Papers

Neighboring seeds in a cluster algebra of affine type are seeds that are as close as possible to the boundary of the g-vector fan. This short note highlights the characterization of neighboring seeds given in a recent paper of Reading, Rupel, and Stella and applies that characterization in two ways. We prove a conjecture on the construction of universal geometric cluster algebras of affine type. We also characterize extended exchange matrices of affine type that are mutation-finite.

The Buchsbaum–Eisenbud–Horrocks conjecture has attracted the attention of many researchers working in Commutative Algebra and Algebraic Geometry in the last fifty years. Quite recently, a variant of this conjecture has been formulated by Lima–Pereira, Nuño–Ballesteros, Orefice–Okamoto and Tomazella in their study of complete intersection singularities. The purpose of this paper is, on the one hand, to exhibit some cases where this conjecture holds. On the other hand, we show that the conjecture fails in general by exhibiting some counterexamples. Finally, we formulate a conjecture that can be regarded as a local variant of the Buchsbaum–Eisenbud–Horrocks' one.

This paper develops a new analytic framework for investigating the Riemann hypothesis. For each fixed integer $\ell \ge 1$, define Euler's $\ell$-totient function $\varphi_\ell$ by \[ \varphi_\ell(n):=n\prod_{\substack{p\ \mathrm{prime}\\ v_p(n)\ge \ell}}\left(1-\frac{1}{p}\right), \] and its summatory function by \[ \Phi_\ell(x):=\sum_{n\le x}\varphi_\ell(n). \] An analytic study of the generalized Euler $\ell$-totient function $\varphi_\ell(n)$ is carried out, including its Euler product representation, meromorphic continuation, and pole structure. For each $\ell$, necessary and sufficient criteria for the Riemann hypothesis are established in terms of the asymptotic behavior of $\Phi_\ell(x)$.

Hedden, Herald and Kirk conjectured that every knot admits a decomposition along a Conway sphere for which the Lagrangian Floer homology of the two associated immersed curves in the pillowcase recovers Kronheimer and Mrowka's reduced singular instanton homology. We prove the conjecture for the hyperbolic pretzel knots $P(-2,3,q)$; it was previously known for two-bridge knots and for some torus knots. The decomposition is obtained from Hedden, Herald and Kirk's decomposition of the torus knot $T(3,5)$ by Dehn twists along the Conway sphere, which act on the pillowcase by a linear shear. Building on our analysis of the torus knots $T(3,n)$, we compute the Floer complex and its $\mathbb{Z}/4$ grading by hand; its differential is nonzero for every member of the family except $P(-2,3,7)$. The same method proves the conjecture for a family of twisted torus knots, under a hypothesis on the chirality of the torus-knot decomposition. On the instanton side, we show that the instanton homology of $P(-2,3,q)$ is free abelian of rank $q+2$ for every odd $q\ge3$, extending Lobb and Zentner's rational computation to the integers by means of Manion's integral Khovanov homology; for the hyperbolic members this also follows from work of Daemi and Scaduto.

For every torus knot, the rank of the differential of the Daemi-Scaduto S-complex of singular instanton homology is determined by the signature and the Alexander polynomial; this follows from theorems of Li-Ye and Daemi-Scaduto. For the torus knots $T(3,n)$ the differential has rank zero or one according to $n$ modulo $6$, and its rank equals Daemi and Scaduto's Frøyshov-type invariant; they proved this in two of the four residue classes and for $n=4,5$, and Hedden, Herald and Kirk computed it for $n\le38$. Following Lewallen, Klassen and Hedden-Herald-Kirk, we describe the generators of the instanton complexes and their $\mathbb{Z}/4$ gradings for two-bridge knots, whose irreducible traceless characters are all binary dihedral, and for $T(3,n)$, where at most one is. We also give a proof through Khovanov homology, using Schütz's computation for three-braids, of Li and Ye's computation of the rank of the instanton homology of $T(3,n)$, and compare the differential of the knot $8_{19}$ with the single pillowcase bigon of Hedden, Herald and Kirk.

We show that any complete shrinking gradient Kähler-Ricci soliton on a resolution of a Kähler cone is necessarily asymptotically conical. From a result of Esparza, it then follows that up to pullback by biholomorphism, there exists at most one complete shrinking gradient Kähler-Ricci soliton on such a resolution. This confirms a special case of the uniqueness part of a conjecture of Song-Zhang. Some other consequences are also discussed.

Graph expansion has long been recognised as an important and desirable property with applications in a wide range of areas in computer science and mathematics. A particular form of expansion known as `sublinear expansion' has recently been used particularly effectively in extremal graph theory, leading to the resolution of many long-standing and notable problems over the last decade and an improved understanding of the structure of sparse graphs. This survey will cover these advances.

We develop a residue-theoretic framework for studying inverse eigenvectors of a square matrix, defined by the nonlinear equation $M\alpha=\alpha^{-1}$. Our main result is an inverse analogue of the spectral theorem: under natural transversality and properness assumptions, the identity operator admits an explicit decomposition into rank-one tensors associated with the inverse eigenvectors. The proof is based on residues of rational differential forms in several complex variables. For real correlation matrices whose off-diagonal entries have modulus strictly less than one, we remove the properness assumption and show that the coefficients in the decomposition are positive and sum to~$1$. Consequently, the inverse eigenvectors support an explicit centered discrete isotropic probability measure and form a weighted tight frame. We extend the construction to arbitrary real Gram matrices, diagonal inverse eigenvectors, and weighted inverse-eigenvector equations. Simple trace and arithmetic–geometric mean arguments yield inverse eigenvectors with controlled Euclidean norm and coordinate product. These estimates lead to short proofs of the strong real polarization inequality and the $n$th real linear polarization inequality, together with weighted and matrix-valued generalizations and further geometric and analytic applications.

Let $d\ge 2$ be an integer and let $c_0(t),\dots, c_{d-2}(t)\in\bar{\mathbb{Q}}[t]$. We consider the family of normalized polynomials $f_\lambda(z):=z^d+\sum_{i=0}^{d-2} c_i(\lambda)\cdot z^i$ parameterized by $\lambda\in\bar{\mathbb{Q}}$; the generic element of our family of polynomials is $f_t(z):=z^d+\sum_{i=0}^{d-2}c_i(t)\cdot z^i\in \bar{\mathbb{Q}}[t][z]$. Also, let $\alpha_1(t),\alpha_2(t),\beta(t)\in\bar{\mathbb{Q}}[t]$, where $\alpha_i(t)$ is not preperiodic under the action of $f_t(z)$ for each $i=1,2$. Under some natural hypotheses, we obtain precise necessary and sufficient conditions for which there exist infinitely many $\lambda\in\bar{\mathbb{Q}}$ with the property that for some $m,n\in\mathbb{N}$ (depending on $\lambda$), we have that $f_\lambda^m(\alpha_1(\lambda))=f_\lambda^n(\alpha_2(\lambda))=\beta(\lambda)$.

For the algebra $\mathbb{R}[x^1,\ldots,x^d]$ of polynomials in $d\geqslant 1$ variables, regard the complete generalised Wronskian $W_d^k$ of differential order $k\geqslant 1$ over $\mathbb{R}^d$ as the $N=\tbinom{d+k}{d}$-ary Lie bracket. Take an $N$-tuple of polynomials, calculate their Wronskian, and keep re-using the newly-created polynomials to produce more of them. The problem is: how fast do their maximal total degrees grow with the number $n$ of iterations of the bracket? Here enter the $N$-bonacci numbers defined by the recurrence $F^{(N)}_n=F^{(N)}_{n-1}+\cdots+F^{(N)}_{n-N}\in \mathbb{N}$. We prove that for any choice of the initial arguments, the sequence of highest total degrees ${\mathsf{D}}^{(N)}_n \geqslant 0$ grows (if at all) asymptotically no faster than the $n$th $N$-bonacci number: $\lim_{n\to+\infty} ({\mathsf{D}}^{(N)}_n/F^{(N)}_n )<\infty$. We show that for $d=1$ and $k$ odd, the highest polynomial degrees do attain the $N$-bonacci bound. Keywords: Differential polynomial, $N$-ary Lie bracket, multivariate Wronskian determinant, Fibonacci numbers, $N$-bonacci numbers, asymptotic growth rate, growth of polynomial degrees, Skolem–Pisot problem.

This paper studies the convergence of stochastic gradient descent (SGD) when the implemented updates are subject to a persistent and state-dependent bias, in which the desired update is scaled by response functions component-wise. Our first contribution is to demonstrate that SGD in this setting implicitly optimizes a penalized problem whose minimizer does not coincide with the true minimizer. To mitigate this convergence failure, we reformulate the original task as an equivalent bilevel optimization problem and propose a gradient-based algorithm, termed Residual Learning. Theoretical analysis shows that Residual Learning finds a solution to the original, unbiased optimization problem despite the hardware imperfections. Beyond exact convergence, we quantify how the response functions affect convergence complexity via the hardware condition number and show that a polynomial dependence on it is unavoidable in general, via a construction of a hard instance. The theoretical results are supported by numerical simulations that demonstrate the effectiveness of the proposed algorithm.

A nonempty subfamily of a $k$-uniform hypergraph is an even cover if every vertex lies in an even number of its hyperedges; for $k=2$ these are edge-disjoint unions of cycles, so the minimum size of an even cover is the natural hypergraph analogue of girth. We prove Feige's 2008 conjecture on the hypergraph Moore bound: there are absolute constants $A$ and $C$ (independent of $k$) such that for every $k\ge3$ and every $1\le\ell\le n$, any $k$-uniform hypergraph on $n$ vertices with more than $C\,n^{k/2}/\ell^{k/2-1}$ hyperedges contains an even cover of size at most $A\,\ell\log(en/\ell)$. Our proof is based on sharp spectral bounds for Kikuchi matrices, which we expect to be of independent interest; we apply them to the refutation of random constraint satisfaction problems in a companion paper.

This paper explores the theory of letter transducers, Mealy automata, and bireversible automata from a combinatorial perspective analogous to the theory of Latin squares. We view the sets of transitions of letter transducers as analogs of orthogonal arrays, and discuss two other combinatorial encodings of Mealy automata analogous to orthogonal pairs of Latin squares and to $(k,n)$-nets. We characterize various classes of automata (Mealy, reversible, invertible, bireversible) in terms of these combinatorial structures. In particular, we represent the inversion and dualization of transducers as parastrophisms. Further, similarly to the notion of the isotopisms of the quasigroups associated to Latin squares, we develop the notion of isotopisms of letter transducers generalizing transducer symmetry and preserving the class of bireversible automata.

The problem of finding the optimal compression rate for a given random variable has been traditionally studied under two main constraints: distortion and perception. The distortion constraint enforces the fidelity of our reconstruction with respect to the observed realization of the random variable, while the perception constraint ensures that the reconstruction is close to a sample from the distribution of the random variable of interest. In this work, we explore the possibility of reconstruction, such that the reconstructed sample is still within a desired fidelity level with our original realization of the random variable, but at the same time, it resembles a sample from a different target distribution. We term this criterion as the deception constraint and find the fundamental tradeoffs of rate-distortion and deception.

In electromagnetic heating and other applications, we need to compute the volume enclosed by an isosurface of the 3D temperature distribution that is numerically represented on a rectangular grid. This situation arises naturally when the temperature distribution is obtained by solving a partial differential equation numerically using a finite difference method. Given the temperature distribution on the 3D grid, the isosurface of a prescribed value is represented approximately by a triangulation, a collection of triangles with vertices on the grid lines. The vertices are determined by a linear interpolation to approximate the locations where the temperature is at the prescribed level. The region enclosed by the isosurface is approximated by that enclosed by the triangulation, which is a set of tetrahedrons. The enclosed volume is approximated by summing those of tetrahedrons. This volume approximation is analogous to approximating a curve using line segments and is limited in accuracy. In this study, we combine extrapolation with the triangulation approximation to develop a more accurate method for computing the volume enclosed by an isosurface of the 3D temperature distribution on a rectangular grid.

We consider the two-species particle system introduced by Stevens (2000) related to the doubly parabolic Keller-Segel equation. It consists of $N$ cells and of a varying number of chemoattractant particles. Cells diffuse in the plane and follow the (mollified) empirical gradient of concentration of chemoattractant. Chemoattractant particles are produced by cells at some constant rate, diffuse and disappear at some constant rate. We show that when the sensitivity of cells to the chemoattractant is small enough, under some rather weak condition on the family of mollifiers, this system approximates the parabolic-parabolic Keller-Segel equation as $N\to \infty$. We also prove that when $N$ is fixed and when the production rate of chemoattractant particles tends to infinity, this system approximates the (non-Markovian) one-species system introduced Talay-Tomašević (2020) and further studied by the authors (2023).

For a finite subgroup $G\subset \operatorname{SL}(3,\mathbb{C})$, the Cautis–Logvinenko conjecture states that for each nontrivial irreducible representation $\rho$ of $G$, the image of the sheaf $\mathcal{O}_0\otimes \rho$ under the derived equivalence of Bridgeland–King–Reid is a pure sheaf on the $G$-Hilbert scheme. We prove a strong form of this conjecture in complete generality, and in doing so, we compute the relevant sheaf explicitly whenever its support is of dimension one. Our main result implies that a matrix defining the Gale dual of the linearisation map is sign-coherent, thereby allowing us to read off the support and cohomological degree of the pure sheaves directly from the matrix.

We study the quasilinear Liouville equation \[ -\Delta_n u=e^u \] on complete noncompact Riemannian manifolds with nonnegative Ricci curvature. Our first result shows that, if a solution $u$ satisfies the optimal logarithmic lower bound \[ u(x)\ge -\frac{n^2}{n-1}\log r(x)+o(\log r(x)) \quad \text{as }r(x)\to+\infty, \] then the underlying manifold is isometric to the Euclidean space and $u$ is a standard bubble solution. Both the leading coefficient and the remainder term in the assumption are sharp. The key ingredient in the proof is the connection between the logarithmic lower bound and a sharp upper bound on the total volume of the solution. We also formulate a conjecture concerning the interaction between the sub-logarithmic decay of solutions and the underlying geometry, and prove it for $n=2$, as well as for $n\ge 3$ under a strengthened assumption.

The Strong Szegő Limit Theorem is a theorem about the asymptotics of the determinants of large Toeplitz matrices. It can be reformulated as a probabilistic statement about eigenvalue statistics of random unitary matrices. We prove a multivariate generalization of the theorem in this latter form, replacing a single unitary with a system of independent random unitaries. It turns out that the standard proofs of the classical theorem do not generalize to this setting, and instead we must import tools from random matrix theory, free probability, and noncommutative function theory. In addition, we obtain results about the averaged determinants of random unitary pencils, as the size of the unitaries tends to infinity; and an auxiliary result giving a formula for the spectral radius of a matricial sum of free Haar unitaries.

For a ladder-system space $X_L$ with carrier $S\subseteq E^{\omega_1}_\omega$, the finite-label uniformization property $M_{<\omega}$ characterizes countable metacompactness, and countable metacompactness is equivalent to the $\Delta$-property. Both equivalences are known for stationary carriers. For arbitrary carriers, an active-tail formulation gives a direct proof that $M_{<\omega}$ is equivalent to the $\Delta$-property and leads to a support-finite decomposition theorem, together with club-smallness and trace criteria that avoid explicit ladder-position thresholds. A club-gap argument, combined with Fodor's lemma, shows that finite and countable tail multiplicity determine the same carrier ideal, namely $\mathrm{NS}\restriction S$. Subsets of the isolated part that meet each ladder in only finitely many points have clopen remainder traces, and these traces form a generalized Boolean algebra. All such traces are disjoint from the carrier part of the remainder. Finally, the subcarriers whose restricted spaces are $\sigma$-closed discrete form an ideal $\mathcal{C}_L$ containing $\mathrm{NS}\restriction S$. If $X_L$ is a $\Delta$-space, a threshold-based gluing argument shows that $\mathcal{C}_L$ is a $\sigma$-ideal. Whether this holds for every ladder system remains open.

We connect modal set theory with Boolean-valued models by developing an internal Kripke semantics for modal formulas whose atomic propositions are set-theoretic sentences. Given a complete Boolean algebra $B$, we view its elements as “local perspectives on truth” inside the Boolean-valued universe $V^{(B)}$ and interpret the modal operators using an accessibility relation $R$ on $B$ defined by co-consistency (equivalently, Boolean compatibility): $aRb$ iff $a\wedge b\neq 0$. Our central conceptual point is that, for set-theoretic sentences $p$, the internal modality $\Diamond p$ holds at $b$ iff there is an ultrafilter $U$ of $B$ containing $b$ such that the classical quotient $V^{(B)}/U$ satisfies $p$. We compute several general and algebra-dependent modal validities, and analyze the special behavior of complete atomic Boolean algebras. Finally, adopting a translation-based semantics on the nonzero part $B^+=B\setminus\{0\}$, we prove a soundness-and-completeness theorem: the normal logic $\KTB$ is exactly the set of modal formulas valid in all translated co-consistency models with parameters.

We have previously given a factorization of a symplectic column under the action of the parity involution which enabled to explicitly have written the inverse of the reduction map in the quantum Littlewood-Richardson bijection. Watanabe has written the reduction map as a composition of several maps, among them, combinatorial $R$-matrices on single columns and a reduction map on a shorter column with rank reduced by one. We now use this approach to write the inverse of the reduction map on a symplectic column by detecting the corresponding symplectic column of rank reduced by one and thus avoiding going through several map compositions.

Algebraic concordance of knots can be understood from the perspective of Seifert matrices, Blanchfield forms, and homology surgery. We initiate a systematic study of algebraic concordance for links from each of these viewpoints. The present article is concerned with algebraic concordance from the perspective of homology surgery and Blanchfield forms, whereas a companion article by the third named author focuses on C-complexes and generalised Seifert matrices. The outcome of the present work consists of two obstructions to $\mu$-component links being concordant. The first obstruction, called the homology surgery invariant, takes values in the Witt group of hermitian forms over the field of fractions $Q$ of $\mathbb{Z}[\mathbb{Z}^\mu]$. The second obtruction, called the Blanchfield invariant, takes values in a Witt group of $Q/\mathbb{Z}[\mathbb{Z}^\mu]$-valued hermitian linking forms. For $\mu\le 2$, we describe these invariants in terms of generalised Seifert matrices.

Influence matrices encode the action exerted on local subsystems by the rest of an extended quantum many-body system during their evolution. Thus, knowledge of the influence matrix facilitates computationally efficient simulations of local dynamics. Here we propose a new systematic approach to generating quantum circuits with complex dynamics for which the influence matrices can be written down exactly. In contrast to previous frameworks of this kind, such as dual-unitary circuits, the resulting influence matrices are non-Markovian, exhibiting nontrivial temporal correlations. We explicitly construct a broad family of circuits of this kind, based on dressing free-fermion (matchgate) circuits with appropriately chosen interaction terms. We show that, contrary to previous solvable instances, these circuits produce patterns of correlations that closely resemble that of typical many-body systems. Our approach can be directly interpreted in terms of an error correction scheme where the terms breaking the solvability of the influence matrices play the role of errors.

Singular Value Decomposition (SVD) underlies matrix factorisation tasks across many fields, with imaging applications demanding real-time processing. Yet SVD algorithms are inherently sequential, constraining real-time GPU throughput and limit online deployment in imaging pipelines. This study introduces a fully parallelized matrix factorization framework for GPUs by enforcing matrix orthogonality on left singular vectors via Lie-parametrised algebra and recovering the remaining components through soft constraints. This asymmetric constraint design enables an efficient parallel and provably valid decomposition, achieves high reconstruction fidelity and substantially accelerates computation relative to the exact SVD, with real-time throughput exceeding standard video frame rates. Performance is evaluated on multiple imaging tasks spanning complementary computational regimes: (1) spatio-temporal background subtraction for ultrasound localisation microscopy, requiring high-dimensional matrix separation, (2) Mueller matrix polarimetry for neurosurgical tissue characterisation, requiring massive batch processing of small matrices, and (3) an MNIST denoising benchmark at an intermediate scale with known ground truth. Across regimes and instruments, the proposed framework demonstrates robust domain transfer at various matrix scales, sufficient for live image-guided workflows that classical solvers cannot currently support in these settings. By prioritising downstream reconstruction fidelity over exact spectral recovery, the proposed SVD framework makes structured matrix factorisation practical for real-time processing.

We prove a series of new results in Hamiltonian dynamics on a general closed symplectic manifold $(M, \omega)$, including: 1. If $M$ admits a Hamiltonian diffeomorphism which is either a pseudo-rotation or has finite order, then $M$ is geometrically uniruled. This resolves a variant of Problem 24 in McDuff–Salamon's list, which predicts an obstruction to the existence of such special Hamiltonian diffeomorphisms in terms of genus zero numerical invariants. 2. If a Hamiltonian possesses a periodic orbit in a non-torsion homology class, then it has infinitely many simple periodic points. The same conclusion holds if the manifold is not geometrically uniruled and the diffeomorphism is minimal for rational Floer homology. These two general results complement the known cases of the Hofer–Zehnder conjecture. 3. If a Hamiltonian diffeomorphism possesses a symplectically degenerate maximum, then it has infinitely many simple periodic points. This resolves an open question which stems from the work of Ginzburg and Gürel. We also establish new cases of the generic Conley conjecture: infinitely many periodic points for generic Hamiltonian diffeomorphisms. The proofs rely on a systematic application of the integral Hamiltonian Floer theory package developed by the first and fourth author, a Künneth isomorphism in equivariant Floer homology, and new quantitative analysis of quantum power maps, which is of independent interest.

We prove Pólya's conjectured lower bound for the Neumann eigenvalue counting function of Euclidean balls. If $B_R^d\subset\mathbb R^d$ is the ball of radius $R$, then, for every $d\ge2$, $R>0$, and $E\ge0$, $$ N_{B_R^d}^{<}(E) \ge \frac{\omega_d}{(2\pi)^d}|B_R^d|E^{d/2} = \frac{(R\sqrt E)^d}{2^d\Gamma(\frac d2+1)^2}, $$ where $\omega_d$ is the volume of the unit $d$-ball and $N_{B_R^d}^{<}(E)$ counts Neumann eigenvalues strictly below $E$. Combined with the Dirichlet theorem for balls, this settles both Pólya inequalities for Euclidean balls in every dimension $d\ge2$. In the disk case, the proof replaces a computer-assisted finite-frequency step by explicit Rayleigh–Ritz estimates. In dimensions $d\ge3$, the radial Neumann condition is a Dini condition rather than a derivative-zero Bessel condition. A strict comparison with an auxiliary Robin problem transfers a derivative-zero Bessel phase estimate to the physical Neumann spectrum. The problem then becomes a comparison between a multiplicity-weighted phase staircase and an integral equal to the Weyl term. Variational trial spaces control low frequencies; finitely many radial levels and beta-integral estimates cover the intermediate range; and a uniform phase estimate treats high frequencies. All finite computations for $2\le d\le6$ are printed in the paper. For $d\ge7$, one compact two-parameter estimate is verified in exact rational arithmetic by the ancillary program.

An antichain is a family of rectangles in which no member contains another. Given a family $\mathcal{E}$ of rectangles, let $h_{\mathcal{E}}$ be the sum of the indicator functions of its members. We show that there exist constants $c, C > 0$ such that for every sparse antichain $\mathcal{E}$ of dyadic rectangles in $\mathbb{R}^2$ one has $\int_E \exp(c h_{\mathcal{E}}) \leq C|E|$, where $E$ is the union of all the rectangles in $\mathcal{E}$. For general sparse families without the antichain condition, the estimate requires replacing $h_{\mathcal{E}}$ by $h_{\mathcal{E}}^{1/2}$, so antichains behave as if they lived in one dimension fewer. We give two applications. First, the dyadic Zygmund conjecture holds in dimension three: the maximal operator associated to dyadic rectangles with sidelengths $2^{m_1} \times 2^{m_2} \times 2^{\Phi(m_1,m_2)}$, where $\Phi$ is monotone increasing in each variable, is weak-type $L \log L$. This recovers a theorem of A. Córdoba. Second, the maximal operator of an arbitrary antichain of dyadic rectangles in the plane is bounded on $L^p$ with norm $O(p')$, which is the growth of the one-parameter maximal function. This bound is sharp, and removing the antichain condition forces a constant that grows like $(p')^2$ instead. The proofs proceed through bounds on $k$-fold intersections: for sparse antichains in the plane, the $k$-wise intersection sums grow at most geometrically in $k$, which we prove through an $L^2$ estimate for the Gram matrix of the normalized indicators of the family. We also show that, in every dimension, the analogous exponential estimate for antichains is equivalent to a $k$-wise intersection bound, and implies the corresponding case of Zygmund's conjecture.

Multi-warehouse inventory allocation is typically formulated as a mixed-integer programming (MIP) problem, yet no single formulation consistently matches heterogeneous instance-level regimes induced by demand concentration, inventory imbalance, replenishment scale, service constraints, and forecast volatility. We study this issue as instance-wise operations research (OR) formulation selection, where each allocation instance is assigned to a solver-executable formulation from a candidate OR expert library. We propose a solver-guided large language model (LLM) framework for OR formulation selection, in which each OR expert corresponds to a MIP formulation encoding a distinct allocation priority. To train the selector, the framework first constructs balanced expert-conditioned supervised fine-tuning (SFT) records for schema learning, and then uses MIP solver evaluation on historical instances to convert solver-evaluated allocation-quality gaps into margin-weighted identity preference optimization (IPO) preferences and per-instance expert-score metadata for reward lookup during group relative policy optimization (GRPO) to assign rewards to sampled responses. Experiments on multi-warehouse inventory allocation instances from JD$\mathord{.}$com, one of China's largest e-retailers, demonstrate that GRPO substantially improves expert-selection accuracy relative to the SFT+IPO selector and, more importantly, produces higher realized allocation quality than both the preference-trained selector and the best fixed formulation. With GRPO, Hit Ratio@1 and Hit Ratio@2 increase from 21.45% to 50.42% and from 70.47% to 82.31%. The resulting selector achieves an allocation accuracy gain of 12.57 percentage points over the incumbent baseline, outperforming both the SFT+IPO selector and the best fixed OR expert, and reduces the gap to the ex-post oracle to 4.85 percentage points.

We establish necessary and sufficient conditions for the uniform boundedness of the multi-parameter exponential sums with product Hilbert kernels $$\sum_{1\le|t_1|\le N_1,\cdots,1\le|t_k|\le N_k} \frac{e^{2\pi i P(t_1,\dots,t_k)}}{t_1\cdots t_k},$$ where $P:\mathbb{Z}^k\to\mathbb{R}$ is a polynomial of the form $P(t)=\sum_{\mathfrak{m}\in \Lambda} c_{\mathfrak{m}}\, t^{\mathfrak{m}},$ with real coefficients. The resulting bound is uniform in both the coefficients $c_{\mathfrak m}$ and the truncation parameters $N_1,\ldots,N_k$. To this end, we develop a higher-dimensional version of the multi-parameter circle method. Under the sufficient condition, we further prove $\ell^p$-boundedness of the associated discrete multiple Hilbert transform.