Papers

We study strict ground-level gaps for autonomous nonlinear Dirac equations on noncompact metric graphs. The relevant level at infinity is the autonomous full-line level \(d_\lambda^\infty=d_{\lambda,\mathbb R}\), since a concentrating sequence escaping along a half-line sees the real line after recentering far from all vertices. We prove that, uniformly for \(\lambda\) in a compact subinterval \(I\Subset(-mc^2,mc^2)\), the presence of a sufficiently long pendant path in the essential reduction of the graph implies \[ d_\lambda(\mathcal G)<d_\lambda^\infty. \] The proof uses a half-line restriction of a symmetric full-line ground state, uniform exponential decay, resolvent localization for massive Dirac-Kirchhoff spectral projections, and a compactness argument for generalized Nehari fibres. We also identify elementary graph classes for which no such gap can hold: pure transmission graphs whose essential reduction is isometric to the full line satisfy \(d_\lambda(\mathcal G)=d_\lambda^\infty\).

Let $\varphi\in L^\infty(\D)$. We study compactness criteria for \(T_\varphi\) on the Bergman space $A^2(\D)$. Axler and Zheng~\cite{AZ1998} established a necessary and sufficient condition for compactness in terms of the Berezin transform. For a general bounded measurable function $\varphi$, however, its Berezin transform $\tilde{\varphi}$ does not readily reveal the intrinsic properties of $\varphi$. Motivated by a characterization in terms of the symbol itself, Zhu~\cite{ZhuSlides} proposed a conjecture on compact Toeplitz operators. In this paper, we characterize compactness of $T_\varphi$ on the unweighted Bergman space in terms of local averages of the symbol. We prove that compactness is equivalent to the vanishing of averages over Bergman disks of any prescribed fixed radius. We also establish an equivalent criterion in terms of Carleson box averages that tend to zero uniformly in the angular variable. Finally, we construct a nonnegative bounded symbol whose Carleson box averages tend to zero at every fixed angle, although the associated Toeplitz operator is not compact.

We prove that every proper subdomain of the Euclidean space equipped with its quasihyperbolic metric is CAT(2). The CAT property leads to a positive resolution of Väisälä's uniqueness, prolongation, quasihyperbolic convexity, and local geodesic conjectures on quasihyperbolic geodesics affirmatively on all dimensions, extending the two-dimensional results by Herron. The sectional curvature bound is optimal as demonstrated by the complement of two points in dimensions at least three. We also show that in any quasihyperbolic domain, quasihyperbolic spheres up to radius $\pi/\sqrt{2}$ are $\mathcal{C}^{1,\frac{1}{2}}$-diffeomorphic to the Euclidean sphere. Consequently, we answer a question by Gehring and Vuorinen on the regularity of such spheres. Similar techniques lead to a positive resolution of Väisälä's conjecture on the Euclidean convexity of quasihyperbolic balls up to radius $\arctan(\sqrt{2})/\sqrt{2}$. We establish that in all dimensions, the quasihyperbolic metric is CAT(0) on convex domains. This leads to a positive answer to a question by Martio and Väisälä on the quasihyperbolic convexity of quasihyperbolic balls on such domains. Finally, we prove that a variable Alexandrov curvature lower bound for a quasihyperbolic domain self-improves to a global lower bound of $-1$ and is equivalent to the concavity of the domain.

Problem definition: We study dynamic pricing of substitutable products with finite, product-specific inventories. Customer substitution couples pricing decisions across products, while the inventory state makes exact dynamic programming intractable at realistic scale. Methodology / results: We develop two MNL-guided policy-learning approaches that replace the dynamic program with a statistical mapping from inventory states to pricing decisions. The first learns prices directly, while the second learns inventory opportunity costs and converts them into prices using the optimal MNL pricing rule. Both policies are trained using decision-focused learning. We also propose an efficient method to generate anticipative customer-choice targets to train our policies. Managerial implications: We conduct an extensive numerical evaluation of our approaches. On small instances for which the optimal dynamic program can be computed, the learned policies achieve average optimality gaps below 0.4%. On larger airline-motivated instances, they consistently improve on the tested revenue-management benchmarks. The comparison between the two architectures also highlights the role of model structure: using the MNL pricing characterization is particularly effective when demand is well described by MNL, while directly learning prices provides greater flexibility under heterogeneous mixed-MNL demand.

We study $p$-absolutely Cesàro bounded operators, with particular emphasis on self-improvement, weighted shifts, and linear dynamics. Our first main result shows that, for $1<p<\infty$, every positive absolutely Cesàro bounded operator on $L^p(\Omega)$ is automatically $p$-absolutely Cesàro bounded. We then characterize $q$-absolute Cesàro boundedness of backward shifts on weighted $\ell^p$-spaces for $q\geq p$ and use this characterization to construct examples exhibiting a wide range of possible growth rates of the powers. In the dynamical direction, we obtain new obstructions to absolute and strong Cesàro boundedness; in particular, no strongly Cesàro bounded operator on a nonzero Banach space is chaotic or upper frequently hypercyclic. We also introduce Cesàro ratio-boundedness and compare it with absolute Cesàro boundedness. Finally, a general Baire-category principle yields a genericity theorem for the failure of $p$-absolute Cesàro boundedness.

We study the Cauchy problem for the Boltzmann-BGK model in the three-dimensional whole space with nontrivial collision frequency $\nu(f)=\rho^\alpha T^{\beta}$, where $\alpha\in(1/3,1]$, $\beta\in[0,1]$, and $3\alpha+2\beta\geq 2$. Existing well-posedness results for such models have largely been confined to near-equilibrium regimes and stationary problems. For nonnegative initial data with finite mass and energy that are sufficiently small in suitable polynomial-weighted $L^{\infty}$ norms, we establish global existence and uniqueness of mild solutions near vacuum. No uniform positive lower bound on the macroscopic density or temperature is imposed. The analysis relies on two complementary mechanisms. First, a phase-space weight invariant along free-transport characteristics yields dispersive estimates for the collision frequency and the gain term. The resulting time-integrable decay controls the nonlinear growth and closes the global weighted estimates. Second, we establish a Lipschitz estimate for the relaxation operator in a weighted $L^1$ space controlling mass and energy, with a Lipschitz constant depending only on weighted upper bounds for the distribution functions. The relaxation operator $\nu\mathcal{M}$ has a more favorable structure for Lipschitz estimates than the local Maxwellian $\mathcal{M}$ alone: the collision frequency compensates for the singular dependence on the macroscopic fields, allowing us to establish Lipschitz continuity without uniform positive lower bounds for the density or temperature. Together, these estimates yield a unique global mild solution with uniform weighted bounds.

Let $X$ be a compact Kähler manifold, let $D=\sum_iD_i$ be a simple normal crossing divisor, and fix $\beta_i\in(0,1)$. We introduce a parabolic structure $TX_*$ on $TX$, given along each $D_i$ by the filtration $0\subset TD_i\subset TX|_{D_i}$ with weight $w_i=1-\beta_i$ on the normal quotient. If $X\setminus D$ carries a Kähler–Einstein metric $\omega$ with cone angle $2\pi\beta_i$ along $D_i$, we show that $TX_*$ is parabolic polystable with respect to $[\omega]$. When the Einstein constant $\lambda$ is non-zero, we introduce parabolic structures on the Tian extension ($\lambda>0$) and on Simpson's canonical Higgs bundle ($\lambda<0$), and prove that these are polystable too. Combined with the Bogomolov–Gieseker inequality for parabolic bundles, this yields a Miyaoka–Yau inequality for these pairs.

We study the computational complexity of fundamental algorithmic problems – membership testing, separation, valid-inequality testing, and linear optimization – over polytopes and cones arising from cooperative games (also known as pseudo-Boolean functions). A central obstacle in the study of such problems is that a general cooperative game on $n$ players requires $2^n$ values, so the input size is $2^n$ for a game with $n$ players, making these computational tasks theoretically trivial. Restricting to $k$-additive games reduces the input size to $O(n^k)$, making such games a natural target for meaningful questions about the existence of efficient algorithms. On the positive side, we give an explicit extended formulation of size $O(n^k)$ for the core of $k$-additive $k$-monotone games, allowing all four problems to be solved by a single polynomial-size linear program – in particular, circumventing the ellipsoid method that is needed when building from earlier tractability results of Deng and Papadimitriou, or of Edmonds. For the cone of $k$-additive $(k{-}1)$-monotone games, we give a complete characterization of its extreme rays and derive the same $O(n^k)$ bound on extension complexity, yielding a geometry-based proof and generalization of a result of Billionnet and Minoux. On the negative side, we show that for $l \leq k-2$ the cone of $k$-additive $l$-monotone games is computationally intractable: membership testing is not in NP (unless NP\,=\,coNP), valid-inequality testing is NP-complete, and extension complexity is at least $1.5^n$. Our hardness results yield, as a special case, a result of Crama and of Gallo and Simone. Furthermore, our hardness results also explain the lack of any good characterization of the extreme rays of the cone of $k$-additive $(k{-}2)$-monotone games.

We propose a deterministic smoothed maximum a posteriori (SMAP) framework for high-dimensional multiple-input multiple-output (MIMO) detection. The discrete constellation prior is approximated by an i.i.d. Gaussian mixture, yielding a differentiable MAP objective that retains the constellation structure while enabling continuous optimization. Unlike sampling-based approaches, SMAP formulates detection as a deterministic optimization problem and can be initialized by the output of an arbitrary detector, allowing it to operate either as a standalone detector or as a refinement stage. For hard detection, we introduce SMAP-KR, which complements the continuous solution with a local $K_R$-neighbor search evaluated according to the original maximum-likelihood metric. Annealed continuation improves robustness for higher-order constellations, while a two-neighbor approximation reduces the cost of evaluating the mixture prior. We further develop a soft-output SMAP method in which a local Gaussian approximation based on the curvature of the smoothed posterior provides symbol probabilities and bit log-likelihood ratios without posterior sampling or explicit counterhypothesis lists. Numerical results for critically loaded MU-MIMO systems show that SMAP-KR provides an increasingly favorable performance-complexity tradeoff as the system dimension grows and can also effectively refine the output of existing detectors. For coded transmission, soft SMAP provides substantial block-error-rate gains over both LMMSE- and K-best-based soft detection.

We give a complete classification of algebraically primitive Teichmüller curves in $\Omega\mathcal M_g(g-1,g-1)^{\mathrm{hyp}}$ for $g>2$. These curves are precisely those generated by the Veech $2p$-gons, where $p=2g+1$ is prime.

Given a smooth projective curve $X$ over $\mathbb{C}$, we show that the twisted character variety of $X$ is equivalent to a component of the character variety of a stacky curve $\mathcal{X}$, constructed by adding an orbifold point to $X$. This allows to interpret the non abelian Hodge correspondence in degree $d$ over $X$ as a statement about the geometry of the Hodge moduli space of $\lambda$-connections on $\mathcal{X}$. Following Shende's approach (see arXiv:1411.4975), we exploit this interpretation to compute the weights of the tautological classes on the cohomology of the twisted character variety, by replacing $\mathcal{X}$ with a simplicial scheme $\Delta_{\mathcal{X}}$ obtained from a triangulation.

We consider the integer version of Alkan–Gale's model on stability in a two-sided market, called the stable generalized allocation one. It is given by a triple $(G,b,C)$, where $G=(V,E)$ is a finite bipartite graph with nonnegative integer capacities $b(e)\in{\mathbb Z}_+$ of edges $e\in E$, and for each vertex (“agent”) $v\in V$, the preferences on the set $E_v$ of its incident edges depend on a choice function $C_v$. The latter acts on the set of vectors in ${\mathbb Z}_+^{E_v}$ bounded by the capacities and obeys the standard axioms of substitutability and size monotonicity. Alkan–Gale's prominent theorem implies that the stability problem in this case always has a stable solution $x\in{\mathbb Z}_+^E$ and, moreover, the set ${\cal S}_{G,b,C}$ of these solutions (“stable generalized allocations”) forms a distributive lattice. However, this lattice is rather intricate to construct and work with, and we wonder whether it can be represented via a “simpler” stability model. Answering this issue, we arrange a sort of splitting techniques to embed ${\cal S}_{G,b,C}$, as a sublattice, in the lattice of stable matchings and, more compactly, in the lattice of stable allocations (as in Baiou–Balinski's stability model). This generalizes Fleiner's result on a detachment in the special case with all-unit capacities. Keywords: stable marriage, stable allocation, choice function, rotation, distributive lattice, poset representation

In this paper, we give an $A_\infty$-enhancement of the triangulated orbit category $(\mathcal{T}/[1])_{tr}$ when $\mathcal{T}$ is an algebraic triangulated category. We then use this enhancement to give a description of $(\text{per}(A)/[1])_{tr}$ in terms of differential modules and Cohen-Macaulay modules when $A$ is a finite dimensional $\mathbb{Z}$-graded algebra of finite global dimension.

In this paper we study conditions for the existence of a geometric quantization of a complete hyperKähler four manifold with an effective tri–Hamiltonian circle action and we show that the vanishing of the third coefficient $a_3$ associated to these metrics is equivalent to flatness.

We study the following general situation. Let $R\to S$ be a finite flat morphism of regular rings of zero (resp. prime) characteristic. For $P\in Spec R$, put $A=R/P, C=S/PS,$ and $B=S/\sqrt{PS}=C_{\mathrm{red}}.$ The basic question is whether a property of $B$ forces the same property of $A$. The main point is that ordinary finite-flat descent applies naturally to $A\to C$, whereas the passage $C\to C_{\mathrm{red}}$ may destroy nilpotent information.

The proliferation of 3D printing raises new security and forensic challenges, including the risk of unauthorized fabrication of untraceable firearms and other regulated items. To enable traceability, we consider single-fragment forensic coding, in which a unique identifier is embedded into a printed object and must be recoverable from any fragment containing an axis-parallel box of prescribed minimum volume and coordinatewise thickness, even in the presence of substitution errors. To address this problem, we introduce and study multidimensional cyclically permutable codes (CPCs), for which every cyclic translate uniquely determines both the original codeword and the applied translation. By applying periodic lifting to a CPC base array, every complete period contained in a fragment corresponds to an unknown cyclic translate of a codeword, thereby reducing the forensic alignment problem to multidimensional cyclic synchronization. We establish theoretical bounds and develop explicit constructions in both the noiseless and substitution-error settings. For fixed dimension $d$ and alphabet size $q$, we obtain the optimal redundancy $d\log_q k+o(1)$ for noiseless $d$-dimensional CPCs of side length $k$. For fixed $t$, the optimal redundancy of $t$-substitution-correcting $d$-dimensional CPCs lies between $(t+1)d\log_q k+O(1)$ and $(2t+1)d\log_q k+O(1)$. For binary alphabets and $d\geq2$, an explicit robust row-anchor construction achieves $(t+1)d\log_2 k+O(\log\log k)$ redundancy, matching the optimal leading term. In the thick-fragment regime $h=cM^{1/d}$, where $M$ and $h$ lower-bound the volume and the side lengths of a box contained in the fragment, periodic lifting yields single-fragment forensic codes of rate $c^d-o(1)$. In particular, the rate approaches one when $c=1-o(1)$.

Let $G$ be a finite group. We prove that the order of the abelianization $G/G'$ divides the order of $H^{2m}(G,\mathbb{Z})$ for every $m\geq1$. As a consequence, for a field $K$, nonvanishing of $Ext^1_{KG}(K,K)$ implies nonvanishing of $Ext^n_{KG}(K,K)$ for every $n\geq1$. This answers a conjecture by Erdmann, Klász and Marczinzik. We use this to show that the Hochschild cohomology of $KG$ is nonzero in every nonnegative degree whenever the characteristic of $K$ divides $|G|$.

Over the past decade, a growing body of research has shown that $\gamma$-weak submodularity broadly arises in numerous subset selection tasks, including feature selection, neural network pruning, and video summarization. Despite its prevalence, maximizing a $\gamma$-weakly submodular function subject to a general matroid constraint remains challenging. To date, the only known approximation guarantee is the conservative $(1+1/\gamma)^{-2}$ factor established by \citet{chen2018weakly}. To improve upon this result, this paper proposes a novel algorithm called \MGPE, which repeatedly performs maximum-gain local exchanges through careful control of a non-homogeneous Poisson clock, and proves that this \MGPE\ can attain an approximation ratio arbitrarily close to $\rho_\gamma=1-\left(\gamma/(2-\gamma)\right)^{ \frac{\gamma^2}{2(1-\gamma)} }$. In sharp contrast to the previous guarantee, our obtained factor $\rho_\gamma$ not only strictly improves upon $(1+1/\gamma)^{-2}$ for every $\gamma\in(0,1]$, but also can asymptotically approach the optimal $(1-1/e)$-approximation for submodular maximization as $\gamma\to1$. Furthermore, we surprisingly find that when the matroid constraint reduces to a cardinality or the objective satisfies the stronger notion of $\alpha$-weak DR-submodularity, \MGPE\ can automatically recover the tight approximation ratios of $1-e^{-\gamma}$ and $1-e^{-\alpha}$, respectively. Here, $\alpha\in(0,1]$ denotes the DR ratio.

We prove that a bounded shortened region initiates decoding throughout multidimensional spatially coupled regular LDPC and MacKay–Neal (MN) codes over the binary erasure channel (BEC). In every fixed finite dimension, uniform hypercube coupling permits the coupling width and shortened region to be chosen independently of the total number $V$ of spatial positions. At fixed widths and dimension $d>1$, replacing a shortened slab by a bounded hypercube reduces the shortening fraction from order $V^{-1/d}$ to order $V^{-1}$, with the same improvement in the shortening term of the check-count rate bound. Regular LDPC codes decode below their uncoupled potential threshold. MN codes achieve capacity for every integer degree choice $\ell>r\geq2$, $g\geq2$: their actual transmitted rates tend to $r/\ell$ and their average bit-erasure probabilities under sum-product decoding vanish below $1-r/\ell$. The proof combines an endpoint-potential identity, removal of an auxiliary constraint, finite-time estimates uniform in direction, and a curvature comparison that transfers flat-boundary progress to expanding balls. For MN codes, elementary inequalities establish fixed-point positivity for all these degrees. Two-dimensional density-evolution examples illustrate the dependence on the initial shortened region; the general sufficient constants are not evaluated numerically.

For every closed oriented smooth four-manifold $M$ admitting a metric of strictly negative sectional curvature, we prove that $\dim_{\mathbb{Q}}H_1(T(M);\mathbb{Q})=\dim_{\mathbb{Q}}H_1(T_{\mathrm{TOP}}(M);\mathbb{Q})=\infty$ where $T(M)$ and $T_{\mathrm{TOP}}(M)$ denote the Torelli groups. This extends a theorem of Budney–Gabai for real hyperbolic manifolds.

We introduce the Endless Exam, a benchmark spanning fourteen parameterised families of mathematical construction problems, with verifiable scores that distinguish progress before and beyond published mathematical frontiers. Each submitted object is checked automatically for validity and assigned a relative quality score against a published frontier or construction baseline, without capping improvements at 1. The benchmark draws long-term challenges from open mathematical problems and generates larger instances by varying their parameters. Compact certificates allow large constructions to be verified without listing every element. Across nine models evaluated on 69 distinct instances, continuous quality scores distinguish performance even though none of the 30 published-frontier references is surpassed. Size-quality curves show how construction quality changes as problem size increases. We release the generators, verifiers, references, model responses and analysis to support continued measurement before and beyond human frontiers.

We prove the Global Attractor Conjecture for complex balanced mass-action reaction networks whose reachable siphons satisfy two structural conditions, allowing multiple linkage classes. The proof proceeds in two steps. A structural condition relating the stoichiometric space to the reactions active within a boundary face ensures that a stoichiometric compatibility class contains at most one boundary equilibrium with any prescribed zero set. Finiteness of the possible zero sets and connectedness of the $\omega$-limit set then imply that any boundary limit set consists of a single equilibrium. To exclude convergence to such an equilibrium, we consider the embedded reaction network obtained by projecting onto the vanishing species and freezing the concentrations of the surviving species at a positive limit. The structural condition guarantees complex balance of this embedded network, while a further condition on its minimal active linkage classes ensures that an explicit Chetaev function is strictly increasing near the boundary. This excludes the boundary point as an accumulation point whenever the surviving concentrations converge. Consequently, every positive trajectory converges to the unique positive equilibrium in its stoichiometric compatibility class. Examples illustrate the hypotheses and their relation to strong endotacticity.

Dirac-harmonic maps are a mathematical version of the supersymmetric non-linear sigma model of quantum field theory. Since they arise as critical points of an unbounded energy functional, it is a challenging task to establish general existence results. Within this manuscript we mainly prove a general existence result based on index theory for manifolds with boundary and apply it in various situations. Due to the presence of a boundary there are additional contributions in the Atiyah-Patodi-Singer index formula allowing for a non-vanishing index, which is the key argument in our proofs. Further explicit examples based on twistor spinors on surfaces are also presented.

We prove interior Hessian estimates in arbitrary dimensions for admissible solutions of Hessian quotient equations with constant right-hand side, when the numerator and denominator differ in degree by one or two. Counterexamples show that this degree range is optimal. The proof combines a new nonlinear comparison function, a doubling maximum principle, and a Pogorelov estimate.

We study affine stochastic Volterra equations on the cone of symmetric positive semidefinite matrices. For scalar kernels acting entrywise on the matrix dynamics, we establish weak existence by exploiting stochastic invariance results for Volterra equations on convex domains and derive a conditional Fourier–Laplace transform formula characterized by matrix-valued Riccati–Volterra equations. As an application, we extend the Gibson–Schwartz commodity model by replacing its variance-covariance structure with a Volterra–Wishart process. The resulting model allows for memory in the variances and for stochastic instantaneous correlation, while retaining affine tractability. Its joint Fourier–Laplace transform admits an exponential-affine representation governed by a matrix Riccati–Volterra equation. While the existence theory considered here excludes kernels that are singular at the origin, shifted fractional kernels remain admissible and provide a tractable specification with power-law memory.

We study finite-time singularity formation for a one-dimensional Euler–Poisson system in Lagrangian coordinates with Cattaneo heat conduction and quadratic heat-flux corrections in the pressure and internal energy. Under the structural choice $\kappa=\kappa_0v$, two different breakdown mechanisms are obtained. First, an explicit affine solution reaches $v=0$ in finite time, so the Eulerian density $\rho=1/v$ diverges while the remaining state variables stay finite at each fixed spatial point. Second, for $a'(1)<0$, the Poisson field is localized near equilibrium by $F=\phi_x/v$, producing a strictly hyperbolic balance law subject to the propagated constraint $F_x=1-v$. We compute the characteristic speeds and an explicit genuine-nonlinearity coefficient and show that genuine nonlinearity fails for all propagating families only when two explicit algebraic degeneracy conditions hold simultaneously. For neutral compactly supported data satisfying the Poisson constraint exactly, a narrow compressive wave generates only an $O(\varepsilon^2\eta)$ zero-speed Poisson mode. A constraint-compatible John–Hörmander–Bärlin bootstrap and a Riccati comparison then yield finite-time gradient catastrophe while the solution remains uniformly close to equilibrium and $F_x$ stays bounded. The two results exhibit density collapse and small-amplitude gradient blow-up as distinct breakdown mechanisms within the same Euler–Poisson–Cattaneo model.

In this paper we prove that there are infinitely many knots that cannot be embedded in the 1-skeletons of the finite iterations of the Sierpinski tetrahedron fractal. We do this by proving that such an embedding induces a sphere decomposition of weight at most 6. There are infinitely many knots with spherewidth greater than this.

We discuss local properties of the classical Fibonacci and Lucas sequences, characterising in both cases those primes~$p$ for which the~$p$-part satisfies the Dold congruences or is realizable. We calculate an exact `repair' multiplier in each case and determine the relative prime densities of the Dold, realizable, almost Dold, and almost realizable prime sets. In that order, the four densities are~$(\frac34,\frac13,1,\frac13)$ for the Lucas sequence and~$(\frac12,\frac12,1,1)$ for the Fibonacci sequence.

We report a proof found by an artificial intelligence system for the following statement: The hyperbolicity cone of every strictly hyperbolic polynomial is spectrahedral. This establishes the generalized Lax conjecture for the generic case, while the full conjecture remains open.

We prove that Chapoton's $q$-Zeta numerator has nonnegative coefficients for the positive-root poset of every finite reduced irreducible crystallographic root system, graded by root height minus one and normalized by Chapoton's prescribed denominator. The same holds for every interval of the root poset with zero adjoined. The key observation is that adjoining a minimum changes the numerator only by the substitution $t\mapsto qt$. The numerator thereby becomes a nonnegative combination of flag $h$-numbers of a lower-semimodular lattice, and a classical labelling of the dual lattice gives a positive descent expansion. To make the coefficients explicit, we construct signed edge-lexicographic (EL) labellings of Ferrers-cell posets with nonincreasing boundaries. They give ballot-word formulas in types $A$, $B$, and $C$, which prove an observation of Chapoton in type $B$, and a coloured path formula in type $D$. For trapezoidal boundaries, one-boundary major-index enumerators evaluate the path sums in Gaussian-binomial form.