Papers

We compute the geometric monodromy factorization of Xiao's genus-two Lefschetz fibration directly from its branched-cover construction, following Moishezon's braid-monodromy method. The complete quadrangle determines the motion of the six branch points and gives four nonseparating and three separating vanishing cycles. At each of the three $3+3$ degenerations, the branch motion determines the spherical mapping class, and Xiao's local holomorphic model determines its genus-two lift. After fixing a distinguished system of paths, we obtain an ordered positive factorization of type $(4,3)$ and give Artin coordinates for the seven factors.

We study computable error bounds and certified early stopping for regularized inverse problems, where a data-fidelity term is traded against a regularizer. The analysis relies on an exact duality-gap identity that splits the total gap of $F(\Phi\mu)+\lambda R(\mu)$ into a data-fidelity Fenchel–Young loss and a regularizer Fenchel–Young loss, $ \Delta(\mu,h)=L_F(\Phi\mu\parallel h)+\lambda L_R(\mu\parallel\eta),\qquad \eta=-\Phi^\star h/\lambda, $ valid for any primal point $\mu$ and any dual point $h$. The data-fidelity term $F$ is strictly convex, so wherever $F^\star$ is differentiable the loss $L_F(\Phi\mu\parallel h)$ is the Bregman divergence of~$F$ between the prediction $\Phi\mu$ and $\nabla F^\star(h)$, and it vanishes exactly at Mirror Alignment $h=\nabla F(\Phi\mu)$. Evaluated at a dual-feasible point $\tilde h$, the gap~$\Delta(\mu,\tilde h)$ is computable and oracle-free, meaning that it uses no knowledge of the solution, and it bounds the suboptimality of $\mu$. Under the source condition, the same Fenchel–Young losses give a priori bounds on the estimation and prediction errors. Their scale is the irreducible model and noise error $L_F(\Phi\mu^\star\parallel h^\star)$, which vanishes exactly when Mirror Alignment holds at the certificate. This gives an early-stopping rule: run the algorithm until the regularizer Fenchel–Young loss falls below a tolerance $\epsilon$. A constructive version of the Brøndsted–Rockafellar theorem then turns the current pair into an exact dual-feasible one, and this proxy lifts to an exact primal certificate. We build the proxy by a proximal step in the geometry of the fidelity, with Bregman kernel $F^\star$ and tilted by the prediction $\Phi\mu$: it recovers the Euclidean step of Carlier when $F$ is the squared error, and it reduces the duality gap by the regularizer Fenchel–Young loss, up to a second-order remainder that vanishes in the quadratic case. Our running example is the Generalized Beurling–Lasso (GBL), where $R$ is the total-variation norm on signed measures. It contains the classical Beurling–Lasso, obtained with the squared error, and also covers robust, logistic, entropic and inverse-optimal-transport losses. The same duality gap certifies deep-learning optimizers such as Lion-K and Muon, in their proximal form, as solvers of the regularized program. A companion paper by the same authors builds on these error bounds to establish exact support recovery for the GBL under a non-degenerate source condition.

Overconstrained rational single-loop linkages are efficient, compact, and low-cost custom mechanisms, yet their deployment in industrial settings is limited. In simulations, rigid body formulations fail due to redundant constraints. This study presents a flexible multibody modeling framework based on the floating frame of reference formulation, and delivers an overall accuracy analysis of assembled linkages prototypes. The approach is validated against 3D-printed PLA prototypes of a Bennett four-bar mechanism, including variants with intentional joint-axis misalignment, which theoretically, from the rigid body point of view, cannot be assembled. A supplementary contribution is delivered in the form of a kinematic parameter identification methodology suited for this type of mechanism with ill-conditioned Jacobian. The experimental and simulation results are compared and reveal that these overconstrained mechanisms exhibit a self-assembling tendency – structural compliance drives the assembly toward the ideal geometric configuration, distributing constraint stress throughout the structure. Additional qualitative demonstrations using cardboard tubes and bamboo sticks as link building blocks confirm that functional mechanisms can be realized from low-cost, unconventional materials with limited manufacturing accuracy. The proposed modeling pipeline is fully algorithmic and enables design optimization in the future.

We study an $\varepsilon$-differentially private synthetic measure for $n$ points in $[0,1]^d$ by applying the existing PrivTree algorithm to construct an adaptive binary partition and then privately releasing its leaf masses. We consider the worst-case data model without any sampling or population-distribution assumption. The 1-Wasserstein error of the synthetic measure is $\widetilde O_d((\varepsilon n)^{-1/d})$ for $d\ge2$, which is optimal compared to the minimax lower bound up to a logarithmic factor. Moreover, for $d\ge3$ and $2<s\le d$, if the data set has covering number at most $Ar^{-s}$ over the relevant finite range of scales $r$, the expected error improves to $\widetilde O_{d,s}((\varepsilon n)^{-1/s})$. Thus the rate depends on a finite-scale intrinsic dimension rather than the ambient dimension, without requiring the recovery of a low-dimensional manifold. We also introduce a shifting technique to further avoid the exponential dependence of the constant on the ambient dimension $d$.

We study uniform high-frequency localization for the Laplacian on compact metric graphs through the least $L^2$-mass that eigenfunctions must place in a prescribed measurable observation set. We first identify this asymptotic localization constant with the minimum of a linear functional over the attainable edge-intensity set; in the generic standard-Kirchhoff setting, this set is governed by the regular Gauss image of the secular manifold. Primitive cycles and exterior-to-exterior paths consequently determine the positivity threshold, but not, in general, the positive numerical value. We introduce a boundary-aware singular-completion cone and prove that it contains all regular secular edge-energy vectors for trees, unicyclic graphs, and closed graphs of cycle rank two. We then construct a cycle-rank-two graph with two Dirichlet leaves for which this completion principle fails. An exact rational separator, combined with a validated Krawczyk enclosure, yields a nonsingular scalar secular state lying outside every boundary-compatible singular sector. A positive radial derivative identity and recurrence in the compact orbit closure convert this local separation into an exact high-frequency eigensequence for a single fixed metric. For a suitable measurable observation set, the true high-frequency localization constant $C_\infty(\omega;\ell)$ and its singular-completion counterpart $C_{\mathrm{sing}}(\omega;\ell)$ satisfy $$C_\infty(\omega;\ell)<1/2<C_{\mathrm{sing}}(\omega;\ell)$$. Thus, singular completion captures the quantitative localization geometry in several low-complexity classes but does not, in general, determine the high-frequency variational problem on a fixed quantum graph.

For any fixed $k$, we prove a lower bound on the $k$th largest modulus of an eigenvalue of the non-backtracking matrix $B$. Specifically, consider any deterministic or random family of graphs that converges locally to the unimodular Galton-Watson tree with root degree distribution $D$, and set $\kappa:=\mathbb E[D(D-1)]/\mathbb E[D]$. Given $\kappa>1$ and an exponential-moment bound on the empirical degree distributions, we show that $|\lambda_k(B)|\geq\sqrt{\kappa}-o_N(1)$, where $N$ is the number of vertices. When restricted to locally tree-like regular graphs, this recovers a well-known consequence of the Ihara-Bass formula. In the specific case where the graph is generated through the Erdős-Rényi model with expected degree $d>1$, this proves a conjecture of Bordenave, Lelarge, and Massoulié. To do this, we show that the normalized log-determinant of the Bethe-Hessian of the graph is bounded by that of the Bethe-Hessian of its local limit. This bound is violated if the eigenvalues of the non-backtracking matrix are too small. We establish this using an effective-conductance interpretation of the tree Green's function recursion.

Consider a branching random walk (BRW) of depth $n$ with standard Gaussian increments, conditioned on all endpoints lying in an interval of radius $r$ (which may depend on $n$), i.e. a small ball event. We prove that this conditioning results in exponential decay of correlations between the endpoints, with correlation length $O(r)$. Our proof is based on the renormalization group, which in this context captures the effect of the endpoint conditioning on earlier steps in the walk. This effect can be separated into so-called relevant and irrelevant parts, and in our setting the relevant part makes the conditioned BRW behave like a branching Ornstein-Uhlenbeck walk for the first $n-O(r)$ steps, leading to the correlation decay. The irrelevant part shrinks in a nontrivial manner, and the simplest argument to bound it uniformly only yields correlation length $O(r^2)$. To achieve the optimal $O(r)$ bound, we establish that the irrelevant part shrinks horizontally in space before decaying uniformly. The BRW is a prototypical example of a log-correlated Gaussian field, and it forms the basis of many hierarchical field theories in the mathematical formulation of quantum field theory. As such, we present our results in a general form which is applicable to other hierarchical field theories with confining potentials, of which the small ball conditioned BRW is an example. We apply our general result to two other models: first, the sinh-Gordon model which has a strictly convex potential, so at the level of covariances the model behaves like one with a quadratic potential, i.e. the model has a positive mass. Second, we consider generalizations of the $\phi^4$ model with any power $\alpha \geq 2$; as $\alpha$ increases from $2$ to $\infty$, these power potentials naturally interpolate between the massive (quadratic) potential and the infinite square well potential of the small ball event.

We prove an equivariant Landweber exact functor theorem for abelian compact Lie groups $G$. For a graded module $N$ over the $G$-equivariant Lazard ring $L_G$, we give necessary and sufficient algebraic conditions on $N$ for the functor \[ X\longmapsto (MU_G)_*(X)\otimes_{L_G} N \] to define a homology theory on the category $\mathrm{Sp}^G$ of genuine $G$-spectra. Our criterion is obtained by varifying the Landweber exactness on each stratum indexed by a closed subgroup of $G$ and then gluing the local flatness data via Euler classes. As an application, we show that for any non-equivariant Landweber exact ring spectrum $E$, $MU_G\wedge_{MU} E$ is equivariantly Landweber exact, thereby proving a conjecture of Wisdom.

A BV structure is constructed on the space of zero modes of a quadratic elliptic higher abelian gauge theory. The associated BV integral is shown to be independent of the choice of zero-mode Lagrangian under mild hypotheses. The construction yields explicit zero-mode partition functions for several $p$-form abelian gauge theories.

Barycentric Rényi divergences were introduced in [Mosonyi, Bunth, Vrana, Linear Algebra and its Applications, 2024] as an alternative to standard Kubo-Ando constructions to define multivariate quantum Rényi divergences. They are defined via a variational expression and depend on a finite collection of quantum relative entropies $D^{q_x}$. When all the relative entropies are monotone under CPTP maps then so are the corresponding barycentric Rényi divergences, and when all the relative entropies are additive then the corresponding barycentric Rényi divergences are subadditive under tensor product. Additivity has only been established before for the case where all $D^{q_x}$ are chosen to be the Umegaki relative entropy, which is also the only case where the barycentric Rényi divergence (called the minimal one) was known to admit an explicit expression. Here we settle the problem of additivity by showing that for any choice of monotone quantum relative entropies, the regularized barycentric Rényi divergence coincides with the minimal barycentric Rényi divergence on invertible inputs. This in turn implies that if at least two of the $D^{q_x}$ are strictly larger than the Umegaki relative entropy (in a precise sense), then the resulting barycentric Rényi divergence is not even weakly additive.

We study the numerical geography and tangent geometry of very amply polarized Calabi–Yau threefolds $(X,H)$ through the positivity of the first jet bundle $J^1(H)$. Writing $d=\int_X H^3$, $c=\int_Xc_2(X) H$, and $e=\int_X c_3(X)$, we exploit two different positivity properties of this single bundle. Mixed intersections on $\mathbb{P}(J^1(H)^*)$ give $e\ge-5d-c-c^2/(4d)$, while a volume estimate for a perturbed tautological class gives $e\ge40d\left[(1-\frac{c}{10d})^{3/2}-1\right]$; in particular $e+6c\ge0$, improving Sun's inequality $e+10c\ge0$. As consequences, we obtain the uniform Hodge bounds $-4d-80\le h^{1,1}(X)-h^{2,1}(X)\le173d/66$, the lower bound $\mathrm{deg}X^\vee\ge78$ for the dual hypersurface, and, in the critical case $X\subset\mathbb{P}^6$, the upper bound $d\le34$. We also prove that, for every $m\ge2$, the tangent-incidence morphism associated with $|mH|$ is the normalization of the tangent variety, conjecture tangent birationality for complete embeddings $X\subset\mathbb{P}^N$ with $N\ge7$, and verify it for several families.

We prove that every smooth properly embedded free-boundary minimal annulus in the unit ball is congruent to the critical catenoid. We also classify positive solutions of $(\Delta_{\mathbb{S}^2}+2)u=0$ on smooth spherical annuli with $u=0$ and $|\nabla u|=1$ on the boundary; their one-homogeneous extensions are the axially symmetric Alt-Caffarelli cones. Lastly, we also treat the case of free-boundary minimal annuli in spherical caps.

We study the singular limit $p \downarrow 1$ for positive radial entire solutions $u_p$ of $$ -\Delta_p u_p=F_p\left(|x|, u_p,\left|\nabla u_p\right|\right) u_p^{-\beta_p} \quad \text { in } \mathbb{R}^n, $$ where $n \geq 2,1<p<\min \{2, \sqrt{n}\}$, and $0 \leq \beta_p \leq p-1$. Assuming that $$ F_p \rightarrow F_1 \quad \text { locally uniformly }, \quad \frac{\beta_p}{p-1} \rightarrow \mathbf{c} \in[0,1] \quad \text { as } p \downarrow 1, $$ we prove that, after passing to a subsequence, $u_{p_j} \rightarrow u$ in $C_{\text {loc }}\left(\mathbb{R}^n\right)$ and $\nabla u_{p_j} \stackrel{*}{\rightharpoonup} \nabla u$ in $L_{\text {loc }}^{\infty}\left(\mathbb{R}^n ; \mathbb{R}^n\right)$, where $u \in W^{1, \infty}\left(\mathbb{R}^n\right)$ is radial, and solves $$ -\Delta_1 u=\mathtt{K}(|x|) \quad \text { in } \mathbb{R}^n . $$ On $\left\{x \in \mathbb{R}^n \mid \mathtt{G}(|x|)<1\right\}$, we prove that $\mathtt{K}(|x|)=w^{-\mathbf{c}} F_1(|x|, u(x), 0)$ and that $\nabla u=0$ a.e., where $u_{p_i}^{p_i-1} \rightarrow w$. On the other hand, we show that if $\rho \mapsto F_1(r, t, \rho)$ is affine for every $r$ and $t$, then $\mathtt{K}(|x|)=$ $w^{-\mathbf{c}} F_1(|x|, u(x),|\nabla u(x)|)$ a.e. in $\mathbb{R}^n$. For $n=1$ and $1<p<2$, we study $$ \frac{\mathrm{d}}{\mathrm{~d} x}\left(\left|u_p^{\prime}\right|^{p-2} u_p^{\prime}\right)=\mathtt{F}_p\left(|x|, u_p,\left|u_p^{\prime}\right|\right) u_p^{-\beta_p} \quad \text { in } \mathbb{R} $$ and obtain analogous results as $p \downarrow 1$.

Let $\mu$ be a finitely supported probability measure on $\mathrm{SL}_2(\mathbb{R})$ whose support generates a Zariski-dense discrete subgroup. We prove that its stationary probability measure on $\mathbb{P}^1(\mathbb{R})$ is singular with respect to the $\mathrm{SO}(2)$-invariant probability measure. The proof compares winding variances on finite covers of a closed hyperbolic surface and uses a topological obstruction.

We show that for all negative $L$-space knots and framings $n\geq2g(K)$, if a certain cobordism map between cables of $K$ is non-vanishing, then the Floer lasagna module of the knot trace of $K$ with framing $-n$ is infinite dimensional. Further, for the maps on the hat flavour of link Floer homology induced by band maps, quasi-stabilisations and pair of pants cobordisms we give a description of the map on the $E^{\infty}$-page of Ozsvath and Szabo's spectral sequence for forgetting components.

Let $C$ be a connected nodal curve of genus $g$ over an algebraically closed field $k$ of characteristic zero. Let \[ A(C)=\Big(J(C), {\overline{P^{g-1}_C}}, \Theta(C) \Big) \] be the stable semi-abelic pair (in the sense of Alexeev) associated with its canonical compactified Jacobian $\overline{P^{g-1}_C}$ in degree $g-1$. The canonical triple of $C$ consists of the closure of its canonical image together with two sets of distinguished points encoding the information of the images of the nonseparating nodes and of the branch data of the canonical map. We prove a functorial reconstruction: given connected nodal curves $C$ and $C'$ over $k$, every isomorphism $A(C)\cong A(C')$ of stable semi-abelic pairs canonically induces a projectivity between the corresponding canonical triples of $C$ and $C'$. Passing to isomorphism classes, this recovers the reconstruction statement in the compactified Torelli theorem of Caporaso-Viviani. Our proof extends Andreotti's Gauss-map strategy. The branch locus of the projection from the normalization of the Gauss graph recovers the nonlinear canonical components and branch data by projective biduality; boundary strata of the compactified Jacobian recover the node-images; and incidence arguments recover the linear components. Along the way, we repair technical gaps in the classical treatments of Andreotti's proof of Torelli's theorem for smooth curves.

Let G be a non-trivial finite subgroup of SL_2(C) and let p : X \to C^2/G be the minimal resolution. We prove the C^*-equivariant Cohomological Crepant Resolution Conjecture for the ADE orbifold [C^2/G]. More precisely, after specializing the quantum parameters at suitable roots of unity, we prove that the Bryan-Gholampour transformation induces an isomorphism between the C^*-equivariant quantum cohomology of X and the C^*-equivariant Chen-Ruan cohomology of [C^2/G]. Our proof is direct and is based on the McKay correspondence, ADE root systems, and character theory. Following the character-theoretic form of the Bryan-Gholampour change of variables, we use the McKay correspondence to diagonalize the transformation and to reduce the compatibility of the products to a uniform root-system identity. In type A, this reduction admits a discrete Fourier realization. In type D, the cyclic subgroup of the binary dihedral group leads instead to finite sine transforms, while the exceptional cases E_6,E_7,E_8 are treated by exact matrix computations over the corresponding cyclotomic fields. As a preliminary result, for a symplectic complex vector space V and a finite subgroup G of Sp(V), we give an explicit presentation of the C^*-equivariant Chen-Ruan product of [V/G] and identify the resulting algebra with the Rees algebra of the center of the group algebra with respect to the age filtration. In particular, the equivariant Chen-Ruan algebra interpolates between the center of the group algebra and its associated graded algebra, the latter recovering the ordinary Chen-Ruan cohomology.

Sorting By Transpositions (SBT) seeks the minimum number of transpositions required to sort a permutation $\pi$ on $n$ symbols into the identity $\iota$. Let $N=n+1$. A cyclic-target pair $(\omega,\beta)$ consists of an even permutation $\omega$ and an $N$-cycle $\beta$ for which $\rho=\omega\beta$ is an $N$-cycle. An SBT instance is the special case $(\bar{\iota}{\bar{\pi}}^{-1},\bar{\pi})$, where $\bar{\pi}$ and $\bar{\iota}$ encode $\pi$ and $\iota$, and $\bar{\iota}{\bar{\pi}}^{-1}\bar{\pi}=\bar{\iota}$. For a prescribed fixed-point-free cycle type, fixed-content words encode $\omega$, with colors distinguishing cycles and ranks recording their orientations relative to $\beta$. A word is realizable exactly when $\rho=\omega\beta$ is an $N$-cycle. Permutations of equal-part colors and shifts of rank origins form auxiliary symmetries that, together with word rotation and position reflection coupled to rank inversion, define a twisted dihedral action. Its orbits are twisted bracelets, and its realizable orbits correspond bijectively to extended-toric equivalence classes of cyclic-target pairs, where reflection is adjoined to classical toric equivalence. The correspondence yields a Burnside identity, and our method generates one encoding word for each such class. The transposition diameter $TD(n)$ is the largest transposition distance in $S_n$. Combining fixed-point contraction and reductions of the ambient instance space based on cycle structure with exhaustive verification of every remaining twisted bracelet, we prove $TD(16)=9$, closing a twenty-five-year gap. This result also yields $TD(19)=11$ and, for every $n\equiv1\pmod{3}$ with $n\geq16$, $TD(n)\leq\left\lfloor(2n-2)/3\right\rfloor-1$, improving the previous general upper bound by one for these $n$.

In this work we propose a new existence theorem for numerical-flux functions for supercritical fluids that are pressure-equilibrium-preserving (PEP). In particular, we characterize the existence of consistent numerical-flux functions that satisfy an algebraic PEP property. Our theory links the existence of PEP schemes to geometric properties of the equation of state describing the thermodynamics of the fluid. When these geometric properties fail for a pair of states on the same isobar, no PEP schemes of the considered form can exist on a domain containing those states. In our analysis the equation of state itself can be fully general only needing to satisfy some fundamental thermodynamic principles. Recently, PEP compatibility conditions for general equations of state have been derived [1] that rely on the existence of certain thermodynamic derivatives which are not guaranteed to be defined under fundamental thermodynamic principles. The geometric conditions in our theory do not depend on the existence of these derivatives and recover the recent compatibility conditions in the case that these derivatives are defined. Finally, using numerical experiments we demonstrate for two supercritical fluids that our existence conditions are restrictive and thus that no PEP schemes of the form we consider exist on the domain we specify for these fluids. Using our geometric perspective, we also shed light on mechanisms by which PEP schemes can develop numerical issues, which we also demonstrate using numerical experiments.

Modern large-scale machine learning tasks often require multiple workers, devices, CPUs, or GPUs to compute stochastic gradients in parallel and asynchronously to train model weights. Theoretical results typically distinguish between two settings: (i) the homogeneous setting, where all workers have access to the same data distribution, and (ii) the heterogeneous setting, where each worker operates on different data distributions. Known optimal time complexities in these settings reveal a significant gap, with far more pessimistic guarantees in the heterogeneous case. In this work, we investigate whether these pessimistic optimal time complexities can be overcome under different assumptions. Surprisingly, we show that improvement is provably impossible under widely used first- and second-order similarity assumptions for any randomized algorithm. We then turn to the interpolation regime and demonstrate that the weak interpolation assumption alone is also insufficient. Finally, we introduce a minimal combination of irreducible assumptions, strong interpolation and the local Polyak-Lojasiewicz condition, to derive a new time complexity bound that matches the dependence on worker computation times in the best-known result in the homogeneous setting, without requiring identical data distributions.

Kemeny's constant quantifies the expected time for a random walk to reach a randomly chosen vertex, capturing global properties of a Markov chain. We develop a matrix-analytic framework for bounding Kemeny's constant of a finite connected weighted graph using degree-weighted compressions of the normalized adjacency matrix, pinching inequalities, and eigenvalue interlacing. Our main partition theorem gives lower bounds in terms of the compressed spectrum, with complete equality characterizations, and converts structural graph information into spectrally computable estimates. For instance, when applied to proper color partitions, the proposed method yields a sharp lower bound on Kemeny's constant in terms of the chromatic number \[ K(G)\ge n-2+\frac{1}{\chi(G)}, \] which extends a bipartite bound of Ciardo, Dahl, and Kirkland (2022) to arbitrary chromatic number, and which is incomparable with the normalized Hoffman bound by Chung (1997). For unweighted graphs, we also characterize all equality cases of this bound. We further illustrate the power of the proposed matrix framework by deriving spectral bounds for NP-hard graph problems involving normalized cuts and conductance. Our results include an asymptotically sharp conductance bound, two-sided interlacing bounds from principal submatrices and quotient matrices, and estimates for the change in Kemeny's constant under connectivity-preserving deletion of multiple edges.

We consider a possible approach to the Lindelöf hypothesis for Dirichlet $L$-functions. It is based on a special form of the functional equation for the corresponding theta functions. To estimate $L_\chi(0.5+it)$ we need to solve certain systems of linear equations. The entries to the corresponding matrices are formed by the summands to the series for theta functions and their derivatives. Numerical data suggest that the inverse matrices have a deep structure and allow us to state a number of conjectures. In particular, it seems that for a character modulo $q$ certain entries to the inverse matrices tend to finite limits when the sizes of the matrices run over arithmetical progressions with step $2q$. Moreover, these limits belong to the real cyclotomic field $\mathbb{Q}(\cos(\pi/q))$ (up to a scaling factor of $\sqrt{q}$).

Large language model (LLM) distillation aims to transfer the capabilities of a powerful teacher to a smaller student. Direct imitation, however, can also transfer the teacher's systematic bias and errors. This challenge is particularly pronounced under covariate shift, when the teacher's reliability on target questions is uncertain and target-domain reward feedback is unavailable. We propose Coupled Calibration and Learning (CCL), an LLM distillation algorithm that couples teacher calibration with student updates through token-level branching, using reward feedback only on source questions. Each iteration calibrates the teacher using source feedback and then uses the calibrated teacher to train the student on target questions. The updated student, in turn, informs subsequent calibration. In an autoregressive policy framework, we prove that the output student's expected average Kullback-Leibler divergence to the oracle student converges to zero at a polynomial rate in the number of iterations. The oracle maximizes the true reference-regularized target reward within the student class, which need not represent the unrestricted optimal policy. Our analysis quantifies the progress of projected student gradient updates while controlling the error in teacher calibration. We further establish a separation from regularized direct matching: its error relative to the oracle student can remain bounded away from zero even when the teacher achieves higher regularized target reward than every student policy. These results demonstrate that LLM distillation can overcome persistent teacher bias and recover the optimal student through coupled calibration and learning, without target-domain reward feedback.

We bound the difference between two stochastic Volterra processes with identical Lipschitz coefficients but different kernels. For non-convolution kernels, we establish estimates in $C^0([0,T];L^p(\Omega))$, $p\geq 2$, and for convolution kernels in $L^p(\Omega;L^q(0,T))$, $q \in [1,p]$, and $C^\beta([0,T];L^p(\Omega))$, $L^p(\Omega;C^\beta([0,T]))$, where the range of the Hölder exponent $\beta \in (0,1]$ is the maximal permitted by the regularity of the processes. For the fractional kernel, we then construct Markovian approximations whose error we show to decay as $e^{-a\sqrt{N}}$ in the aforementioned norms, using an $N$-node quadrature based on sinc methods. Numerical experiments for the fractional Brownian motion verify our findings.

Explicit computations of generic infinitesimal Newton–Okounkov bodies are difficult even for varieties with simple product structure. We give an explicit formula in arbitrary dimension for positive-degree external tensor products on products of smooth projective curves. Writing $d^\downarrow=(d_1^\downarrow,\ldots,d_n^\downarrow)$ for the decreasing rearrangement of the degree vector and setting $d_{n+1}^\downarrow=0$, the body admits the explicit Minkowski decomposition $\sum_{j=1}^n(d_j^\downarrow-d_{j+1}^\downarrow)S_j^{(n)}$, where the $S_j^{(n)}$ are the embedded simplices defined below. Using this description, we give a sharp criterion for equality in the Minkowski inclusion. A simultaneous relabeling argument also allows finitely many such bodies to be realized on a common very general locus of flags after independent decreasing rearrangements of their degree vectors.

Balogh, Bollobás and Narayanan conjectured that among all finite simple $r$-uniform $d$-regular hypergraphs, the number of weak independent sets is maximized by a natural quasi-bipartite construction $H_{r,d}$. We give three types of evidence for this conjecture. For every fixed $r$, we prove the conjectured asymptotic exponential rate whenever the twin quotient has maximum pair codegree $o(d)$. The proof uses the hypergraph container method. For hypergraphs with no cross-edges, the occupancy method gives the sharper error bound $O_r(\log d/d)$. We show that a stronger version of the conjecture in terms of the so-called occupancy fraction is not true, by providing a counterexample for every $r\ge 3$. We also prove exact cases of the conjecture when the hypergraph is $2$-regular. Using an entropy decomposition in the dual edge-cover problem, we settle every odd $r$ and the case $(r,d)=(4,2)$.

We show that the universal rigid monoidal category on one object has an abelian envelope. Since it was proven by Coulembier-Etingof-Ostrik-Pauwels (2023) that this category cannot have an abelian envelope with the quotient property, this disproves the conjecture in [op.cit.] saying that every abelian envelope has the quotient property. Monoidal Ringel duality (Flake-Gruber arxiv:2512.19558) yields a candidate envelope as a lower finite highest weight category. Our proof that this is an abelian envelope relies on Coulembier-Etingof's (2024) continuants to show the required universal property. AI was used to find these results.

Let $n \geq 5$. We prove that the helical maximal operator is bounded on $L^p(\mathbb{R}^n)$ for some $p<2n-4$. This improves the previously known range $p>2n-4$ obtained by Gan-Maldague-Oh. As observed by Beltran-Hickman, the exponent $2n-4$ is the natural limit of the standard local smoothing method.

The (massless) sinh-Gordon model is defined in terms of a continuum massless Gaussian free field perturbed by a cosh interaction. For the hierarchical version of the model, we prove existence of the infinite volume limit, a uniform log-Sobolev inequality, and a mass gap. Our results hold for all $b^2 \in (0,1)$ and simplify for $b^2 \in (0,1/2)$ for which the Gaussian multiplicative chaos has two moments. In an appendix, our renormalization group analysis is compared with the conjectures and controversies in physics which are mostly based on formal analytic continuation of conjectures for the sine-Gordon model and we raise some questions.

A random unitary quantum circuit is expected to be incompressible for exponentially long times. We show that the constant-error circuit complexity of a random unitary circuit grows almost linearly with time as $\Omega(T/\log T)$. The bound holds for all $2\leq T\leq 4^n$ where $n$ is the system size, and involves no other $n$-dependence. This improves previous lower bounds derived from spectral gaps and unitary designs by a factor of $\mathrm{poly}(n)$. Drawing on insights from stochastic calculus, geometric functional analysis, and randomized linear algebra, our approach exploits the circuit's response to variations of individual gates and requires no control over convergence to high-order unitary designs.