Papers

We study the fractional Navier-Stokes equations with cubic damping in the specific Lei-Lin-Gevrey space $\mathcal{X}^0_{a,\sigma}(\mathbb{R}^3)$ (with $a\geq0$ and $\sigma\geq1$). We establish local and small-data global well-posedness, quantitative blow-up criteria, Gevrey loss-of-radius estimates, and perturbative stability in the critical endpoint case $\gamma=\frac12$ of the usual Navier-Stokes system.

We formalise in Lean 4, on top of Mathlib, the solvability of the Dirichlet problem for second-order linear elliptic operators in divergence form. The machine-verified results, with no sorry in the development, include the Poincaré inequality, the existence of weak solutions by the Lax-Milgram theorem, Rellich-Kondrachov compactness, the Fredholm alternative, the spectral theorem, interior regularity estimates, and the Sobolev embedding theorem. From these results we obtain a formalisation of classical solvability for sufficiently regular coefficients and data. Our Lean library includes a self-contained theory of Sobolev spaces developed independently of existing formalisations. Throughout the paper we associate each prose statement with the named machine-checked Lean declaration that discharges it.

We study entropy-regularized linear programs with sparse affine constraints and compare two exact dual representations induced by whether a redundant normalization constraint is retained or eliminated. Eliminating it yields a partially separable sum-exp dual with sparse affine factors but unbounded curvature. Our main result is an accelerated randomized block method whose acceleration preserves sparse data-access cost. The key ingredient is a global entropy-specific Hessian–gap bound that remains valid for signed constraint matrices, admits computable block-overlap refinements, and provides deterministic curvature certificates before sampling. This geometry supports square-root importance sampling and an exact lazy implementation in which sparse arithmetic cost is weighted by local curvature rather than by a worst-block factor. We further introduce a gap-dependent curvature dimension that connects the generalized-smooth regime to the classical effective-rank intuition for coordinate methods. Retaining the normalization constraint yields a log-sum-exp dual with globally bounded higher derivatives, enabling gradient-regularized and cubic-regularized Newton methods. Experiments compare the resulting methods in terms of coordinate epochs, arithmetic operations, and wall-clock time, separating the effects of acceleration, touched-data sparsity, and problem-specific structure.

Recurrent fractal interpolation extends the usual fractal interpolation framework by introducing a recurrence structure. Fractal histopolation, on the other hand, is concerned with the area-matching properties of fractal functions. In this paper, we combine these two approaches and introduce a recurrent fractal histopolants.

Motivated by recent developments in generalized Kähler geometry, we study morphisms of double Lie groupoids from a simplicial viewpoint. We show that the codiagonal functor $\Wbar$ extends from objects to horizontal and vertical principal bibundles and to square-shaped $(1,1)$-morphisms. The latter determine 2-dimensional morphisms of square and globular shape between anafunctors of Lie 2-groupoids. We illustrate these constructions through two applications: First, we revisit the equivalence between two models of nonabelian gerbes: groupoid bundle gerbes and principal 2-bundles. We show that a principal 2-bundle and its associated groupoid bundle gerbe determine equivalent anafunctors from a manifold to the corresponding structure Lie 2-group. Second, we refine the integration of Manin triples in transitive Courant algebroids. We show that integrations associated with different choices of suitably transverse Manin triples in the same Courant algebroid are related by symplectic Morita equivalences. Consequently, the resulting integration of the background Courant algebroid is well defined up to symplectic Morita equivalence.

The theory of polyptych lattices seeks to incorporate, into a single combinatorial framework, the piecewise-linear bijections (mutations) that appear in many theories that generalize toric geometry, such as Newton-Okounkov bodies, toric degenerations, cluster varieties, and other areas. The link to algebraic geometry comes from the accompanying notion of detropicalizations of polyptych lattices, and their associated compactifications. The main result of this manuscript gives a concrete construction of a strict dual pair $(\mathcal{E}, \mathcal{F})$ of polyptych lattices, which we call a Gorenstein PL cone extension. To define our construction, we introduce the notion of Gorenstein PL cones, which are a polyptych analogue of a Gorenstein cone in the classical setting. If the original polyptych lattices are detropicalizable, then the new polyptych lattices $\mathcal{E},\mathcal{F}$ are also detropicalizable, via a simple explicit formula. Our Gorenstein PL cone extensions give a rich source of examples of strict dual pairs of polyptych lattices. In particular, any pair of Gorenstein-Fano polytopes $\Delta,\Delta'$ of dimension $n,n'$ respectively, gives rise to a strict dual pair $(\mathcal{E}(\Delta,\Delta'), \mathcal{F}(\Delta,\Delta'))$ of detropicalizable polyptych lattices of rank $n+n'+1$. Other examples can be built from cluster data.

In the nonlinear source of the pressure Poisson equation the fluctuating enstrophy and strain nearly cancel. We ask where the cancellation is complete, which we call silent, and how the source escapes it. Pointwise the source depends only on the eigenvalues of the velocity gradient. Pure shear has none, so it is silent. It is the origin of the normalised invariant plane, whose second-invariant coordinate is the normalised source. Let $f$ be the shear fraction of the gradient and $m$ its distance from that origin. We prove that $m\le1-f\le3m$ at every point of every incompressible flow. Wherever the source nearly vanishes and the third invariant is small, the gradient is nearly pure shear. Otherwise the cancelling gradient is a balanced vortex, a swirl its own strain cancels at any strength. At a no-slip wall the silent object is exact: the wall shear-stress fluctuation, a rank-one sheet. Away from the wall we assume the same sheet, dressed with a small residual, and derive how it escapes. The escape passes through one element of the residual, the along-sheet variation of the sheet-normal velocity. Its amplitude is the geometric mean of sheet and residual, and its sign selects swirl or strain. A rank-two silent object has the same source through the same gates but balanced eigenvalues, so the assumption fixes the escape's topology, not its source. The shear fraction is thus a pointwise proxy for cancellation by shear, and a model of the source must be built on what escapes.

This paper addresses fundamental questions about the existence and structure of isotropic and anisotropic spherical random fields and the properties of composite transformations of isotropic fields. It introduces a hierarchical structure comprising several classes of anisotropic random fields, using geometric and spectral approaches, and provides a geometric–spectral classification of second-order anisotropy on $\mathbb{S}^2$. We characterize feasible isotropic and anisotropic scenarios and identify those that, in principle, cannot occur, in contrast to the case of random fields in Euclidean space. Furthermore, we establish connections between geometric invariance properties and the structure of the spectral covariance matrix of spherical random fields. The obtained results lay the foundations for applications in statistics and data analysis, providing guidance on feasible models, their spectral representations, and appropriate sampling schemes for parameter estimation or anisotropy hypothesis testing for spherical data.

In this note, we consider the advection-diffusion equation on the $d$-dimensional torus, where $d$ is arbitrary. First, for the associated transport equation, we provide an explicit example of a space-time smooth, divergence-free, random velocity field that is a universal exponential mixer. Second, we set the molecular diffusivity to a (small) positive value, and we show that the same velocity field is dissipation enhancing with optimal rate. Our construction is based on the Random Dynamical Systems viewpoint, and relies on the uniform ergodicity of the two-point Markov process associated with the two-dimensional Pierrehumbert model.

Difference-in-means steering requires activations recorded while a model shows the desired behavior, which a sandbagging model withholds by deliberately underperforming. We introduce Activation Flow (ActFlow), which manufactures these activations from $k$ correct labels without fine-tuning. ActFlow sets target logits that rank each labeled item's correct answer first, and moves the logits toward them by adding one vector $x$ to all $k$ residual streams at one layer. ActFlow is a family of ordinary differential equations for $x$, one for each rule that maps the required logit change to the velocity of $x$. The smallest-norm rule lands exactly on the targets, while the others keep only the top singular directions of the Jacobian. We test ActFlow on three instruction-tuned models, each locked by a sandbagging prompt and by a password-locked LoRA. At $k=40$, ActFlow keeping five singular directions raises the mean held-out ARC-Easy accuracy over the six locked models from $0.05$ to $0.85$, against $0.88$ for rank-$16$ LoRA fine-tuning, which trains about $3{,}000$ times as many parameters, and $0.92$ for the honest models. Furthermore, it scores higher than the smallest-norm rule in 16 of the 18 combinations of locked model and $k$, and its steering direction is nearly orthogonal to the honest difference-in-means direction. It also unlocks two LoRA locks where the honest direction fails. Surprisingly, with one labeled item, the single-step smallest-norm rule raises the Qwen2.5-7B prompt lock from $0.04$ to $0.59$.

Let $m\geq1$ be fixed. We consider the space $\mathcal{P}_m(\mathbb{R}^n)$ of real $m$-homogeneous polynomials on $\mathbb{R}^n$, endowed with the standard Gaussian measure $\gamma_{m,n}$ associated with the Bombieri norm. We study how well the norm of a typical polynomial on the unit ball of $\ell_\infty^n$, namely the cube $[-1,1]^n$, can be recovered from its values at the vertices. For $P\in\mathcal{P}_m(\mathbb{R}^n)$, set \[ M(P)=\max_{x\in[-1,1]^n}|P(x)|, \qquad V(P)=\max_{\varepsilon\in\{-1,1\}^n}|P(\varepsilon)|. \] If $P_n$ is chosen according to $\gamma_{m,n}$, we prove that the relative loss \[ 1-\frac{V(P_n)}{M(P_n)} \] is of order at most $n^{-1/2}$ in probability. Consequently, for every $0<\beta<1/2$, \[ \gamma_{m,n}\left\{ P\in\mathcal{P}_m(\mathbb{R}^n): V(P)\geq (1-n^{-\beta})M(P) \right\} \longrightarrow1 \] as $n\to\infty$. Thus, with respect to the Bombieri Gaussian measure, the vertices of the cube are asymptotically norming for homogeneous polynomials of fixed degree.

Fix a square-free monomial $m = x_{i_1} \cdots x_{i_s}$ in $S = \K[x_1, \dots, x_n]$, where $\K$ is a field. The ideal generated by all square-free monomials that are Borel moves of $m$ is referred to as the square-free principal Borel ideal, denoted by $\mathrm{sfBorel}(m)$. While the Waldschmidt constant of $\mathrm{sfBorel}(m)$ has been partially studied in the literature, we consider the remaining cases and provide bounds for the Waldschmidt constant of $\mathrm{sfBorel}(m)$. In some cases, we explicitly compute its exact value. Furthermore, we study this invariant for square-free Borel and square-free lex-segment ideals, establishing both bounds and exact values for special cases.

The current deep learning training paradigm employs end-to-end backpropagation, regardless of the training stage, i.e. pre-training or fine-tuning. However, backpropagating through the entire model is neither biologically plausible nor memory efficient, since learning inside the brain is highly localized. Therefore, we propose LocalProp, a training procedure that locally updates the weights of a model. Our neuro-localized weight updates follow the "pre-training then fine-tuning" paradigm, where the pre-training is based on I-JEPA. After locally updating the weights, a pruning operation is performed, followed by a short final fine-tuning phase. Pruning helps by sending the learning signal from higher blocks to lower blocks. We perform experiments on several datasets, including large-scale benchmarks such as ImageNet, and empirically show that LocalProp reaches good performance at a fraction of GPU peak memory. By varying the number of jointly optimized blocks, we identify gradient-propagation span as a practical control over the accuracy-memory trade-off.

We prove an exponential strong converse for the unassisted private capacity of every finite-dimensional degradable quantum channel. We use a single error criterion: the infidelity between the actual message-estimate-environment state and an ideal state consisting of uniformly distributed, perfectly correlated messages that are independent of the environment. At every rate strictly above the channel's maximum coherent information, the optimized fidelity to this ideal state decays exponentially in the number of channel uses. The same bound holds when the ideal environment state is fixed to the actual environment marginal. The proof applies to arbitrary mixed-state encoders and arbitrary collective decoders. Its main operator ingredient is an overlap bound for privacy tests: two receiver placements with a common subsystem of dimension $d_C$ have overlap at most $d_C/M$, where $M$ is the key size, independently of both shield dimensions. We combine this bound with the signed averaging construction presented in arXiv:2608.01308 and a single-filter reduction to symmetric states. A Gaussian filter replaces the common receiver subsystem by a system of controlled dimension from which the original correlations can be approximately recovered using quantum side information. The filter preserves exact exchange symmetry, and a uniform smooth-entropy estimate bounds its output dimension for arbitrary correlated channel inputs. We also obtain an exponential strong converse at zero for antidegradable channels under the same joint criterion and, consequently, for the private capacity of the quantum erasure channel throughout its parameter range.

Let $r\geq11$ be a prime. We show that there are infinitely many integers $C$ for which the Fermat-type equation $$ x^r+y^r=Cz^p $$ has no non-trivial primitive solutions for all sufficiently large (in terms of $r$ and~$C$) prime exponents~$p$. The proof uses several Frey curves to force simultaneous Frobenius trace equalities at the primes above~$3$; a new trace separation argument shows that these equalities imply $3\mid x+y$, after which a further Frey curve and level lowering give a contradiction with the Ramanujan bound.

This work considers strong and weak stochastic Galerkin approximations of random domain problems for the elliptic random partial differential equation (PDE). The random domain problems are realized by sufficiently regular random domain mapping, allowing a transformation on deterministic reference domain. A traditional numerical method for solving the resulting high-dimensional coupled stochastic Galerkin systems is replaced by deep learning techniques. We compare a physics-informed neural network approach, based on the strong stochastic Galerkin residual, with a Deep Ritz approach, based on the weak stochastic Galerkin system, reformulated as Ritz energy minimization. The neural networks serve as surrogates for the deterministic spectral coefficients of the respective stochastic Galerkin solution. The stochastic parameters are not used as neural network input parameters and the stochastic domain is not sampled during training. The efficiency of the methods is demonstrated on a randomly stretched interval and a randomly deformed annulus in two spatial dimensions.

In this paper, we show that a knot with a nontrivial Jones polynomial of span at most 11 does not admit purely cosmetic surgery.

We give a complete classification of proper biharmonic Lagrangian $H$-umbilical submanifolds in complex space forms. Such submanifolds exist only in complex projective space and are locally congruent to an explicit model obtained via the Hopf fibration. In particular, their nonexistence in complex Euclidean space provides a partial affirmative answer to Chen's conjecture.

We present a fully GPU-oriented SVD pipeline based on polar decomposition, motivated by iterative methods that rely solely on matrix multiplications, such as the Newton-Schulz iteration. We show that this approach enables up to a $2\times$ speedup compared to standard implementations. Furthermore, we derive a numerically stable backward pass for the polar decomposition and leverage it to obtain a fully differentiable SVD. Our methods are released as open-source implementations in both PyTorch and JAX: https://github.com/fallnlove/cans_svd.

The planar loop equations of a Hermitian one-matrix model leave finitely many low moments undetermined. Hankel positivity constrains these moments but does not reveal how a fixed multicut family is embedded in moment space. We introduce a finite-plane diagnostic using the Cauchy kernels $f_n(x)=(z_n-x)^{-1}$ which we use as spectral probes. Their Gram matrix is the positive-semidefinite Pick matrix $P_{mn}$, whose positivity defines a finite-node bootstrap analogous to the Hankel bootstrap. For a fixed regular multicut topology, the nonbranching double zeros of the spectral discriminant, which we call dressed saddles, impose transverse constraints on the corresponding moment-space locus, while its remaining directions are filling fractions. Writing $z=E+i\eta$, $E$ selects a spectral region and $\eta>0$ sets a continuous resolution scale. We use $P(z,z)$ as a finite-plane spectral response and interpret $\nabla_{\mathbf m}P(z,z)$ as a moment-space susceptibility. Near a regular real dressed saddle, this susceptibility is enhanced as $\eta^{-2}$ and aligns with a conormal to the filling-fraction manifold, whereas the leading enhancement cancels along tangent deformations. Responses near several independent saddles can therefore reconstruct the conormal space and, through their common kernel, its tangent space. Combining Pick positivity with reality and analyticity of the resolvent, we obtain alternative analytic derivations of known planar results in quartic and sextic models. The asymmetric quartic makes the one-dimensional filling-fraction geometry explicit, while sextic models exhibit independent conormal directions. These results clarify how local consistency conditions constrain the low moments before global period matching and the equilibrium variational inequality select the equilibrium measure.

We study the survival of a branching random walk in a supercritical Galton-Watson tree subject to a deterministic, superlinear killing barrier with heavy-tailed displacements. For a fixed $m>0$, we consider a family of such barriers and kill particles whose ancestral paths fall below the barrier. Under some standing assumptions, we establish a sharp phase transition at $m=4$: the process becomes extinct almost surely for $m<4$, while survives with positive probability for $m>4$. Notably, a natural first-moment heuristic suggests the threshold $m=2$; the true critical value $4$ arises from a deterministic constraint along infinite surviving rays. At the critical value $m=4$, we provide examples showing that the same standing assumptions do not determine the survival behavior. The proofs are based on analysis of infinite surviving rays and embedded trees.

We study smoothed online prediction with bounded adversarial responses. This widely studied framework bridges i.i.d. sampling and adversarial covariate selection through a smoothness parameter $\textsf{C}_{\textsf{cov}}$, which bounds conditional covariate densities relative to a fixed, unknown base measure. We propose Hedge-Cover, an information-theoretic algorithm that achieves sublinear regret $\widetilde{O}(\sqrt{\text{Pdim}(\mathcal{F}) \textsf{C}_{\textsf{cov}} T})$ for function classes with bounded pseudo-dimension. The algorithm aggregates a carefully constructed family of experts using Hedge, with a prior that links regret to the number of disagreements between a consistent selector and a target function. We bound this number by exploiting covariate smoothness. This answers an open problem posed in \cite{blanchard2025agnostic} on the minimax optimal adaptive regret of the smoothed online regression problem. We establish a matching lower bound for the class of linear predictors. The main intricacy of the lower bound lies in explicitly constructing a challenging sequential covariate distribution supported on mutually orthogonal hyperplanes. This construction may be of independent technical interest. Finally, we revisit the well-specified setting and quantify the effect of response noise. For conditionally $\nu^2$-subGaussian responses, we extend the existing lower bound under realizable responses by showing that the minimax expected regret is $\Omega((1\vee \nu)\sqrt{(\textsf{C}_{\textsf{cov}}-1)dT})$ for a function class of VC dimension $d$. A corresponding upper bound for ERM matches this dependence on $\nu$.

Let $P_q$ be the graded polynomial algebra $\mathbb F_2[x_1,x_2,\ldots ,x_q]$ over the prime field of two elements, $\mathbb F_2$, with the degree of each $x_i$ being 1. We study the hit Peterson problem of finding a minimal set of generators for $P_q$ as a module over the mod-$2$ Steenrod algebra, $\mathcal{A}.$ In this note, we present the minimal sets of $\mathcal{A}$-generators for $P_5$ in the cases where the degree are sufficiently general.

We show that the constraint satisfaction problem of any finite conservative structure that can not pp-construct st-Con (whose CSP is NL-complete) or 3Lin2 (whose CSP is Mod2L-complete) is solved by a symmetric linear Datalog program and therefore in L.

We show that optimally pseudorandom $K_4$-free graphs of order $n$ and degree $d = \Theta(n^{4/5})$ exist by constructing a graph in the split Cayley hexagon, matching the known upper bound. This resolves the first open case for $K_k$-free graphs after $k=3$ for which Alon gave a tight construction in 1994. This has a variety of implications for $K_4$-free pseudorandom graphs. Furthermore, it implies an explicit lower bound on the Ramsey number of $r(4, t) \geq t^{1.\overline{6} - o(1)}$, improving the previous record by Kostochka, Pudlák, and Rödl of $r(4, t) \geq t^{1.6-o(1)}$.

Motivated by the modern characterization of the Markoff spectrum via mechanical words and the Markoff Uniqueness Conjecture, we study the Lagrange constants of continued fractions generated by a parameterized family of substitutions $\phi_{a,b}$ defined by $0 \mapsto aa$ and $1 \mapsto bb$ with $b = a + 1$. Based on a three-fold arithmetic classification of pairs of rational numbers $0 \le x < y \le 1$, we investigate the order relations between the corresponding Lagrange constants $\mathcal{L}([\phi_{a,a+1}(G(x))])$ associated with the mechanical words $G(x)$. While an inequality is established for pairs of the second type (type (2)) whenever $a \ge 1$, the comparisons for type (1) and type (3) pairs hold for $a \ge 2$. Consequently, for any integer $a \ge 2$, where the order relations across all three types are completely determined, we establish an analogue of the Markoff Uniqueness Conjecture: the map $x \mapsto \mathcal{L}([\phi_{a,a+1}(G(x))])$ is injective on $\mathbb{Q} \cap [0, 1]$. Furthermore, supported by extensive numerical computations, we propose a general conjecture on the order relations for arbitrary $b > a$, highlighting a sharp phase transition in ordering behavior at the boundary $b = a^2 + 2$.

In sampling, different flavours of annealing, tempering, or other successive approximation approaches are widely used and studied. In this work we investigate annealed underdamped Langevin sampling for logconcave distributions of the form $\pi(x) \propto e^{-\Psi(x)}$ for a potential $\Psi:\mathbb{R}^d \rightarrow \mathbb{R}$. That is, we assume access to the score of a general approximating family of distributions $(\pi_{\tau})_{\tau\in [0,1]}$ with $\pi_0=\pi$. This family is used in a time-inhomogeneous underdamped Langevin process with moving target $\pi_{\tau(t)}$ and an appropriate annealing schedule $t\mapsto \tau(t)$ such that $\tau(t)\downarrow 0$ as $t\to T$ for some $T>0$. While the degeneracy of the Brownian motion prohibits conventional Girsanov arguments, we are able to circumvent this issue by relying on an appropriately designed auxiliary process. To the best of our knowledge these are the first (quantitative) results for sampling using time-inhomogeneous underdamped Langevin dynamics. Moreover, we obtain an iteration complexity of $\mathcal{O}(\epsilon^{-2})$ in total variation distance in comparison to $\mathcal{O}(\epsilon^{-6})$ for annealed overdamped Langevin sampling. While we currently assume convexity, our proof strategies are amenable to generalization for non-logconcave targets and enable for several future research directions.

For area-$4\pi$ metrics on the two-sphere, we obtain quantitative refinements of classical eigenvalue inequalities and study the joint behavior of the first two Laplace eigenvalues. We prove a sharp quadratic improvement of Hersch's reciprocal-sum inequality in terms of the first eigenvalue, together with a simultaneous quadratic estimate for the first three eigenvalues and a geometric stability result for admissible measures. We also establish a quantitative refinement of Nadirashvili's second-eigenvalue inequality with sharp exponential scale. Finally, using explicit spectral families due to Petrides, a degree argument, and a refined finite-dimensional matrix method, we obtain nontrivial inner and outer bounds for the joint range of $(\lambda_1,\lambda_2)$.

[An extended abstract is provided in-text] Motivated by topological insulators, this paper analyzes classification schemes for projection-valued maps on $d$-dimensional tori, $d \le 2$, subject to Altland-Zirnbauer-Cartan (AZC) symmetries (time-reversal, particle-hole, and chiral). The main objective is to compare three notions of equivalence for these maps: Murray-von Neumann, unitary, and homotopy equivalence. The analysis relies on constructing symmetric bases for the projection ranges under a pseudo-periodicity constraint, where select basis vectors pick up a phase when looped around the torus. If $d=1$, fully periodic symmetric bases exist across all AZC classes. If $d=2$, topological obstructions appear as non-trivial phases characterized by $\mathbb{Z}$ (Chern number) or $\mathbb{Z}_2$ (Fu-Kane-Mele) invariants. These serve as absolute invariants that classify both Murray-von Neumann and unitary equivalences. For homotopy equivalence, the behavior depends on the ambient Hilbert space: if this has non-minimal dimension, homotopy equivalence reduces directly to unitary equivalence, while in minimal ambient spaces additional relative homotopy invariants emerge. Their precise values change under unitary conjugation, leaving only relative topological phases between maps well-defined. Ultimately, this provides a unified geometric framework for gapped quantum systems in low dimensions, demonstrating how distinct equivalence relations generate specific absolute and relative topological invariants.

Many learning problems couple a high-dimensional block of parameters with a handful of adversarial or dual variables: worst-group risk over a few groups, or learning under a few constraints. We consider $\min_{x\in X}\max_{y\in Y} f(x,y)$ with $X\subseteq\mathbb{R}^n$, a compact convex set and strongly convex with condition number $\kappa$, and we count the oracle calls for $x$ and for $y$ separately. The textbook approach runs a cutting-plane method in $y$ and solves every inner problem to accuracy $\varepsilon$ with an accelerated method; it needs $O(m\log(1/\varepsilon))$ calls for $y$ but $O(m\sqrt{\kappa}\log^2(1/\varepsilon))$ calls for $x$. We show that the inner problems need not be solved accurately. Our method, certificate transport, keeps a strongly convex lower model of a single slice and uses it as a prior for a short accelerated run on the next slice. By concavity in $y$, every call for $y$ then either cuts the localizer or moves the lower model to a mixture of the two slices with a certified increase of the lower bound. For $m=1$ this gives an $\varepsilon$-saddle point after $O(\sqrt{\kappa}\log(1/\varepsilon))$ calls for $x$, up to an initialization term, and $O(\log(1/\varepsilon))$ calls for $y$; both counts are optimal, even though $f(x,\cdot)$ is only assumed to be concave and Lipschitz. For general $m$, with centers of gravity of the localizer treated as computable, the method needs $O((m+\sqrt{m\kappa})\log(1/\varepsilon))$ calls for $x$ and the optimal $O(m\log(1/\varepsilon))$ calls for $y$. Under strong concavity in $y$, a two-point accelerated method makes the number of calls for $x$ independent of $m$.