Papers

Late-time wave tails decay at different rates along future null infinity and along timelike worldlines at finite radius. A compactified numerical evolution must represent both the slower decay at null infinity and the faster interior decay, producing an increasingly sharp transition between the two regimes. We address this difficulty for semilinear wave equations in Minkowski spacetime using homothetic hyperboloidal coordinates adapted to the scaling structure of the tail. In these coordinates, the tail approaches a smooth radial profile with the same decay rate at every compactified radius. The formulation therefore avoids the steepening of the radial profile seen in stationary hyperboloidal evolutions, and it reaches late times in a number of steps that grows only logarithmically with retarded time. We demonstrate this approach using pseudospectral simulations in 3+1 dimensions and reproduce the generic decay rates conjectured by Rinne. We also provide numerical evidence consistent with a nongeneric codimension-one cancellation of the leading tail coefficient at null infinity, resulting in a faster decay rate.

Many robust planning problems are solved by checking a family of ordinary optimization problems, one for each uncertainty threshold. Repeatedly building and solving those relaxations can dominate runtime. We show that, when a decision chooses exactly one option from each group, the entire threshold family of multiple-choice knapsack relaxations can instead be bounded together. A cancellation removes threshold-specific baselines and reduces each group's contribution to two simple envelopes. After sorting, one multiplier is evaluated across all thresholds in time linear in their number and nearly linear in the number of options. The resulting interval bound is valid, matches the ordinary relaxation on a single feasible threshold, and is at least as strong as the matched group-clique comparator. A certified one-dimensional search controls multiplier error, while exact checks resolve numerically ambiguous feasibility and comparison cases. In an otherwise identical adaptive search, the method wins all 60 instance-level median comparisons and achieves a 2.37-fold geometric-mean speedup. Additional ablations, independent linear-program checks, and nine instances transformed from a published archive test the mechanism beyond the internal generator. A separate integer audit shows the boundary: faster relaxation bounds need not help when integer search dominates.

For p in $[1,\infty]$, we show that the persistence landscape gives 1-Lipschitz embeddings of metric spaces of countable persistence diagrams and barcodes with p-Wasserstein distances into an $L^p$ space.

In a previous work we computed the number of integral forms, over its field of definition, of an irreducible representation of a dihedral group. Here we apply the theory of Bruhat-Tits buildings to give similar formulas for a wider family of metacyclic groups that have representations of arbitrarily large dimension. Occasionally, these formulas can be extended to groups containing a subgroup in that family.

For $0 \le \alpha \le 1$ and $-1 \le r < 0$, let $\#_{r,\alpha}$ denote the weighted Kubo-Ando power mean of order $r$ and weight $\alpha$. Ando proved that, for every unitarily invariant norm, $\| A \ \#_{r,\alpha}\ B \| \le \| A \|\ \#_{r,\alpha}\ \| B \|$. We show that this inequality remains valid also for the Schatten $q$-quasi-norms with $0 < q < 1$. More generally, our main result resolves a problem recently posed by Hiai and Lim in the stronger form of weak log-majorization for multivariable Lim-Pálfia power means of negative order. It yields analogous Schatten quasi-norm inequalities for every finite family of positive definite matrices.

Boumal, Criscitiello and Rebjock (BCR) proved that if $M$ is a contractible, connected and complete Riemannian manifold, then every smooth function $f\colon M\to R$ satisfying the global Polyak–Łojasiewicz inequality (PŁI) is necessarily of the form $f = f^* + \|\phi\|^2$ with $\phi$ a submersion. Informally, minimizing such a function amounts to solving a nonlinear least-squares problem in new coordinates. The global PŁI hypothesis fails, however, in many problems of interest, among them continuous-time LQR policy optimization in optimal control and a standard formulation of logistic regression. A hierarchy of weakened PŁ inequalities has been introduced in order to cover such problems, and more generally to study the effect of noise and adversarial perturbations on gradient flows. This note shows that, with minor modifications, the same reduction to a nonlinear least-squares problem holds under a substantially weaker hypothesis, “semiglobal” PŁI, which is satisfied in both of the examples just mentioned. That condition asks that $f$ satisfy an estimate $\|\nabla f(x)\| \ge \alpha\bigl(f(x)-f^*\bigr)$ for all $x$, with $\alpha$ merely positive definite and bounded below by a positive multiple of $\sqrt{s}$ for small $s>0$.

For a finite group $G$, denote by $k'(G)$ and $L(G)$ the number of conjugacy classes of subgroups and the subgroup lattice of $G$, respectively. Let $d'(G)=\frac{k'(G)}{|L(G)|}$ and $d^*(G)$ be the minimum value of $d'(S)$, when $S$ runs through all sections of $G$. In this paper we deduce some criteria on the nature of $G$. We show that if $d^*(G)>\frac{9}{59}$, then $G$ is solvable, while if $d^*(G)>\frac{1}{2}$, then $G$ is a supersolvable group. The last criterion is also valid when replacing “supersolvable" with “CLT".

We consider the evolution of a viscous incompressible fluid in three-dimensional distorted pipes, of finite length, modeled through the Navier-Stokes equations with mixed boundary conditions. Specifically, the inflow is given by an arbitrary datum, the outflow is subject to a dynamic condition including a directional do-nothing boundary condition; standard no-slip assumptions are imposed on the remaining walls of the domain. We introduce a new functional framework that accommodates the dynamic boundary condition and the incompressibility constraint. Within this framework, we establish the existence of weak solutions. Moreover, under a smallness assumption on the data, we prove the existence and uniqueness of global strong solutions. The newly introduced functional setting enables us to deal, at first, with the associated Stokes problem, and then with the full Navier-Stokes system, via the Galerkin method implemented with a basis of eigenfunctions of a suitably modified Stokes operator.

Given a graph $G$ with degree sequence $d_{1},\ldots,d_{n}$ and a positive real number $p$, let $e_{p}(G)=\sum_{i=1}^{n} d_{i}^{p}$. For a fixed family of graphs $\mathcal F$, let $ex_{p}(n, \mathcal F)$ denote the maximum value of $e_{p}(G)$ over all $\mathcal F$-free graphs $G$ on $n$ vertices. In 2000, Caro and Yuster introduced the following Turán-type problem: For a positive integer $p$ and a fixed graph $F$, determine $ex_{p}(n, F)$, and characterize the extremal graphs $G$ on $n$ vertices that attain $ex_p(n, F)$. Recently, Gao, Liu, Ma and Pikhurko proved that $ex_{p}(n, \mathcal F)=(\tau(\mathcal F)-1+o(1))n^p$ for real $p>\frac{1}{1-\alpha}$, where $\mathcal F$ is a degenerate family of graphs with classical Turán number $ex(n, \mathcal F)=O(n^{1+\alpha})$ for some $\alpha\in[0,1)$, and $\tau(\mathcal F)$ is the minimum size of an independent vertex cover over all bipartite graphs $F\in\mathcal F$. Based on their method, we obtain a stability result for $ex_{p}(n, \mathcal F)$, and prove that all extremal graphs must contain the complete bipartite graph $K_{\tau(\mathcal F)-1,n-\tau(\mathcal F)+1}$ when $n$ is sufficiently large. Our results can be used to deduce all previously known results about $ex_{p}(n, F)$ when $F$ is a bipartite graph and $n$ is sufficiently large. We also obtain several new exact results for $ex_{p}(n, F)$, namely, when $F$ is an even cycle, a complete bipartite graph, a discrete hypercube, a caterpillar forest, and a spider forest.

In this paper, we study the regions of asymmetric extensions of the Catalan and Shi arrangements. We first determine the characteristic polynomials using the finite field method, thereby obtaining explicit formulas for the number of regions. We then introduce two new classes of combinatorial objects: $\mathbf{c}$-labelled Dyck paths and $\mathbf{c}$-building functions, which generalize classical Dyck paths and parking functions, respectively. We prove that the regions of the asymmetric $\mathbf{c}$-Catalan arrangement are in natural bijection with $\mathbf{c}$-labelled Dyck paths, while those of the asymmetric $\mathbf{c}$-Shi arrangement are in natural bijection with $\mathbf{c}$-building functions. This resolves a problem proposed by Theo Douvropoulos during the 2022 Oberwolfach Workshop on Enumerative Combinatorics. In particular, our results extend Stanley's celebrated correspondence between $k$-parking functions and the regions of the $k$-Shi arrangement by introducing new combinatorial models and developing entirely different proof techniques.

We show that in any Haken 3-manifold $M$, the dimensions of $\mathcal{ML}(M)$ and $\mathcal{ML}_0(M)$ can be calculated using normal surfaces. As a corollary, we prove the number of closed orientable essential surfaces of Euler characteristic at least $-2n$ in a closed orientable hyperbolic 3-manifold with $H_2(M,\mathbb{Z}/2\mathbb{Z}) = 0$ is asymptotic to $n^{\dim \mathcal{ML}_0(M)}$.

Cells often estimate chemical concentrations from only a few diffusing molecules, whose stochastic motion limits sensing precision. We consider a perfect monitoring instrument that records the instantaneous molecular density throughout a finite region without perturbing the concentration field or distinguishing molecular identities. The standard Berg-Purcell estimator weights all positions uniformly. Here we derive the optimal spatial weighting within a large class of unbiased estimators for both concentration and gradient sensing. Variance minimization maps to an electrostatic problem. Surprisingly, although the instrument monitors the entire volume, the optimal estimator assigns all weight to its boundary, reducing the uncertainty in both concentration and gradient sensing. We extend the results to arbitrary geometries and spatial dimensions.

We extend a finite-support $q$-Hermitian radial calculus to Clifford-valued coordinate polynomials. For $N$ bosonic Hermitian vector labels and finite fermionic support, the stable range $m\ge 2N$ admits a canonical Gram–harmonic normal form by Howe separation. A label–Capelli inverse normalizes the Clifford contractions, while a finite Wick–Chevalley conjugation lifts the divided-power scalar gauge to the full Clifford PBW module. Coupling the normalized contractions to contractible fermionic seed complexes gives two conjugate polynomial-preserving $q$-Hermitian coordinate families. They are square-zero, anticommute within each polarization, restrict exactly to the radial PBW calculus, have scalar boundary trace $[2m-2n]_q$, and recover the classical Hermitian super Dirac pair as $q\to1$. For the mixed polarizations we introduce a relative triangular transport. The Grassmann relations for the normalized Capelli contractions hold on the full coordinate module and yield a labelwise factorization. Hence all higher-filtration mixed terms are finite subset products of explicit local defects, and the mixed polarization algebra closes for arbitrary finite label sets and finite fermionic support. The classical complex structure is common to both polarizations, whereas the two full transported Clifford–Weyl products are distinct for $0<q<1$. We also show that the fermionic Cartan denominator is forced in the natural polynomial first-order Berezin–Weyl class and determine the minimal factorwise Ore localization needed for the conjugate all-charge homotopies. The $q$-dependent Cartan orbit is nonresonant for $m\ge\max\{2N,n\}$. Finally, fixed finite-order differential and finite nonzero-shift realizations on the undeformed coordinate algebra are ruled out.

A differential calculus for causal variational principles is developed, which generalizes the exterior calculus of differential forms and some of the associated differential topological structures to non-smooth spaces. Our calculus includes the exterior derivative, de Rham cohomology, glueing constructions (restrictions and extensions of differential forms, Mayer-Vietoris sequence), a Künneth formula and Poincaré's lemma. Moreover, we prove versions of Stokes' theorem and the Gauß divergence theorem. The constructions and results are illustrated by several examples.

We establish a local Lewy-type theorem for \(p\)-harmonic function with non-zero gradient in dimension three space $R^3$. Let \(1<p<\infty\), and let \(u\in W^{1,p}_{\mathrm{loc}}(\Omega)\) be a weak \(p\)-harmonic function in a domain \(\Omega\subset\mathbb R^3\), assume it satisfies \(|Du|>0\), we prove that a locally homeomorphic gradient map \(Du\) must have non-vanishing Hessian determinant. Hence \(Du\) is a local diffeomorphism.

We prove the $\Delta$-conjecture, which dates back to Gurvich's 1978 thesis. Specifically, let the edges of a complete graph be colored with colors $1,\ldots,d$, and for each $i$ let $G_i$ be the graph on the same vertex set formed by the edges of color $i$. We prove that if every choice of a maximal stable set $S_i$ of $G_i$, one for each $i\in[d]$, has nonempty intersection, then the coloring contains no rainbow triangle. Together with a result of Andrade, Boros, and Gurvich, this characterizes CIS $d$-graphs as precisely the Gallai $d$-graphs whose chromatic components are ordinary CIS graphs. We also show that every factor in the canonical modular decomposition of a CIS $d$-graph is a CIS $d$-graph whose edge-coloring uses at most two colors.

Kemeny's constant for a connected graph $G$, denoted by $\mathcal{K}(G)$, is the expected time for a random walk to reach a randomly chosen vertex $u$, regardless of the choice of the initial vertex. Recently, Kim et al. (2026) proposed two conjectures on Kemeny's constant. The first conjecture asserts that if $G$ is a connected graph of order $n$ and diameter 2, then $\mathcal{K}(G) = O(n)$. The second conjecture asserts that if $G$ be a graph of order $n$, then $\min\{\mathcal{K}(G), \mathcal{K}(\overline{G})\} = O(n)$, and if both $G$ and $\overline{G}$ are connected, then $\mathcal{K}(G)\mathcal{K}(\overline{G}) = O(n^4)$, where $\overline{G}$ denotes the complement of $G$. In this paper, we confirm both conjectures. For the first conjecture, we prove that if $G$ is a connected graph of order $n$ and diameter 2, then \[ \mathcal{K}(G) \leq (3 + \sqrt{5})(n - 1). \] For the second conjecture, we prove that for any $n$-vertex graph $G$, \[ \min\{\mathcal{K}(G), \mathcal{K}(\overline{G})\} \leq (8 + 2\sqrt{5})n - (10 + 2\sqrt{5}). \] Moreover, if both $G$ and $\overline{G}$ are connected, then \[ \mathcal{K}(G)\mathcal{K}(\overline{G}) \leq \frac{3 + \sqrt{5}}{2}n^4. \] Our proof relies on effective estimates on resistance distances and spectral gaps of graphs.

Our first main purpose is to establish a framework for initial value problems for time-fractional evolution equation of order $\alpha \in (0,1)$ in Banach space $X$: $$ \pppa (u(t)-a) = Au(t) + F(t), \quad 0<t<T. \eqno{(*)} $$ Here $u: (0,T) \rrrr X$ is an $X$-valued function defined in $(0,T)$, and $a \in X$ is an initial value. The operator $A$ satisfies a decay condition of resolvent which is the same as a generator of analytic semigroup. Based on $X$-valued Laplace transforms, we establish a solution formula yielding the well-posedness for (*). In particular, we can directly treat a case $X=L^p(\OOO)$ over a bounded domain $\OOO$ and a uniform elliptic operator $A$. Our theory is feasibly applicable to other topics such as regularity of solutions, inverse problems and control problems.

For $k\geq2$, let $S_r(k;\ell)$ be the smallest $n$, if exists, such that every $r$-coloring of $\{1,2,\ldots,n\}$ has a monochromatic solution $\mathcal{S}$ to the equation \[ x_1+x_2+\cdots+x_k=x_{k+1} \] such that the number of distinct integers in $\mathcal{S}$ is exactly $\ell+1$. We prove that, if $\ell\geq2$ is fixed, then \[ S_2(k;\ell)= k^2+\left[\frac{(\ell+1)(\ell-2)}{2}+2\right]k+\ell(\ell-2) \] for all large enough $k$. In particular, we have $S_2(k;2)=k^2+2k$ for all $k\geq3$.

We propose a theory describing the structure of minimal locally concave functions on an arbitrary subdomain of $\mathbb{R}^d$. We introduce the notion of an extremal set, that is, a set on which the function is affine, and prove that extremal sets foliate the domain. We show that extremal sets are covered by simplices with vertices on the boundary of the domain, and that these simplices themselves lie in the extremal set. We also prove that one-dimensional extremal sets can approach the free boundary only tangentially, not transversally.

Motivated by the analogy with the Bismut connection in Hermitian geometry, we study Sasaki with torsion manifolds. In particular, we characterize co-Kähler-like and flat Sasaki with torsion manifolds, and we introduce the notion of a $\nabla$-Einstein manifold as the odd-dimensional analogue of the Bismut Hermite-Einstein condition. We provide non-compact examples and we study compact $\nabla$-Einstein manifold in dimension $5$ and $7$. We also develop a general framework for geometric flows of almost contact metric structures. In particular, we derive a flow for Sasaki with torsion structures that preserves the strong condition, that is, the closure of the torsion. Furthermore we prove that such flow is gauge-equivalent to the generalized Ricci flow and it is gauge-equivalent to the pluriclosed flow, after performing a trivial product with $S^1$.

Starting from the $(E_8,T_1)$ Nahm sum for the vacuum character of $T_{10}M_{\rm eff}(11,2)$, we construct a three-dimensional (3d) $\mathcal N=2$ $U(1)^8$ Chern–Simons matter theory $\mathcal T$ with level matrix $C_{E_8}$. Its $A$-twisted vacuum half-index reproduces the character exactly, while four distinguished Wilson loop half-indices agree with the remaining characters to the orders checked. Applying particle–vortex duality and gauging yields a second 3d $\mathcal N=2$ Chern–Simons matter theory $\mathcal T^\vee$ with level matrix $C_{E_8}^{-1}$. Exact superconformal-index matching supports the claim that the two theories flow to 3d $\mathcal N=4$ rank-zero mirror SCFTs. For the $(\mathcal D,D_c)$ boundary condition, the mirror map between the $A$-twisted Wilson loop half-indices of $\mathcal T$ and the $B$-twisted Wilson loop half-indices of $\mathcal T^\vee$ coincides componentwise with the Zagier transformation, realizing a Zagier duality between the complete $(E_8,T_1)$ and $(T_1,E_8)$ Nahm systems. Using the Bethe-equation formalism, we show that the corresponding $A/B$-twisted TQFT data agree up to orientation reversal and the $E_8$ framing phase $e^{-2\pi \mathrm{i}/3}$. The duality interface gives the torus bilinear $Z_{\mathcal I}(\tau)=\sum_i\chi_i(\tau)\chi_i^\vee(\tau)=\chi_{(E_8)_1}(\tau)$, where $\{\chi_i\}$ and $\{\chi_i^\vee\}$ are the corresponding five-component character bases. Thus Zagier duality is realized as part of a concrete 3d mirror and interface structure.

An expert who supplies examples of a quantity often also signals how plausible each one is; when is that signal worth using? We study eliciting a Bayesian prior from an expert who provides example points together with their approximate likelihoods. We propose fitting the prior by least squares–minimizing the squared discrepancy between a parametric density and the elicited likelihoods–which defines an M-estimator that remains well posed even for families whose moments do not exist. We establish consistency and asymptotic normality, and prove–under explicit regularity conditions, comprising a well-separation and a uniform-concentration requirement that we verify for the families considered–a non-asymptotic O(1/sqrt(n)) Berry–Esseen bound on its sampling distribution, uniform and nonuniform, by extending a result of Pinelis for maximum-likelihood estimators to the M-estimation setting. We then relax the assumptions that most limit the method in practice. Experts need not report on the density's own scale: an unknown reporting scale can be profiled out in closed form and estimated jointly, at no asymptotic cost for location families. The theory extends to multivariate parameters, where a directional Berry–Esseen bound follows from the multivariate delta method applied to a smooth implicit proxy for the estimator. An additive error floor in the noise model removes a degeneracy in the optimal design, making optimal designs interior. Simulations for normal and beta families confirm the predicted n^-1 error rate and the accuracy of the normal approximation at moderate sample sizes. Finally, we compare the estimator with the sample-only maximum-likelihood baseline, derive an explicit threshold on the expert's reporting noise below which the elicited likelihoods provably reduce estimation error, and calibrate that threshold against eleven datasets of human frequency judgments.

We develop a response geometry for smooth parameterized families of Einstein metrics equipped with canonical probe data. The differential of the probe observations defines a response bundle morphism; pulling back a fixed metric on the observation bundle gives a positive-semidefinite response tensor whose kernel is exactly the space of first-order invisible parameter directions. On the complete-response locus this tensor is Riemannian and yields local rigidity and quantitative reconstruction estimates whenever the response is locally realized by an observation map. We study rank-defect loci, the associated quotient geometry, Gram and conditioning operators, response volume, and covariant variation formulas for simple response eigenvalues. Under an explicit realizability hypothesis we also show that a single scalar observation can be chosen to detect every direction of a finite-dimensional parameter space. These constructions are applied to the harmonic-probe package arising from the Einstein Detection Principle. A finite-dimensional interval-matrix example illustrates the quantitative formulas, while marked unit-volume flat two-tori provide a fully explicit model: three harmonic-energy observations reconstruct the marked metric globally, the induced response metric admits an explicit positively curved hyperboloid realization, and its conditioning deteriorates toward the cusp.

A class of elliptic solutions to the matrix CKP equation is studied. Equations of motion for their poles and matrix coefficients at the poles are obtained. As in the scalar case, they are of the first order, in contrast to what takes place in the KP and BKP hierarchies, where the equations of motion are of the second order.

We present a systematic study of finite, non-split 2-group symmetries with a non-trivial Postnikov class, focusing on the simplest example with a $\mathbb{Z}_2$ 0-form symmetry and a $\mathbb{Z}_2$ 1-form symmetry, intertwined together by the non-trivial Postnikov class in $H^3(B\mathbb{Z}_2;\mathbb{Z}_2)\cong\mathbb{Z}_2$, denoted by $\mathcal{G}$. We classify the anomalies of this 2-group symmetry for physical theories in $d$-dimensional spacetime, by computing the oriented bordism groups $\Omega_{d+1}^{\rm SO}(B\mathcal{G})$ and the spin bordism groups $\Omega_{d+1}^{\rm Spin}(B\mathcal{G})$ for $d\leq 5$. For the case of $d=3$, we investigate the Symmetry TFT/TO of the 2-group symmetry $\mathcal{G}$ using the language of fusion 2-categories, as well as (3+1)D TQFT actions, for the cases without or with the 2-group anomaly classified by $\operatorname{Hom}\left(\widetilde{\Omega}_4^{\rm SO}(B\mathcal{G}),U(1)\right) \cong H^4(B\mathcal{G};U(1))\cong\mathbb{Z}_2$. We classify the minimal topological and physical boundary conditions of the Symmetry TFTs, and carry out the categorical Landau paradigm for such a non-split finite 2-group.

We investigate the asymptotic behavior of the rectangle partition functions $p(n,n)$ and $p(3,n)$. The function $p(n,n)$ counts partitions of the square $n\times n$ into rectangular blocks with integer sides, while $p(3,n)$ counts such partitions of the rectangle $3\times n$. Two rectangle partitions are identified when they contain the same multiset of rectangle types, and a block $a\times b$ is identified with a block $b\times a$. Our main results are $$ p(n,n) = \exp\left( \left(\tfrac{\pi}{\sqrt3}+o(1)\right)n\sqrt{\log n} \right) $$ and $$ p(3,n) = \exp\left( \pi\sqrt{\tfrac{11n}{3}}+O(\log n) \right). $$ We also present simpler upper and lower bounds for $p(n,n)$ and an independent upper bound for $p(3,n)$.

We prove a sharp spatial regularity theorem for the Itô and Stratonovich cross integrals of continuous semimartingales in separable Hilbert spaces. Without global moment assumptions, both integrals admit versions that are almost surely jointly continuous in the two time variables, with values in the little weak-trace ideal. This regularity is optimal even for Wiener processes with deterministic trace-one covariance. Given any positive sequence tending to zero, we construct such a process for which the singular values of its Lévy area, multiplied by their indices, tend to zero but have termwise ratios to the prescribed sequence tending to infinity almost surely. The same property holds simultaneously for both self-integrals. The upper bound follows from a rank-one cancellation in Itô's formula for a determinant, together with finite-rank approximation. For the lower bound, conditional Gaussian estimates and orthogonal time functions yield finite-dimensional spectral bounds, which are combined through a trace-one covariance to obtain lower bounds for all sufficiently large indices.

We study the distribution of the two-replica overlap $R_{1,2}$ in the Ising and spherical Sherrington-Kirkpatrick models at the critical inverse temperature $\beta = 1$. Our main result shows that in both models, $R_{1,2}$ has scale $N^{-1/3}$, and the quenched distribution of $N^{1/3} R_{1,2}$ converges to an explicit random probability measure defined in terms of the reflected $\mathrm{Airy}_1$ point process. As a consequence, we characterize the limiting value of $N^{2/3} \mathbb{E} \langle R_{1,2}^2 \rangle$, answering a question of Talagrand \cite{talagrand2011mean2}. For the spherical SK model, we obtain the limit by representing the Gibbs measure as an anisotropic Gaussian on $\mathbb{R}^N$ conditioned to have norm $\sqrt{N}$, and then passing to the $\mathrm{Airy}_1$ scaling limit at the GOE spectral edge. For the Ising SK model, the proof is based on a sphere-to-cube comparison principle showing that the quenched distributions of $N^{1/3} R_{1,2}$ under the spherical and Ising Gibbs measures asymptotically coincide. This paper is a companion to \cite{du2026fluctuations}, where we introduced a related comparison principle to identify the limiting fluctuations of the SK free energy. Most of the arguments in this paper were generated using GPT-5.6 Pro, with the aim of exploring further consequences of the ideas developed in that work.

We prove a sharp isoperimetric inequality for the harmonic mean of the first $n$ nonzero Neumann eigenvalues of the Witten-Laplacian on origin-symmetric Lipschitz domains in space forms, endowed with radial log-concave measures. The main novelty is that we establish the sharp harmonic mean inequality under general radial log-concave measures, without assuming the weight function to be non-increasing. This extends previous results that were restricted to specific or more restrictive weighted settings. The proof relies on a refined analysis of the first eigenfunction on geodesic balls, a monotonicity property derived from a convexity condition on the radial weight, and a matrix trace inequality.