Papers

Bayesian Additive Regression Trees (BART) have shown state-of-the-art performance in both prediction and causal inference problems. Previous theoretical work has attempted to explain BART's superior performance by establishing posterior contraction rates for standard BART models, but these rates depend strongly on the number of covariates. Here, we take a different approach and study the behavior of BART as the number of trees grows towards infinity. We show that in this regime, BART converges to a Gaussian process (GP) with a particular kernel. The kernel and its corresponding reproducing kernel Hilbert space (RKHS) have favorable inferential properties that help explain BART's excellent performance. We introduce random tree features as an approximation to this limiting GP, and establish minimax-optimal learning rates for ridge regression on these random features that depend only logarithmically on dimension. In addition to providing insight into the empirical success of BART, random tree features offer a computational benefit over traditional MCMC estimation. The random-features approximation also allows practitioners to easily incorporate BART into any model which has a linear predictor, expanding the applicability and flexibility of BART.

For a plurisubharmonic function $\varphi$ on $\Omega\subset \mathbb C^n$ we defined $\alpha$-Hölder integrability exponent $c_{\alpha}(\varphi)$ and obtained a lower bound in terms of Lelong number $\nu(\varphi)$ and intersection numbers $e_j(\varphi)$. This lower bound improves the earlier result of Kaufmann \cite{Kf} on the integrability of plurisubharmonic functions with respect to Monge-Ampére measure with $\alpha$-Hölder continuous potentials. We also give a sharp lower bound for $\alpha$-Hölder integrability exponent of certain type of plurisubharmonic functions.

Complexity one $T$-spaces are Hamiltonian $T$-spaces $(M,\omega,\Phi)$ such that $\frac{1}{2}\dim M -\dim T=1$. The skeleton of a complexity one $T$-space is an important invariant in the classification and encodes the information about non-generic orbits. In this paper, we prove that the moment image of the skeleton of a compact, connected maximally short complexity one $T$-space, which is in fact a GKM space, is connected. The proof relies on the well-known fact that each connected component of regular values of a proper moment map is a convex locally polyhedral set. We also gave an elementary proof of that fact along the way. Then we use the connectedness result to estimate the number of symplectic toric $(T \times S^1)$-manifolds whose underlying complexity one $T$-space is the same as the given maximally short complexity one $T$-space.

The Watts-Strogatz random graph model on $n$ vertices with parameters $K$ (a positive even integer) and $p \in [0, 1]$ is constructed in two steps. First, one starts with a ring lattice on $n$ vertices, where each vertex is connected to its $K/2$ nearest neighbors on each side. Each edge in turn is then independently rewired with probability $p$ by replacing one endpoint with a uniformly chosen vertex not already adjacent to it. We study the empirical eigenvalue distribution of the adjacency matrix for this model, whose entries are highly dependent due to the rewiring construction. In the regime where both $K$ and $pK$ grow to infinity with the vertex size $n$, we show that, after appropriate scaling, the empirical eigenvalue distribution converges to the semicircle law. The proof is based on a novel coupling argument that approximates the adjacency matrix by a sum of two independent random matrices, one a sparse Wigner matrix and the other a random band matrix. In the case where $K$ and $p$ remain fixed, we propose conjectural formulas for the first five moments of the limiting eigenvalue distribution. These conjectures are supported by a convergence result relating the Watts-Strogatz model to another random graph model, together with numerical simulations.

We present three results on the multiset dimension of graphs, resolving one conjecture and two open questions from the literature. First, we disprove the conjecture of Simanjuntak, Siagian and Vetrík (2017) that every graph $G$ of order $n(G)$ with finite multiset dimension satisfies $\dim_m(G) \le n(G)-1$: an exhaustive computation over all 1,018,690,328 connected graphs of orders 2 through 11 shows that exactly eight graphs attain $\dim_m(G)=n(G)$, all of order 11, so 11 is the smallest order at which the trivial upper bound is attained. This also answers a question from the recent survey of Farhan, Klavžar, Kuziak and Yero. Second, we prove that $\dim_m(P_n \boxtimes P_n)=4$ for every $n \ge 5$, answering a question of Hakanen and Yero: after a $45^\circ$ change of coordinates the Chebyshev metric of the king grid becomes half the Manhattan metric on a parity sublattice, and four boundary inequalities reduce every potentially resolving three-landmark set to two geometric cases, in each of which we exhibit an explicit collision. Third, on king strips the parameter grows linearly: $\dim_m(P_3 \boxtimes P_n)=n$ for $n \ge 6$ (with the small cases determined exactly), where the lower bound rests on three local separation conditions and a finite min-plus transfer certificate whose equality case yields a finite automaton with a 19-state recurrent core, and the upper bound is an explicit landmark pattern of period three that works for every height. Combined with a blindness lower bound, $\dim_m(P_h \boxtimes P_n) = \Theta(n)$ for every fixed $h \ge 3$, so the constant answer on square king grids requires both dimensions to grow.

Freed and Hopkins developed an ansatz for classifying interacting SPT phases using invertible field theories with a natural Free to Interacting (FTI) map from free fermion phases. This ansatz has been generalized to include crystalline phases and a crystalline equivalence principle (CEP). However, motivated by failure of the CEP for weak free fermions and the FTI, here we generalize the original Freed and Hopkins ansatz to a fully equivariant version for symmorphic crystallographic symmetries and show there is a natural equivariant FTI map from symmorphic crystalline weak free fermions. We further discuss why this equivariant ansatz is both mathematically and physically more natural than the spatial symmetry extension by Freed and Hopkins and why full equivariance should hold over the CEP.

We prove that the dimer constant of the cubic lattice is bounded above by $0.452130$, improving the previous best upper bound of $0.457547$ obtained by Lundow (2001) using the transfer-matrix method of Ciucu (1998). Our construction instead uses diagonal transfer layers and an inequality of Csikvári (2017) to relate the spectral radius of the resulting matrix to the dimer constant, rather than exploiting symmetry special to the rectangular torus. We reduce the dimension of the transfer matrix using a compression method similar to Lundow's. Finally, we use a variant of the Collatz–Wielandt formula, derived from an asymptotic result of Friedland–Schneider (1980), to obtain a monotone sequence of upper bounds converging to the true spectral radius, each of which can be computed without forming the full matrix.

In finite-volume methods on embedded-boundary meshes, arbitrarily small cut cells can produce coefficients that scale as O(alpha^{-1}) when the time step is chosen for the regular grid. For a one-dimensional periodic upwind discretization, we show that the resulting instability is carried by an eigenmode concentrated at the cut cell. Blending the unstabilized update with the volume-weighted state obtained by merging the cut cell with its left neighbor yields the leading redistribution threshold s_0(L)=(L-2)/(L-1) for fixed cut-cell Courant number L>2. We prove that the cell-merging correction aligns with the unstable mode as alpha tends to zero and establish a block-matrix result showing that diverging cut-cell rows generate a finite cluster of unbounded eigenvalues whose invariant subspace approaches the span of the cut-cell coordinates. For MUSCL with a minmod limiter, analysis of the piecewise-linear region containing the localized mode gives s_M(alpha,L)=(L-2)/(L-1)-L(L-2)alpha/[4(L-1)^2]+O(alpha^2), with the SSPRK2 update reaching unit amplification at the same crossing on that branch. Numerical tests show lower error with this reduced redistribution than with the tested published weighting specializations and full two-cell merging. A two-dimensional 45-degree embedded-boundary channel likewise reaches spectral radius at most one with blending parameters below full merging. These results show that full redistribution is not always necessary to control the unstable cut-cell mode, and that using a smaller blending parameter can reduce numerical error in the tested problems.

The symmetric Dyson exclusion process (SDEP) is an exclusion process on the lattice with a long-range logarithmic Coulomb-type interaction. It appears in several equivalent forms: as symmetric random walkers conditioned, in the Doob-transform sense, not to collide; as the maximal-activity limit of a conditioned SSEP; and as a ground-state transform of the spin- 1/2 XX chain. In this work, we exploit this latter representation to obtain exact evolution formulas from deterministic initial configurations. The resulting determinantal kernel is expressed through a finite interpolation expression in terms of Lagrange polynomials, leading to explicit density evolution in finite and infinite lattices. For the melting of a densely packed block, we show that the full hierarchy of density moments is governed by a finite-dimensional polynomial algebra, and we identify Catalan numbers in the leading time coefficients. We also derive the Euler-scale density profile and the arctic curve separating frozen and liquid regions, and show that they coincide with conjectured hydrodynamic results obtained in a previous work.

We investigate generalized Favard lengths associated to smooth families of nonlinear projections. Under suitable regularity and transversality assumptions, we prove that generalized projections are locally comparable to orthogonal projections on sufficiently small scales. This yields a comparison principle that transfers quantitative upper bounds for classical Favard length to broad classes of nonlinear projection families. As a consequence, known upper bounds for the Favard length of purely unrectifiable self-similar 1-sets yield corresponding upper bounds for their generalized Favard lengths. We also prove that the union of circles with centers in a purely unrectifiable self-similar 1-set has Lebesgue measure zero whenever the radii vary sufficiently slowly. More generally, the same method yields measure estimates for unions of curves arising from suitable level-set families.

Mathematics is often valued for its role as the language of science, but it is equally important as a human practice that teaches us how to think and cultivates creativity, problem-solving skills, attention, and judgment. The rapid rise of modern AI systems, which can solve many standard problems and produce fluent mathematical explanations, forces a reconsideration of the nature of mathematical understanding and how it is developed. This article reflects on the impact of AI across three intertwined domains: mathematics education, mathematical research, and the broader role of mathematics as a way of learning to think. In education, AI changes how students interact with mathematical material, making high-quality assistance widely available while weakening the traditional link between assigned work and demonstrated understanding. The challenge concerns not only academic integrity but also the risks of cognitive offloading and the illusion of explanatory depth. In research, AI can accelerate exploration in various ways, including generating conjectures and examples, performing symbolic manipulation, suggesting proofs, and assisting in the communication of ideas. It has also begun to contribute to discoveries on problems that had resisted human efforts. Yet its limitations remain serious, especially in long arguments requiring global structure, careful verification, and the synthesis of multiple ideas. The central claim is that in an era when routine tasks become easier to automate, the most human aspects of mathematics, including conceptual insight, problem framing, genuine understanding, and creative judgment, become more visible and more valuable.

In \cite{GNS} we proved that, for every $\alpha>0$, the potential of three positive point charges has at most $12$ nondegenerate equilibrium points. We also observed that the same method would give the sharper bound $6$ if a certain auxiliary polynomial system $Q=R=0$ had at least four solutions, counted with multiplicity, in each open quadrant of the $(f,g)$-plane. Here we prove this four-solution statement. The main new ingredient is a separation argument at the unique saddle point of a separated-variable first integral. Consequently, the upper bound for three charges improves from $12$ to $6$.

Let $Y$ be a cubic 7-fold. When $Y$ is smooth, its Kuznetsov component is a Calabi-Yau category of dimension 3, and we construct an embedding of the Kuznetsov component into the derived category of Clifford modules over $\mathbb{P}^4$. When $Y$ is a general cubic 7-fold singular along a line, we construct an explicit weakly crepant categorical resolution of its Kuznetsov component by the derived category of a smooth Calabi-Yau 3-fold, recovering a result of Favero-Kelly. In this construction, we express the resolution functor explicitly as a composition of geometric functors and mutations, and identify its kernel with a category equivalent to a pull-back of the derived category of a stacky curve.

Recent attempts to train AI models to recognize knots have produced millions of “hard” unknot diagrams resistant to simplification by Reidemeister moves, pass moves, or random walks on the Reidemeister graph. Most are easy for the methods based on triangulations of the knot complement found in Regina and SnapPy, but more difficult for simplifiers based on diagrammatic moves. We present ReAPR (Re-embedding And Pass Rerouting), a semi-diagrammatic simplifier which alternates pass-move reduction on a knot diagram with a geometric re-embedding step. The re-embedding minimizes the total variation of a height function on the diagram subject to crossing constraints. We show that for an n-crossing diagram, the minimum total variation is 2(n-k), where k is the least number of crossings one must virtualize to make the diagram virtually alternating; this is a combinatorial invariant of the diagram. Reprojecting the resulting embedding from a new viewpoint reveals previously hidden simplifications. ReAPR successfully simplifies every published hard-unknot example we are aware of, as well as several new collections (~2.6 million examples in total) in under 30 seconds of total CPU time, making it as effective as the best non-diagrammatic unknot recognition method (SnapPy) and about 60x faster. ReAPR is just as effective at simplifying diagrams of random knots, where it is regularly used to simplify diagrams with tens of millions of crossings.

For a bounded Lipschitz domain $\Omega\subset\mathbb R^2$, let $T(\Omega)=\int_\Omega u_\Omega\, dx$ denote its torsional rigidity, where $-\Delta u_\Omega=1$ in $\Omega$ and $u_\Omega=0$ on $\partial\Omega$. We prove a quantitative Pólya–Szegő inequality for tangential polygons. Let $N\ge3$, let $P$ be a tangential $N$-gon, set $A=|P|$, and let $R_N$ be the regular $N$-gon of area $A$. Writing $L(\cdot)$ for perimeter, we obtain the explicit deficit estimate \[ T(R_N)-T(P)\ge \frac{A^2}{8N\tan(\pi/N)} \left(1-\frac{L(R_N)^2}{L(P)^2}\right)^2.\]Thus, at fixed area, the torsional deficit is controlled from below purely by the perimeter ratio. In particular, the regular $N$-gon is the unique maximizer of torsional rigidity among tangential $N$-gons of prescribed area; for $N=3$ this gives the classical triangular Pólya–Szegő theorem with a quantitative estimate. The same perimeter estimate yields an explicit positive lower bound for $T(R_{N+1})-T(R_N)$ for equal-area regular polygons, and hence a rather short alternative proof of the strict monotonicity of torsional rigidity in $N$. Combined with the Kohler–Jobin inequality, it also gives an explicit sufficient condition for the polygonal Faber–Krahn inequality within the tangential class. Our full quantitative inequality is stronger: it contains, in addition, a nonnegative angular Jensen deficit, which yields quantitative angular stability away from degenerate configurations.

Fomin Zelevinsky quiver mutation is an essential tool in the theory of cluster algebras. In this paper, we extend this framework to quivers with relations by introducing an involutive mutation operation on admissible ideals of path algebras. Briefly our mian motive, is to develop systematic framework for producing new admissible and gentle bound quiver algebras from given ones. So, for every cluster structure we associate classes of bound and gentle algebras. Finally we also suggest several directions for future work, including the study of mutation classes of gentle algebras, the relation with derived equivalence, and the possibility of developing a broader decorated mutation theory for bound quivers with relations.

We construct examples of totally c-symplectically aspherical near-symplectic manifolds with non-trivial second homotopy group.

We make progress on the question of constructing explicit evaluation places for leakage-resilient Shamir's secret sharing, over composite order fields. Previously, Maji et al. (EUROCRYPT 2024) showed that random evaluation places yield Shamir's secret sharing over the composite order field $\mathbb{F}_{p^d}$ that is statistically secure against physical-bit leakage. Later, Nguyen (EUROCRYPT 2025) established a dichotomy that linear code-based secret-sharing scheme over the field $\mathbb{F}_{p^d}$ is either statistically secure or completely insecure against such leakage. Building upon Nguyen's dichotomy, we present a partial derandomization of evaluation places, improving upon the Maji et al. result for a restricted regime of parameters. We replace the random choice of $n$ independent evaluation places by the iterates $x_j = \Phi^j(x_0)$ of a simple fixed rational function $\Phi$, where the initial point $x_0 \in \mathbb{F}_{p^d}^*$ is randomly chosen. The randomness in the evaluation places thus drops from $nd \log p$ bits to $d\log p$ bits. Our construction is valid for the regime $n = O(d/\log_p d)$, and any reconstruction threshold $k \ge 2$; in fact, the scheme attains perfect security (statistical distance exactly zero) against single-block leakage. Our technique is a partial fraction nondegeneracy argument that exploits the distinct poles of the rational iterates.

Let $K$ be a finite unramified extension of $\mathbb{Q}_p$ and $D$ the non-split quaternion algebra over $K$. We compute the associated graded modules of the duals of the predicted representations of $D^\times$ occurring in the Jacquet–Langlands correspondence.

For any finite family of Poisson multiple integrals and any finite $p\geq2$, we construct a common increasing filtration generated by finitely many exact Poisson counts such that the associated conditional expectations converge in $L^p$, remain in their original chaoses, and have bounded step kernels with finite-measure support. This finite-count martingale core allows regular fixed-chaos identities and estimates to be extended under the sole assumption of a finite fourth moment. In particular, if $F$ lives in a Poisson chaos with unit variance and finite fourth moment, we prove that the Kolmogorov distance between $F$ and a standard normal is bounded by $15.6(\mathbb{E}[F^4]-3)^{1/2}$. This removes Assumptions $\mathbf A$ and $\mathbf A^{\textbf{loc}}$ from the Kolmogorov bound of Döbler and Peccati (Ann. Probab., 2018). We also obtain quantitative $L^4$ estimates for all iterated Malliavin derivatives and, for $F$ in a Poisson chaos, the fourth moment assumption of $F$ forces the $L^4$-integrability of its kernel.

We construct an infinite family of $q$-ary primitive narrow-sense BCH codes whose minimum distance strictly exceeds the Bose distance; in fact, the gap between the two can be arbitrarily large as the length of the code tends to infinity. The key idea is to embed these BCH codes in a suitably large punctured generalized Reed–Muller code, whose codeword weights obey divisibility conditions supplied by Ax's theorem. This divisibility forces the minimum distance of the BCH codes far above the Bose distance. In particular, our family disproves a longstanding conjecture of Charpin asserting that this difference is at most four.

This proceedings contribution elaborates on the findings of arXiv:2605.26234v2: a joint work with Marco Usula, where we introduced a machine learning framework based on physics-informed neural networks (PINNs), aimed at constructing near-minimal discs in hyperbolic space asymptotic to a prescribed knot at infinity. We used this method to provide numerical evidence for a conjecture of Joel Fine relating minimal surfaces in $H^{4}$ to the coefficients of the HOMFLY polynomial. This is a methodological companion to that paper, based on a presentation given at the 2026 edition of the workshop "DANGER: Data, Numbers, and Geometry". Rather than reviewing the results, which are presented extensively in the preprint above, we discuss the two aspects of the framework which, in our experience, determined whether the method worked at all. First, the geometry of the problem must be encoded in the architecture of the model, so that the boundary condition and asymptotics at infinity hold exactly for every value of the learnable parameters - leaving us with a single-component loss function; second, the evaluation of the PDE residual must be engineered with care to ensure that complete trainings can be performed in a reasonable time. On the latter point, we describe two implementation techniques which are not spelled out in detail in the original paper: replacing nested reverse-mode automatic differentiation with the forward propagation of second-order jets, and compiling the computational graph of the residual once instead of rebuilding it at every optimisation step. Together, on identical hardware, these two changes reduce the cost of a training step by a factor of roughly forty to fifty. We hope these methodological discussions can be useful for researchers in differential geometry and geometric analysis who wish to deploy PINNs on problems of their own.

We investigate the long-time limit of an elephant random walk conditioned to avoid zero and show that the resulting process admits a Skorokhod-type embedding into a three-dimensional Bessel process. Using this embedding, we further establish several limit theorems for the conditioned elephant random walk, including a scaling limit to a deterministically rescaled and time-changed three-dimensional Bessel process, and a law of the iterated logarithm.

We provide a complete enumeration of the finite polyhedra that are noble, that is, polyhedra that are vertex transitive and facet transitive, through a computer-assisted proof which relates the possible existence of noble polyhedra to the roots of certain univariate or bivariate cubic polynomials. This is done through a parametrization of the orbits of point groups in $\mathbb{R}^3$, which naturally leads to a notion of criticality for certain orbits of a given point group, and a related equivalence relation $\equiv_c$ on the set of orbits. We show that within a given equivalence class of orbits under this relation, noble facetings are equivalent, and furthermore that there are a finite number of such equivalence classes. This gives a finite number of test cases, which may be tested through computer search. In total, we find that there are exactly 146 noble polyhedra in addition to the previously known infinite sets of stephanoids (crown polyhedra) and disphenoids.

The paper deals with the existence and multiplicity of nontrivial solutions for the doubly elliptic problem $$\begin{cases} -\Delta u=f(u) \qquad &\text{in $\Omega$,}\\ \phantom{-}u=0 &\text{on $\Gamma_0$,}\\ -\Delta_\Gamma u +\partial_\nu u =g(u)\qquad &\text{on $\Gamma_1$,} \end{cases} $$ where $\Omega$ is a bounded open domain of $\mathbb{R}^N$ ($N\ge 2$) with $C^1$ boundary $\Gamma=\partial\Omega$, with $\Gamma=\Gamma_0\cup\Gamma_1$, $\Gamma_0\cap\Gamma_1=\emptyset$, $\Gamma_1$ being nonempty and relatively open on $\Gamma$, $\mathcal{H}^{N-1}(\Gamma_0)>0$. The terms $f$ and $g$ are subcritical with respect to Sobolev embeddings, respectively in $\Omega$ and on $\partial\Omega$. We prove that, under suitable assumptions, the problem admits nontrivial solutions at the depth of the potential well energy level, which is the minimum energy level for nontrivial solutions. We also prove that the problem has infinitely many solutions at higher energy levels.

We study the convergence properties of the random-sweep Gibbs sampler for Gaussian graphical models with a thin-membrane prior. We demonstrate that the convergence rate of the Gibbs sampler is significantly accelerated in the dual model, which is obtained by applying the Fourier transform to the local factors of the normal factor graph representing the original model. In both domains, we derive the exact convergence rates for homogeneous $k$-regular graphs. We prove that, for all homogeneous models whose graphical representations contain cycles, the convergence rate in the dual domain is universal and independent of the underlying graph topology. Moreover, we show that the effective convergence rate in the dual domain is governed by the algebraic connectivity of the graph, providing an additional acceleration without increasing the computational complexity per sweep. We further establish an explicit algebraic relation between the covariance structures of the primal and dual models, enabling marginal statistics of the primal model to be recovered directly from those of the dual model. Finally, numerical experiments on several graph families confirm our theoretical results and demonstrate substantial improvements in the convergence rates in various settings.

We prove that (1) the weighted composition operator \(Tf(z)=c f(Bz)\) on the Fréchet space \(H(\mathbb{C}^d)\) of entire functions on \(\mathbb{C}^d\) (\(d\in\mathbb{N}\)) with the compact-open topology has the shadowing property whenever \(0<|c|<1\) and \(B\in\operatorname{GL}_d(\mathbb C)\) has spectral radius less than \(1\); (2) every bilateral weighted forward shift on \(\mathbb K^{\mathbb Z}\) with nonzero weights has the shadowing property; and (3) no continuous linear operator on a countably infinite product of nonzero finite-dimensional normed spaces is metrically positively expansive for any compatible metric, and no linear homeomorphism on such a product is metrically expansive for any compatible metric. These results answer the \(H(\mathbb C)\)-part of \cite[Problem A]{BCDFP} and the \(\mathbb K^{\mathbb Z}\)-part of \cite[Problem B]{BCDFP}.

Let $T_n$ be the lower-triangular prefix-sum matrix and let $c_{\mathrm{F}}(T_n)$ and $c_2(T_n)$ be the factorization costs that govern the mean and maximum per-coordinate squared error of the Laplace matrix mechanism under pure $\varepsilon$-differential privacy, for $\varepsilon>0$. We prove $c_{\mathrm{F}}(T_n),c_2(T_n)=\Theta((\log(n+1))^{3/2})$ with no sign, sparsity, or squareness restriction and with arbitrary finite inner dimension. Consequently, within the pure-$\varepsilon$-DP matrix-mechanism class, the optimized maximum and mean squared errors are both $\Theta(\varepsilon^{-2}\log^3(n+1))$. Under the factorization contract of Arkhipov and Kalinin (arXiv:2607.08963v1), who prove the matching lower order for factors with entries in $\{0,1\}$ and state the arbitrary-factor extension as open, the theorem below establishes the order for arbitrary real factors. The lower bound runs through a $p$-nuclear obstruction: an aggregate column-width estimate $D_k(T_n)\asymp n^{3/2}k^{-1/2}$, valid in the low-rank range $1\leq k\leq n/16$, for the prefix chain, fed into the classical approximation-space conversion of Pietsch and Hinrichs–Pietsch, becomes harmonic at the critical exponent $p=2/3$, and Hölder's inequality transfers it to both factorization costs. The same computation determines $\mathfrak{n}_p(T_n)$ for each fixed $0<p<1$: order $n$ below $2/3$, $n\log n$ at $2/3$, and $n^{3p/2}$ above. A Fenwick interval factorization supplies matching upper bounds. The claims are confined to pure-$\varepsilon$-DP Laplace matrix mechanisms and the two stated squared-error criteria; they do not cover non-matrix continual mechanisms, approximate-DP sensitivity, or expected maxima across coordinates.

For every $b\ge1$, let $Q_{2,b}$ be the ordered graph obtained from a transitive ordered triangle by attaching a monotone tail of length $b$ at its rightmost vertex. We prove $\rho_{<}(Q_{2,b})=\frac12$ for all $b\ge1$. Thus the previously isolated case $Q_{2,2}$ is one member of an exact infinite triangle-tail family. The lower bound is the sharp forward-template identity $\lambda(Q_{2,b})=1/2$, and it is realized already inside binary-level hosts: a parity-cut construction gives $Q_{2,b}$-free binary-level graphs with density exactly $1/2$ on every level. The upper bound uses the binary rich-level reduction. Its key input is the intrinsic decomposition of a $Q_{2,b}$-free graph into tail-starting vertices $T_b$ and the complement $R$: there are no forward edges from $R$ to $T_b$, $G[T_b]$ is ordered-triangle-free, and $G[R]$ is monotone-$\vec{P}_{b+1}$-free. We control these pieces by weighted binary $\vec{P}_{b+1}$ smoothing and weighted binary Mantel smoothing, the latter following from a binary ultrametric cut-domination theorem. We also record exact path-blow-up values, giving a reusable template/rich-host calculus for ordered relative densities.

This paper concerns the study of the existence of a unique equilibrium state for a Hölder continuous function under the dynamics of a holomorphic correspondence defined on the Riemann sphere. We mainly work with the correspondence restricted on the support of the Dinh-Sibony measure and identify topologically interesting correspondences, namely distance expanding ones. Further, we consider Hölder continuous potentials defined on the support of the Dinh-Sibony measure, for which we prove the uniqueness of equilibrium state. Along the way, we also prove some interesting topological results related to holomorphic correspondences. Finally, we establish a result connecting the Ruelle operator for holomorphic correspondences and the unique equilibrium state under a suitable hypothesis. The concluding part of the paper is devoted to some discussion related to the hypothesis involved and providing some examples.