Sparse counting lemma for $K_4$

The sparse analogue of Szemerédi's regularity method has played a central role in the development of extremal results for random graphs. While the sparse embedding lemma (the KLR conjecture) has been resolved, the corresponding sparse counting lemma remains widely open. The conjecture, formulated by Gerke, Marciniszyn, and Steger, states that for every fixed graph $H$ and any $β>0$, there exists $\varepsilon>0$ such that the following holds. Consider a balanced blow-up of $H$ with vertex classes of size $n$, where each pair corresponding to an edge of $H$ forms an $(\varepsilon)$-regular bipartite graph with exactly $m$ edges. Assume that $m$ is above the natural threshold $m \gg n^{2-1/m_2(H)}$, then all but a $β^m$ proportion of such graphs contain at least $(1-δ)$ times the expected number of copies of $H$.   At present, among the complete graphs, the conjecture is known only for $H=K_3$. In this paper, we establish the $H=K_4$ case of the conjecture.


On lower bounds for the number of ideal and finite vertices of right-angled hyperbolic polyhedra in dimensions from 5 to 12

We investigate lower bounds for the number of ideal and finite vertices of right-angled hyperbolic polyhedra of finite volume. We use a geometric method of orthogonal gluings to establish new bounds in low dimensions, specifically $v_\infty(P^5) \ge 3$ and $v_{fin}(P^7) \ge 4$. By combining these initial bounds with double counting arguments and recurrence relations, we obtain improved lower bounds for both types of vertices in all higher dimensions up to $n=12$, the maximal dimension where polyhedra of this class exist.


Two Littlewood identities for fully inhomogeneous spin Hall-Littlewood symmetric rational functions

Fully inhomogeneous spin Hall-Littlewood symmetric rational functions $F_λ$ arise as partition functions of certain path configurations in the $\mathfrak{sl}_2$ higher spin six vertex models. They are multiparameter generalizations of the classical Hall-Littlewood symmetric polynomials. We establish two new generalizations of the classical Littlewood identity, where we express a weighted sum of $F_λ$'s over all partitions $λ$ as a product of the Littlewood kernel and another simple product in one case, and a product of the Littlewood kernel and a Pfaffian in the other case. As a corollary we obtain a novel Littlewood identity for Hall-Littlewood symmetric polynomials.   We also elaborate on the newly established connection between the fully inhomogeneous spin Hall-Littlewood symmetric rational functions $F_λ$ and the modified Robbins polynomials, the latter being multivariate generating functions for alternating sign matrices. This connection allowed us to discover the two generalizations of the Littlewood identity and we provide a bijection between the underlying combinatorial models in the case where $λ$ is strictly decreasing.


Reciprocity of Skew Hall-Littlewood-Schubert Series

Carnevale, Schein and Voll proved self-reciprocity of the generalized Igusa functions, and Maglione and Voll did the same for the Hall-Littlewood-Schubert series. We introduce a simultaneous generalization and refinement of these two rational functions, and prove that it satisfies a self-reciprocity property. This answers a problem posed by Maglione and Voll. Our method of proof is elementary, avoiding the use of $p$-adic integration.


On Lexicographic Product and Multi-Word-Representability

Benny George Kenkireth
Gopalan Sajith
Sreyas Sasidharan

We investigate the relationship between the lexicographic product of graphs and their multi-word-representation number. Although the lexicographic product of two word-representable graphs need not itself be word-representable, a precise characterization has not previously been established. We provide a complete characterization, showing that for word-representable graphs $G_1$ and $G_2$, the lexicographic product $G_1 \circ G_2$ is word-representable if and only if $G_2$ is a comparability graph. For lexicographic powers, we prove that $G^{[k]}$ is word-representable if and only if $G$ is a comparability graph. The multi-word-representation number $μ$ for lexicographic powers and products satisfies the following bounds. If $G$ is a non-comparability graph, then $μ(G^{[k]}) \le k$, whereas if $G$ is the union of two comparability graphs, then $μ(G^{[k]}) = 2$. More generally, for graphs $G_1$ and $G_2$ with $μ(G_1) = k_1$ and $μ(G_2) = k_2$, the lexicographic product $H = G_1 \circ G_2$ satisfies the upper bound $μ(H) \le k_1 + k_2$. This bound is tight, with equality $μ(H) = k_1$, when $k_1 \ge k_2$ and $G_2$ is the union of $k_1$ comparability graphs. Moreover, if $G_1$ and $G_2$ are minimal non-word-representable graphs, then $μ(G_1 \circ G_2) \le 3$. Finally, we study the function $τ(n)$, which measures the size of the largest word-representable induced subgraph guaranteed in every $n$-vertex graph. By constructing extremal graphs via lexicographic powers, we establish a sublinear upper bound, showing that $τ(n) \le n^{0.86}$ for sufficiently large $n$.


Seymour-tight orientations

Krystal Guo
Ross J. Kang
Gabriëlle Zwaneveld

We investigate `almost counterexamples' to Seymour's second neighbourhood conjecture. In what we call Seymour-tight orientations, the size of the first neighbourhood of each vertex equals the size of its second neighbourhood. We give several examples and constructions. Specifically, we prove that the class of Seymour-tight orientations is closed under taking (generalized) lexicographic products. Moreover, the lexicographic product of a putative counterexample to Seymour's second neighbourhood conjecture and a Seymour-tight orientation is again a counterexample.   Using lexicographic products, we show that if the conjecture is false, then there exist counterexamples that are close to regular tournaments, and moreover that any digraph occurs as an induced subgraph of a counterexample. We then use this same machinery to construct special putative counterexamples to Sullivan's conjecture.   The inherent symmetry of these orientations give access to an algebraic perspective. Seymour-tight orientations that are also Cayley digraphs correspond to special pairs of critical sets in groups, which connects potentially to additive combinatorics. We use Kemperman's theorem to characterize those Seymour-tight orientations that are the Cayley digraph of an abelian group.


A Finite-State Proof of the Well-Definedness of a Perturbed Hofstadter Sequence

We prove that the perturbed Hofstadter-type sequence Q(1)=1, Q(2)=1, and Q(n)=Q(n-Q(n-1))+Q(n-Q(n-2))+(-1)^n is well-defined for all n>=1, in the sense that all recursive arguments remain positive. This contrasts with the classical Hofstadter Q-sequence, for which global well-definedness remains open.   The proof reduces the infinite recursion to a finite combinatorial constraint system. We introduce a symbolic encoding of local configurations, compute the finite set of admissible contexts, and construct a compatibility relation that captures all valid local transitions. We then show that valid assignments split into two global modes, which reduces all potential obstructions to a finite critical core. A complete finite verification excludes these obstructions and establishes global well-definedness.   More generally, the argument shows that certain meta-Fibonacci recursions admit a finite-state description whose global consistency can be decided by exhaustive combinatorial analysis.


Critical dimensions and small cycle dominance from all-orders asymptotics of $d$-matrix theory

Supersymmetric sectors of $\mathcal{N}=4$ super-Yang-Mills theory motivate the study of the partition function for the counting of gauge-invariant functions of $d=2,3$ matrices transforming under the adjoint action of $U(N)$. The partition function $ \mathcal{Z}_d ( x) $ in the large $N$ limit has a known Hagedorn phase transition at $ x = d^{-1} $ which provides a simple model for the phase structure of the thermal partition function of SYM. We study the all-orders asymptotic expansion of $ \mathcal{Z}_d(x)$ based on a geometric picture of concentric circles of poles in the complex plane accumulating in a natural boundary at $|x| =1$. We find that the order by order structure has a precise combinatorial interpretation organized in terms of increasing cycle size of permutations arising in the enumeration of the invariants. We refer to this organization as small-cycle dominance, and find that it extends to refined versions of the partition functions depending on several complex variables. An analysis of the coefficients in the asymptotic expansion of $ \mathcal{Z}_d(x) $ using the modular property of the Dedekind eta function reveals that the asymptotic expansion is actually convergent for $d\ge d_{ \rm crit } = 13$. A fermionic version of $\mathcal{Z}_d (x)$ has an analogous critical dimension of $ d_{ \rm crit} = 7$. This distinction indicates that the partition functions of the matrix models can be completely reconstructed from their high-energy (UV) limit for $d\ge d_{ \rm crit}$ whereas additional input is required to reconstruct the exact coefficients of the low-energy (IR) expansion for $2\le d \le d_{ \rm crit } -1 $.


Permutation modules for Ramsey structures

Suppose $R$ is a commutative ring and $G$ is a group acting on a set $W$. We consider the $RG$-module $RW$ in the case where $G$ is the automorphism group of an $ω$-categorical structure $M$ and $W$ is, for example, $M^n$ (for $n \in \mathbb{N}$). We develop methods which may provide information about two questions in the case where $R$ is a field $F$: whether $FW$ has a.c.c. on submodules; and in the case where $M$ is finitely homogeneous, whether $FW$ is of finite composition length. In the case where $M$ is a Ramsey structure and so $G$ is extremely amenable, we give a simple `decision procedure' for membership in a submodule of $RW$ specified by a given generating set. If $F$ is a field, we show that there is a duality between submodules of $FW$ and the topological $FG$-module of definable functions from $W$ to $F$.


Resolving problems on the polynomial identity characterization of daisy cubes

Let $X\subseteq\{0,1\}^n$ be a set of binary strings of length $n$. The daisy cube $Q_n(X)$ is the subgraph of the hypercube $Q_n$ induced by the union of the intervals $I(x,0^n)$ for $x\in X$. As a subclass of partial cubes, it generalizes Fibonacci cubes and Lucas cubes. For a graph $G$ and a vertex $u\in V(G)$, we consider the cube polynomial $C_G(x)$, the distance cube polynomial $D_{G,u}(x,y)$, and the polynomial $W_{G,u}(x)$, which count $k$-cubes, $k$-cubes at distance from $u$, and vertices at distance $k$ from $u$, respectively.   In this paper, we prove that for a partial cube $G$ with a vertex $u\in V(G)$, $G$ is a daisy cube and $u=0^n$ if and only if one of the following equivalent conditions holds: (1) $C_{G}(x)=W_{G,u}(x+1)$; (2) $D_{G,u}(x,y)=W_{G,u}(x+y)$; (3) $D_{G,u}(x,y)=C_{G}(x+y-1)$. In particular, conditions (1) and (3) give affirmative answers to two open problems posed by Klavžar and Mollard [European J. Combin., 80 (2019) 214--223]. Further, we obtain that for arbitrary partial cube $G$, $D_{G,u}(x,y)\leq W_{G,u}(x+y)$ and $C_{G}(x)\leq W_{G,u}(x+1)$. Besides, another bound for $C_G(x)$ due to Xie et al. [J. Graph Theory, 106 (2024) 907--922] is given by the clique polynomial $Cl_{G^\#}(x+1)$ of the crossing graph of $G$. We also compare these two bounds and show that the simplex graphs form the unique class of graphs for which the two bounds coincide.


Randomstrasse101: Open Problems of 2025

Afonso S. Bandeira
Daniil Dmitriev
Kevin Lucca
Petar Nizić-Nikolac
Almut Rödder

Randomstrasse101 is a blog dedicated to Open Problems in Mathematics, with a focus on Probability Theory, Computation, Combinatorics, Statistics, and related topics. This manuscript serves as a stable record of the Open Problems posted in 2025, with the goal of easing academic referencing. The blog can currently be accessed at randomstrasse101.math.ethz.ch


A SAT-based Filtering Framework for Exact Coverings of K33 by Cliques of Order 3, 4 or 5

We investigate the minimum number of cliques of orders $3$, $4$, and $5$ needed to cover the edges of $K_{33}$ with zero excess. General covering results yield the lower bound 57. The main result of the paper is that no decomposition of $K_{33}$ into $57$ blocks from $\{K_3,K_4,K_5\}$ exists.   Our approach is algorithmic and relies on a layered exact-search pipeline rather than a single monolithic solver. We combine symmetry reduction, enumeration of local signatures, arithmetic profile restrictions, geometric tests for partial configurations, SAT realisation on reduced instances, and final decoding checks. The benchmark comparison shows that this structured approach is substantially more effective than direct ILP, DLX, or SAT formulations on the full problem.   As a consequence, we obtain $C^ξ(33,\{3,4,5\},2)\ge 58$. A short additional counting argument further strengthens this to $C^ξ(33,\{3,4,5\},2)\ge 59$. We also give new compressed proofs for the known exceptional cases $K_{18}$ and $K_{19}$ in the setting of $\{K_3,K_4\}$-decompositions, illustrating the same combination of theoretical reduction and exact computation.   Finally, we explain the relevance of the $K_{33}$ result to the open packing problem of determining the packing number $D(33,5,2)$. A packing of $51$ copies of $K_5$ in $K_{33}$ would leave a $4$-regular graph on $9$ vertices, and our exclusion already rules out two natural candidate leave structures.


On the combinatorics of the refined 1-leg DT/PT correspondence

Davide Accadia
Danilo Lewański
Sergej Monavari

We provide a new proof of a result of Bessenrodt on the relation among the generating series of reversed plane partitions and skew plane partitions, motivated by the geometric DT/PT wallcrossing formula for local curves recently proved by the third author. This also recovers a result of Sagan.   We moreover establish various new closed formulas for the weighted enumeration of reversed and skew plane partitions, proving a result dual to a theorem by Gansner, we find a new identity on the generating series counting internal and external hooks of a given Young diagram, and we combine the latter with Bessenrodt's theorem. Finally, we interpret our results as identities in the Fock space via the bosonic/fermionic formalism.


Binomial determinants: some closed formulae

This paper is intended to give closed formulae for binomial determinants with consecutive or almost consecutive rows or columns, as well as calculating the generator of left nullspaces defined by some binomial matrices. In the meantime, we reprove, by different means, the positivity of binomial determinants shown by Gessel and Viennot.


A short proof of a perturbation inequality for the spectral radius

Let $G$ be a simple graph, and denote by $λ(G)$ its spectral radius. Sun and Das (2020) established that for any non-isolated vertex $v$ with degree $d(v)$, \[ λ(G)\leq \sqrt{λ(G-v)^2 + 2d(v) - 1}, \] which is a conjecture original posed by Guo, Wang, and Li (2019). Sun and Das's proof uses several tools from spectral graph theory. In this short note, we provide a concise and self-contained proof of this inequality using matrix analysis.


Graph Eigenvalues and Projection Constants

Let $λ_1(G)\ge λ_2(G)\ge \cdots \ge λ_n(G)$ denote the adjacency eigenvalues of a graph $G$ of order $n$. We prove that for every $k\geq 2$ and every graph $G$ on $n\geq k$ vertices, $$ λ_k(G)\le \frac{λ_{\mathbb{R}}(k-1)}{2(k-1)}\,n-1, $$ where $$ λ_{\mathbb{R}}(r)=\sup_{N\ge r}\frac1N \max_{Q\in \mathcal P_r(N)}\sum_{i,j=1}^N |q_{ij}| $$ and $\mathcal P_r(N)$ denotes the set of rank-$r$ orthogonal projections in $\mathbb{R}^{N\times N}$. In Banach space theory, $λ_{\mathbb{R}}(r)$ is well known as the maximal absolute projection constant, which has been shown to equal the quasimaximal absolute projection constant $μ_{\mathbb{R}}(r)$.   This yields a new conceptual connection: universal upper bounds on $λ_k(G)$ are controlled by the real maximal absolute projection constant $λ_{\mathbb{R}}(k-1)$. In dimensions where $λ_{\mathbb{R}}(k-1)$ is known explicitly, this gives explicit coefficients. In particular, for $k=3$ this recovers Tang's recent sharp bound $λ_3(G)\le n/3-1$. For $k=4$, using $λ_{\mathbb{R}}(3)=\frac{1+\sqrt5}{2}$ together with Linz's closed blowups of the icosahedral graph, we obtain the result $$ λ_4(G) \leq \frac{1+\sqrt5}{12}n-1. $$ The method allows us to transfer known upper bounds on $λ_{\mathbb{R}}(k-1)$ to match the best known upper bounds on $λ_k(G)$ for other values of $k$, such as $k=5$.


Multicolor $K_r$-Tilings with High Discrepancy

Henry Chan
Daniel Cheng
Lior Gishboliner
Xiangyu Li

We study the minimum degree threshold $δ_{r,q}$ guaranteeing the existence of $K_r$-tilings of high discrepancy in any $q$-edge-coloring. Balogh, Csaba, Pluhár and Treglown handled the 2-color case, proving that $δ_{r,2} = \frac{r}{r+1}$ for all $r \geq 3$. Here we determine $δ_{r,q}$ for all $q$ large enough, namely $q \geq \binom{r}{2}$. For example, we show that for $r \geq 4$, $δ_{r,q} = \frac{r}{r+1}$ for $\binom{r}{2} \leq q \leq \binom{r+1}{2}$ and $δ_{r,q} = \frac{r-1}{r}$ for $q \geq \binom{r+1}{2}+2$. Thus, $δ_{r,q}$ has a phase transition at $q = \binom{r+1}{2}$, where it drops from $\frac{r}{r+1}$ and then stabilizes at the existence threshold $\frac{r-1}{r}$. We also show that $δ_{r,q} \leq \frac{r}{r+1}$ for all $r,q$, supplementing and giving a new proof for the result of Balogh, Csaba, Pluhár and Treglown.


Unexpected toric Richardson varieties

Eugene Gorsky
Soyeon Kim
Melissa Sherman-Bennett

We prove that an open Richardson variety in the complete flag variety for $\mathrm{GL}_n$ is isomorphic to a torus if and only if the corresponding closed Richardson variety is toric. Such toric varieties can be classified in terms of the combinatorics of Bruhat intervals, and include many varieties of dimension larger than $n-1$. We give a combinatorial description of the corresponding polytopes, and compute several explicit examples.


Bourgain's projection theorem over normed division algebras

We give a simple and self-contained proof of an extension of a projection theorem of Bourgain over the reals to division algebras over local fields of zero characteristic.


New Lower Bounds for C4-Free Subgraphs of the Hypercubes Q7 and Q8

We establish new lower bounds ex$(Q_7,C_4) \ge 304$ and ex$(Q_8,C_4) \ge 680$ for the maximum number of edges in a $C_4$-free subgraph of the 7- and 8-dimensional hypercubes. Both bounds are witnessed by explicit constructions whose $C_4$-freeness is certified by exhaustive enumeration of all four-cycles (672 for $Q_7$, 1792 for $Q_8$). The $Q_7$ bound improves the previously known lower bound of approximately 300.2 by 3.8 edges; the $Q_8$ bound improves the previously known lower bound of approximately 674.9 by 5.1 edges. For $Q_7$ we additionally found 19866 $C_4$-free subgraphs with pairwise distinct edge sets (not quotiented by automorphism), all achieving 304 edges and all locally maximal. The consistent failure of 305-edge searches (minimum $C_4$ violation never reaching 0 over 1076 independent trials) supports the conjecture ex$(Q_7,C_4)=304$. Edge lists and verification code are publicly available at https://github.com/minamominamoto/c4free-hypercube


Beyond the Laurent phenomenon

In a cluster algebra, a subset of initial cluster variables can be specialised in such a way that all elements of the resulting algebra become polynomial in the remaining variables.


Graphs of group actions and group actions on tree

Florian Lehner
Christian Lindorfer
Rögnvaldur G. Möller
Wolfgang Woess

Bass-Serre theory provides a powerful framework for studying group actions on trees. While extremely effective for structural questions in group theory, it is less suited to the systematic construction of group actions with prescribed local behaviour. Motivated by local-to-global constructions such as the Burger-Mozes universal groups and local action diagrams, we develop an analogue of Bass-Serre theory for group actions.   The central object of study in our are graphs of group actions, combinatorial structures similar to graphs of groups from Bass-Serre theory, encoding compatible local permutation actions on a base graph. From these we can construct groups which act on tree-like graphs called scaffoldings and hence also on trees.   We prove uniqueness and universality results for the resulting groups and show that our framework unifies and generalises (among other known constructions) both graphs of groups and local action diagrams. Remarkably, we are able to encapsulate the full generality of the former while still allowing for efficient construction of groups with certain local properties like in the latter.


Hypergeometric Bernoulli Polynomials Defined on Simplicial $d$-Polytopic Numbers

We introduce an ${\rm S}_d$-analogue of the hypergeometric Bernoulli polynomials and study their properties. To achieve this goal, we introduce a calculus defined on the simplicial $d$-polytopic numbers. Two definitions of the ${\rm S}_d$-derivatives are given. These two definitions allow us to derive an identity relating Kummer confluent hypergeometric function and Touchard polynomials. This calculus is closely related to the $d$-Hoggatt binomial coefficients. ${\rm S}_d$-analogs of the exponential function and the hypergeometric functions are given.


Composition of random functions and word reconstruction

Given two functions $\mathbf{a}\!:\! [n] \rightarrow [n]$ and $\mathbf{b}\!:\! [n] \rightarrow [n]$ chosen uniformly at random, any word $w=w_1w_2\dots w_k\in \{a,b\}^k$ induces a random function $\mathbf{w}\!:\! [n] \rightarrow [n]$ by composition, i.e. $\mathbf{w}=φ_{w_k}\circ \dots \circ φ_{w_1}$ with $φ_a=\mathbf{a}$ and $φ_b=\mathbf{b}$. We study the following question: assuming $w$ is fixed but unknown, and $n$ goes to infinity, does one sample of $\mathbf{w}$ carry enough information to (partially) recover the word $w$ with good enough probability?   We show that the length of $w$, and its exponent (largest $d$ such that $w={u}^d$ for some word ${u}$) can be recovered with high probability. We also prove that the random functions stemming from two different words are separated in total variation distance, provided that certain ``auto-correlation'' word-depending constant $c(w)$ is different for each of them. We give an explicit expression for $c(w)$ and conjecture that non-isomorphic words have different constants. We prove that this is the case assuming a major conjecture in transcendental number theory, Schanuel's conjecture.


The Priority Lattice

We introduce the priority lattice, a structure arising from the priority search algorithm on rooted trees and forests. We prove bijectively that its maximal chains are labeled by parking functions, and that the maximal chains of its principal ideals are labeled by partial parking functions. We establish that it is a graded lattice and compute its Möbius function and characteristic polynomials.


Odd Hadwiger number and graph products

Henry Echeverría
Andrea Jiménez
Suchismita Mishra
Daniel A. Quiroz
Mauricio Yépez

The Odd Hadwiger number of a graph $G$ is the largest integer $r$ such that $G$ has a clique of size $r$ as an odd minor. In this paper, we investigate how large is the Odd Hadwiger number of the product of two graphs, when considering any of the four standard graph products: Cartesian, direct, lexicographic, strong. We provide an optimal lower bound in the cases of the strong and lexicographic products.


Upper bound on the $k$-th eigenvalue of a graph

We prove a general upper bound on the $k$-th adjacency eigenvalue of a graph. For $k\ge 2$, we show that \[ λ_k(G)\le \frac{(k-2)\sqrt{k+1}+2}{2k(k-1)}\,n-1 \] for every graph $G$ on $n$ vertices. We build on a recent approach that addresses the case $k=3$ and generalize the upper bound for all $k \geq 3$ by using the positivity of Gegenbauer polynomials. The upper bound is tight for $k \in \{2,3,4,8,24\}$. We also highlight the close relation of $λ_k(G)$ to questions about equiangular lines.


Optimal bounds for an Erdős problem on matching integers to distinct multiples

Let $f(m)$ be the largest integer such that for every set $A = \{a_1 < \cdots < a_m\}$ of $m$ positive integers and every open interval $I$ of length $2a_m$, there exist at least $f(m)$ disjoint pairs $(a, b)$ with $a \in A$ dividing $b \in I$. Solving a problem of Erdős, we determine $f(m)$ exactly, and show $$ f(m)=\min\bigl(m,\lceil 2\sqrt{m}\,\rceil\bigr) $$ for all $m$. The proof was obtained through an AI-assisted workflow: the proof strategy was first proposed by ChatGPT, and the detailed argument was subsequently made fully rigorous and formally verified in Lean by Aristotle. The exposition and final proofs presented here are entirely human-written. [This paper solves Problem #650 on Bloom's website "Erdős problems".]


A Gray code for arborescences of tournaments

Marthe Bonamy
Michael Hoffmann
Clément Legrand-Duchesne
Günter Rote

We consider the following question of Knuth: given a directed graph $G$ and a root $r$, can the arborescences of $G$ rooted in $r$ be listed such that any two consecutive arborescences differ by only one arc? Such an ordering is called a pivot Gray code and can be formulated as a Hamiltonian path in the reconfiguration graph of the arborescences of $G$ under arc flips, also called flip graph of $G$. We give a positive answer for tournaments and explore several conditions showing that the flip graph of a directed graph may contain no Hamiltonian cycles.


Enumeration of general planar hypermaps with an alternating boundary

Valentin Baillard
Ariane Carrance
Bertrand Eynard

In this paper, we extend the enumerative study of planar hypermaps with an alternating boundary introduced in an earlier work of Bouttier and the second author. In that article, an explicit rational parametrization was obtained for the associated generating function in the case of m-constellations, using a variant of the kernel method. We develop here a new strategy to obtain an algebraic equation in the general case, which includes maps decorated by the Ising model, through a classical many-to-one correspondence. One of the main steps of our strategy is the simultaneous elimination of two catalytic variables. We then apply this strategy to the case of Ising quadrangulations, where we obtain an explicit rational parametrization. As a consequence, we show that some notable properties of the constellations case are no longer satisfied in general.


Zeros in the character table of the symmetric group

Computations of Miller and Scheinerman suggest that the vast majority of the zeros appearing in the character table of the symmetric group are of a certain special type. While we cannot prove this, we resolve a conjecture arising in their paper concerning these zeros, and address a related question of Stanley.


On Generalizations of Maiorana-McFarland and $\mathcal{PS}_{ap}$ Functions

Sezel Alkan
Nurdagül Anbar
Athina Avrantini
Erroxe Etxabarri-Alberdi
Tekgül Kalaycı
Beatrice Toesca

We study generalizations of two classical primary constructions of Boolean bent functions, namely the Maiorana-McFarland ($MM$) class and the (Desarguesian) partial spread ($\mathcal{PS}_{ap}$) class. The construction of bent functions lying outside the completed $MM$ class has attracted considerable attention in recent years. In this direction, we construct families of generalized Maiorana--McFarland bent functions that are not equivalent to any function in the classical $MM$ or $\mathcal{PS}_{ap}$ classes, and hence lie outside their completed classes. As a second contribution, we investigate the decomposition of generalized $\mathcal{PS}_{ap}$ functions. We prove that when the degree is sufficiently small relative to the size of the underlying finite field, such functions do not, in general, admit a decomposition into bent or semibent functions. Consequently, they cannot be obtained from known secondary constructions based on concatenation. Finally, we present a secondary construction of Boolean bent functions arising from the concatenation of components of vectorial generalized $\mathcal{PS}_{ap}$ functions. Our constructions and proofs rely on classical results concerning second-order derivatives of bent functions and their duals. In addition, we employ methods from the theory of algebraic curves and their function fields.


The free tracial post-Lie-Rinehart algebra of planar aromatic trees for the design of divergence-free Lie-group methods

Adrien Busnot Laurent
Hans Munthe-Kaas
Venkatesh G. S

Aromatic Butcher series were successfully introduced for the study and design of numerical integrators that preserve volume while solving differential equations in Euclidean spaces. They are naturally associated to pre-Lie-Rinehart algebras and pre-Hopf algebroids structures, and aromatic trees were shown to form the free tracial pre-Lie-Rinehart algebra. In this paper, we present the generalisation of aromatic trees for the study of divergence-free integrators on manifolds. We introduce planar aromatic trees, show that they span the free tracial post-Lie-Rinehart algebra, and apply them for deriving new Lie-group methods that preserve geometric divergence-free features up to a high order of accuracy.


Nonvanishing $k$-flats of Boolean and vectorial functions

$k$th-order sum-free functions are a natural generalization of APN functions using the concept of (non)vanishing flats. In this paper, we introduce a new combinatorial technique to study the nonvanishing flats of Boolean functions. This approach allows us to determine the number of nonvanishing flats for an infinite family of Boolean functions. We moreover use it to show that any $k$th-order sum-free $(n,n)$-function of algebraic degree $k$ gives rise to an $(n-k)$th-order sum-free $(n,n)$-function of algebraic degree $n-k$. This implies the existence of millions of $(n-2)$th-order sum-free functions.


Private neighbors, perfect codes and their relation with the $\vt$-number of closed neighborhood ideals

Delio Jaramillo-Velez
Hiram H. López
Rodrigo San-José

In this work, we investigate the connections between dominating sets, private neighbors, and perfect codes in graphs, and their relationships with commutative algebra. In particular, we estimate the $\vt$-number of closed neighborhood ideals in terms of minimal dominating sets and private neighbors. We show how the $\vt$-number is related to other graph invariants, such as the cover number, domination number, and matching number. Moreover, we explore the relation with the Castelnuovo-Mumford regularity, proving that the $\vt$-number is a lower bound for the regularity of bipartite and well-covered graphs. Finally, drawing from the relation between efficient dominating set and perfect codes, we use the redundancy of Hamming codes to present lower and upper bounds for the $\vt$-number of some special family of graphs.


Cyclic sieving phenomena on parabolic classes of faces of the cluster complex

The cyclic sieving phenomenon was introduced by Reiner, Stanton and White in 2004 as a generalization of Stembridge's $q=-1$ phenomenon. In a paper from 2008, Eu and Fu studied many occurrences of this phenomenon on the faces of the generalized cluster complex with the action of the Fomin-Reading rotation in the classical types $A_n$, $B_n$, $D_n$ and $I_2(k)$. There was yet no known uniform $q$-analogue of the $k$-face numbers of these complexes. In a more recent paper from 2023, Douvropoulos and Josuat-Vergès provided a refinement of the enumeration of the faces of the generalized cluster complex using a uniform formula. For a parabolic subgroup $W_X \subset W$ of the associated Coxeter group $W$, their formula factorises nicely under the assumption that $N_W(W_X)/W_X$ acts as a reflection group on $X$, which is very often the case. Using this condition, we provide a uniform refinement of these cyclic sieving phenomena using a $q$-analogue of their main formula with a type by type proof based on the classification of finite irreducible Coxeter groups.


Off-diagonal Rado number for $x+y+c=z$ and $x+y+k=z$

Rajat Adak
Yash Bakshi
L. Sunil Chandran
Saraswati Girish Nanoti

The study of Ramsey-type problems for linear equations originated with Schur's theorem and was later placed in a systematic framework by Richard Rado. In the off-diagonal setting, one fixes a pair of distinct linear equations $(\mathcal{E}_1, \mathcal{E}_2)$ and asks for the least integer $N$ such that every red--blue coloring of $\{1, 2, \dots, N\}$ must yield either a red solution to $\mathcal{E}_1$ or a blue solution to $\mathcal{E}_2$. This threshold integer is referred to as the off-diagonal Rado number of the system $(\mathcal{E}_1, \mathcal{E}_2)$. In this work, we study the discrete and continuous off-diagonal Rado number for non-homogeneous linear system of equations $x+y+c=z$ and $x+y+k=z$ where $c\le k$. We determine the exact two-color discrete and continuous off-diagonal Rado number $R_2(c,k)$ associated with this system of equations.


Limit Laws for the Distance to Fréchet Means of Random Graphs

This paper investigates the Fréchet mean of the Erdős-Rényi random graph $G_{n,p}$ with respect to the Frobenius distance on graph Laplacians, a metric that captures global structural information beyond local edge flips. We first characterize the Fréchet mean set as consisting of quasi-regular graphs (i.e., graphs where all vertex degrees differ by at most one). We then analyze the asymptotic behavior of the Frobenius distance $F_n=d_{\mathrm{F}}(G_{n,p},R)$ as $n\to\infty$, where $R$ is any Fréchet mean. Closed-form expressions for the mean and variance of $F_n^2$ are derived, which are invariant to the choice of $R$. Leveraging these results, we establish several weak convergence laws for the Frobenius distance over all regimes of $p \in (0,1)$ as $n \to \infty$. Finally, under the scaling condition $n^2 p(1-p) \to \infty$ we prove the asymptotic normality of this distance, which exhibits a phase transition governed by the growth rate of $np(1-p)$. Our results reveal how metric selection fundamentally shapes Fréchet mean geometry in random graphs.


Towards Pósa's Conjecture for $3$-graphs

Debmalya Bandyopadhyay
Allan Lo
Richard Mycroft

We prove that every $3$-graph $H$ on $n$ vertices with minimum codegree $δ_2(H) \geq 7n/9 + o(n)$ contains the square of a tight Hamilton cycle. This strengthens a theorem of Bedenknecht and Reiher that $δ_2(H) \geq 4n/5 + o(n)$ is sufficient. The central novelty of our arguments is an improved understanding of the connectivity structure of $3$-graphs with large minimum codegree.


Simplicial shells and thickness in the partition graph

For each positive integer $n$, let $G_n$ be the graph whose vertices are the partitions of $n$, with edges given by elementary transfers of one unit between parts, followed by reordering. We study the local simplex dimension in the clique complex $K_n=\Cl(G_n)$ as a geometric thickness invariant of $G_n$. For a partition $λ\vdash n$, let $τ_n(λ):=\dim_{\mathrm{loc}}(λ)$ be its simplicial thickness. This gives threshold thick zones $T_{\ge r}(n)=\{λ: τ_n(λ)\ge r\}$ and, relative to the boundary framework of $G_n$, a shell/core decomposition into outer shells $Sh_r(n)$ and inner cores $Core_r(n)$. Using local-morphology results established earlier in the series, we work with simplicial thickness as a local invariant. We prove that it is preserved by conjugation, that the induced thick zones, shells, and cores are conjugation-invariant, and that the antennas remain strictly one-dimensional in the simplicial sense and are excluded from all nontrivial thick zones. The first shell order at which a nontrivial shell can occur is therefore $2$, and the corresponding shell $Sh_2(n)$ is the triangular skin, while higher simplicial regimes form nested higher-order shells inside the triangular regime. We also develop a complete finite computational atlas for $1\le n\le 30$, giving first-occurrence tables for the regimes $T_{\ge r}(n)$ and supporting a finite-range rear-central thickening pattern.


Some new results on Andrews' and Warnaar's q-identities

In this paper, by the technique of inverse relations and comparing coefficients, we establish some generalized forms of Andrews' q-series identity and two new Bailey pairs and q-identities closely related to Andrews-Warnaar's sum identity for partial theta functions.


A note on irreducible representations of symmetric groups and Sergeev superalgebras

We provide an explicit construction and a closed dimension formula in terms of hook lengths for the irreducible representations for the symmetric groups $\mathfrak{S}_p$ and the Sergeev superalgebras $\mathcal{Y}_p$ over an algebraically closed field $\mathbb{F}$ of characteristic $p>0$.


Cycles of length $2$ modulo odd $k$ and admissible cycles in graphs

In 1976, Burr and Erdős conjectured that if $k\mathbb{Z} + \ell$ contains an even integer, there exists a constant $c$ such that every graph with at least $cn$ edges contains a cycle of length $\ell \pmod k$. This conjecture was settled for odd $k$ by Bollobás, while Thomassen completed the proof for all $k$ by resolving the case for even residues. Let $c_{\ell,k}$ denote the smallest constant such that every $n$-vertex graph with at least $c_{\ell,k}n$ edges contains a cycle of length $\ell \pmod{k}$. $k$ cycles are said to be admissible if they form an arithmetic progression of length $k$ with common difference one or two. The exact value of $c_{\ell,k}$ remains unknown for most $(\ell,k)$. Recently, Gao, Huo, Liu, and Ma showed that every graph with minimum degree at least $k+1$ contains $k$ admissible cycles. In this paper, we provide a sharp size version of their result. As a corollary, we show that $c_{2,k}=k$ for all odd $k$. In 2016, Verstraëte conjectured that every $n$-vertex graph $G$ containing no $k$ cycles of consecutive even lengths has at most $(2k+1)(n-1)/2$ edges, with equality only if every block of $G$ is a clique of order $2k+1$. We prove this conjecture for $2k+2\leq n\leq 4k+1$, and in fact obtain a stronger result in this range.


A characterization of graphs with no $K_{3,4}$ minor

A complete structural characterization of graphs with no $K_{3,4}$ minor is obtained, and the following consequences are established. Every $4$-connected non-planar graph with at least seven vertices and minimum degree at least five contains both $K_{3,4}$ and $K_6^-$ as minors, thereby proving a conjecture of Kawarabayashi and Maharry in a strengthened form. Moreover, every $4$-connected graph with no $K_{3,4}$ minor is hamiltonian-connected, extending a theorem of Thomassen, and admits an embedding on the torus.


Note on the thickness of the Cartesian product of a complete graph and a path

We determine the thickness of the Cartesian product $K_{6p+4} \square P_2$ for $p \ge 0$ and of the Cartesian product $K_8 \square P_m$ for $m \ge 1$, where $K_n$ and $P_m$ denote the complete graph on $n$ vertices and the path on $m$ vertices, respectively.


Determining the normal subgroups of the automorphism groups of some ultrahomogeneous structures via stabilisers

Thomas Bernert
Rob Sullivan
Jeroen Winkel
Shujie Yang

We show the simplicity of the automorphism groups of the generic $n$-hypertournament and the semigeneric tournament, and determine the normal subgroups of the automorphism groups of several other ultrahomogeneous oriented graphs. We also give a new proof of the simplicity of the automorphism group of the dense $\frac{2π}{n}$-local order $\mathbb{S}(n)$ for $n \geq 2$ (a result due to Droste, Giraudet and Macpherson). Previous techniques of Li, Macpherson, Tent and Ziegler involving stationary weak independence relations (SWIRs) cannot be applied directly to these structures; our approach involves applying these techniques to a certain expansion of each structure, where the expansion has a SWIR and its automorphism group is isomorphic to a stabiliser subgroup of the automorphism group of the original structure.


Log-concavity from enumerative geometry of planar curve singularities

We propose a log-concavity conjecture for BPS invariants arising in the enumerative geometry of planar curve singularities, identified with the local Euler obstructions of Severi strata in their versal deformations. We further extend this conjecture to ruling polynomials of Legendrian links and to E-polynomials of character varieties. We establish these conjectures for irreducible weighted-homogeneous singularities (torus knots) and for ADE singularities, and prove a multiplicative property for ruling polynomials compatible with log-concavity.


Geometry of ample/lopsided sets

Hans--Jürgen Bandelt
Victor Chepoi
Andreas Dress
Jack Koolen

Lopsided sets were introduced by Jim Lawrence in 1983 when he studied the subsets of $\{-1,+1\}^E$ that encode the intersection pattern of a convex set $K$ with the orthants of ${\mathbb R}^E$. Lopsided sets have been independently rediscovered by several other authors, in particular by Andreas Dress in 1995, who called them \emph{ample} sets. Dress defined ample sets as the set families satisfying equality in a combinatorial inequality, which holds for all set families.   In a previous article we characterized ample sets in various combinatorial and graph-theoretical ways. In this paper we study geometric realizations of ample sets as cubihedra (cube complexes), which yields several new characterizations. One such characterization establishes that the cubihedra of ample sets endowed with the intrinsic $\ell_1$-metric are exactly the isometric subspaces of $\ell_1$-spaces (which we call, weakly convex sets). We also view the barycenter maps of faces of cubihedra of ample sets as collections of $\{ \pm 1, 0\}$-sign vectors and, in analogy with the characterization of oriented matroids by the covectors and the cocircuits. Moreover, we characterize the collections of $\{ \pm 1, 0\}$-sign vectors corresponding to barycenter maps of all faces and all maximal faces of an ample set. Furthermore, we show that any ample set $\covectors\subseteq \{ -1,+1\}^E$ is realizable as the intersection pattern of a weakly convex set $K$ with the orthants of ${\mathbb R}^E$. All this testifies that the concept of ample sets is quite natural in the context of cube complexes.


Unboundedness of the Heesch Number for Hyperbolic Convex Monotiles

We provide a resolution of the Heesch problem for homogeneous (also known as semi-regular) tilings, and as a corollary, for tilings by convex monotiles in the hyperbolic plane. We also provide the first known example of weakly aperiodic convex monotiles arising from the dual of homogeneous tilings.


Buffon Discrepancy and the Steinhaus Longimeter

Let $Ω\subset \mathbb{R}^2$ be a convex set. We study the problem of distributing a one-dimensional set $S$ with total length $L$ so that for any line $\ell$ in $\mathbb{R}^2$ the number of intersections $\#(\ell \cap S)$ is proportional to the length $\mathcal{H}^1(\ell \cap Ω)$ as much as possible; we use the term Buffon discrepancy for the largest error. A construction of Steinhaus can be generalized to prove the existence of sets with Buffon discrepancy $\lesssim L^{1/3}$. We also show that the unit disk $\mathbb{D}$ admits a set with uniformly bounded Buffon discrepancy as $L \rightarrow \infty$.


Graphs with core(G) = nucleus(G)

Vadim E. Levit
Eugen Mandrescu
Kevin Pereyra

Let $G$ be a finite simple graph. An independent set $I$ of $G$ is critical if $\left|I\right|-\left|N(I)\right|\ge\left|J\right|-\left|N(J)\right|$ for every independent set $J$ of $G$. A critical independent set is maximum if it has maximum cardinality. The $core$ and the $nucleus$ of $G$ are defined as the intersection of all maximum independent sets and the intersection of all maximum critical independent sets, respectively. In 2019, Jarden, Levit, and Mandrescu posed the problem of characterizing the graphs satisfying $core(G)=nucleus(G)$. In this paper, we provide a complete solution to this problem. Using Larson's independence decomposition, which partitions any graph into a König--Egerváry component $L_G$ an a $2$-bicritical component $L_G^c$, we establish that $core(G)=nucleus(G)$ holds if and only if $core ({L_G^c})=\emptyset$ and no vertex of $corona(G)$ lies in the boundary between $L_G$ and $L_G^c$. We also show that the same boundary condition is equivalent to the identity $diadem(G)=corona(G) \cap L(G)$. Several consequences and related structural properties are also derived.


Truncated Plethystic Exponentials Preserve Power Sum Constraints

Given an arbitrary sequence $(α_1, \ldots, α_n) \in \mathbb{C}^n$, we show that the degree-$n$ truncation of the formal exponential $\exp\bigl(-\sum_{k=1}^{\infty} \frac{α_k}{k} x^k\bigr)$ produces a polynomial whose roots $ρ_1, \ldots, ρ_n$ satisfy $\sum_{i=1}^n ρ_i^{-k} = α_k$ exactly for $k = 1, \ldots, n$. This truncation-exactness property is an algebraic identity in the ring of formal power series, proved by coefficient matching. It defines a natural embedding of sequences into multisets of complex numbers and yields an $O(n^2)$ algorithm for computing the polynomial from the prescribed power sums. We apply the result to the polylogarithm family $α_k = k^{1-s}$, where the associated exponential $\exp(-\mathrm{Li}_s(x))$ produces factorial-integer coefficient sequences for $s \leq 0$ and encodes values of the Riemann zeta function through $\lim_{n\to\infty} P_n^{(s)}(1) = \exp(-ζ(s))$ for $\mathrm{Re}(s) > 1$.


An infinite family of non-extendable MRD codes

Daniele Bartoli
Alessandro Giannoni
Giuseppe Marino
Alessandro Neri

In the realm of rank-metric codes, Maximum Rank Distance (MRD) codes are optimal algebraic structures attaining the Singleton-like bound. A major open problem in this field is determining whether an MRD code can be extended to a longer one while preserving its optimality. This work investigates $\mathbb{F}_{q^m}$-linear MRD codes that are non-extendable but do not attain the maximum possible length. Geometrically, these correspond to scattered subspaces with respect to hyperplanes that are maximal with respect to inclusion but not of maximum dimension. By exploiting this geometric connection, we introduce the first infinite family of non-extendable $[4,2,3]_{q^5/q}$ MRD codes. Furthermore, we prove that these codes are self-dual up to equivalence.


On four network monitoring parameters in graphs and their gaps

Let \( G \) be a finite simple undirected graph. Four graph parameters related to network monitoring are the \emph{geodetic set}, \emph{edge geodetic set}, \emph{strong edge geodetic set}, and \emph{monitoring edge geodetic set}, with corresponding minimum sizes, denoted by \( g(G), eg(G), seg(G) \), and \( meg(G) \), respectively. These parameters quantify the minimum number of vertices required to monitor all vertices and edges of \( G \) under progressively stricter path-based conditions. As established by Florent \textit{et al.}\ (CALDAM 2023), these parameters satisfy the chain of inequalities: \( g(G) \leq eg(G) \leq seg(G) \leq meg(G). \) In 2025, Florent \textit{et al.}\ posed the following question: given integers \( a, b, c, d \) satisfying \( 2 \leq a \leq b \leq c \leq d \), does there exist a graph \( G \) such that \( g(G) = a, \quad eg(G) = b, \quad seg(G) = c, \quad \text{and} \quad meg(G) = d? \) They partially answered this affirmatively under three specific hypotheses and gave some constructions to support it. In this article, we first identify quadruples of values that cannot be realized by any connected graph. For all remaining admissible quadruples, we provide explicit constructions of connected graphs that realize the specified parameters. These constructions are modular and efficient, with the number of vertices and edges growing linearly with the largest parameter, providing a complete and constructive characterization of such realizable quadruples.


Sets of subspaces with restricted hyperplane intersection numbers

Let $\mathcal{X}$ be a set of $(h-1)$-dimensional subspaces of $\mathrm{PG}(kh-1,q)$ with the property that every hyperplane contains at most $t$ elements of $\mathcal{X}$. We prove the upper bound $|\mathcal{X}| \leq (t-k+2)q^h + t$, and characterise the structure of $\mathcal{X}$ in the case of equality. We call sets attaining this bound \emph{length-maximal}. For $k=3$, such sets are known as maximal arcs and have been well-studied. They are known to exist for $t<q^h$ if and only if $q$ is even and $t$ divides $q^h$. For $k=4$ and $q>2$, we show that any length-maximal set must satisfy $t = q^h+1$ and that every hyperplane is either a $t$-secant or a $1$-secant. For $k \geq 5$ and $q>2$, no length-maximal set exists. In the language of additive codes, these results assert that additive two-weight codes over $\mathbb{F}_{q^h}$ attaining the natural Griesmer-type bound do not exist when the code dimension is $5$ or more and $q>2$.


Exact Algorithms for Edge Deletion to Cactus

We study two related problems on simple, un-directed graphs: Edge Deletion to Cactus and Spanning Tree to Cactus. Edge Deletion to Cactus has been known to be NP-hard on general graphs at least since 1988. We show improved exact algorithms for the former and a polynomial time algorithm for the latter.


Growth-rate distributions at stationarity

We propose new analytical tools for describing growth-rate distributions generated by stationary time-series. Our analysis shows how deviations from normality are not pathological behaviour, as suggested by some traditional views, but instead can be accounted for by clean and general statistical considerations. In contrast, strict normality is the effect of specific modelling choices. Systems characterized by stationary Gamma or heavy-tailed abundance distributions produce log-growth-rate distributions well described by a generalized logistic distribution, which can describe tent-shaped or nearly normal datasets and serves as a useful null model for these observables. These results prove that, for large enough time lags, in practice, growth-rate distributions cease to be time-dependent and exhibit finite variance. Based on this analysis, we identify some key stylized macroecological patterns and specific stochastic differential equations capable of reproducing them. A pragmatic workflow for heuristic selection between these models is then introduced. This approach is particularly useful for systems with limited data-tracking quality, where applying sophisticated inference methods is challenging.


FcsIT: An Open-Source, Cross-Platform Tool for Correlation and Analysis of Fluorescence Correlation Spectroscopy Data

FcsIT is a platform-independent, open-source tool for calculating the correlation and fitting fluorescence correlation spectroscopy data. The software is written in Python and uses a powerful Dear PyGUI engine for its interface. It provides reading and correlating the TTTR data, as well as TCSPC filtering of the photon time-trace data. The circular-block bootstrap method applied to the calculation of correlation data and its variance results in data quality comparable to that obtained with commercially available software. An intuitive fitting interface provides efficient analysis of large datasets and includes nine predefined mathematical models for fitting correlation curves. Moreover, it allows users to add their own models in a user-friendly manner. Validation of the FcsIT tool against simulated FCS data and real FCS experiments confirms its usability and potential appeal to a wide variety of FCS users.


A systematic approach to Covariance matrix formulation in charged particle activation experiments

This work presents a detailed covariance and correlation matrix analysis for experimentally measured cross sections obtained using the activation technique. Both statistical and systematic contributions to the covariance matrix were explicitly calculated using sensitivity coefficients. The detector efficiency was determined by refitting standard source data with an exponential function, and the associated covariance matrix of the fitted parameters was propagated to estimate the uncertainty in efficiency at the relevant $γ$-ray energy. The cross sections and the corresponding experimental parameters, such as beam flux, target thickness, $γ$-ray intensity, and decay corrections, were taken from previously published measurements and are used here for the purpose of illustrating the covariance formalism. The resulting covariance and correlation matrices provide a comprehensive representation of uncertainties and their interdependencies. This formalism demonstrates the importance of including correlated uncertainties for reliable interpretation and comparison of experimental cross section data.


AI Cosplaying as Astrophysicists: A Controlled Synthetic-Agent Study of AI-Assisted Astrophysical Research Workflows

Large Language Models (LLMs) are now widely used in astrophysics, but do they actually make our lives easier, or do they merely invent new physics with enough confidence to hide a minus sign? In a specialized field where checking fluent hallucinations is itself labor-intensive, AI assistance can demand as much work as the task it claims to simplify. To evaluate where AI genuinely improves scientific workflows, we bypassed human trials and instead forced AI agents to cosplay as astrophysicists. We simulated 144 synthetic researchers, varying in career stage, AI awareness, and willingness to verify outputs, across 2,592 daily astrophysics research assignments. Comparing solo work against four styles of AI assistance produced 12,960 scored episodes. No assisted policy universally outperformed unassisted work in the primary Qwen production run. Instead, performance depends strongly on the task, the style of AI use, and the identity of the actor. While cautious assistance helps on creative, extractive, and critique-oriented tasks, it can fail catastrophically on derivation-heavy physics. A full actor-swap DeepSeek rerun changes that picture materially: verification-heavy use becomes the strongest assisted policy, two assisted modes enter the higher-utility/lower-risk quadrant, and the derivation-heavy fragility that dominates the Qwen production run largely disappears. In its current form, AI is useful, but only conditionally, its value is uneven, task-specific, and shaped jointly by workflow, usage policy, and which LLM you are using.


The Closure Challenge: a benchmark task for machine learning in turbulence modelling

Ryley McConkey
Tyler Buchanan
Tess Smidt
Abigail Bodner
Richard Dwight
Paola Cinnella

We introduce a field-wide benchmark challenge for machine learning in Reynolds-averaged Navier-Stokes (RANS) turbulence modelling. Though open-source datasets exist for training data-driven turbulence closure models, the field has been notably lacking a standard benchmark metric and test dataset. The Closure Challenge is a curated collection of open-source datasets and evaluation code that remedies this problem. We provide a variety of high-fidelity training data in a standardized format, including mean velocity gradients. The test cases (periodic hills, square duct, and NASA wall-mounted hump) evaluate Reynolds number and geometry generalization, two key issues in the field. We present results from three early submissions to the challenge. This is an ongoing challenge, intended to continuously spur innovation in machine learning for turbulence modelling. Our goal is for this benchmark to become the standard evaluation for new machine learning frameworks in RANS. The Closure Challenge is available at https://github.com/rmcconke/closure-challenge-benchmark.


Declarative bespoke modelling: A new approach

DBM Collaboration
David Komanek
Vaclav Pavlík
Santiago Jimenez
Rhys Taylor

Modern numerical models are increasingly complex, opaque, and computationally expensive, yet frequently fail to predict even qualitative features of observed phenomena. We propose a new paradigm, Declarative Bespoke Modelling, in which the modeller explicitly declares the relationship between model inputs and outputs. We demonstrate that this approach achieves perfect predictive accuracy, unconditional numerical stability, and complete interpretability. It represents a natural endpoint of contemporary modelling practice and near-zero CO2 emission.


Compressing Transformer Language Models via Matrix Product Operator Decomposition: A Case Study on PicoGPT

Younes Javanmard
Tanmoy Pandit
Masoud Mardani

Transformer-based language models achieve strong performance across NLP tasks, but their quadratic parameter scaling with hidden dimension makes deployment on resource-constrained hardware expensive. We study Matrix Product Operator (MPO) decomposition as a principled compression method for transformers. MPO factorises weight matrices into chains of low-rank cores, with approximation quality controlled by the bond dimension chi. We replace every nn.Linear layer in PicoGPT, a GPT-2-style character-level language model with about 1M parameters, with an MPOLinear module parameterised as an MPO chain. Cores are initialised either by TT-SVD from pretrained dense weights or from random initialisation, and trained using standard PyTorch autograd without a custom backward pass. We derive balanced factorisation schemes for the five distinct weight shapes in PicoGPT and evaluate bond dimensions chi in {4, 8, 16, 32} on Tiny Shakespeare. MPO compression achieves up to 13x compression per transformer block at chi = 4. At chi = 16, the model uses 191,872 parameters instead of 1,020,224 while retaining 97.7% of baseline token accuracy (51.6% vs 52.8%). Reconstruction error follows the expected trend and is lower for three-site than two-site factorisations at the same bond dimension. The chi = 8 model gives the best accuracy per parameter, exceeding the dense baseline by 2.7x on this metric. These results show that MPO parameterisation is a practical and theoretically grounded alternative to low-rank methods and unstructured pruning for transformer compression.


Solving the inverse problem of X-ray absorption spectroscopy via physics-informed deep learning

Suyang Zhong
Boying Huang
Pengwei Xu
Fanjie Xu
Yuhao Zhao
Jun Cheng
Fujie Tang
Weinan E
Zhong-Qun Tian

Resolving transient atomic configurations in non-crystalline or dynamic environments remains a fundamental bottleneck in the physical sciences. While X-ray absorption spectroscopy (XAS) is a premier probe of local structure, inverting spectra into structural descriptors is a notoriously ill-posed problem due to inherent many-to-one mapping. Here, we present the Spectral Pattern Translator (SPT), a physics-informed deep learning framework that establishes a robust bridge between large-scale theoretical datasets and experimental reality. Our strategy exploits the Fourier duality between spectral energy oscillations and spatial scattering paths to overcome the "simulation-to-experiment" gap. By decomposing spectra into frequency domains, SPT effectively isolates robust structural coordination signals from the destabilizing noise inherent in experimental data. Trained on a massive library of diverse atomic environments, this approach achieves state-of-the-art accuracy in resolving continuous phase transitions in battery cathodes and deciphering local order in amorphous materials. With millisecond-scale latency, SPT removes the primary computational barrier to autonomous materials discovery, establishing a robust, noise-resilient engine for closed-loop robotic chemistry.


Deep brain microelectrode signal: $q$-statistical approach

Ana Luiza Souza Tavares
Henrique Santos Lima
Artur Pedro Martins Neto
Bruno Duarte Gomes
Constantino Tsallis

We characterize the amplitude statistics of intraoperative microelectrode recordings (MERs) obtained during deep brain stimulation (DBS) surgery in 46 patients with Parkinson's disease, using 184 recordings equally balanced between inside and outside the subthalamic nucleus (STN). The probability density of every recording is quantitatively well described by a $q$-Gaussian (grounded on a nonadditive entropic functional), $ρ(x) \propto [1 + β(q-1) x^2]^{-1/(q-1)}$, with $q > 1$ in all cases, reflecting persistent long-range temporal correlations inconsistent with Gaussian dynamics. Within the superstatistics framework, the slowly fluctuating local variance visible in the raw MER signals is a physical mechanism that directly generates the $q > 1$ form. Beyond individual fits, $q$ and $β$ collapse across all 184 recordings onto the single functional constraint $q = 3 - 1.85\,β^{-0.33}$ ($R \approx -0.91$), a reduction to one effective degree of freedom that is the quantitative hallmark of near-critical dynamics, previously identified in scale-free network growth and in acoustic precursors of material fracture. The index $q$ is statistically indistinguishable across the STN boundary ($\langle\bar{q}_\text{out}/\bar{q}_\text{in} \rangle = 1.03$), while the inverse-widthparameter shows a modest systematic difference ($\langle\barβ_\text{out}/\barβ_\text{in} \rangle = 1.18$). Since $q > 1$ is expected for any brain structure exhibiting long-range correlations, healthy or pathological, it is the tight $q(β)$ coupling, not $q > 1$ per se, that constitutes the candidate near-criticality signature of the parkinsonian cortico-basal-ganglia-thalamocortical loop.


Bayesian estimation of optical constants using mixtures of Gaussian process experts

We propose modeling absorption spectrum measurements as mixtures of Gaussian process experts. This enables us to construct a flexible statistical model for interpolating and extrapolating measurements, facilitating statistical integration of Kramers-Kronig relations to estimate the whole complex refractive index. Additionally, we statistically model the anchoring points used in subtractive Kramers-Kronig relations to account for possible measurement errors of the anchor point. In addition to flexible statistical modeling, the mixtures of Gaussian process formulation enables automatic selection of measurement points to use for extrapolation. We apply the method to experimental absorption spectrum measurements of gallium arsenide, potassium chloride, and transparent wood.


Vision Transformers and Graph Neural Networks for Charged Particle Tracking in the ATLAS Muon Spectrometer

The identification and reconstruction of charged particles, such as muons, is a main challenge for the physics program of the ATLAS experiment at the Large Hadron Collider. This task will become increasingly difficult with the start of the High-Luminosity LHC era after 2030, when the number of proton-proton collisions per bunch crossing will increase from 60 to up to 200. This elevated interaction density will also increase the occupancy within the ATLAS Muon Spectrometer, requiring more efficient and robust real-time data processing strategies within the experiment's trigger system, particularly the Event Filter. To address these algorithmic challenges, we present two machine-learning-based approaches. First, we target the problem of background-hit rejection in the Muon Spectrometer using Graph Neural Networks integrated into the non-ML baseline reconstruction chain, demonstrating a 15 % improvement in reconstruction speed (from 255 ms to 217 ms). Second, we present a proof-of-concept for end-to-end muon tracking using state-of-the-art Vision Transformer architectures, achieving ultra-fast approximate muon reconstruction in 2.3 ms on consumer-grade GPUs at 98 % tracking efficiency.


Increasing trends in the severity of Australian fire weather conditions over the past century

Understanding how weather and climate influence fire risk is important for many purposes, including climate adaptation planning and decision-making in sectors such as emergency management, finance, health and infrastructure (e.g., for energy and water availability). In this study, bias-corrected 20CRv2c reanalysis data are used to investigate the climatology and long-term trends of weather conditions associated with landscape fires in Australia. The McArthur Forest Fire Danger Index (FFDI) is used here as a broad-scale representation of weather conditions known to influence fire behaviour based on wind speed, humidity, temperature and rainfall measures. In particular, using this reanalysis dataset allows analysis over a longer time period than previous studies, from 1876 to 2011. Another novel aspect is that trends are examined using several different approaches, including a method to help account for the influence of interannual drivers of climate variability not previously used for fire weather analysis. Results show increases in mean and extreme seasonal FFDI values throughout Australia in general, with all statistically significant trends being positive in sign for individual climate zones. Humidity and temperature trends, attributable to human-caused climate change, are shown to be the main cause of the increase in dangerous weather conditions for fires. These findings build on previous studies, with the novel data and methods used adding confidence to the overall understanding of fire risk factors in a changing climate.


Chiral moments make chiral measures

Emilio Pisanty
Nicola Mayer
Andrés Ordóñez
Alexander Löhr
Margarita Khokhlova

We develop a family of chiral measures to quantify the chirality of a distribution and assign it a handedness. Our measures are built using the tensorial moments of the distribution, which naturally encode its spatial character, not only via its angular shape consistently with existing multipolar-moment approaches, but also its radial dependence. We combine these tensorial moments into a rotationally-invariant pseudoscalar using a newly-defined cross product and triple product for arbitrary symmetric tensors. We analyze these measures for a variety of toy-model distributions, providing intuition for the geometry and guiding the choice of chiral measure optimal for a given distribution. We also apply our measures to a physically-motivated example coming from photoionization in polychromatic chiral light. Our work provides a robust, flexible, intuitive, highly geometrical, and physically-driven framework for understanding and quantifying the chirality of a wide variety of distributions, together with an open-source software package that makes this toolbox readily applicable for the analysis of numerical or experimental data.


Short-Term Turbulence Prediction for Seeing Using Machine Learning

Mary Joe Medlej
Rahul Srinivasan
Simon Prunet
Aziz Ziad
Christophe Giordano

Optical turbulence, driven by fluctuations of the atmospheric refractive index, poses a significant challenge to ground-based optical systems, as it distorts the propagation of light. This degradation affects both astronomical observations and free-space optical communications. While adaptive optics systems correct turbulence effects in real-time, their reactive nature limits their effectiveness under rapidly changing conditions, underscoring the need for predictive solutions. In this study, we address the problem of short-term turbulence forecasting by leveraging machine learning models to predict the atmospheric seeing parameter up to two hours in advance. We compare statistical and deep learning approaches, with a particular focus on probabilistic models that not only produce accurate forecasts but also quantify predictive uncertainty, crucial for robust decision-making in dynamic environments. Our evaluation includes Gaussian processes (GP) for statistical modeling, recurrent neural networks (RNNs) and long short-term memory networks (LSTMs) as deterministic baselines, and our novel implementation of a normalizing flow for time series (FloTS) as a flexible probabilistic deep learning method. All models are trained exclusively on historical seeing data, allowing for a fair performance comparison. We show that FloTS achieves the best overall balance between predictive accuracy and well-calibrated uncertainty.


Learning Response-Statistic Shifts and Parametric Roll Episodes from Wave--Vessel Time Series via LSTM Functional Models

Parametric roll is a rare but high-consequence instability that can trigger abrupt regime changes in ship response, including pronounced shifts in roll statistics and tail risk. This paper develops a data-driven surrogate that learns the nonlinear, causal functional mapping from incident wave--motion time series to vessel motions, and demonstrates that the surrogate reproduces both (i) parametric roll episodes and (ii) the associated statistical shifts in the response. Crucially, the learning framework is data-source agnostic: the paired wave--motion time series can be obtained from controlled experiments (e.g., towing-tank or basin tests with wave probes and motion tracking) when a hull exists, or from high-fidelity simulations during design when experiments are not yet available. To provide a controlled severe-sea demonstration, we generate training data with a URANS numerical wave tank, using long-crested irregular seas synthesized from a modified Pierson--Moskowitz spectrum. The demonstration dataset comprises 49 random-phase realizations for each of three sea states, simulated at a fixed forward speed selected to yield encounter conditions under which parametric-roll episodes can occur. A stacked LSTM surrogate is trained on wave-elevation time series and evaluated on held-out realizations using time-domain accuracy and distributional fidelity metrics. In the most severe case, the model tracks the onset and growth of large-amplitude roll consistent with parametric excitation, and captures the corresponding changes in roll probability density functions (PDFs). We further compare loss-function choices (MSE, relative-entropy-based objectives, and amplitude-weighted variants) and show how they trade average error for improved tail fidelity relevant to operability and risk assessment.


Ising noise filter: physics-informed filtering for particle detectors

We present the Ising noise filter, a highly portable, graph-based pre-filtering algorithm for early-stage background suppression in particle accelerators and astrophysical detectors. Standard noise rejection methods relying on track fitting suffer from severe combinatorial explosion. Our method bypasses this by mapping individual detector hits to a network of binary spins and minimizing an energy functional. The interaction kernels are physics-informed, tailored to the underlying physics and geometry of the experiment. We demonstrate the efficacy of this approach in two distinct experimental regimes. Applied to the Baikal-GVD neutrino telescope the filter yields fast, standard-quality noise rejection with 96.8\% recall for astrophysical neutrinos. For the SPD detector at the NICA collider the filter attains recall of 97\% on a toy Monte Carlo sample. Furthermore, when combined with a Peterson--Hopfield network for track finding, our physics-informed coupling improves the TrackML score from 0.5 to 0.95.


Machine vision with small numbers of detected photons per inference

Shi-Yuan Ma
Jérémie Laydevant
Mandar M. Sohoni
Logan G. Wright
Tianyu Wang
Peter L. McMahon

Machine vision, including object recognition and image reconstruction, is a central technology in many consumer devices and scientific instruments. The design of machine-vision systems has been revolutionized by the adoption of end-to-end optimization, in which the optical front end and the post-processing back end are jointly optimized. However, while machine vision currently works extremely well in moderate-light or bright-light situations -- where a camera may detect thousands of photons per pixel and billions of photons per frame -- it is far more challenging in very low-light situations. We introduce photon-aware neuromorphic sensing (PANS), an approach for end-to-end optimization in highly photon-starved scenarios. The training incorporates knowledge of the low photon budget and the stochastic nature of light detection when the average number of photons per pixel is near or less than 1. We report a proof-of-principle experimental demonstration in which we performed low-light image classification using PANS, achieving 73% (82%) accuracy on FashionMNIST with an average of only 4.9 (17) detected photons in total per inference, and 86% (97%) on MNIST with 8.6 (29) detected photons -- orders of magnitude more photon-efficient than conventional approaches. We also report simulation studies showing how PANS could be applied to other classification, event-detection, and image-reconstruction tasks. By taking into account the statistics of measurement results for non-classical states or alternative sensing hardware, PANS could in principle be adapted to enable high-accuracy results in quantum and other photon-starved setups.


Searching for Anomalies with Foundation Models

Foundation models have the potential to extend the discovery reach for anomaly detection searches. When studying the large OmniLearned foundation model on data from the CMS experiment, unexpected behavior was observed in a mass sideband. The purpose of this paper is to perform a full analysis, including a complete background estimate, on the phase space picked out by the large model. We find that the background estimation describes the data well in validation regions, but is unable to accurately model the signal region. We invite further scrutiny of these events and our methods.


Beyond the Central Limit: Universality of the Gamma Distribution from Padé-Enhanced Large Deviations

The central limit theorem provides the theoretical foundation for the universality of the normal distribution: under broad conditions, the asymptotic distribution of a sum of independent random variables approaches a Gaussian. Yet, physical systems described by positive random variable -- from earthquakes to microbial growth to epidemic spreading -- consistently exhibit gamma rather than Gaussian statistics -- what leads to field-specific mechanistic explanations that are non robust to small changes in the model details. We show that gamma distributions emerge naturally from large deviation theory when Padé approximants replace polynomial expansions of the derivative of the scaled cumulant generating function, respecting positivity constraints that the central limit theorem violates. Gamma universality thus emerges as the constrained analog of Gaussian universality, providing a mechanism-free explanation for its pervasive appearance across different disciplines.


Accelerate Vector Diffusion Maps by Landmarks

Sing-Yuan Yeh
Yi-An Wu
Hau-Tieng Wu
Mao-Pei Tsui

We propose a landmark-constrained algorithm, LA-VDM (Landmark Accelerated Vector Diffusion Maps), to accelerate the Vector Diffusion Maps (VDM) framework built upon the Graph Connection Laplacian (GCL), which captures pairwise connection relationships within complex datasets. LA-VDM introduces a novel two-stage normalization that effectively address nonuniform sampling densities in both the data and the landmark sets. Under a manifold model with the frame bundle structure, we show that we can accurately recover the parallel transport with landmark-constrained diffusion from a point cloud, and hence asymptotically LA-VDM converges to the connection Laplacian. The performance and accuracy of LA-VDM are demonstrated through experiments on simulated datasets and an application to nonlocal image denoising.


Construction of the Global $χ^2$ Function for the Simultaneous Fitting of Correlated Energy-Dependent Cross Sections

Linquan Shao
Haoyu Yan
Yingjun Chen
Jiaxin Pi
Xingyu Zhou

In this paper, the global $χ^2$ function for the simultaneous fitting of correlated energy-dependent cross sections is constructed, where the correlations between the measured cross sections of different processes and/or at different center-of-mass energy points, as well as the contributions from the integrated luminosity measurement and the center-of-mass energy measurement, are taken into account.


Sparse Weak-Form Discovery of Stochastic Generators

The proposed algorithm seeks to provide a novel data-driven framework for the discovery of stochastic differential equations (SDEs) by application of the Weak-formulation to stochastic SINDy. This Weak formulation of the algorithm provides a noise-robust methodology that avoids traditional noisy derivative computation using finite differences. An additional novelty is the adoption of spatial Gaussian test functions in place of temporal test functions, wherein the use of the kernel weight $K_j(X_{t_n})$ guarantees unbiasedness in expectation and prevents the structural regression bias that is otherwise pertinent with temporal test functions. The proposed framework converts the SDE identification problem into two SINDy based linear sparse identification problems. We validate the algorithm on three SDEs, for which we recover all active non-linear terms with coefficient errors below 4%, stationary-density total-variation distances below 0.01, and autocorrelation functions that reproduce true relaxation timescales across all three benchmarks faithfully.


Enhancing Neutrinoless Double-Beta Decay Sensitivity of Liquid-Xenon Time Projection Chamber with Augmented Convolutional Neural Network

E. Aprile
J. Aalbers
K. Abe
M. Adrover
S. Ahmed Maouloud
L. Althueser
B. Andrieu
E. Angelino
D. Antón Martin
S. R. Armbruster
F. Arneodo
L. Baudis
M. Bazyk
L. Bellagamba
R. Biondi
A. Bismark
K. Boese
R. M. Braun
G. Bruni
G. Bruno
R. Budnik
C. Cai
C. Capelli
J. M. R. Cardoso
A. P. Cimental Chávez
A. P. Colijn
J. Conrad
J. J. Cuenca-García
V. D'Andrea
L. C. Daniel Garcia
M. P. Decowski
A. Deisting
C. Di Donato
P. Di Gangi
S. Diglio
K. Eitel
S. el Morabit
R. Elleboro
A. Elykov
A. D. Ferella
C. Ferrari
H. Fischer
T. Flehmke
M. Flierman
R. Frankel
D. Fuchs
W. Fulgione
C. Fuselli
R. Gaior
F. Gao
R. Giacomobono
F. Girard
R. Glade-Beucke
L. Grandi
J. Grigat
H. Guan
M. Guida
P. Gyorgy
R. Hammann
A. Higuera
C. Hils
L. Hoetzsch
N. F. Hood
M. Iacovacci
Y. Itow
J. Jakob
F. Joerg
Y. Kaminaga
M. Kara
S. Kazama
P. Kharbanda
M. Kobayashi
D. Koke
K. Kooshkjalali
A. Kopec
H. Landsman
R. F. Lang
L. Levinson
A. Li
I. Li
S. Li
S. Liang
Z. Liang
Y. -T. Lin
S. Lindemann
M. Lindner
K. Liu
M. Liu
F. Lombardi
J. A. M. Lopes
G. M. Lucchetti
T. Luce
Y. Ma
C. Macolino
J. Mahlstedt
F. Marignetti
T. Marrodán Undagoitia
K. Martens
J. Masbou
S. Mastroianni
V. Mazza
A. Melchiorre
J. Merz
M. Messina
A. Michel
K. Miuchi
A. Molinario
S. Moriyama
K. Morå
M. Murra
J. Müller
K. Ni
C. T. Oba Ishikawa
U. Oberlack
S. Ouahada
B. Paetsch
Y. Pan
Q. Pellegrini
R. Peres
J. Pienaar
M. Pierre
G. Plante
T. R. Pollmann
A. Prajapati
L. Principe
J. Qin
D. Ramírez García
A. Ravindran
A. Razeto
R. Singh
L. Sanchez
J. M. F. dos Santos
I. Sarnoff
G. Sartorelli
J. Schreiner
P. Schulte
H. Schulze Eißing
M. Schumann
L. Scotto Lavina
M. Selvi
F. Semeria
F. N. Semler
P. Shagin
S. Shi
H. Simgen
Z. Song
A. Stevens
C. Szyszka
A. Takeda
Y. Takeuchi
P. -L. Tan
D. Thers
G. Trinchero
C. D. Tunnell
K. Valerius
S. Vecchi
S. Vetter
G. Volta
C. Weinheimer
M. Weiss
D. Wenz
C. Wittweg
V. H. S. Wu
Y. Xing
D. Xu
Z. Xu
M. Yamashita
J. Yang
L. Yang
J. Ye
M. Yoshida
L. Yuan
G. Zavattini
Y. Zhao
M. Zhong
T. Zhu

Dual-phase time projection chamber (TPC) that employs a multi-ton-scale liquid xenon (LXe) target mass is a pioneering detector technology to search for dark matter. Beyond its advantage in dark matter direct detection efforts, the natural xenon target allows it to search for the neutrinoless double-beta decay ($0νββ$) process, which would violate lepton number conservation and indicate that neutrinos are Majorana particles. However, such $0νββ$ searches have been limited by gamma-ray backgrounds originating from the detector materials. In this work, we designed an augmented convolutional neural network (A-CNN) model to extract additional event-topology information from detector data. Using simulation and calibration data from XENONnT, a leading LXe TPC experiment, our model achieved over 60% background rejection while maintaining 90% signal acceptance. This rejection power improves XENONnT's projected sensitivity of the $^{136}$Xe $0νββ$ search by about 40%. The implementation of A-CNN in the data analysis of future liquid xenon observatories, such as XLZD, will further enhance their sensitivities for $0νββ$ with $^{136}$Xe.


From Data to Laws: Neural Discovery of Conservation Laws Without False Positives

Conservation laws are fundamental to understanding dynamical systems, but discovering them from data remains challenging due to parameter variation, non-polynomial invariants, local minima, and false positives on chaotic systems. We introduce NGCG, a neural-symbolic pipeline that decouples dynamics learning from invariant discovery and systematically addresses these challenges. A multi-restart variance minimiser learns a near-constant latent representation; system-specific symbolic extraction (polynomial Lasso, log-basis Lasso, explicit PDE candidates, and PySR) yields closed-form expressions; a strict constancy gate and diversity filter eliminate spurious laws. On a benchmark of nine diverse systems including Hamiltonian and dissipative ODEs, chaos, and PDEs, NGCG achieves consistent discovery (DR=1.0, FDR=0.0, F1=1.0) on all four systems with true conservation laws, with constancy two to three orders of magnitude lower than the best baseline. It is the only method that succeeds on the Lotka--Volterra system, and it correctly outputs no law on all five systems without invariants. Extensive experiments demonstrate robustness to noise ($σ= 0.1$), sample efficiency (50--100 trajectories), insensitivity to hyperparameters, and runtime under one minute per system. A Pareto analysis shows that the method provides a range of candidate expressions, allowing users to trade complexity for constancy. NGCG achieves strong performance relative to prior methods for data-driven conservation-law discovery, combining high accuracy with interpretability.


From the Stochastic Embedding Sufficiency Theorem to a Superspace Diffusion Framework

Carolina Garcia
Lucía Perea Durán
Agnese Venezia
Alex Conradie

A generalisation of Takens' delay-coordinate embedding theorem to stochastic systems, the Stochastic Embedding Sufficiency Theorem, is an inverse methodology enabling non-parametric recovery of both drift and diffusion fields from scalar time series without prior assumptions about the governing physics.   A blind protocol, receiving only raw time series and sampling interval, is applied identically to nine domains: classical mechanics, statistical mechanics, nuclear physics, quantum mechanics, chemical kinetics, electromagnetism, relativistic quantum mechanics, quantum harmonic oscillator dynamics, and quantum electrodynamics. Fundamental constants (the Boltzmann constant, the Planck constant, the speed of light, the Fano factor, and the Van Kampen scaling exponent) emerge in both drift and diffusion channels without prior specification. The recovered diffusion coefficients, viewed across domains, constitute an empirical pattern, the $σ$-continuum, in which $k_B$, $\hbar$, and $c$ play structurally distinct roles. The Gravitational Diffusion Theorem, derived from the fluctuation-dissipation theorem, massless mode structure of linearised gravity, and gravitational self-coupling via the equivalence principle, determines the gravitational diffusion coefficient as one Planck length per square root of Planck time.   Four canonical axioms formalise the framework, within which the noise character, drift, covariance operator, and fluctuation amplitude are uniquely determined by theorem, yielding the superspace diffusion hypothesis:   $\mathrm{d}g_{ij} = \mathcal{D}_{ij}[g]\,\mathrm{d}τ+ \ell_P\,\mathrm{d}W_{ij}$   where all coefficients are non-parametric, first-principles consequences of the axioms. Coarse-graining of the superspace Fokker-Planck equation via Mori-Zwanzig projection yields predictions for galactic-scale gravitational acceleration testable against kinematic data.


Detecting the 3D Ising model phase transition with a ground-state-trained autoencoder

Ahmed Abuali
David A. Clarke
Morten Hjorth-Jensen
Ioannis Konstantinidis
Claudia Ratti
Jianyi Yang

We develop a one-class, deep-learning framework to detect the phase transition and recover critical behavior of the 3D Ising model. A 3D convolutional neural network autoencoder (CAE) is trained on ground-state configurations only, without prior knowledge of the critical temperature, the Hamiltonian, or the order parameter. After training, the model is applied to Monte Carlo configurations across a wide temperature range and different lattice sizes. The mean-square reconstruction error is shown to be sensitive to the transition. Finite-size scaling of the peak location for the reconstruction error susceptibility yields the critical temperature $T_c=4.5128(58)$ and the correlation-length critical exponent $ν=0.63(27)$, consistent with results from the literature. Our results show that a one-class CAE, trained on zero-temperature configurations only, can recover nontrivial critical behavior of the 3D Ising model.


VecAmpFit: vectorized amplitude-analysis fitting library

A new library VecAmpFit for multidimensional amplitude analyses in high-energy physics has been developed for an ongoing amplitude analysis at Belle II experiment. It includes a fitter performing likelihood calculation and explicitly-vectorized subprograms for amplitude implementation. The fitter supports explicit gradient calculation and simultaneous fitting of multiple data sets.


Jet flavor tagging with Particle Transformer for Higgs factories

Taikan Suehara
Takahiro Kawahara
Tomohiko Tanabe
Risako Tagami

We study the performance of the Particle Transformer (ParT) for jet flavor tagging using ILD full simulation events (1M jets) as well as fast simulation samples (10M and 1M jets). We perform 3-category ($b/c/d$), 6-category ($b/c/d/u/s/g$), and 11-category trainings (including quark--antiquark separation), incorporating multivariate hadron particle identification information from $dE/dx$ and time-of-flight. For $b$/$c$ tagging, we observe a factor of 5--10 improvement over previous BDT-based taggers, and we obtain reasonable performance for strange tagging and quark/antiquark separation. The 10M-jet fast simulation study indicates that further gains are possible with higher training statistics.


ALABI: Active Learning for Accelerated Bayesian Inference

We present Active Learning for Accelerated Bayesian Inference (\texttt{alabi}): an open-source Python package for performing Bayesian inference with computationally expensive models. Given a forward model and observational data to construct a likelihood and priors, \texttt{alabi}\ uses a Gaussian Process (GP) surrogate model trained to predict posterior probability as a function of input parameters, and employs active learning to iteratively improve GP predictive performance in high-likelihood regions where the GP is most uncertain. \texttt{alabi}\ provides a uniform interface for using Markov chain Monte Carlo (MCMC) with different packages, including the affine-invariant sampler \texttt{emcee}, and nested samplers \texttt{dynesty}, \texttt{multinest}, and \texttt{ultranest}. This approach facilitates accurate estimation of the desired posterior distribution, while reducing the number of computationally expensive model evaluations required by factors of thousands. We demonstrate the performance of \texttt{alabi}\ on a variety of test cases, including where inference is challenging due to complex posterior structure or high dimensionality. We show that \texttt{alabi}\ offers a substantial improvement for likelihood functions with evaluation times $\gtrsim 1$\,s, speeding up MCMC computations by a factor of $10-1000\times$ when tested on problems with up to 64 dimensions.


Long-term outburst activity of comet 17P/Holmes and constraints on ejecta size distributions

Maria Gritsevich
Marcin Wesołowski
Josep M. Trigo-Rodríguez
Alberto J. Castro-Tirado
Jorma Ryske
Markku Nissinen
Peter Carson

A quantitative understanding of cometary outbursts requires robust constraints on the size distribution of ejected particles, which governs outburst dynamics and underpins estimates of released gas and dust. In the absence of direct measurements of particle sizes, assumptions about the size distribution play a central role in modelling dust-trail formation, their dynamical evolution and observability, and the potential production of meteor showers following encounters with Earth. We analyse brightness amplitude variations associated with outbursts of comet 17P/Holmes from 1892 to 2021, with particular emphasis on the exceptional 2007 mega-outburst. During this event the comet underwent a rapid and substantial brightening: at its peak, the expanding coma reached a diameter exceeding that of the Sun and briefly became the largest object in the Solar System visible to the naked eye. We constrain the size distribution and total mass of porous agglomerates composed of ice, organics, and dust ejected during the outburst. The inferred particle size distribution is consistent with a power law of index q, yielding effective particle sizes between 10^-6 m for q = 4 and 5 x 10^-3 m for q = 2. Accounting for effective particle size, sublimation flux, and bulk density, we find that the total number of ejected particles increases with both q and sublimation flux. These results place constraints on the physical properties of outburst ejecta and provide physically motivated initial conditions for long-term dust-trail evolution modelling. They further indicate that cometary outburst brightness is determined primarily by the number of particles and their size distribution, rather than by the total ejected mass alone, with direct implications for the origin and evolution of meteoroid streams and the interplanetary dust population.


An approximate graph elicits detonation lattice

This study presents a novel algorithm based on graph theory for the precise segmentation and measurement of detonation cells from 3D pressure traces, termed detonation lattices, addressing the limitations of manual and primitive 2D edge detection methods prevalent in the field. Using a segmentation model, the proposed training-free algorithm is designed to accurately extract cellular patterns, a longstanding challenge in detonations research. First, the efficacy of segmentation on generated data is shown with a prediction error 2%. Next, 3D simulation data is used to establish performance of the graph-based workflow. The results of statistics and joint probability densities show oblong cells aligned with the wave propagation axis with 17% deviation, whereas larger dispersion in volume reflects cubic amplification of linear variability. Although the framework is robust, it remains challenging to reliably segment and quantify highly complex cellular patterns. However, the graph-based formulation generalizes across diverse cellular geometries, positioning it as a practical tool for detonation analysis and a strong foundation for future extensions in triple-point collision studies.


Extreme-Value Criticality and Gain Decomposition at the Integer Quantum Hall Transition

Extreme-value fluctuations at quantum critical points remain poorly understood in the presence of strong correlations and openness. At the integer quantum Hall transition in the open Chalker--Coddington network, we show that the maximal wave-function amplitude separates into a global gain and an intrinsic extreme component, $|ψ|_{\max}=A\,|\tildeψ|_{\max}$. We introduce extreme-moment scaling for $|ψ|_{\max}$ and observe an approximately parabolic exponent function $τ_{\max}(q)$ over moderate $q$, while $\ln|ψ|_{\max}$ displays an almost Gaussian bulk over the studied sizes. The gain factor is close to log-normal and largely controls the raw extremes. Gain normalization reorganizes the statistics: $\tildeτ_{\max}(q)$ changes qualitatively and $|\tildeψ|_{\max}$ does not support a single-parameter generalized extreme-value collapse under standard centering/scaling in the accessible size window. Extreme observables thus provide a robust probe of correlated criticality in open quantum systems.


Automatic Termination Strategy of Inelastic Neutron-scattering Measurement Using Bayesian Optimization for Bin-width Selection

Kensuke Muto
Hirotaka Sakamoto
Kenji Nagata
Taka-hisa Arima
Masato Okada

Currently, an excessive amount of event data is being obtained in four-dimensional inelastic neutron-scattering experiments. A method for automatic bin-width optimization of multidimensional histograms has been developed and recently validated on real inelastic neutron-scattering data. However, measuring beyond the equipment resolution leads to inefficient use of valuable beam time. To improve experimental efficiency, an automatic termination strategy is essential. We propose a Bayesian-optimization-based method to compute stopping criteria and determine whether to continue or terminate the experiment in real time. In the proposed method, the bin-width optimization is performed using Bayesian optimization to efficiently compute the optimal bin widths. The experiment is terminated when the optimal bin widths become smaller than the target resolutions. In numerical experiments using real inelastic neutron-scattering data, the optimal bin widths decrease as the number of events increases. Even the optimal bin widths for data downsampled to 1/5 are comparable with the resolutions limited by the sample size, choppers, and so on. This implies excessive measurement of the inelastic neutron experiments for the moment. Moreover, we found that Bayesian optimization can reduce the search cost to approximately 10% of an exhaustive search in our numerical experiments.


A complex network approach to characterize clustering of events in irregular time series

Ambedkar Sanket Sukdeo
K. Shri Vignesh
Sachin S. Gunthe
T Narayan Rao
Amit Kumar Patra
R. I. Sujith

In complex systems, events occur at irregular intervals that inherently encode the underlying dynamics of the system. Analyzing the temporal clustering of these events reveals critical insights into the non-random patterns and the temporal evolution. Existing techniques can effectively quantify the overall clustering tendency of events using global statistical measures. However, these macroscopic approaches leave a critical gap, as they do not attempt to investigate the dynamics of individual clusters. Analyzing individual clusters is essential, as it helps comprehend the local interactions that actively drive the system dynamics, which may be obscured by global averaging, while simultaneously revealing the time scales involved. To address these limitations, we propose a complex network-based framework for analyzing clustering of events occurring at irregular intervals. The framework establishes connections using arrival times, transforming the time series into a network. Network properties are then used to quantify the clustering. Further, a community detection algorithm is used to identify individual clusters in time series. We illustrate the method by applying it to standard arrival processes, such as the Poisson process and the Markov-modulated Poisson process. To further demonstrate its scope, we apply the method to two diverse systems: the time series of droplet arrivals in turbulent flows and the R-R intervals in electrocardiogram (ECG) signals.


py5vec: a modular Python package for the 5-vector method to search for continuous gravitational waves

Luca D'Onofrio
Federico Muciaccia
Lorenzo Mirasola
Matthew Pitkin
Cristiano Palomba
Paola Leaci
Francesco Safai Tehrani
Francesco Amicucci
Lorenzo Silvestri
Lorenzo Pierini

We present \texttt{py5vec}, a Python package for implementing and extending the 5-vector method, used to search for continuous gravitational wave (CW) signals. We also provide a comprehensive theoretical review of the 5-vector method and extend the relative likelihood formalism by marginalizing over the noise variance, resulting in a more robust Student's t-likelihood, and over the initial phase to account for pulsar glitches. \texttt{py5vec} provides a modular architecture that separates data representation, signal demodulation, and statistical inference into independent abstract stages. It supports multiple input data formats and interoperates with existing Python software, providing a bridge between different approaches. For example, using a \texttt{bilby}-based interface, \texttt{py5vec} implements Bayesian parameter estimation within the 5-vector formalism for the first time. The modular design also allows for making exact multi-level and direct comparisons between other software, such as \texttt{cwinpy} and \texttt{SNAG} in MATLAB. In \texttt{py5vec}, we implement a multidetector targeted search for known pulsars, validated using LIGO data from the O4a run and hardware injections, demonstrating consistent reconstruction of signal parameters. This package therefore provides a flexible platform for current targeted searches and for future extensions to other CW search strategies.


A Deep-Learning-Boosted Framework for Quantum Sensing with Nitrogen-Vacancy Centers in Diamond

Changyu Yao
Haochen Shen
Zhongyuan Liu
Ruotian Gong
Md Shakil Bin Kashem
Stella Varnum
Liangyu Li
Hangyue Li
Yue Yu
Yizhou Wang
Xiaoshui Lin
Jonathan Brestoff
Chenyang Lu
Shankar Mukherji
Chuanwei Zhang
Chong Zu

Nitrogen-vacancy (NV) centers in diamond are a versatile quantum sensing platform for high sensitivity measurements of magnetic fields, temperature and strain with nanoscale spatial resolution. A common bottleneck is the analysis of optically detected magnetic resonance (ODMR) spectra, where target quantities are encoded in resonance features. Conventional nonlinear fitting is often computationally expensive, sensitive to initialization, and prone to failure at low signal-to-noise ratio (SNR). Here we introduce a robust, efficient machine learning (ML) framework for real-time ODMR analysis based on a one-dimensional convolutional neural network (1D-CNN). The model performs direct parameter inference without initial guesses or iterative optimization, and is naturally parallelizable on graphics processing units (GPU) for high-throughput processing. We validate the approach on both synthetic and experimental datasets, showing improved throughput, accuracy and robustness than standard nonlinear fitting, with the largest gains in the low-SNR regime. We further validate our methods in two representative sensing applications: diagnosing intracellular temperature changes using nanodiamond probes and widefield magnetic imaging of superconducting vortices in a high-temperature superconductor. This deep-learning inference framework enables fast and reliable extraction of physical parameters from complex ODMR data and provides a scalable route to real-time quantum sensing and imaging.