Papers

Weak-to-strong generalization is the phenomenon where a strong student model trained with labels produced by a weak teacher model is able to generalize better than the teacher. In this paper, we study this phenomenon in two-layer random feature networks where the model strength is determined by its width. Using tools from random matrix theory, we derive deterministic equivalents for the population errors of an optimally trained teacher and a student trained with gradient flow. For ReLU activation and a pure spherical harmonic target, we obtain sharp asymptotics under a Gaussian universality assumption, showing a quadratic improvement: the student error scales as the square of the teacher error. These results attain the general lower bound of Medvedev at al (2025). We also analyze how the student behaves under more general stopping times and targets supported on multiple harmonic degrees, characterizing the regimes in which weak-to-strong generalization occurs and identifying the transition between quadratic, non-quadratic, and no improvement.

We study which graph classes determine every graph with simple spectrum up to isomorphism by homomorphism counts from their members. We show that such a class has unbounded treewidth and Euler genus and contains a graph with the complete graph on seven vertices as a minor. To prove these conditions, we construct pairs of non-isomorphic cospectral graphs with simple spectrum, each bipartite, nonsingular and of maximum degree at most six. The construction attaches rooted trees to Cai-Furer-Immerman graphs and preserves homomorphism indistinguishability over every minor-closed class. We then prove that subcubic homomorphism counts determine every graph whose adjacency matrix has kernel dimension at most one. This extends the known result for nonsingular graphs and applies to all graphs with simple spectrum. We also prove that two graphs have the same subcubic homomorphism counts if and only if they have the same number of vertices and there is a bijective linear isometry between the images of their adjacency matrices that intertwines the adjacency operators and preserves the coordinatewise trilinear forms. Finally, we prove that homomorphism counts from the single-vertex graph and from subdivisions of any sequence of connected cubic graphs have the same distinguishing power as subcubic homomorphism counts if every complete graph is a minor of some member of the sequence.

We study long-time mean-field large deviations for granular-media diffusions. Under a dissipativity assumption, for arbitrary $T_N\to\infty$, the occupation measure of the $N$-particle empirical measure satisfies a large deviation principle on $\mathcal{P}(\mathcal{P}(\mathbb{R}^d))$ with speed $NT_N$ and rate function $\mathcal{I}(\Gamma)=\int_{\mathcal{P}(\mathbb{R}^d)}\mathfrak{i}(\mu)\Gamma(d\mu)$. Here $\mathfrak{i}(\mu)$ is one quarter of the free-energy dissipation: $\mathfrak{i}(\mu)=\frac14\left\|\nabla\frac{\delta\mathcal{F}}{\delta\mu}(\mu)\right\|_{L^2(\mu)}^2$, where $\mathcal{F}$ is the free energy of the model. The local cost $\mathfrak{i}(\mu)$ can be interpreted as the per-unit-time Dawson–Gärtner action of the constant path at $\mu$, and $\mathcal{I}$ averages these costs according to $\Gamma$. The main difficulty in the upper bound is that empirical measures are atomic, so their free energy is infinite. We address this by analyzing a regularized free-energy functional along the empirical-measure flow of controlled granular-media diffusions. The lower bound uses approximation by finitely supported occupation measures whose atoms have smooth densities with Gaussian tails. For quadratic potentials, we show that the variational problem associated with an atypically large overall space-time variance exhibits a transition between a Dirac minimizer at a Gaussian profile and non-Dirac minimizers supported on Gaussian profiles with different centers.

Pairwise comparisons are widely used to infer latent scores and identify top-ranked items. Although sharp statistical guarantees are available under uniform sampling, real data often induce irregular comparison graphs with heterogeneous observation counts across pairs. In this paper, we study an arbitrary fixed comparison graph under the Bradley–Terry–Luce model with minimal assumptions. We prove that both the standard maximum likelihood estimator and Rank Centrality achieve a high-probability entrywise error rate of order $1/\sqrt{\lambda_{\mathcal{D}}}$ up to logarithmic and dynamic-range factors, where $\lambda_{\mathcal{D}}$ is the algebraic connectivity of the count-weighted observation graph. This guarantee is graph-monotone since $\lambda_{\mathcal{D}}$ cannot decrease when additional comparisons are added. Under heterogeneous sampling, where comparison pairs are sampled independently with unequal probabilities, our guarantee improves existing error bounds or requires weaker assumptions. We further extend our analysis to show that both estimators are robust against outcome-adaptive augmentation, where an adversary can choose additional comparison pairs after observing the initial outcomes.

In this paper, we construct and analyze a stochastic model for infectious disease spread in service systems. Our model considers a $M/M/1$ queue where a proportion, $p$, of the arriving population is infectious and the remaining population is susceptible. When an infectious person overlaps in the queue with a susceptible customer for more than $\ell$ units of time, then the susceptible person becomes infected. We provide both a personalized perspective of how an individual susceptible customer is infected and a community perspective of how one infectious customer can infect the others around them. From the personalized perspective, we derive the probability that a susceptible person becomes infected and the probability they are infected by at least $k$ different people. From the community perspective we compute the mean and higher moments of the number of susceptible customers that are infected by one infectious individual. Finally, we use stochastic simulation to verify our analytical results and show that our new formulas accurately describe the behavior our stochastic model of infection and to expand our model into the consideration of different arrival and service regimes.

Exponential directed last passage percolation is a random maximization model connected to tandem queues and interacting particle systems. We study the line-to-point model of the percolation, in which the starting point is allowed to vary inside a line boundary, conditioned on an atypically large last-passage time to a fixed endpoint. Extending recent results for the point-to-point model where the starting point is fixed, we develop heuristic predictions for the conditional limit shape and prove corresponding fluctuation results in certain regimes through steepest-descent analysis of exact multi-point formulas. Our calculations suggest that the conditional fluctuations near diagonal are described by Brownian bridges coupled through an additional shared Gaussian component.

We prove that every affine coordinate subspace of positive dimension and positive codimension fails to be of strong Khintchine type for convergence. Combining this with Huang's criterion for Khintchine-type affine subspaces gives a sufficient Diophantine condition for an affine coordinate subspace to be of Khintchine type but not of strong Khintchine type for convergence.

Function-correcting codes (FCCs) protect the value of a function $f$ of the message against $t$ errors. In the original formulation of Lenz et al. (2023), the encoding is systematic, and the lower bound of $2t$ on the redundancy relies on this form. Recently, D. Ho (arXiv, 2026) showed, for linear functions and linear encodings, that systematicity can cost redundancy. We begin with the OR function to show that the cost is not confined to the linear setting: a non-systematic encoding attains redundancy $1$ while every systematic encoding needs $2t$. For a linear function $f$ and a fixed linear code $C$, let $d_f$ denote the minimum distance between codewords of messages with different function values. Different generator matrices of $C$ assign different codewords to the messages and can give different values of $d_f$. We study which generator matrix of $C$ gives the largest $d_f$. We give an algorithm that constructs an optimal generator matrix and determines the optimal value as the weight of a codeword in a greedy basis of $C$. We then characterize, in terms of information sets, when a generator matrix in systematic form attains this optimum, and give a necessary condition that is checked on the low-weight codewords of $C$ alone. For two-valued functions, we drop both linearity and systematicity. Using the minimality of initial segments of the simplicial order with respect to Hamming neighbourhoods, we show that a non-systematic $(f,t)$-FCC of length $n$ exists if and only if a condition depending on $f$ only through the size of its smaller preimage holds. For $t=1$, redundancy $1$ is sufficient for every nonconstant two-valued function on $\mathbb{F}_2^k$ with $k\ge 10$. For general $t$, redundancy $1$ suffices for all two-valued functions once $k$ is large enough, and for each $s<2t$ we give an upper bound on the threshold in $k$ beyond which redundancy $s$ suffices.

We establish global Harnack inequalities for positive viscosity solutions of several classes of nonlinear parabolic equations. The equations include positively homogeneous uniformly parabolic flows on Euclidean space and flat tori, normalized weighted $p$-Laplacian heat flows, and homogeneous isotropic diffusions on closed smooth metric measure spaces. For the normalized infinity-Laplacian heat flow, no curvature assumption is needed. For the ordinary weighted $p$-Laplacian, we obtain pressure estimates under nonnegative Bakry-Émery Ricci lower bounds. The proofs combine comparisons at two independent space-time points using the maximum principle for semicontinuous functions with weighted geodesic variations. The corresponding Li-Yau inequalities then follow in the first-order viscosity sense.

We study the partially fractional Grushin-type operator \begin{equation} \mathcal L_{\alpha,\beta}u=(-\Delta_x)^\alpha u-(1+\beta)^2|x|^{2\beta}\Delta_y u, \qquad (x,y)\in\mathbb{R}^n\times\mathbb{R}^m,\quad 0<\alpha<1,\ \beta\ge 0, \end{equation} which couples fractional diffusion in $x$ with degenerate local diffusion in $y$. Working with the anisotropic dilations $(x,y)\mapsto(\rho x,\rho^{\alpha+\beta}y)$ and an adapted quasi-distance, we prove a uniform fractional Grushin-Poincaré inequality on anisotropic cylinders and, via a Riesz-type potential built from a telescoping sum over dyadic cylinders, a global Sobolev inequality at the sharp critical exponent $2^*_{\alpha,\beta}=2Q_{\alpha,\beta}/(Q_{\alpha,\beta}-2\alpha)$, where $Q_{\alpha,\beta}=n+(\alpha+\beta)m$. We then establish a concentration-compactness principle for the associated energy space. For $\beta>0$ the weight $|x|^{2\beta}$ destroys translation invariance in $x$, and we show that concentration is confined to the degenerate hyperplane $ \Sigma= \{(0,y):y\in\mathbb{R}^m\}$; this localization yields the existence of extremals for the sharp constant $S_{\alpha,\beta}(n,m)$.

We prove that for a geometrically connected modular curve $X_H$ of prime level larger than $311$, all automorphisms are modular, i.e., they arise from the moduli interpretation of $X_H$. The proof relies on a general criterion involving the gonality of $X_H$ and the dimension of the CM part of the Jacobian of $X_H$. For arbitrary level, we give algorithms to compute the modular automorphisms and, in many cases, to prove that there are no other automorphisms. Finally, we apply these algorithms to the curves in the database LMFDB.

While offline reinforcement learning (RL) enables policy optimization from static datasets without costly online interaction, it remains bottlenecked by the risk of executing out-of-distribution (OOD) actions. Recent approaches mitigate this by learning a behavior-cloning policy through flow matching and then performing RL within its constrained latent space. However, naively optimizing the latent policy can easily cause the policy to collapse into a brittle mode or exploit sharp artifacts of the learned critic. In this work, we find that entropy regularization is essential in latent-space RL for addressing these challenges. We introduce LASER, a novel offline RL algorithm that applies latent-space adjoint matching to achieve entropy-regularized latent-space RL with expressive flow policies while avoiding backpropagation through time. Through comprehensive experiments on 40 challenging OGBench tasks with varying dataset qualities, we show that LASER achieves state-of-the-art performance. Notably, LASER uses fixed method-specific hyperparameters across all tasks and outperforms the evaluated baselines, including those with task- and dataset-specific tuning, which highlights the robust applicability of LASER. Project website: https://mit-realm.github.io/laser/.

The paper studies learning in reservoir computing systems through the lens of the internal model principle. The key idea is to model training signals as outputs of an exogenous signal generator, thus connecting reservoir computing to output regulation, observer theory, and frequency-domain interpolation. For linear signal generators, we characterize the steady-state response of a linear reservoir through invariant subspaces, establishing connections with Luenberger observers and classical frequency-domain interpolation. For nonlinear signal generators, we obtain analogous results using invariant manifolds, connecting synchronization to Kazantzis-Kravaris/Luenberger observers and readout training to nonlinear moment matching. Case studies illustrate how our perspective explains possible instability of trained predictors and informs reservoir design.

We prove that every countable Borel equivalence relation containing a finite index Borel treeable subequivalence relation is Borel treeable. This answers a long-standing question of Jackson–Kechris–Louveau. Our proof is based on a median subalgebra filtration construction on products of trees, together with an extension of the median graph treeing algorithm of Chen–Poulin–Tao–Tserunyan to certain well-behaved increasing unions.

Toeplitz matrices are among the simplest structured models of translation invariance, arising naturally in convolution, stationary covariance models, and discretizations of translation-invariant operators. For a pair of real matrices, the Böttcher–Wenzel form (BW form) is a nonnegative quartic measuring the gap in the corresponding commutator inequality. László conjectured that, when both matrices are Toeplitz, this quartic is a sum of squares for every matrix order. We show that the conjecture is false. More precisely, we identify an explicit family of corner-supported Toeplitz pairs, retaining only the outermost diagonals on the two sides, for which the associated BW form is not a sum of squares at all sufficiently large orders. The proof reduces the complete family of Gram representations to a finite complex test: its Gram-invariant part develops a strictly negative limiting direction, while the remaining Gram-dependent contribution vanishes. We also give an explicit sufficient threshold for the matrix order. The negative result is complemented by two positive regimes. If either Toeplitz factor is symmetric or skew-symmetric, then the corresponding BW form is SoS for every order, with the other factor allowed to be an arbitrary real Toeplitz matrix. For unrestricted Toeplitz pairs, we further prove SoS representability for every $2\le N\le50$, with formal Lean~4 verification through order 20. The large-order non-SoS theorem, including the explicit sufficient threshold, has also been formally verified in Lean 4.

Experience with invariant finite-difference schemes for one-dimensional hydro\-dynamic-type equations for a function $u(t,x)$ suggests that nonlinear polynomial schemes on orthogonal uniform meshes with the 9-point stencil cannot simultaneously preserve difference analogues of the energy and momentum conservation laws, with the multipliers given by the standard difference approximations of $u_t$ and $u_x$. For the 9-point stencil this hypothesis is proved for polynomials of arbitrary degree. On larger stencils it is investigated by means of the difference analogue of the direct method: for polynomials of bounded degree, such schemes are found to exist only on sufficiently large stencils, and the second-order equations they approximate are of Monge–Ampère type. As a by-product, a scheme for the homogeneous Monge–Ampère equation is obtained that has five conservation laws, admits most of the symmetries of the differential equation, and is invariant under the discrete dihedral group $D_4$. Numerical tests on exact solutions confirm its second order of approximation and show that the conservation laws are satisfied on these solutions with the same order of accuracy.

We prove a spectral multiplier theorem for the standard sub-Laplacian on the Engel group under a scale-invariant local $H^s$ condition with $s>2$. The multiplier operators $F(\mathcal L)$ are of weak type $(1,1)$ and bounded on $L^p$ for every $1<p<\infty$. Together with the known lower bound, this identifies the sharp threshold as $2$, half the topological dimension. The proof combines a weighted Plancherel estimate for quartic oscillators with a decomposition into rank-two spectral subspaces, which controls parameter derivatives uniformly as one well escapes to infinity. For fixed smooth multipliers compactly supported in $(0,\infty)$, we also determine the leading term and first correction for the localized kernels. Whether the weak type $(1,1)$ estimate holds under the local $H^2$ condition remains open.

We develop a computationally tractable statistical framework for angular data supported on a recursively embedded curved torus $\mathbb{T}^n\hookrightarrow\mathbb{R}^{n+1}$. Due to the recursive structure we build an exact uniform sampler by generalizing the process from the curved 2-torus. We generalize projected normal distributions to the curved $n$-torus and we develop inference theory for these distributions. We utilize both maximum-log-likelihood estimates and energy-based Fréchet estimates for the projected normal distribution for this purpose. This is done by establishing identifiability for isotropic projected normal distributions and by applying theory of maximum likelihood and intrinsic statistics for the curved tori. We numerically verify the geometry, sampler, density transformation, and energy solver. Lastly, we demonstrate the applicability of the proposed model using real data.

We prove the convergence of the isoparametric Barrett–Garcke–Nürnberg (BGN) method for mean curvature flow of closed surfaces. For finite elements of degree $k\ge6$ and time steps $\tau=O(h^2)$, the numerical surfaces converge to the exact surfaces in the manifold distance with order $h^2$. The proof relies on two ingredients. First, we construct approximation flows by an expansion in time, whose interpolations satisfy the BGN method with an improved defect. Second, we derive a refined estimate for the tangential error from an algebraic identity for tangential test functions.

It is known that completeness of the coalitional ($C$-)desirability relation in a simple game does not imply the existence of a weighted-majority representation. We show that weightedness is implied in an approximate sense. Call an $n$-player simple game $\varepsilon$-weighted if, for some weight vector in the $(n-1)$-simplex, any losing coalition weight exceeds any winning coalition weight by at most $\varepsilon$; let $\varepsilon^{\ast}$ be the smallest such $\varepsilon$. We establish an upper bound on $\varepsilon^{\ast}$ in a general $C$-complete game, and deduce that, if all classes of equally desirable players become large asymptotically but the number of classes is bounded, then $\varepsilon^{\ast}=O(1/n_{\min})$ for the minimal player class size $n_{\min}$, and the bound is tight. Importantly, the minimal attainable fraction $\delta$ of coalitions mislabeled by a weighting rule also tends to $0$ when $n_{\min}\rightarrow\infty$ and the number of player classes is bounded, with $\delta=O(1/\sqrt{n_{\min}})$.

Let N(T) count the nontrivial zeros of the Riemann zeta function up to height T with multiplicity, N_d(T) the distinct ones, and N_0^s(T) those that are simple and on the critical line. We prove liminf N_d(T)/N(T) >= 1645064/1960733 = 0.83900...; earlier work proves 0.83699..., and a report we have not verified claims 0.83805.... Hence more than 67.80% are simple. Also liminf N_0^s(T)/N(T) >= 0.67353...; earlier work proves 0.67250..., and reports we have not verified claim 0.673492. Two estimates of Lamzouri are improved: at least 88.93% of the zeros are simple or on the critical line, and the proportions of simple zeros and of zeros on the line average at least 83.67%. By the unconditional form of Montgomery's pair-correlation theorem the energy, a sum of a test function over pairs of zeros, is known asymptotically. A zero of multiplicity d contributes d^2 to it and nearby zeros contribute too, so a lower bound for such pairs leaves less energy for multiple zeros. The proof has three steps. First, for zeros at least a fixed fraction of the mean spacing apart, a large sieve inequality lets their pairs be used in full, with multiplicities. Second, their contribution is bounded below by a computer-assisted inequality for seven or eight consecutive zeros which distinguishes simple and double zeros. Its correction terms telescope, as increments of a storage function. Third, the test function is chosen to make this contribution large, at the cost of more energy. We also prove upper limits for what such inequalities can give with the two main test functions. The lower bounds and these limits are proved in Lean 4, analytic inputs included. The proofs use three axioms beyond those of Mathlib, one per inequality. Each records that a search program returned true; Lean's kernel does not check the run itself. This work is an experiment in AI-assisted mathematical research.

We propose an algorithm for the numerical solution of the time-dependent Schrödinger equation for N >> 1bosons interacting via a two-body interaction potential in the mean-field scaling regime. Unlike existing numerical algorithms for quantum dynamics, the proposed algorithm has a computational complexity that is independent of the number N of particles. It is based on an ansatz for the wave function inspired by the Bogoliubov theory, which is an asymptotic theory for large N. In a first part, we follow Bogoliubov theory in evolving the condensate function by the Hartree equation, but we use the Dirac–Frenkel time-dependent variational principle to derive nonlinear equations of motion for the excitation functions in the Bogoliubov ansatz. By carefully exploiting the bosonic symmetry, the so obtained approximate wave functions are shown to have derivatives bounded independently of N apart from a time-dependent global phase factor which rotates with a frequency proportional to N. In the second part, we numerically approximate the wave function (modulo the phase) by a composite algorithm, which employs an appropriate splitting of the equations of motion and uses a dynamical low-rank approximation of the fundamental two-particle excitation function by linear combinations of products of single-particle functions. The low-rank excitation function is evolved in time by a basis update and Galerkin (BUG) integrator that is robust to small singular values of the coefficient matrix and allows for rank adaptivity. Numerical results are provided to assess the performance of the proposed algorithm.

For a finite irreducible rational transition matrix $P=(P_{ij})$ on $n$ states, the hunger game introduced by Li and Propp is a greedy chip-firing process in which each state $i$ carries a real-valued hunger $h_i$. At each step, a state $i$ of maximal hunger is selected, with ties resolved by a fixed order on the states. Firing $i$ subtracts one from its hunger and then adds $P_{ij}$ to the hunger of each state $j$. Let $\pi$ be the stationary distribution of $P$, and let $T$ be the least positive integer such that $T\pi$ is integral. We prove that every periodic orbit has least period $T$, with state $i$ firing $T\pi_i$ times, and that the set of periodic states of total hunger zero tiles the corresponding hyperplane by lattice translations. These results settle two conjectures of Li and Propp. We also prove an analogous least-period theorem for nontrivial periodic orbits of fixed-priority chip-firing on finite strongly connected digraphs, another greedy chip-firing model. We then find a connection between the hunger game and the chairman assignment problem: the greedy rule that always chooses the state furthest below its proportional target is precisely a rank-one hunger game. Motivated by this application and related allocation rules, we generalize the hunger game by assigning each state a score that is a nondecreasing function of its hunger and firing a state of maximal score each time. For these monotone-score hunger games, we prove that every periodic orbit still has least period $T$ and give a sufficient condition for eventual periodicity from every initial state.

In quantum reading, classical messages are encoded in sequences of quantum channels and recovered by probing the channels and processing their outputs. We study the reading capacity of a finite family of finite-dimensional channels under an average constraint on the probing energy, allowing arbitrary adaptive operations between channel uses. Using an input-dependent chain rule for the Belavkin-Staszewski relative entropy, we derive a converse bound expressed as an optimization involving channel Choi operators and the operator relative entropy. This formulation admits semidefinite approximations and reduces, without an energy constraint, to a channel information radius. To obtain achievable rates, we construct bilinear semidefinite lower approximations to a standard non-adaptive reading bound. Alternating optimization produces feasible probe states and encoding distributions, whose Holevo information gives a directly evaluable achievable rate. For jointly classical-quantum channels, we derive a separate converse based on the Umegaki relative entropy. Without an energy constraint, this converse matches a non-adaptive achievable rate, recovering a recent result of Pascual Abraldes and Winter: non-adaptive protocols suffice to achieve the unconstrained reading capacity of jointly classical-quantum channels.

We construct a separated logarithmic compactification of a divisor in the moduli of polarised irreducible holomorphic symplectic (IHS) varieties of generalised Kummer type in dimension at least four. This compactification carries a universal family of logarithmic IHS varieties, and its tropicalisation carries a universal family of tropical generalised Kummer varieties. The compactification is built from a more general construction of explicit, projective, and logarithmically smooth models for type~II and type~III degenerations of generalised Kummer varieties of arbitrary dimension and over base logarithmic schemes of any dimension. Over a base of any logarithmic rank, these degenerations come with a tropicalisation map to a family of integral affine manifolds with singularities, and with boundary in the type II case. This tropical family is constructed through a tropical generalised Kummer construction. Our approach uses logarithmic Hilbert schemes to extend the conventional generalised Kummer construction to families of logarithmic abelian varieties. We describe fibres of our degenerations, and thus every point in the boundary of our compactification. Proper models correspond to polyhedral structures on the associated tropical generalised Kummer variety, with the resulting polyhedral complex serving as the dual intersection complex; each irreducible component is constructed by applying the generalised Kummer construction to a product of logarithmic group compactifications of semi-abelian varieties.

Modern MIMO systems are evolving toward higher-dimensional and more flexible architectures, such as distributed and holographic MIMO, offering substantial benefits while giving rise to increasingly complex channel correlation structures. This growing architectural diversity makes it difficult to characterize the fundamental limits of different MIMO systems on a case-by-case basis, motivating a unified analytical framework applicable across a broad range of architectures. This paper addresses this research gap by investigating a generally correlated Rayleigh channel whose vectorized form follows a general Gaussian distribution with an arbitrary covariance matrix. For that purpose, we first prove that the spectral norm of the channel matrix is bounded in the asymptotic regime where the numbers of transmit and receive antennas grow proportionally. We then derive a deterministic approximation for the ergodic mutual information, with an explicit convergence rate governed by the structure of the channel correlation. The approximation is characterized by a pair of matrix-valued self-consistent equations, for which we establish the existence and uniqueness of the solution and propose an iterative numerical algorithm. Furthermore, we develop a framework based on positive linear maps to analyze the stability of these equations, which can facilitate the spectral analysis of broader classes of random matrices. The proposed characterization unifies many existing results for structured MIMO channels as special cases while remaining applicable to a broad class of general MIMO architectures that are difficult to evaluate using existing analytical frameworks. To demonstrate its utility, we apply the developed theory to two representative systems, namely downlink distributed MIMO and uplink heterogeneous MIMO. Numerical results confirm the accuracy of the derived deterministic approximations.

We introduce novel scalar generating potentials to compute three and four point cosmological correlators in orthogonal Grassmannian space. These correlators are obtained by applying appropriate directional derivatives on these potentials, combined with helicity-dependent minors. We emphasize that, this is not a spin-raising analog, but a much more powerful construction. The four point generating potential is associated with the geometry of a projective triangle. First-order directional derivatives generate gluon correlator, while the Hessian of the same scalar potential produces the four-graviton correlators, thus realizing a differential double copy. Thus, double copy is a natural consequence of the underlying scalar potential as the origin of the gluon and graviton four point correlators. This framework extends naturally to any spin-$s$. We exhibit a geometric origin for the gluon colour identities: the projective homogeneity of the scalar potential implies Kleiss-Kuijf identity, while the kinematic Jacobi identity follows as a consequence of the closure of the triangle under directional derivatives. We also uncover a connection between the BCFW bridge deformation of the scalar potential and directional derivatives that produces the spin-s correlator. Thus the generating potential provides a universal geometrical interpretation for any spin-$s$ four point Grassmannian correlator, unlike \cite{Bala:2026lvw} where geometric interpretation was only shown for the gluons.

Laurent phenomenon algebras were introduced by Lam and Pylyavskyy as a generalisation of cluster algebras. For a family associated with directed graphs, they showed that every cluster variable is a Laurent polynomial in the variables of any cluster and conjectured that these expansions have nonnegative coefficients. We prove this conjecture for all directed graphs without loops or multiple edges with respect to an arbitrary seed cluster.

We compute the higher-rank translation ($\mathcal{T}$) and rotation ($\mathcal{R}$) matrices associated with a genus-one spectral curve. Using these data, we reconstruct the worldsheet correlators of vertex operators in the corresponding supersymmetric Landau–Ginzburg (LG) theory. This provides a generalization of the genus-zero, one-cut spectral-curve construction to higher-genus geometries. The matrix-model correlators are mapped to integrals over the moduli space of Riemann surfaces for the symmetric sextic double-well potential, which has a double-cut solution. We further identify the rational and elliptic contributions to the worldsheet integrated vertex operators and show that each of the sector precisely matches the corresponding correlators of single-trace operators in the matrix model. Altogether, our construction showcases an explicit dictionary between the higher-genus spectral-curve data, the associated Cohomological field theory (CohFT), and the worldsheet description of the corresponding matrix-model observables.

We give a proof of optimal pointwise linear inviscid damping and vorticity depletion for a large class of spectrally stable shear flows $b(y)$ on the rectangular torus. This extends the linear inviscid estimates in \cite{CJWZ25} for Kolmogorov flow $b(y) =\cos y$ using completely different methods. Crucially, both methods remove an apparent logarithmic correction to the depletion and inviscid damping rates. Our method combines pointwise bounds for a parametrix with an explicit solution of a simpler inner problem which makes the removal of the logarithm clear.