Papers

The discretization of continuous frames in Hilbert spaces is well established, while the corresponding problem for Hilbert $C^*$-modules presents additional difficulties arising from the $C^*$-algebra-valued inner product and the structure of compact operators on Hilbert modules. In this paper, we develop an operator-selection framework for the discretization of continuous frames in Hilbert $C^*$-modules over finite-dimensional $C^*$-algebras. Our main result is a trace-controlled selection theorem for positive compact operators acting on countably generated Hilbert $C^*$-modules over finite-dimensional $C^*$-algebras. In particular, we establish a module version of the binary selection principle for positive trace-controlled compact operators. The proof combines finite-rank approximation, the matrix representation of finite-dimensional $C^*$-algebras, and operator estimates adapted to the $C^*$-module setting. We then apply these selection results to the discretization of continuous Hilbert $C^*$-module frames. Under suitable assumptions on the underlying metric measure space, with the coefficient algebra $\mathcal A$ finite-dimensional, and assuming that every closed submodule of $\mathcal H$ is orthogonally complemented. The resulting sampling family is a Hilbert $C^*$-module frame with explicit frame bounds and controlled separation.

While the maximal coding rate reduction (MCR$^2$) objective has become a useful principle for learning compact and discriminative representations, a theoretical understanding of its optimization geometry remains limited, especially regarding why first-order methods often exhibit fast convergence near its local maximizers. In this work, we show that the MCR$^2$ objective satisfies an error-bound property in a neighborhood of its local maximizers. This property guarantees that the distance to the local maximizer set can be bounded above by the gradient norm at any point in the neighborhood, directly implying regularity conditions such as the Polyak–Łojasiewicz inequality and quadratic growth. Leveraging this property, we prove that first-order methods converge linearly to a local maximizer of the MCR$^2$ objective under mild conditions. Numerical experiments validate the predicted local convergence behavior and provide empirical evidence that optimizing the MCR$^2$ objective improves representation geometry while maintaining competitive downstream accuracy.

A backbite on a Hamiltonian path of a graph $G$ adds an edge of $G$ from an endpoint $u$ of the path to a vertex $v$ and deletes an edge of the path at $v$ so that the result is again a Hamiltonian path. We study the connectivity of the reconfiguration graph $R(G)$, whose vertices are the Hamiltonian paths of $G$, with two adjacent when a single backbite carries one to the other. We prove that $R(G)$ is connected whenever $G$ contains the square of a Hamiltonian path as a spanning subgraph, and that any two Hamiltonian paths can be reconfigured using $O(n^2)$ backbites. Combining the theorem with known results on spanning squared paths gives connectivity for every graph with $\delta(G)\ge\lceil\frac{2n-1}3\rceil$, for the circulants $\mathrm{Circ}(n;S)$ with $a,2a\in S$ for some $a$ coprime to $n$, and, with high probability, for the random graph $G(n,p)$ with $p\ge c/\sqrt n$, for a sufficiently large constant $c$. In the other direction, graphs with arbitrarily high connectivity may have disconnected reconfiguration graphs. For every $n\ge8$ and every $1\le m\le\lfloor n/2\rfloor-3$ we construct a graph on $n$ vertices with $\kappa=\lambda=\delta=m$ whose reconfiguration graph is disconnected, with at least $(n-2m-4)!$ components. We conjecture that $\delta(G)\ge n/2$ suffices for connectivity.

In 1949, De Bruijn and Erdős considered infinite stick-breaking sequences. In this work, we consider finite sequences and prove that an infinite sequence that is due to De Bruijn, Erdős, and Toulmin is in some sense unique.

Secure device-to-device (D2D) caching requires users to generate delivery signals from their own caches and to recover their requested files without learning the other files. We construct schemes that meet both requirements when communication is limited to nearby users. The network consists of $K$ users on a cycle, a library of $N$ independent files, and a reception radius $r$. Delivery uses one round of noiseless orthogonal local broadcasts, with no server transmission. Each user's cache has capacity $M$ file units. An outer secret-sharing code assigns file shares to senders, and a secure local caching scheme at each sender delivers the requested shares. The cache budget includes the sender's library representation, random state, and delivery keys. We prove that each receiver's cache and all audible broadcasts jointly reveal no information about nonrequested files, despite the dependence between the outer shares. The converse counts transmissions crossing a contiguous user block and uses secrecy to remove one user's observation from the entropy bound. Let $h=2r$ and $K\ge h+1$. The private coded caching (PCC) instantiation attains the minimum peak sum load $K/h$ for every $M\ge Nh$ when $N>1$. For $N\ge K$, its memory-sharing envelope is within a factor $15$ of the optimum throughout $M\in[N/h+2,Nh]$. We also apply lifting without secrecy constraints, achieving the optimal load on $M\in[hN/(h+1),N]$ and a factor-$9/2$ guarantee on $M\in[N/(h+1),N]$ when $N\ge K$.

This paper develops a mesh-free numerical method for the one-phase Mullins-Sekerka flow, in which the interface bounding a set moves as the Wasserstein gradient flow of its perimeter. One difficulty in computing this flow is that two pieces of the interface with opposite normals form a hidden boundary when components merge. Our approach represents the interface by an oriented point cloud, which we treat as an oriented varifold. We introduce a quantity that measures how much nearby normals cancel each other and use it to detect the hidden boundary. Based on this quantity, we construct a perimeter estimator, prove that its expected error decays at an algebraic rate in the number of points, also in the presence of a hidden boundary, and use it in a minimizing movement scheme for the flow in two dimensions. Numerical experiments show that the scheme computes mergers of components through the hidden boundary, while the perimeter term in the scheme decreases at every step and the area is nearly conserved. As hidden boundaries also appear in other perimeter-driven interface problems, we expect our oriented representation to be applicable to them as well.

Let $f(d)$ be the smallest integer such that every connected bridgeless graph of diameter $d$ admits an orientation with directed diameter at most $f(d)$. Chvátal and Thomassen (1978) proved $\left\lceil{d^2}/{2}\right\rceil+d\le f(d) \le 2d^2+2d$. In this paper, we prove that $f(d)\le \left\lceil d^2/2\right\rceil+6d$, which shows $f(d)= d^2/2+\Theta(d)$ and determine the leading quadratic coefficient of $f(d)$. The key method of our proof is to construct a central subgraph $H$ such that it admits a strong orientation of diameter $O(d)$ and the distance between $H$ and every vertex outside $H$ is at most $ d/2 $.

Pointwise Mutual Information (PMI) is a core measure of testing word association, and Skip-gram with Negative Sampling (SGNS) is essentially a method that implicitly factorizes a shifted PMI matrix. However, a systematic and well-rounded characterization of finite-sample uncertainty in PMI estimation remains absent and imperative to venture into. We provide a statistical framework for PMI estimation and its connection to SGNS. We prove consistency, asymptotic unbiasedness, and asymptotic normality of the empirical PMI estimator, derive its variance via the Delta method, and, applying stochastic approximation theory, obtain a variance decomposition for SGNS-based PMI estimation that separates data variance from optimization variance. Simulation experiments validate the Delta method approximation. Real-data experiments on the Brown Corpus (d = 100) reveal that SGNS systematically deviates from the theoretical relationship PMI + log K. The empirical relationship shows an attenuated PMI coefficient, an amplified log K effect, and a positive intercept, indicating systematic bias. Word analogy validation confirms the models are effective. The failure to validate the variance decomposition under low-dimensional conditions does not diminish its theoretical value; rather, it identifies the unbiasedness assumption as the key bottleneck and clarifies the gap between asymptotic theory and practice, providing implications for both practice and theory.

The classical measure concentration phenomenon on the standard sphere $\mathbb{S}^n$ asserts that the volume of any tubular neighborhood of a totally geodesic equator $\mathbb{S}^{n-1}$ concentrates almost the entire volume of $\mathbb{S}^n$ as the dimension $n$ grows to infinity. In this paper, we extend this \lq\lq Fat Equator Effect” to arbitrary closed minimal hypersurfaces $\Sigma^{n-1} \subset \mathbb{S}^n$. By combining the analytic framework of Brownian motion and its Mean Exit Time (MET) function with the geometric tool of Schwarz symmetrization, we establish that minimality alone is sufficient to guarantee measure concentration around $\Sigma^{n-1}$ in high dimensions.

We construct a quasi-Banach space with a semi-greedy Markushevich basis which is not quasi-greedy, and hence not almost greedy. This gives a negative answer to the question of whether the equivalence between semi-greediness and almost greediness for Markushevich bases in Banach spaces extends to the general quasi-Banach setting.

We define a renormalization operator for pairs of commuting orientation preserving circle diffeomorphisms of class $C^3$. The operator is driven by the accelerated Brun multidimensional continued fraction algorithm. For rotation vectors of simultaneously bounded type, under a summable strong convergence condition satisfied by all eventually periodic types with Pisot period matrix, the renormalizations converge exponentially fast in $C^2$ to pairs of rigid rotations. The pair is simultaneously $C^{1+\gamma}$ conjugate to the pair of rotations, with $\gamma>0$ depending only on the bounded type constant. A fast renormalization extends exponential convergence and $C^1$ rigidity to a class of rotation vectors of full Lebesgue measure, containing explicit vectors outside bounded type. We show that the operator is topologically hyperbolic. The partition of the renormalized pairs by rotation vector is an invariant lamination whose leaves are the topological conjugacy classes. All results hold for pairs of class $C^{2+\alpha}$.

A second-order exponential low-regularity integrator is proposed for the incompressible Navier–Stokes equations. The method requires only linear solves and satisfies a discrete energy-decay property. Second-order temporal convergence is established under a uniform $H^4$ solution bound, with an error constant that remains bounded as the viscosity tends to zero. For the Fourier–Galerkin discretization, the optimal spatial convergence rate is established for fixed viscosity. The method can also be extended to conforming finite element methods while retaining the discrete energy-decay structure. Numerical results confirm the convergence orders and robustness with respect to the viscosity.

We present an oracle-optimal method for finite-sum monotone inclusions under mean-square Lipschitz continuity. Our switching regularization method finds a point $y$ and a certificate $g\in G(y)$ with $(\mathbb{E}\|F(y)+g\|^2)^{1/2}\le\varepsilon$ using $\mathcal{O}(n+\sqrt{n}LR/\varepsilon)$ expected component evaluations and resolvent evaluations. It removes the additive $n\log n$ cost of restarting a variance-reduced solver at every regularization stage by switching to a centered stochastic proximal iteration at regularization strength $L/\sqrt{n}$. Carrying an operator estimate between the remaining stages limits their total cost to $\mathcal{O}(n)$. A matching $\Omega(n+\sqrt{n}LR/\varepsilon)$ lower bound holds for randomized linear-span component-oracle algorithms with adaptive stopping and expected query budgets. Thus, for $0<\varepsilon\le LR/2$, our method attains the optimal worst-case expected component complexity in this oracle model, up to universal constants.

We prove weak approximation for Campana points on the projective line with log Fano boundary, and on projective $n$-space with either $n+3$ hyperplanes in general position or two transverse smooth quadrics, each of coefficient $1/2$.

The two-block odd partition function \(a(n)\) is the signed enumeration of partitions into exactly two distinct part sizes, each occurring an odd number of times. By separating the underlying representations according to the \(2\)-adic valuations of the two part sizes, we obtain a signed decomposition, completed by two classical theta-function evaluations, which explains the structural relation \(a(2^km)=a(m)+(2^{k-1}-1)\sigma(m)\), where \(m\) is odd, \(k\ge1\), and \(\sigma(m)\) is the sum-of-divisors function. Combined with a previously established arithmetic formula for \(a(n)\), this relation yields congruences along the dyadic progressions \(2^k(4n+3)\), as well as the families \(a\!\left(2^\alpha3^\beta(12n+11)\right)\equiv0\pmod3\) and \(a\!\left(2^\alpha(18n+15)\right)\equiv0\pmod3\). We also use a factorization of the generating function of \(a(n)\) involving \(\operatorname{pod}(n)\), the number of partitions in which odd parts are distinct and even parts are unrestricted, to establish a uniform reduction modulo every odd prime of the associated auxiliary series. The resulting theta-weighted sums involving \(\operatorname{pod}(n)\) are congruent to \(a(n)\) and therefore admit explicit evaluations involving the sum-of-divisors function and the character divisor sum associated with the nonprincipal Dirichlet character modulo \(4\). As special cases, we obtain infinite families of weighted congruences modulo \(3\), \(5\), and \(13\).

We study the commuting scheme of the exceptional Lie algebra $\mathfrak{g}_2$ via computational methods. Over a field of characteristic zero, we show that the commuting scheme of $\mathfrak{g}_2$ is reduced, Cohen-Macaulay and normal. Our computations give an explicit homogeneous system of parameters, determine the Hilbert series and degree, identify the first syzygies of the commuting ideal and show that it is not Gorenstein. We extend some of the above results to positive characteristic.

We give an alternative proof of the $5/4$ result of Guth-Iosevich-Ou-Wang on the Falconer distance conjecture: if a compact set $E$ in the plane has Hausdorff dimension $>5/4$, then there exists $x\in E$ such that the pinned distance set $$\Delta_x(E):=\{|x-y|: y\in E\} $$ has positive Lebesgue measure. Moreover, we are able to beat the $5/4$ barrier if the packing dimension of $E$ is $<2$. More generally, we obtain sufficient conditions involving both the Hausdorff and packing dimensions of $E$. These conditions improve the $5/4$ threshold when the packing dimension of $E$ is $<2$, and recover the second author's result for $\dim_H E=\dim_P E>1$. The proof is based on $L^2$ ball inflation and does not use decoupling inequalities.

In many scientific studies, evaluating the association between two responses requires adjusting for shared covariates. The expected conditional covariance measures this adjusted association and is used in conditional independence testing and treatment effect estimation. Researchers often estimate it using fitted conditional means, but removing the bias caused by the estimation errors of these conditional means remains a difficult task. This difficulty can increase when the true covariate density is unknown and restricted only by known upper and lower bounds. In this article, we study how accurately the expected conditional covariance can be estimated under these density bounds. We propose a multiresolution estimator that reduces bias without estimating the density. Theoretical results show that, for bounded responses and rough regression functions, an unknown density slows the optimal worst-case rate by a power of sample size compared with a known density. We establish that the optimal rate under density bounds alone strictly improves upon the previously conjectured polynomial rate by a sub-polynomial factor, resolving a long-standing conjecture. Our estimator attains this rate up to a logarithmic factor. Through simulations and a real data resampling experiment on diamonds, we show that our method can yield smaller mean squared errors than standard uncorrected estimators.

We prove the Baker–Rumely conjecture that the tau invariant of a metrized graph admits a positive lower bound proportional to its total length, with an absolute constant. We also construct simple cubic metrized graphs whose normalized tau invariants tend to $59/7260<1/108$, disproving the proposed universal constant $1/108$. The lower bound is independent of the genus, the number of edges, and the distribution of edge lengths. Its proof combines a second-moment inequality for Euclidean lattices with a partition of the edge coordinates of a cycle lattice into three independent sets. The counterexamples have only two edge lengths and admit an elementary resistance calculation. Through the tropical moment identity, the lower bound also gives a uniform estimate for the non-archimedean terms in height formulas for Jacobians.

For a positive integer $n$, let $$ S_1(n)=\{n\bmod k:1\leq k\leq\lfloor n/2\rfloor\}, \qquad s_1(n)=|S_1(n)|. $$ Baraskar and Vukusic proved $s_1(n)=\Xi n+O(n/(\log n\log\log n))$ for an explicit positive constant $\Xi$, and asked for a sharper error term. Their computations suggested that an $O(n^{1/3})$ bound might hold. We prove that, for every $\alpha>\sqrt2$, $$ \limsup_{n\to\infty} \frac{s_1(n)-\Xi n}{n\exp\{-\alpha\sqrt{\log n\log\log n}\}} =\infty, $$ and the corresponding limit inferior is $-\infty$. In particular, no fixed power saving is possible. We also prove the complementary upper estimate $$ s_1(n)=\Xi n+O\left( n\exp\{-\alpha\sqrt{\log n\log\log n}\} \right) $$ for every fixed $\alpha\in (0,\sqrt{2})$, sharpening the previous $O(n/(\log n\log\log n))$ bound. Thus, the constant $\sqrt2$ is the threshold for bounds of the form $n\exp\{-\alpha\sqrt{\log n\log\log n}\}$.

We study when to buy a share that will later be sold optimally, when the price follows a geometric multi-skew Brownian motion whose skew levels model support and resistance. The reward for buying is the exit premium of the liquidation problem solved in a companion paper. This premium is strictly $r$-subharmonic inside the exit continuation region, so buying is optimal only at levels where the price is pushed upward, and the entry problem reduces to a finite one whose value is an upper concave hull of finitely many points. With one support and one resistance, the optimal rule is to buy exactly at the support, in every exit regime, with an explicit value.

Observing a prediction error at one node can help correct predictions elsewhere, but the benefit depends on the residual dependence between nodes. A graph suggests where that dependence might occur, yet does not establish its sign or strength. We develop a Bayesian model of residual covariance to determine how revealed errors should update a fixed feature-based predictor. For a fixed set of revealed nodes, we derive an exact identity for the change in expected squared error under linear residual propagation. The identity characterizes the optimal linear update and expresses the excess error of any other update as an exact quadratic. Under covariance uncertainty, the Bayes-optimal fixed linear update uses the posterior mean covariance. We average over four positive-semidefinite covariance families representing no dependence, positive dependence, negative dependence, and dependence whose sign can alternate with graph distance. Across eight graph benchmarks, four show posterior support within the distance-profile family for negative covariance between neighbors and alternating signs with distance. Among the evaluated methods, ours achieves the smallest worst-case recall gap to the best method on each graph. The method improves squared error over residual propagation on five of eight graphs, but performs substantially worse on one dense heterophilous graph. In these experiments, selecting the highest-posterior family performs similarly to model averaging.

The ideal magnetohydrodynamic (MHD) system that governs a conducting plasma—the Maxwell equations together with the conservation laws of mass, momentum and energy—is the same local balance structure that describes hot accretion flows around black holes. We introduce a uniform four-dimensional spacetime dilation \[ \Phi_L:\;(t',x',y',z')\mapsto (t,x,y,z)=(Lt',Lx',Ly',Lz'),\qquad L>1, \] and show that it is a smooth, orientation-preserving $C^\infty$ diffeomorphism of flat spacetime. Pulling the MHD conservation laws back through $\Phi_L$, we prove that the closed ideal-MHD system is form-invariant (covariant), with the current density rescaling as $\bm j=\bm j'/L$, the magnetic diffusivity as $\eta'=\eta/L$, and external sources rescaled by $L$. The transformation realises three physically distinct effects on the perturbed flow: a linearization to the tangent-space (linearised MHD) system, a movie slow-motion dilation of dissipative and dynamical timescales, and an electron-microscope magnification of small-scale structure. Taking the formal limit $L\to\infty$ localises the dynamics on a neighbourhood of a horizon: resistivity vanishes, the asymptotic time derivative freezes, and all transient multipolar structure of the infalling plasma redshifts away. The remaining stationary, axisymmetric, asymptotically flat electrovacuum is uniquely the Kerr–Newman geometry, so that the exterior of the compact object is characterised only by its mass $M$, angular momentum $J$ and charge $Q$. The result provides a local, conservation-law-based perspective on the black hole no-hair theorem that is complementary to the classical Israel–Carter– Robinson theorems and to contemporary gravitational-wave and Event Horizon Telescope tests.

The Riemann hypothesis is equivalent to $F(x)$ belonging to the Laguerre–Pólya class. Brändén [J. Reine Angew. Math., 2011] proved that if an entire function in the Laguerre–Pólya class has nonnegative Taylor coefficients, then its coefficient sequence is infinitely log-concave. Consequently, the Riemann hypothesis implies the infinite log-concavity of $(\lambda_n)_{n\ge0}$. In this paper, we prove that the sequence $(\lambda_n)_{n\ge0}$ is strictly infinitely log-concave. This resolves a conjecture of Zhu [Math. Z., 2023]. The proof combines explicit complex-analytic estimates for the iterated logarithmic ratios, rigorous interval arithmetic for a finite range of indices, and a global closure argument.

Optimization in machine learning is usually expressed through iterative rules that update model parameters using information from the loss landscape. In this work, we study an alternative viewpoint in which parameter optimization is treated as the evolution of a dissipative dynamical system. The model parameters are regarded as generalized coordinates, the loss acts as a potential energy, and a positive-definite metric defines the local kinetic geometry of parameter space. Starting from a variational formulation, we derive the corresponding Hamiltonian dynamics, the geometric force generated by a position-dependent metric, and a metric-compatible Rayleigh dissipation law. The resulting continuous system satisfies a monotonic energy-dissipation relation, while its discrete form allows the influence of curvature on the optimization trajectory to be studied directly. We illustrate the framework using a controlled CIFAR-10 image-reconstruction problem for which the optimum is known analytically. With the image Hessian used as the metric, the curvature dependence of the quadratic modal dynamics is removed. We then reparameterize the same image as a matrix product state, producing a genuinely position-dependent metric and a nonzero geometric force. Poincaré return maps provide a complementary phase-space view of the resulting contraction dynamics. These examples establish HAMLET as a geometric, energy-based framework for studying optimization as dissipative motion in parameter space. On a five-seed MNIST MLP benchmark, HAMLET attains the highest mean test accuracy (97.95%) and the lowest mean test negative log-likelihood (0.0696) among the three evaluated optimizers.

Let $d\ge3$ and $2\le r<\infty$. We consider the strong maximal operator associated with Euclidean surface averages over the $\ell^r$ unit sphere and independent coordinate dilations. We prove that this operator is bounded on $L^p(\mathbb R^d)$ if and only if \[ p>\max\left\{\frac{d+1}{d-1},\frac{r}{d-1}\right\}. \] We also obtain $L^p\to L^q$ bounds for the corresponding local maximal operator. For each fixed $p$ in certain ranges, the resulting range of $q$ is optimal up to endpoints. The proof uses a dyadic surface decomposition and frequency-localized $L^2$ estimates. We establish these estimates by combining exact multiplier factorization via iterated stationary phase with scalar Fourier inversion and $TT^*$ arguments. In dimension three, these estimates are combined with quantitative multiparameter local smoothing.

We develop an equivariant K-theoretic vertex formalism for a colored, abelianized higher-rank sector on smooth projective toric threefolds $X$. Nonzero first Chern classes are encoded as magnetic fluxes and absorbed into line-bundle twists, after which the virtual character is redistributed into finite perturbative, vertex, and edge contributions. We compute the resulting (unreduced and trace-free) virtual dimensions and analyze the interaction terms relevant to a possible reduction to rank one. We emphasize that, on a compact toric threefold, colored partitions do not classify all torus-fixed stable higher-rank sheaves: the resulting series is therefore a proposed abelianized/framed vertex series unless an additional comparison theorem with the full stable-sheaf moduli problem is supplied. When the threefold obstruction theory is not available in the desired form, we pass to the local Calabi–Yau fourfold $Y=\operatorname{Tot}(K_X)$ and formulate the corresponding colored DT4 vertex. A $K_X$-twisted tautological insertion removes fixed points with nontrivial fiber layers, reducing solid partitions to plane partitions at the level of the local vertex. In rank one we give a scheme-theoretic explanation: the Hilbert functor of subschemes of $X$ is canonically isomorphic to the constrained Hilbert functor of subschemes of $Y$ lying scheme-theoretically in the fixed zero section. Under the hypotheses of the DT4 K-theoretic Lefschetz/virtual-pullback formalism, the associated support insertion reduces the rank-one fourfold theory to the threefold theory. We keep this global rank-one statement logically distinct from the higher-rank colored-vertex reduction.

In Max-$k$-Cut games, optimal colorings maximize social welfare, but are they stable against coordinated deviations that strictly benefit every participating player? A recurring conjecture in the literature asserts that, for unweighted graphs, every optimal coloring has this stability property, known as strong equilibrium. We disprove the conjecture for every $k\ge3$ by exhibiting a finite, simple, connected, unweighted graph with an optimal coloring that is not a strong equilibrium. The profitable coalitional deviation leads to another optimal coloring: every coalition member gains, while players outside the coalition bear the corresponding losses and social welfare remains unchanged. We also give an exact identity relating coalitional gains, changes in cut value, and internal conflicts, recovering the known stability guarantee for optimal colorings when the maximum degree is at most $2k-1$. For strong deviations from general Nash equilibria, we establish a sharp upper bound on the number of monochromatic edges remaining within the coalition and show that this number can exceed the coalition size.

Let $\lambda_1(n,\Delta)$ be the maximum adjacency spectral radius among connected nonregular simple graphs of order $n$ and maximum degree $\Delta$. Using effective resistance bounds, explicit comparison graphs and a one-dimensional Wirtinger inequality, we prove that, for every fixed integer $\Delta\ge3$, $\lambda_1(n,\Delta)=\Delta-c_\Delta\pi^2/(4n^2)+O_\Delta(n^{-5/2})$ as $n\to\infty$, where $c_\Delta=\Delta-1$ for odd $\Delta$ and $c_\Delta=2(\Delta-2)$ for even $\Delta$. This proves the asymptotic conjecture posed by Liu [J. Combin. Theory Ser. B 169 (2024), Conjecture 7.1].

We characterize, up to logarithmic factors, the minimax expected query complexity of finite-sum nonconvex-strongly-concave optimization under mean-squared averaged smoothness. For randomized zero-respecting incremental first-order algorithms, the complexity is $\tilde{\Theta}(n+\min\{\sqrt{n}\,\kappa,n^{3/4}\sqrt{\kappa}\}\,L\Delta\varepsilon^{-2})$ throughout $\kappa=L/\mu\ge1$, under exact dual initialization and for $0<\varepsilon^2\le c_0L\Delta$, where $c_0>0$ is universal. This characterization is our main result. A dual-chain construction provides the lower bound and matches the existing Catalyst upper bound for $\kappa\ge\sqrt{n}$. For $1\le\kappa\le\sqrt{n}$, we introduce recursive proximal descent-ascent (RPDA), which retains a recursive gradient estimator across proximal subproblems and attains $\tilde{O}(n+\sqrt{n}\,\kappa L\Delta\varepsilon^{-2})$ with a fixed query budget. Both upper bounds guarantee an expected squared primal gradient at most $\varepsilon^2$. Under individual smoothness, we also prove $\Omega(n+\min\{\kappa,\sqrt{n\kappa}\}\,L\Delta\varepsilon^{-2})$ for every $\kappa\ge1$. It matches the polynomial upper rate for a bilinear subclass when $\kappa\ge n$; the intermediate-condition-number gap remains open.