proportional to $e^{-W}$, where $\mathrm{D}^{2}W\succeq-\rho\mathrm{Id}$. We prove that the Brenier map $T$ from $\mu$ to $\nu$ has Lipschitz constant $\operatorname{Lip}(T)\leq\sqrt{\Lambda/(2\pi)}\int_{-D/2}^{D/2}e^{\rho t^{2}/2}\,\mathrm{d}t$, which is sharp in every fixed dimension. For log-concave targets, the optimal coefficient of $\sqrt{\Lambda}\,D$ is $1/\sqrt{2\pi}$. For sources with finite second moment and density proportional to $e^{-V}$ for which $x\mapsto\frac12\langle Qx,x\rangle-V(x)$ is convex, where $Q$ is symmetric positive definite, and log-concave targets with compact support $K$, the Brenier potential $\Phi$ satisfies $\partial_{vv}\Phi\leq C_{1}\sqrt{\langle Qv,v\rangle}\,w_{K}(v)$ in the sense of distributions, where $w_{K}(v):=\sup_{y\in K}\langle y,v\rangle-\inf_{y\in K}\langle y,v\rangle$ is the width of $K$ along $v\in\mathbb{R}^{d}$ and $C_{1}<0.463$. This removes Kolesnikov's $\sqrt d$ loss. For log-concave sources with $\mathrm{D}^{2}V\preceq\Lambda\mathrm{Id}$ and the target $\nu$, the Brenier map $T$ satisfies $\operatorname{Lip}(T)^{2}\leq\Lambda\int_{0}^{D^{2}/4}e^{\rho t}\,\mathrm{d}t$, so $\operatorname{Lip}(T)$ has the optimal exponential rate $1/8$ in $\rho D^{2}$.

Fix $I = (0,\rho)$, where $0<\rho\leq\infty$, and let $\mathbb{P}_n(I)$ be the set of positive semidefinite $n\times n$ matrices with entries in $I$. A longstanding problem in matrix theory is to characterize the functions $f: I \to \mathbb{R}$ for which the entrywise calculus $f[A] = [f(a_{ij})]_{i,j = 1}^{n}$ preserves positive semidefiniteness for all $A \in \mathbb{P}_n(I)$. We characterize these functions exactly: if $f\in C^{2n-2}(I)$ and $\mathcal{E} = x\frac{d}{dx}$, then this holds if and only if $$ f^{(k)}(x)\geq 0 \quad (0\leq k\leq n-1) \qquad\text{and}\qquad \bigl[\mathcal{E}^{i+j}f(x)\bigr]_{i,j = 0}^{n-1}\succeq 0 $$ for every $x\in I$. Regularization then removes all a priori smoothness: for $n\geq2$, every preserver belongs to $C^{2n-4}(I)$ and the same characterization holds by interpreting the last two derivatives in the sense of distributions. As applications, we recover classical results of FitzGerald–Horn and Vasudeva, and obtain a complete classification of generalized polynomials with prescribed real exponents and arbitrary coefficients. We also determine optimal constants in entrywise domination inequalities under finite regularity, extend the sharp finite-sum thresholds of Belton–Guillot–Khare–Putinar and Khare–Tao to positive mixtures of powers, and answer a question of Khare and Tao by showing that no finite collection of matrices with entries strictly inside $I$ can detect positivity preservation on $\mathbb{P}_n(I)$.

Let $K$ be a number field of degree $D$ with maximal order $\mathcal{O}_K$. We show that under certain conditions on $K$, which in particular are always satisfied if $D$ is odd or if $D \geq 3$ and $K$ is primitive, the set of positive integers $N_K$ that can be expressed as a sum of two units in $\mathcal{O}_K^*$ is a finite effectively computable set. This result partially resolves an open problem posed by Tinková, Yatsyna, and the first author. We illustrate our method by explicitly computing $N_K$ for the smallest totally real quintic field $K$ with Galois group $S_5$.

We prove the constant-one discrete nonlinear Hausdorff-Young inequality. As a consequence, by a discrete-to-continuous limiting argument, we obtain $$ \|(\log|a_f|^{2})^{1/2}\|_{L^{p'}(\mathbb{R})} \le \|f\|_{L^{p}(\mathbb{R})},\quad 1\le p<2, $$ for all $f\in L^{p}(\mathbb{R})$, where $a_f$ denotes the transmission coefficient of the nonlinear Fourier transform. In particular, this resolves the Muscalu-Tao-Thiele uniformity problem. The proof uncovers a hidden Hilbert-space structure that reduces the nonlinear inequality to classical interpolation.

We study optimization problems with noisy zeroth-order objective observations and deterministic nonlinear equality constraints with available derivatives. We propose a constrained variant of the adaptive-sampling trust-region derivative-free optimization algorithm—ASTRO-DF. The method builds quadratic local models from estimated objective values at interpolation points within a moving trust region and promotes feasibility through a Byrd–Omojokun composite-step based on linearized constraints, following an SQP-like framework. We prove almost sure convergence using a new constrained criticality test and present numerical results on an equality-constrained stochastic activity network problem.

We prove that algebraic values of bounded degree and polynomially bounded height are sparse on rational translates of the graph of a transcendental entire function. More precisely, for fixed $\theta\in\overline{\mathbb{Q}}\cap\mathbb{R}$, $D\geq1$, $t>0$, and for every $\varepsilon>0$, only $O(Q^{\varepsilon})$ rationals $r\in[0,1]$ of height at most $Q$ can satisfy simultaneously $[\mathbb{Q}(f(\theta+r)):\mathbb{Q}]\leq D$ and $H(f(\theta+r))\ll H(r)^t$. Consequently, if these bounds hold for every rational $r$ of sufficiently large height, then $f\in\overline{\mathbb{Q}}[z]$ and $\deg f\leq t$. As applications, we obtain rigidity results for entire functions taking rational or bounded-degree algebraic values with polynomially controlled arithmetic height. In particular, this excludes the polynomial-denominator scenario that arises naturally in connection with Mahler's problem on Liouville numbers. The proof combines Pila's bounded-degree counting theorem with standard height estimates and a simple geometric analysis of transcendental entire graphs.

We introduce monoidal su-categories, an abstract categorical notion of single-input higher-order process over a monoidal category. The definition separates a base category C of lower-order processes from a monoidal category V of holes or supermaps and axiomatizes the compatibility needed for partial application to bipartite processes. For a fixed monoidal base C, monoidal su-categories, monoidal su-functors, and monoidal su-natural transformations form a 2-category MonSuCatC. We then show that the category Optic[C] of coend optics is 2-initial in this 2-category, giving an alternative universal-property characterisation of coend optics as the minimal monoidal theory of single-hole contexts.

We extend the $\mathsf{pmaj}$ statistic of Loehr and Remmel to labelled Dyck paths in the $n \times kn$ grid, and generalize their bijection sending the bistatistic $(\mathsf{dinv},\mathsf{area})$ to $(\mathsf{area}, \mathsf{pmaj})$, proving in this way a new combinatorial formula for $\nabla^k e_n$ ($k \geq 1$). At $k = 1$ we recover the original statistic and the original bijection. Moreover, we provide an explicit description of the recurrent configurations of the sandpile model on a family of graphs $G_{\mu, \nu}^{(k)}$, indexed by an integer $k \geq 1$ and two compositions $\mu$ and $\nu$: at $k = 1$ these are the clique-independent graphs of D'Adderio et al. Finally, we define a $\mathsf{delay}$ statistic on these configurations, and we show that, together with the usual level statistic, it can be used to provide a new combinatorial interpretation of the polynomials $\langle \nabla^k e_n,e_\mu h_\nu \rangle$ from the $(n,kn)$-shuffle theorem. At $k = 1$ we recover the main results of D'Adderio et al.

We prove time-uniform self-normalised Bennett inequalities for a martingale $(M_n)_{n\geq0}$ in a separable Hilbert space, with $M_0=0$ and increments bounded in norm by one. Writing $V_n$ for its predictable covariance process and $h(u)=(1+u)\log(1+u)-u$ for the Bennett rate function, we show that, for every regularisation parameter $\rho>0$, the process \[\exp\left\{\rho\,h\!\left(\frac{\left\lVert (V_n+\rho I)^{-1/2}M_n\right\rVert}{\sqrt{\rho}}\right)-\frac12\log\det\left(I+\rho^{-1}V_n\right)\right\},\qquad n\geq0,\] is a nonnegative supermartingale with initial value one, where $\det$ is the Fredholm determinant. Ville's inequality yields time-uniform Bennett and Bernstein bounds for $\lVert (V_n+\rho I)^{-1/2}M_n\rVert$. The result permits conditional covariance increments of infinite rank; in finite dimensions, the resulting boundaries sharpen existing martingale-transform and determinant-based variational bounds. The same construction extends to compensated marked point processes with bounded jumps. Mixing these supermartingales over $\rho$ gives simultaneous control over the regularisation parameter. Consequences include an upper law of the iterated logarithm for the regularised ellipsoidal radius in separable Hilbert spaces and, in finite dimensions, spectrum-sensitive finite-time bounds and an upper law of the iterated logarithm for the unregularised radius $\lVert V_n^{-1/2}M_n\rVert$, whose constant $1$ is sharp over the class. We also obtain a time-uniform Bernstein inequality for the martingale norm $\lVert M_n\rVert$ with dependence on $\operatorname{tr}(V_n)$ and $\lVert V_n\rVert_{\mathrm{op}}$.

An $n$-tuple of positive integers is $G$-harmonic if there are subgroups of $G$ having those indices whose cosets can be chosen pairwise disjoint, and $\mathbb{Z}$-harmonic if there are pairwise disjoint residue classes with those moduli. Ginosar asked whether every $G$-harmonic tuple is $\mathbb{Z}$-harmonic. Margolis and Schnabel proved this for tuples of length at most $4$, and analysed a particular family of length-$5$ tuples that would yield a counterexample if any member were $G$-harmonic. We show that the bound $4$ is sharp: $(6,6,6,10,15)$ is $A_5$-harmonic but not $\mathbb{Z}$-harmonic. Moreover, the five pairwise disjoint cosets realising this tuple can be extended to a coset partition of $A_5$ using only cosets of indices $6$, $10$, and $15$. The index tuple of this partition is not $\mathbb{Z}$-harmonic; because its indices repeat, this does not contradict the Herzog–Schönheim conjecture. We also prove that no member of the length-$5$ family analysed by Margolis and Schnabel in connection with possible counterexamples is $G$-harmonic for any group $G$.

We prove that every unimodular locally compact group $G$ of noncompact Lie dimension $n$ satisfies the sharp Brunn–Minkowski inequality \[ \mu_G(XY)^{1/n}\ge\mu_G(X)^{1/n}+\mu_G(Y)^{1/n}, \] and establish a general form for arbitrary, possibly nonunimodular, locally compact groups. This fully confirms the nonabelian Brunn–Minkowski conjecture proposed by the present authors and Zhang. As an application, we obtain an isoperimetric inequality on symmetric spaces of noncompact type.

We study the Hausdorff dimension of the difference sets between the set of typical numbers, defined via the Erdős–Rényi law governing the longest run of digits, and the set of numbers normal in base $2$. Although both properties hold for Lebesgue-almost every real number, neither implies the other. The resulting difference sets $\textit{TpcN}\setminus\textit{Normal}$ and $\textit{Normal}\setminus\textit{TpcN}$ are $D_2({\bf \Pi}_3^0)$-complete in the Borel hierarchy. We show that this logical complexity is matched geometrically: both difference sets have full Hausdorff dimension. Unlike the classical Besicovitch-type sets governing single-digit frequencies, normality requires the correct frequency of every finite block simultaneously, and the explicit Moran-set constructions that succeed for simple normality break down under this requirement. We instead establish the dimension via a counting argument bypassing the need for an explicit construction.

We give a random-bit-efficient construction for the inverse star discrepancy. For every fixed $u\in(0,1)$, $k$-wise independent uniform points $\boldsymbol{X}_1,\ldots,\boldsymbol{X}_N$ with $k=O(d(1+\log(1+N/d)))$ satisfy the Monte Carlo bound $D_N^*(\boldsymbol{X}_1,\ldots,\boldsymbol{X}_N) =O(\sqrt{d/N})$ with probability at least $u$. Consequently, $N=O(d\varepsilon^{-2})$ and $k=O(d(1+\log\varepsilon^{-1}))$ suffice to attain discrepancy at most $\varepsilon$. The proof isolates the finitely many moments required by a chaining argument and gives explicit constants. A random vector-valued polynomial over a finite field realizes the required bounded independence on a grid using $O(d^2(1+\log(1+N/d))\log N)$ random bits, rather than the $\Theta(dN\log(dN))$ bits used by independent grid sampling.

We present a combinatorial proof of Simonis type MacWilliams identity for harmonic higher weight distributions of linear codes. Furthermore, we investigate the statistical moments of the harmonic higher weight enumerators for random linear codes. Defining the enumerators via rank functions of the generator matrices of linear codes, we prove that its expectation vanishes for all non-trivial harmonic functions due to the inherent symmetry of random matrices, and we also derive an explicit, non-trivial formula for the covariance.

We recall definitions of linking numbers and Wu–Simon numbers for spatial graphs. We expose a `converse' to the Conway–Gordon–Sachs theorem (i.e. description of linking functions for embeddings $K_6\to\mathbb{R}^3$), and some results on Wu–Simon numbers. We conjecture and discuss a generalization of the Conway–Gordon–Sachs theorem to multiple linking. The exposition is based on plane diagrams, so no knowledge of spatial geometry is required.

In arXiv:1405.4105 it was asked whether an affine Hopf algebra with finite Gelfand-Kirillov dimension is necessarily Noetherian. It is well-known that the converse is not true –take the group algebra of a polycyclic group which is not nilpotent-by-finite. However, it was conjectured in arXiv:2301.04428 that a pointed affine Noetherian Hopf algebra whose group of group-likes is nilpotent-by-finite necessarily has finite Gelfand-Kirillov dimension. In the present paper we conjecture that a post-Nichols algebra over a polycyclic-by-finite group is Noetherian if and only if it is affine and has finite Gelfand-Kirillov dimension. Partial results supporting this conjecture are presented; and their consequences for the preceding questions and conjectures is analyzed.

Let $G$ be a graph together with a total order $\prec$ on its edges. We say that $\prec$ is realizable in $\mathbb{R}^d$ if there is a placement of the vertices of $G$ in $\mathbb{R}^d$ such that the Euclidean lengths of the edges induce exactly the order $\prec$. Almendra-Hernández and Martínez-Sandoval proved that every total order on the edges of the complete graph $K_n$ is realizable in $\mathbb{R}^{n-2}$. We show that the same is not true for the disjoint union of two complete graphs: for every $n\geq 3$ there is a total order on the edges of $K_n\sqcup K_n$ that is not realizable in $\mathbb{R}^{n-2}$, but is in $\mathbb{R}^{n-1}$. Surprisingly, the realizability of an order on a disconnected graph is not determined by its restrictions to the connected components. We also study realizability on the real line: we characterize which disjoint unions of two cycles are realizable, and estimate the largest number of edges an $n$-vertex graph can have while all of its edge-orders remain realizable on the line. In general dimension, we show that the largest number of edges of an $n$-vertex graph all of whose edge-orders are realizable in $\mathbb{R}^d$ is $dn+O\!\left(dn/\ln(dn)\right)$.