Papers

Despite the widespread use of Low-Rank Adaptation (LoRA), little is known about its dynamics in continual learning and the mechanisms by which low-rank updates affect catastrophic forgetting. We provide an asymptotically exact dynamical characterization of LoRA in a solvable two-task teacher-student model. In the high-dimensional online-learning limit, we derive a closed system of ordinary differential equations for a finite set of macroscopic order parameters, yielding exact expressions for the generalization errors throughout both the initial Task 1 learning phase and the subsequent LoRA fine-tuning on Task 2. The theory quantitatively matches finite-dimensional simulations and exposes two characteristic effects of LoRA: low-rank adaptation reduces interference with features learned on the first task, but its initialization slows adaptation to the second task. Building on this mechanistic picture, we analyze a state-dependent masking strategy that freezes hidden units carrying the strongest first-task representations and restricts adaptation to the complementary subspace. This structural partitioning markedly reduces forgetting, while preserving plasticity on the new task. Our framework further clarifies the role of adapter rank: transfer improves only up to the intrinsic dimensionality of the target task and saturates beyond it, while forgetting continues to grow with rank. These results provide a dynamical and geometric account of how low-rank adaptation organizes information across sequential tasks and are qualitatively reproduced on a sequential MNIST benchmark.

While most attention logits can be computed in low precision without degrading numerical stability, current attention kernels fail to exploit this phenomenon. We introduce a novel hardware-algorithm co-design in the form of mixed-precision FlashAttention. Our method accumulates key-query products and evaluates their exponentials in 8-bit formats, then adaptively identifies sensitive sub-blocks and recomputes them in 16-bit formats. We propose the specifications for a dedicated accelerator capable of executing this pipeline efficiently. Simulated experiments with Qwen3 and Gemma 3 show that rerouting a selective minority of sub-blocks to high precision is sufficient to recover the baseline model performance.

Asynchronous SGD is a popular algorithm for distributed learning where each client's gradient update is applied on arrival. This leads to a speed-up, but also an increased vulnerability to attacks, as fast clients can dominate the total update. We introduce Throttle, a Byzantine-robust generalization of asynchronous SGD where the key idea is to exponentially down-weight updates from faster clients by a factor $q$. Both asynchronous SGD ($q=1$) and synchronous Byzantine-robust SGD ($q\to\infty$) correspond to specific settings of Throttle. We provide a theoretical analysis of the convergence rate and validate the robustness to attacks both theoretically and empirically. Remarkably, our experiments show that this down-weighting mechanism can also improve performance over standard asynchronous SGD even in the non-Byzantine setting.

We study the global dynamics of the generalized Boussinesq equation near standing solitary waves. For even-odd perturbations, Maulen [J. Math. Pures Appl. 177 (2023)] constructed an asymptotically stable center-stable manifold M near the solitary wave Q. We complete this local description by proving that any sufficiently small even-odd perturbation of Q lying outside M either scatters in the energy space as t -> +infinity or blows up in finite time. The proof relies on two main ingredients. First, we classify even-odd solutions with energy below the ground-state energy by identifying two invariant regions, associated with scattering and blow-up, respectively. Second, we establish a one-pass theorem in one dimension: a non-scattering solution cannot re-enter a suitably chosen neighborhood of the solitary wave once it has left that neighborhood. Consequently, the solution remains confined to one of the two invariant regions. Together with the classification above, this yields a dichotomy between scattering and blow-up outside M.

We develop a general second-order asymptotic framework for estimators constructed from invariant probability measures of data-dependent Markov processes. The framework yields a universal bias correction for smooth functionals, computed from the same invariant measure used to construct the estimator, without the resampling and repeated estimation involved in bias-correction methods such as the bootstrap and jackknife. The statistical input is represented by a random one-form, allowing differentials of log-likelihoods and general random criteria, as well as estimating functions, to be treated in a common framework. Under local regularity and localization conditions, we derive second-order stochastic expansions for averages with respect to these invariant measures and for decision rules defined through general loss functions. The expansions separate the contributions of the statistical input, temperature, drift, and loss, using precontrast geometry and an O-derivative induced by the loss. The framework includes maximum likelihood and posterior-based estimators and recovers existing bias-reduction methods based on the choice of prior or adjustments to estimating equations. In the likelihood setting, for the Jeffreys posterior $\mu_n$ and its Fisher–Rao Fréchet mean $\widehat z_n$, the corrected estimator $2\gamma(\widehat z_n)-\mu_n(\gamma)$ has frequentist bias $o(a_n^2)$ for every fixed smooth functional $\gamma$, where $a_n$ denotes the estimation rate. For coordinate squared loss, the theory yields a diffusion determined only by the likelihood and Fisher information. The mean of its invariant probability measure estimates the parameter with bias $o(a_n^2)$. This construction remains available even when no bias-reducing prior exists. Numerical results in the gamma shape–scale model illustrate this bias reduction.

Collocation methods based on Birkhoff interpolation at Gaussian-type points are well-conditioned for one-dimensional initial and boundary value problems [Wang et al., SIAM J. Sci. Comput. 36 (2014)], but the underlying construction does not extend directly to multiple dimensions. We address the long-standing ill-conditioning of multidimensional collocation methods for second-order elliptic-type problems. The key observation is that the second-order differentiation matrix and its inverse, the pseudospectral integration matrix (PSIM) constructed from Birkhoff interpolation at Legendre-Gauss-Lobatto points, are both similar to symmetric negative definite matrices. This enables stable diagonalisation of the dense, non-symmetric and ill-conditioned differentiation and integration matrices, even for thousands of collocation points, and leads to efficient multidimensional Birkhoff preconditioners. For variable-coefficient problems, the coefficients are incorporated directly into the diagonalisation and preconditioner construction, which is essential for highly anisotropic, high-contrast, oscillatory and degenerate elliptic operators. We provide spectral analysis and extensive two- and three-dimensional numerical experiments, demonstrating substantial reductions in condition numbers and nearly polynomial-degree-independent GMRES convergence while retaining high-order accuracy. The resulting Birkhoff-collocation schemes make multidimensional spectral collocation methods practical for challenging elliptic problems.

Let $G$ be the graph on the positive integers in which distinct vertices are adjacent if they differ by one or their quotient in one order is prime. We prove that $6\leq\operatorname{diam}(G)\leq7$ and $4\leq\operatorname{rad}(G)\leq5$. Under Dickson's conjecture for two linear forms, we prove that $\operatorname{diam}(G)=6$ and $\operatorname{rad}(G)=4$.

We construct the first known fault-tolerant scheme for injecting states into quantum error-correcting codes with constant space and time overhead. That is, we construct a family of constant-rate quantum error-correcting codes for which a set of bare physical qubits can be injected, i.e. fault-tolerantly encoded, into a code block. Similarly, a code state can be ejected, i.e. fault-tolerantly decoded, back into bare physical qubits. We show that these injection and ejection procedures succeed under circuit-level locally stochastic noise, while incurring just a small constant probability of corrupting each qubit, which is unavoidable for bare physical qubits. We also show how to perform fault-tolerant error correction and code-state preparation under locally stochastic noise. All of our gadgets can be implemented with constant quantum circuit depth (i.e. are single-shot), and with a number of physical qubits growing linearly with the number of logical qubits, assuming the ability to run polynomial-sized noiseless classical circuits on the side. We construct our quantum codes by taking a high-dimensional hypergraph product of classical LDPC codes, which in turn are a simplified version of Spielman's linear-time encodable codes (STOC'95). As our resulting product codes are only resilient to physical errors occurring with non-uniform probabilities across qubits, we then show how to concatenate with inner codes of various sizes to obtain fault-tolerance against uniform noise.

Naff's planarity estimate is a crucial tool for analysing singularities of quadratically pinched mean curvature flows in high codimension, showing that high-curvature regions become asymptotically codimension one. Naff established the estimate throughout the full Andrews–Baker pinching range in dimensions at least $7$, while in dimensions $5$ and $6$ a stronger pinching assumption was required. We close this remaining gap, proving the planarity estimate in dimensions $5$ and $6$ throughout the full Andrews–Baker range. The key new ingredient is a sharp Kato-type inequality relating the twisting of the principal normal to the gradient of the second fundamental form.

This paper studies orthogonal model structures, i.e., model structures such that a lifting in the Lifting axiom is unique. Cofibrant (fibrant) objects are defined without initial (terminal) objects. For an orthogonal model structure on a category with enough cofibrant objects and fibrant objects, it is proved that the homotopy category is equivalent to the full subcategory of cofibrant-fibrant objects. TTF model structures, bi-reflective model structures, and torsion model structures, are introduced. They are all orthogonal. One to one correspondences between TTF model structures and TTF triples in an abelian category, bi-reflective model structures and bi-reflective pairs in any category, and torsion model structures and twin torsion pairs in an abelian category, are established in a constructive way. Torsion model structures on a poset, on the category of $G$-sets, and on the category of topological groups, are also constructed. The homotopy categories of all these model structures are computed. In particular, the homotopy category of a TTF model structure is an abelian category.

Choi, Erdős and Szemerédi showed that there exists an absolute constant $C$ such that for all subsets $A \subseteq \{1, 2, \ldots, 2n\}$ with at least $n+C$ elements, there exist four distinct positive integers whose pairwise sums are all contained in $A$. A proof that one can take $C = 3166$ was recently sketched by the first author, and here we show that we actually have $C = 4$ for all $n \ge 6$, which is optimal. The proof we present was originally conceived of by AI, with the final result proved and formalized by Aristotle, the formal reasoning agent developed by Harmonic.

Let $p\in[1,\infty)$ and $\delta\in(0,n]$. Recently, to characterize the weighted boundedness of maximal operator on Choquet integrals based on Hausdorff content $\mathcal H_\infty^\delta$, a class of Capacity Muckenhoupt weights is introduced, denoted by $\mathcal A_{p,\delta}$, and, for any fixed $\delta\in(0,n]$, this new class of weights is proved to satisfy the self-improving property with respect to the index $p\in(1,\infty)$. In this note, we show that, for every fixed $p\in[1,\infty)$, the class of capacity Muckenhoupt weights $\mathcal A_{p,\delta}$ fails to satisfy the self-improving property with respect to the index $\delta\in(0,n]$.

We study stratification by normal-bundle splitting type in families of unramified rational curves of fixed class in a smooth projective toric variety over an algebraically closed field {of arbitrary characteristic}. Using the Cox construction, we construct an explicit two-term resolution of the relative normal bundle by direct sums of line bundles on the universal curve over the fixed-source parameter space. Then we construct morphisms of vector bundles on the parameter space, which we call relative cohomology morphisms, from which we can investigate generic splitting type and jumping loci of the family. In particular, we obtain sufficient numerical criteria for unbalancedness of the normal bundle. We apply this method to blowups of projective space along linear subspaces, and show that the numerical criteria for unbalancedness are equivalent to those of Cela–Lian \cite{CelaLian2026}. We also obtain explicit formulas for the expected Chow classes of the jump loci.

We focus on solving the Lyapunov equation $AX + XA^T = F$, where $A$, $X$ and $F$ are square matrices, $A$ is symmetric positive definite (SPD) and sparse, and $F$ is symmetric and of a low-rank. The solution $X$ is then also symmetric and in general dense, but it can be approximated by a low-rank matrix. If $n$ is large, $X$ cannot be computed directly, but it is accessible by using the so-called low-rank arithmetics (LRA). The equation is solved by matrix reformulation of the method of conjugate gradients (MCG). We are interested in the behavior of ranks of the solution approximations $X_\ell$, residuals $R_\ell$, and direction vectors $P_\ell$ during iterations.

We prove that the enriched free energy of vector spin glasses converges as the number of sites tends to infinity. The interaction is not assumed to be convex. This is the convergence assumed by Chen and Mourrat (Probab. Math. Phys., 2025) in their representation of the limit as a critical point representation, so that representation holds without further assumptions.

This paper considers a class of multiblock nonconvex and nonsmooth optimization problems arising in many applications. Existing methods construct proximal linearized operators or their variants within standard Euclidean geometry to solve this class of problems, forcing their block variable updates to rely on the standard inner product and its induced norm. Nevertheless, this construction fails to capture the geometric structure of the target problem, leading to low numerical efficiency. To overcome these drawbacks, we propose a generalized geometry proximal linearized operator for updating block variables, and develop the Generalized Geometry Block Proximal Linearized (GGBPL) method based on this operator. Compared with existing proximal linearized operators, the proposed operator allows the block surrogate functions to be constructed using arbitrary inner products and general admissible metrics, thereby enabling the GGBPL method to adapt its updates to the geometric structure of various problems. We also introduce the inertial version of GGBPL, named the inertial GGBPL (iGGBPL) method. We further establish a new unified convergence framework under this generalized geometry, within which we prove that our methods guarantee convergence of the objective function values, establish global convergence of the generated sequence to a critical point, and derive the convergence rate of our methods. We also establish an $\mathcal{O}(\varepsilon^{-2})$ iteration complexity bound for obtaining an $\varepsilon$-stationary point. We apply our methods to two nonconvex and nonsmooth problems: sparse nonnegative matrix factorization with $\ell_0$-constraints and sparse nonnegative CP decomposition with $\ell_0$-constraints. Numerical results demonstrate the superior numerical performance of our proposed methods over several state-of-the-art methods.

Ekedahl's theory relates diagonal complexes to \(F\)-gauges, but turning this equivalence into explicit calculations requires careful control of its constructions. These notes explain the equivalence and clarify the generators-and-relations definition of \(\rmt\). At each finite level, we construct bounded projective resolutions of bounded coherent gauge complexes by finite sums of the standard generators. Morita equivalence expresses \(\rmt\) as a tensor product functor. Comparing the constructions across levels yields an exact fully faithful embedding of the diagonal heart into \(\Ch(R)\). Finally, we compute three distinct domino types arising from abelian threefolds that are all supergeneral.

Let \( G \) be a finite group, and consider its common non-zero graph, which we denote by \( \Gamma_{nv}(G) \). In this graph, the vertices represent the non-linear irreducible characters of \( G \). There is an edge connecting two distinct vertices \( \chi_1 \) and \( \chi_2 \) if there is a vanishing element \( g \in G \) such that \( \chi_1(g) \chi_2(g) \neq 0 \). A finite group \( G \) is described as a strongly \( \mathcal{H}_1' \)-group if the set of vanishing elements of each of its non-linear irreducible character is equal to the set of vanishing elements of \( G \). In this paper, we prove Conjecture 1 from \cite{ourself}, which states that if \( \Gamma_{nv}(G) \) is null (has no edge), then \( G \) must either be a strongly \( \mathcal{H}_1' \)-group or a Frobenius group, whose Frobenius complement is isomorphic to \( Q_8 \). Additionally, we show that if there are no triangles in \( \Gamma_{nv}(G) \), this implies that the group is solvable. We also look at the situation where the common non-zero graph of a finite group \( G \) contains only one edge. This leads us to conclude that \( G \) is either isomorphic to \( S_4 \); or that the single edge in \( \Gamma_{nv}(G) \) is of the form \( \{\theta, \zeta\theta\} \) for some \( \zeta \in \Irr(G/G') \) and \( \theta \in \Irr(G|G') \), $G$ is a $p$-nilpotent group for some prime $p$ and for all characters \( \chi \in \Irr(G|G') - \{\theta, \zeta\theta\} \), we find \( \Van(\chi) = \Van(G) \). Lastly, our study also explores groups whose common non-zero graph forms a star shape, concluding that such configurations can only occur if \( G \) has exactly one non-linear irreducible character.

In this paper, we investigate a viscous regularization for the BGK model of the Boltzmann equation and its hydrodynamic limit. First, we establish rigorous convergence toward the viscous Euler equations, by exploiting bounds derived from spatial Fisher information. We then analyze the formal derivation of the Euler equations with an entropy inequality for both single-and multi-species case, comparing our results against symmetric hyperbolic thermodynamically compatible theory.

An old conjecture in non-Kähler geometry states that if the Chern holomorphic sectional curvature of a compact Hermitian manifold is equal to a constant $c$, then the metric must be Kähler when $c\neq 0$ and be Chern flat when $c=0$. The conjecture is known to be true in dimension two by the work of Balas–Gauduchon and Apostolov–Davidov–Muškarov in the 1980s and 1990s. Recently, Qin and Tian proved the conjecture in complex dimension three when $c\neq 0$, and the $c=0$ case was proved by Chen–Li under the additional assumption that the metric is balanced. In this article we remove this additional hypothesis and complete the confirmation of the conjecture in complex dimension three. The proof relies heavily on the fact that in dimension three the torsion $3$-tensor can be equivalently expressed as a $2$-tensor twisted by the canonical line bundle, plus Bochner type integration formulas. In particular, the method cannot be directly generalized to higher dimensions.

We prove interior curvature estimates for admissible graphical solutions of curvature quotient equations in the range $3\leq k<n$ on the full Gårding cone $\Gamma_k$. No convexity, semiconvexity, or $\Gamma_{k+1}$-admissibility assumption is required. The proof combines a new quantitative compression inequality, a two-surface comparison and doubling argument, and a Pogorelov estimate.

Reinforcement learning and contextual bandit algorithms have become increasingly common in sequential decision-making applications. When these methods are deployed in high-stakes domains, there is growing interest not only in learning effective policies, but also in conducting statistical inference for quantities learned under adaptive data collection. However, classical procedures applied naively in these settings can fail: even when estimators are unbiased, their variance becomes path-dependent and as a result may not be asymptotically normal. A growing literature has emerged to ameliorate this problem, but solutions tend to be problem specific and often rely on correct specification of a working model. In this work, we develop a unified framework for constructing asymptotically valid confidence intervals to cover smooth functionals of nonparametric M-estimands under adaptive sampling. Under Neyman orthogonality, we provide two novel methods for performing inference: (1) a self-normalized statistic based on the realized quadratic variation of the influence function and (2) a statistic using a plug-in estimate of the conditional variance based on reweighted influence function increments. Our results allow for flexible nonparametric estimation of nuisance parameters and remain valid under model misspecification. Our theory is supported by a simulation study for a dynamic pricing application which demonstrates that this method can produce asymptotically valid confidence intervals where standard methods fail.

The Three-Dimensional Trailer Loading Problem (3D-TLP) involves determining the optimal placement and orientation of heterogeneous items within the confined space of a trailer while maximizing volume utilization and satisfying a wide range of complex logistical and safety constraints. The 3D-TLP is NP-hard, rendering exact optimization approaches computationally impractical for large-scale industrial applications. To address this challenge, we propose an enhanced Biased Random-Key Genetic Algorithm (BRKGA) accelerated through a novel island-based parallelization framework, PANGEA. The proposed method combines the search efficiency and robustness of BRKGA with a multi-population evolutionary scheme for genetic algorithms. This island-model strategy promotes population diversity, mitigates premature convergence, and significantly reduces computational times. The proposed solution was validated in a real trailer loading process, providing an effective solution approach for real-world large-scale logistics.

We investigate a stochastic individual-based population model with trait-dependent reproduction, death, competition, variation, and deleterious transfer. Nonlinearities appear in the competition term and the transfer form, which moreover is asymmetric. In the large-population limit, the process converges to a deterministic measure-valued equation. The latter helps us to uncover the past history of living individuals. We replace in the interactions the original stochastic process with the deterministic large population approximation obtaining a nonhomogeneous measure-valued branching process, for which we use a spinal decomposition, generalizing a method that existed only for the stationary case. Reversing time, we identify the law of typical ancestral lineages. This approach encompasses a variety of settings and requires delicate functional analysis. In contrast to earlier approaches in the literature, which relied either on stochastic calculus or on favourable jump structures under stationarity assumptions, our method provides a more direct and more general point of view.

Dynamic Discretisation Discovery (DDD) is a framework for solving a problem by iteratively solving and refining a relaxed version of the problem. DDD is the basis for state-of-the-art algorithms for various problems, including the continuous-time service network design problem (CTSNDP), which we study here. In the refinement step, such algorithms must identify conflicts in the relaxation solution that make it infeasible for the original problem and then apply fixes to the relaxation to prevent them from recurring. We study the optimal refinement problem of selecting from a set of fixes a subset that fixes all conflicts and yields a relaxation of minimum size, with the aim of making it easier to solve. We demonstrate that while the number of conflicts may be exponential in the instance parameters, we can represent them more compactly, using only pseudo-polynomial space and time. Based on our more compact representation, we develop exact methods for solving the refinement problem. The best method allows us to find small relaxations in negligible amounts of time, more than 100 times faster than a natural integer programming formulation for some instance classes. Using this refinement approach in a DDD algorithm for the CTSNDP, we observe reductions in model sizes and running times of about 10 to 20 % compared to existing approaches.

Qiu, Guan, and Yu conjectured that in the even-order Kaleidoscope Yang–Baxter equation every word $\Lambda T^{k_1}\Lambda\cdots T^{k_M}\Lambda$, built from a square-zero matrix $\Lambda$ and integer powers of a shift $T$ of order $N$, vanishes once $M\geq N/2$. We prove this for every even $N\geq4$ and every complex parameter $u$ with $u\neq0$ and $u^N\neq1$, and show that the bound is sharp. For $M=N/2-1$ we determine exactly which words are nonzero; each of them has rank two. After block reduction, the factors associated with the intervening powers of $T$ lower a common flag of length $N/2$ by at least one step. The spectral idempotents of $T$ then give a presentation of the algebra generated by $T$ and $\Lambda$ by paths between levels, in which two paths compose only when the level changes monotonically. This algebra is independent of $u$, has dimension $N^2-N+2$, and is a pullback of two incidence algebras. Its radical is generated by $\Lambda$ and has nilpotency index $N/2+1$, and the algebra has infinite global dimension and infinite representation type. Replacing powers of $T$ by arbitrary functions of $T$ gives exact rank formulas and, for $N\geq6$, rank-one elements in the top radical power. Finally, four conditions on a filtered pair of operators over an arbitrary field recover the threshold and the radical; none of them can be omitted.

We prove that the $(-2,3,7)$ pretzel knot is the only hyperbolic knot of genus at most $5$ with a lens space surgery. Our key technical result asserts that if a genus-$g$ hyperbolic knot $K$ has an elliptic surgery of slope greater than $4g-3$, then the invariant foliations of its monodromy have an orientation that is reversed by the monodromy, and so the opposite of the dilatation must be a root of the Alexander polynomial of $K$. Combined with prior work of Moser, Wu, Bleiler–Litherland, Baker, and Greene, this verifies the Berge conjecture for knots of genus at most $5$.

European option smiles determine the risk-neutral marginal laws of an asset, but not their intertemporal coupling, which is decisive for many applications. The Bass martingale construction selects, among all calibrated martingales, the one closest to Bachelier dynamics; it permits fast calibration at discrete maturities and recovers the Dupire local-volatility (LV) model as the maturity grid is refined. This article develops a modular calibration overlay for existing stochastic and path-dependent volatility models. We sequentially construct a martingale that matches all prescribed marginals exactly while remaining as close as possible, in an adapted Knothe-Rosenblatt sense, to the reference dynamics. As with the Bass LV model, each calibration step is amenable to an efficient Martingale Sinkhorn algorithm. We develop the theoretical foundations, numerical implementation and consider convergence to the SLV model. We also benchmark the method for Heston and Bergomi finite-factor path-dependent volatility dynamics and develop the multi-asset extension.

Given any tracial von Neumann algebra $\mathcal{A}$, the first author has defined a unitary tensor functor $\Phi:{\rm Rep}(\mathcal{Q}_{\rm aut}(\mathcal{A}))\to {\rm Bimod}(\mathcal{A})$ in his paper [Gos24]. In this paper, we consider $\mathcal{A}$ to be the underlying von Neumann algebra of $C^*(S_3)$ and show that given any DQG $\mathcal{Q}$, which contains $C^*(S_3)$ as a quantum subgroup and has an outer action $\tilde{\Phi}$ (in the sense of Definition 3.4}) on ${\rm Bimod}(\mathcal{A})$, which agrees with $\Phi$ when restricted on ${\rm Rep}(C^*(S_3))$, is monoidally equivalent to the DQG $C^*(S_3)$. Furthermore, given any DQG $\mathcal{Q}$ with an outer action $\Gamma:{\rm Rep}(\mathcal{Q})\to {\rm Bimod}(\mathcal{A})$, such that $ {\rm Out}(\mathcal{A})\subseteq {\rm Im}(\Gamma)$, is either monoidally equivalent to Out$(\mathcal{A})$ or monoidally equivalent to $C^*(S_3)$ itself.

We study functional inequalities along Wasserstein geodesics. If $\mu$ 0 and $\mu$ 1 are respectively $\kappa$ 0 -and $\kappa$ 1 -strongly log-concave probability measures on R n , we prove that their quadratic Wasserstein geodesic satisfies for all probability measures $\nu$ on R n . The coefficient is sharp. By linearization, this recovers the Poincaré estimate of Han and Zhu [15]. On the real line, we prove convexity of the square roots of the optimal T 1 and T 2 constants along monotone interpolation between arbitrary probability measures. The argument applies to more general transport entropy inequalities. We also establish convexity of the rescaled L p Poincaré constants for every finite p $\ge$ 1, including the square root of the Poincaré constant and the inverse Cheeger constant. Finally, we construct a planar Wasserstein geodesic whose endpoints satisfy T 2 and all finite-p L p -Poincaré inequalities, whereas every interior interpolant fails these inequalities.