Papers

A decoding method is proposed for polar codes under channels with strong phase noise. In contrast to traditional SC decoding, where recursions are performed on scalar soft information such as probabilities, the proposed decoding recurses over probability density functions defined on the phase-noise variable. To make the density-level recursion computationally tractable, we represent each log-density by its spectrum, and derive approximate formulas for the spectrum recursion, therefore, the proposed method is termed log-density spectrum recursion (LDSR). Numerical results are given under strong phase noise and AWGN, showing that a prominent superiority of error performance is achieved by LDSR in comparison with traditional SC and Kalman-based methods.

This paper studies the closures of the double Bruhat cells of symmetric spacess in its equivariant embeddings. We call them large double Bruhat varieties. Our aim is to establish the double Bruhat stratification as a natural setting for extending the geometry of Richardson varieties to symmetric spaces. For the wonderful compactification of the associated adjoint symmetric space, we show their scheme-theoretic intersections with the unique closed orbit are reduced unions of projected Richardson varieties. This connection with Richardson geometry on partial flag varieties supports large double Bruhat varieties as a natural class of subvarieties in symmetric-space embeddings. In positive characteristic, we prove that every equivariant embedding admits a Frobenius splitting compatible with all large double Bruhat varieties and all boundary divisors. These varieties are strongly F-regular, and globally F-regular when the ambient embedding is projective. In characteristic zero, they are of strongly F-regular type, and of globally F-regular type in the projective case; in particular, they are normal, Cohen-Macaulay, and have rational singularities.

The robustness of two-level overlapping additive Schwarz (OAS) preconditioners for highly heterogeneous elliptic problems hinges on the choice of the coarse space. We propose an algebraic multiscale (AM) coarse space built from the system matrix alone through an algebraic formulation of the multiscale finite element method (MsFEM). The coarse basis functions are energy-minimizing extensions of the solutions of reduced boundary condition (RBC) problems on the subdomain interface, so that coefficient variations along the interface enter the coarse space. By posing the RBC problems on slabs that extend the interface, the construction also applies to general unstructured domain decompositions. Embedding the AM formulation in an energy-minimizing framework extends it to linear elasticity, and we show that existing condition number bounds for OAS with MsFEM coarse spaces carry over to the algebraic setting. For problems with high coefficient inclusions on the coarse nodes, the AM iteration count and condition number are unaffected by increasing the contrast. In 3D scalar diffusion with log-normal coefficients, AM needs about as many iterations as the generalized Dryja–Smith–Widlund (GDSW) coarse space, while solving a coarse problem ten times smaller. For linear elasticity, omitting the coarse basis functions associated with the rotational rigid body modes yields a fully algebraic coarse space that still scales well for homogeneous problems.

For a fixed graph $T$ and a fixed nonempty family $\mathcal{F}$ of graphs, let $\operatorname{ex}(G, T, \mathcal{F})$ denote the maximum number of copies of $T$ in an $\mathcal{F}$-free subgraph of a graph $G$. Alon and Shikhelman~[Algorithmica, 2022] conjectured that if $e(T)> 0$ and no member of $\mathcal{F}$ is a subgraph of a blow-up of $T$, then for every $\varepsilon>0$, approximating $\operatorname{ex}(G, T, \mathcal{F})$ within an additive error $n^{v(T)-\varepsilon}$ is NP-hard for an input graph $G$ on $n$ vertices. The previously established general cases for this conjecture include $\operatorname{ex}(G, K_2, \mathcal{F})$ by Alon, Shapira and Sudakov~[Ann. of Math., 2009], and $\operatorname{ex}(G, K_m, K_k)$ by Alon and Shikhelman. In this paper, we confirm this conjecture for every bipartite graph $T$ and every clique $T$. Our reductions use iterative leaf additions, symmetric rooted extensions and random transversals to encode the maximum number of edges in a $(k-1)$-colorable subgraph of $G$, where $k=\min_{F\in \mathcal{F}}\chi(F)$.

For positive integers $k,m,n$ with $\gcd(m,n)=1$, we prove a $(km,kn)$-square paths theorem: the symmetric function $(-1)^{k-1}\frac{[km]_q}{[k]_q}p_k[-MX^{m,n}]\cdot1$ is a weighted sum over $(km,kn)$-preference functions. This proves a corrected form of the conjecture of Iraci, Pagaria, Paolini, and Vanden Wyngaerd and generalizes the square paths theorem of Sergel. We encode preference functions by row strict tableaux and closed walks on a set of states, compare them with parking functions for each value of the touch statistic, and apply Mellit's compositional $(km,kn)$-shuffle theorem. Finally, we establish a fall Delta square theorem conjectured by Iraci, Pagaria, and Paolini.

An isolated quantum system evolves reversibly, yet a small subsystem can lose nearly all information about its initial state. Conserved quantities may retain some of this information, depending on how strongly they overlap with local observables. Here we consider a subsystem in an arbitrary initial state coupled to a maximally mixed bath. We show that the worst-input mean-square equilibration error, maximised over window origins, is bounded from above and below by the mean square of a complete local autocorrelation, with the stationary part subtracted. A correlation record over a finite interval therefore controls the equilibration error in every interval of the same length. The result allows arbitrary degeneracies of energies and gaps. We apply it to two static Hamiltonian families. For fixed spin subsystems in the one-particle sector of a bounded-range periodic clock, fixed-accuracy thermalisation is guaranteed on windows that grow only slightly faster than linearly with system size, despite exponentially small positive energies. A random commuting model with logarithmic interaction range retains an extensive set of conserved observables. Its expected mean-square error is bounded by a finite-size term and an inverse-time term. The inverse-time dependence is unavoidable because of the initial transient. These examples show how local relaxation can coexist with exact conservation laws and a finely spaced spectrum.

Let $C$ be a smooth projective curve of genus $g$, and let $T_{n,L}$ denote the tautological line bundle on its $n$-th symmetric product $C_n$. We prove effective vanishing criteria for Koszul cohomology $K_{p,q}(C_n,T_{n,B};T_{n,L})$, extending two classical theorems of Mark Green. The main ingredient is a weight-one vanishing theorem proved by induction on $n$, using a filtration of the kernel bundle of $T_{n,L}$ to reduce the required cohomological vanishings to lower symmetric products. As a consequence, when $g\geq1$ and $n\geq2$, $T_{n,L}$ satisfies property $(N_p)$ for $\mathrm{deg}(L)\geq(n+1)g+p$. This bound is sharp for elliptic symmetric squares and also gives evidence for a folklore conjecture on adjoint syzygies.

We establish global existence, uniqueness and quantitative stability for one-dimensional pressureless flow governed by \[ \partial_t\rho+\partial_x\bigl(\rho f(u)\bigr)=0, \qquad \partial_t(\rho u)+\partial_x\bigl(\rho u f(u)\bigr)=0, \] where $\rho$ is the mass density, possibly measure-valued, $u$ is the physical velocity, and $f\in C^1(\mathbb R)$, with $f'>0$, determines the transport velocity $f(u)$. The initial mass has finite positive mass and finite second moment, allowing atoms and unbounded support, and the initial physical velocity is bounded. The difficulty at a cluster collision is that mass and momentum conservation determine a mass-weighted average of physical velocity, whereas transport is governed by the nonlinear law $f$ applied to that average. We construct global entropy solutions by a Bregman variational time discretization whose cost is chosen to reproduce the transport velocity determined by this mass-weighted average, coupled with conditional-expectation updates of physical velocity. For every entropy solution satisfying the Oleinik condition on the transport velocity and initial weak continuity of the kinetic energy measure, we establish a Lagrangian representation and prove uniqueness within this class by comparing flows with possibly different cluster intervals. For two such solutions with the same velocity law and common mass, we obtain a $W_\infty$ estimate for the normalized mass measures in terms of the initial position and physical-velocity discrepancies in mass coordinates, whenever the initial $W_\infty$ distance is finite. The Lagrangian evolution has the semigroup property.

Two basic problems in the theory of integral self-affine tiles are to characterize the digit sets that generate translational tiles and to understand whether such tiles admit periodic translational tilings. We give a complete solution to the first problem: for an expanding integer matrix $\mathbf{A}$ and a digit set $\mathcal{D}\subset\mathbb{Z}^n$ with $\#\mathcal{D}=|\det\mathbf{A}|$, the self-affine set $T(\mathbf{A},\mathcal{D})$ is a tile if and only if, for some $N\geq1$, the iterated digit set $\mathcal{D}_{{\mathbf{A}},N}$ is of skew-product form with respect to $\mathbf{A}^N$. As a consequence, every integral self-affine tile admits a fully periodic self-replicating $q$-fold tiling. Moreover, $q$ divides $|T(\mathbf{A},\mathcal{D})|$. In dimension two we prove more: every integral self-affine tile admits a fully periodic tiling. In particular, the periodic tiling conjecture holds for integral self-affine tiles in $\mathbb{R}^2$.

Motivated by a question of Hochman, we investigate whether two deterministic time measures or time sets can be distinguished almost surely from a single planar Brownian image. Using rescaled correlation integrals, we give a criterion for mutual singularity of the laws of the image measures, even when the time measures have the same dimension. For self-similar time sets satisfying the open set condition, we also recover the natural occupation measure from the compact image. Together, these results allow us to distinguish lattice and non-lattice self-similar sets and their natural measures within this class. For a naturally parametrised family with two contractions, with fixed dimension, each parameter can be distinguished from all but finitely many others. If the dimension is sufficiently small, every distinct parameter can be distinguished. The results are formalised in Lean.

Consider the problems of computing $\ell_p$-Lewis weights, Kelly portfolios, D-optimal designs, general equilibria in Fisher markets, and optimal error exponents for channel coding in information theory. All of them can be formulated as optimization problems over the probability simplex, but except for the last one, the associated objective functions are neither Lipschitz continuous nor smooth, rendering existing first-order optimization algorithms and their guarantees not directly applicable. Nevertheless, first-order algorithms for solving these problems do exist in the literature and, perhaps surprisingly, look very similar, even though their interpretations, derivations, and analyses mostly appear unrelated. In this paper, we develop a framework unifying these results that captures their common problem structure through log-homogeneity and a novel second-order condition on the objective function, and identifies their common algorithmic pattern as multiplicative gradient methods. The framework recovers known algorithms and their optimization error guarantees for computing Kelly portfolios, D-optimal designs, optimal error exponents for channel coding in classical communication, $\ell_p$-Lewis weights with $p < 2$, and general equilibria in Fisher markets. To illustrate the insights provided by the framework, we further show that multiplicative gradient methods yield state-of-the-art performance guarantees for computing $\ell_p$-Lewis weights with $p > 2$ and optimal error exponents in classical-quantum communication. Both results are novel to the best of our knowledge.

Let $G$ be a split real semisimple group of adjoint type. We prove that the cellular decomposition of the totally nonnegative part of its wonderful compactification is a regular CW complex, confirming a conjecture of Galashin, Karp, and Lam. In particular, the closure of every cell is homeomorphic to a closed ball. The proof combines a Bruhat atlas compatible with total positivity and a CL-labeling of the augmented face poset. The atlas models the strata locally with open Richardson varieties in the flag variety of a larger Kac-Moody group and identifies their totally nonnegative cells.

Task-oriented communication improves communication efficiency by transmitting semantic information required for downstream tasks, but may incur redundancy with previously transmitted information. To address this issue, we propose a variational directed information bottleneck joint source-channel coding (VDIB-JSCC) framework that transmits only previously unacquired information. Specifically, we use directed information to characterize the causal flow of task-relevant incremental information and theoretically demonstrate its advantages over mutual information and conditional mutual information. Based on this metric, we formulate a directed information bottleneck (DIB) objective that balances task-relevant information preservation and redundant information compression. We further derive a variational formulation, termed VDIB, whose tractable upper bound serves as the training loss. The encoder integrates a Vision Transformer for frame-wise semantic extraction and a Mamba-based causal state-space model for capturing temporal dependencies and extracting incremental information, while the decoder uses a Mamba-based state-space model for task inference. Experiments on CIFAR-10 and CIFAR-100 video streams demonstrate that VDIB-JSCC achieves higher classification accuracy with less transmitted information than state-of-the-art semantic communication methods. These results show that leveraging historical knowledge enables task completion with less information, improving communication efficiency.

We describe Redhill, a Lean formalisation of the major results in Hölzl, Kleine and Stephan's "Improved Lower Bounds for Strong $n$-Conjectures" (arXiv:2409.13439). Design decisions regarding types and data structures are justified as enabling canonical versions of the paper's claims, and from there the most direct proofs of said claims. The formalisation process also uncovered many errors, omissions and optimisation opportunities in the informal proof, all of which are addressed in the final library. Although Redhill is meant to stand alone, a few lemmas were upstreamed to the primary Lean mathematics library mathlib. We discuss the tension between upstreaming and leaving lemmas in separate libraries. Unlike many other contemporary formalisation projects, Redhill was completed without any assistance from large language models (LLMs). We compare our work with LLM-assisted projects, in particular an extension of arXiv:2409.13439 to $n=4$, aiming to establish design and style guidelines that define good Lean formalisations.

We study Fisher information about the amplitude of a known smooth mean shift from the first Dirichlet eigenvalue and spatial responses of a Liouville random medium. The complete response record determines the speed measure and the Gaussian free field, whereas every finite record strictly loses information; an optimal design exists in each finite-dimensional probe space. We quantify this gap as the known Liouville parameter $\gamma$ tends to zero. If the probe space contains an explicit matching probe, the minimum loss equals $\gamma^2$ times a positive compression error for a deterministic second-chaos operator, up to $o(\gamma^2)$, with designs allowed to depend on $\gamma$. After enlarging the probe dictionary, the optimal $m$-response coefficient is asymptotic to $\|Lg\|_{L^1(D)}^2/(8m)$, where $L=-(2\pi)^{-1}\Delta_D$ on the unit disk $D$. A quadratic correction of the readouts attains the same fixed-design loss to second order.

We construct representations of the symmetry algebra and derive the exact spectrum of a three-dimensional oscillator with two spin-$\tfrac12$ degrees of freedom. The Hamiltonian and its first-order scalar and pseudo-scalar integrals were obtained in earlier classifications. Here we determine their action on the complete spectral basis. At fixed total angular momentum, a quadratic algebra acts on the three triplet orbital channels and admits a three-dimensional deformed-oscillator representation, while an overlapping two-state exchange algebra joins the middle triplet to the singlet. These finite modules determine angular–spin multiplicities but do not quantize the radial energy. A separate positive-discrete-series $su(1,1)$ representation of the common radial Hamiltonian generates the energy sequence. We give the finite matrices, Casimir and structure functions, normalized eigenfunctions, radial ladder actions, the exceptional $j=0$ sector, complete energy-shell multiplicities, and the effect of particle interchange.

We establish sharp estimates for the generalized Zalcman functional corresponding to the index $n=2$ for normalized starlike mappings of order $\alpha$ and normalized quasi-convex mappings of type $B$ in the unit ball of a complex Banach space and on the unit polydisk in $\mathbb{C}^n$. We also derive sharp Fekete-Szegö inequalities for normalized starlike mappings of order $\alpha$ and normalized quasi-convex mappings of type $B$ in the unit ball of a complex Banach space. Our results extend classical coefficient estimates to higher-dimensional holomorphic mappings and provide a partial confirmation of the generalized Zalcman conjecture in higher dimensions.

Frobenius morphisms play a crucial role in algebraic geometry and commutative algebra in positive characteristic. In particular, the Kunz theorem, characterizing non-singularity via Frobenius morphisms, is a monumental work and has a relative version, called the Radu–André theorem. In this paper, we extend the Kunz theorem and the Radu–André theorem to logarithmic algebraic geometry in positive characteristic.

In this paper we deal with the celebrated Gamow liquid drop model, namely, for $m>0$, $N\geq 2$ and $\alpha\in (0,N)$, we study the problem \[ \min\Big\{P(\Omega)+ V_\alpha(\Omega) : \Omega\subset \mathbb{R}^N,\;|\Omega|=m\Big\}\,, \] and focus mostly on the nonexistence of minimizers for large masses. First we show that any possible minimizer in the asymptotic regime must satisfy an optimal thickness property. Eventually, in the simplified setting of sets made up as a tiling of cubes, we show that nonexistence occurs for large masses for the whole range of parameters $N\geq 2$ and $\alpha\in (0,N)$.

In this paper, we study derivation based noncommutative geometry of path algebras arising from quivers by identifying a Lie algebra of derivations from which a differential calculus is constructed. We show that properties of the differential calculus are intimately connected with the structure of the quiver. In particular, by introducing the notion of a separating quiver, we find a condition on the quiver which is equivalent to the module of differential forms being a projective bimodule. For such quivers, one can construct bimodule connections, as well as bimodule forms which serve as noncommutative analogues of the Riemannian metric. As an illustration of the above concepts, we present a class of quivers for which one can construct Levi-Civita connections, i.e. torsion free connections compatible with a bimodule form.

We study the query complexity of finding an an $\varepsilon$-stationary points of a one-dimensional non-convex function whose $p$-th derivative is $\beta_p$-Hölder continuous. We determine the optimal complexity for deterministic and randomized algorithms and for oracles that return only derivatives, only function values, or full local information. Randomness provably helps when only derivatives are available, function values help in all cases, and higher-order derivatives never do. Together with known results in higher dimensions, this shows how these benefits depend on the dimension. Our deterministic lower bound also implies that the optimal accuracy exponents of objective-function-free adaptive regularization of order $p$ (OFFAR$p$) in every dimension $d\ge1$ and of deterministic gradient-only trapping in dimension $d=2$.

We study Fourier extension estimates for the truncated cone in $\R^{n+1}$ with weights satisfying an $m$-dimensional ball condition only at radii between $1$ and $R^{1/2}$, where $0<m\le n-1$. Within the discrete induction framework of Du–Li–Wang–Zhang, we obtain a family of bounds by combining multilinear refined Strichartz inequalities, narrow decoupling, and Lorentz rescaling. We also give lightplank, lattice, and hybrid lattice–Knapp examples.

For any nonarchimedean local field $k$, we exhibit the first examples of Zariski-dense, discrete subgroups of special linear groups over $k$, which are not commensurable to any lattice in the points of a semisimple group over a local field, and contain an infinite index subgroup isomorphic to a lattice in the $k$-points of a higher rank $k$-semisimple group. We prove that any Gromov-hyperbolic double of a non-elementary free group along a cyclic subgroup embeds as a quasi-isometrically embedded subgroup of $\mathsf{SL}_5(k')\times \mathsf{SL}_5(k')$, where $k'$ is a discretely valued field. More precisely, $k'$ is the field of rational functions $k(t)$ equipped with the Gauss valuation and $k$ is any nonarchimedean local field of characteristic not equal to $2$ or $3$.

In this paper, we determine the generalized Hamming weight hierarchies of two classes of twisted Reed–Solomon (TRS) codes. For TRS codes with one twist, we first determine all possible values of the generalized Hamming weights. We then give a complete characterization of the weight hierarchy in terms of the minimum distance of the dual code. Explicit conditions are derived for the minimum distances of the code and its dual, which make the whole weight hierarchy explicitly computable. As special cases, the obtained results recover several known characterizations for TRS codes to be MDS, NMDS codes, and $s$-gap AMDS codes. For TRS codes with two twists, we similarly determine the complete generalized Hamming weight hierarchy. In this case, the higher generalized Hamming weights are determined by the first two generalized Hamming weights of the dual code, for which explicit rank conditions are also obtained. In addition, we further determine the subcode support weight distributions of this class of single-twist TRS codes. More precisely, for each $1 \le r \le k$, we derive an explicit formula for the number of $r$-dimensional subcodes with a prescribed support size, thereby giving a complete description of the support distribution of subcodes.

We study weighted and Sobolev estimates for two fractional Schrödinger operators on $\mathbb{R}^n$, where $n\geq3$ and $0<s<1$. For the additive operator $L=(-\Delta)^s+V$, with a nonnegative, nontrivial potential $V$ belonging to a suitable reverse Hölder class, $RH_q$, we establish a critical-radius form inequality and decay estimates for the semigroup mass. These yield a heat-kernel bound combining polynomial off-diagonal decay with a fixed positive exponent of decay relative to the fractional critical radius $\rho_s$. A decomposition at this scale gives weighted strong-type estimates for the semigroup maximal operator and weighted Hardy–Littlewood–Sobolev estimates for $L^{-\alpha/2}$, $0<\alpha s<n$, together with the corresponding endpoint weak-type estimates. The admissible weights satisfy explicit growth restrictions determined by the kernel decay, and the fractional integration order is $\alpha s$. We also consider the spectral operator $Q=(-\Delta+V)^s$. Assuming $V\in RH_{q_1}$ with $n/(1+s)\leq q_1<n$ and classical critical radius $\rho_1(x)\geq1$, we prove that $\nabla Q^{-1/2}$ maps $H^{1-s,p}(\mathbb{R}^n)$ into $L^p(\mathbb{R}^n)$ for $1<p\leq p_0$, where $1/p_0=1/q_1-1/n$. By duality, its adjoint maps $L^{p^{'}}(\mathbb{R}^n)$ into $H^{s-1,p^{'}}(\mathbb{R}^n)$ for $p_0^{'}\leq p^{'}<\infty$.

We study classical local conserved charges obtained after finite-defect reduction of doubled four-dimensional Chern–Simons theory. At a finite simple zero, the inhomogeneous derivative term in the affine gauge transformation induced by the twist function vanishes, so the residue evaluations of the two Gaudin currents transform by the ordinary adjoint action. For each simple ideal, we use the Evans-compatible invariant tensors of the principal Drinfeld–Sokolov hierarchy as external algebraic input, construct local circle integrals, and treat the same-zero, distinct-zero, and mixed sectors separately, proving pairwise Poisson commutativity. Under the explicit assumptions of Stedman's simple-pole/simple-zero boundary Hamiltonian, the periodic trivial-monodromy sector, and boundary weights independent of phase-space variables, the reduced Hamiltonian decomposes term by term into finite-pole, infinity, finite-zero, and residual first-class-constraint contributions, which yields conservation of the constructed charges. We distinguish the finite-defect reduced space, the extended space carrying the residual constraint, and the physical quotient. The result is a conditional existence statement for a classical local Abelian Poisson subalgebra; we make no claim of functional independence, maximality, or Liouville completeness. Extensions to higher-order zeros and nontrivial monodromy sectors are outside the scope of this work.

In this PhD thesis, we take an axiomatic approach to studying multivariate divergences, which are measures of the information content of a tuple of classical or quantum states. These divergences are studied in the context of large-sample and catalytic convertibility of tuples of states. This type of convertibility is characterized by the ordering of an axiomatically defined family of divergences. In this thesis, we identify this family of divergences explicitly for two scenarios, one classical and one quantum. In the classical setting, we show that the relevant divergences are certain generalizations of the bivariate Rényi divergences. Unlike previous studies, we do not require the distributions to have the same support, yielding interesting new divergences. In the quantum setting, we consider pairs of classical-quantum states with pure component states, and find that majorization is characterized by all $\alpha$-$z$ relative entropies for $\alpha<1$. This demonstrates the first operational interpretation of these relative entropies.

Conway's $99$-graph is a strongly regular graph with parameters $(v,k,\lambda,\mu) = (99,14,1,2)$ and whose existence has long been an open problem. We use both a sum-of-squares program and the fact that a strongly regular graph with $\lambda = 1, \mu = 2$ has no diamond or $K_{2,3}$ subgraph to derive a loose upper bound for the number of triangles that such a graph could contain. This bound is less than the number of triangles that the graph's spectrum asserts, implying that it does not exist.

We establish a functional-analytic framework that justifies the LaSalle convergence argument outlined in [van der Schaft, 2026] on port-Hamiltonian primal-dual gradient dynamics for finite-horizon optimal control of linear systems. In addition to strict convexity of the running cost as in \cite{vanderSchaft2026PDGJuly}, our analysis requires global Lipschitz continuity of the cost gradient. This ensures that the nonlinear gradient operator is well-defined and Lipschitz on the entire underlying space. We establish global well-posedness of the optimizer dynamics: every initial value determines a unique global mild solution, and the associated solution operators form a contraction semigroup. Similar to [Gernandtand Schaller, 2025], this is proved using tools from maximal monotone operator theory. Assuming existence of an optimal solution, we combine a dissipation inequality with an infinite-dimensional LaSalle invariance principle to prove strong convergence of every trajectory with a relatively compact positive orbit to the unique optimal state, control, and costate. Moreover, we present a sufficient condition for compactness through uniform control regularity and show that convergence holds for all initial states. Finally, we present an example with a strictly but not strongly convex running cost for which every mild solution converges strongly to the unique equilibrium, while no global exponential estimate uniformly over initial values can be obtained.

A graph is elementary if its edges can be coloured with two colours so that the two edges of every induced path on three vertices receive different colours. Elementary graphs are one of the two basic classes in the Chvátal–Sbihi decomposition of claw-free perfect graphs; they include all line graphs of bipartite multigraphs and all complements of bipartite graphs. We prove that every elementary graph $G$ satisfies $\ch(G)=\chi(G)=\omega(G)$, confirming for this class the conjecture of Gravier and Maffray that claw-free graphs are chromatic-choosable. The proof combines the Maffray–Reed description of elementary graphs as augmentations of line graphs of bipartite multigraphs, Galvin's theorem, an extremal (edge-maximal) choice of counterexample, and a Hall-type lemma describing how list colourings of a complete graph minus a matching can fail.