Papers

We introduce a mathematical and cryptographic framework for exact recovery of noisy hidden paths in high dimensional discrete path spaces. The work is inspired by the path integral viewpoint, where global quantities arise from contributions over many possible trajectories. Instead of approximating a global path sum, we study the inverse problem of recovering one exact hidden trajectory from incomplete, noisy, projected, and aggregated observables. The hidden object is a planted discrete path whose transitions may include macro steps, microscopic perturbations, and discrete noise. Public information is represented by large observable vectors rather than short hash digests, since excessive compression would bound the effective recovery problem by the digest size. We formalize several recovery notions, including planted exact recovery, arbitrary witness recovery, canonical recovery, quotient recovery, and recovery of derived encodings. The main distinction is that approximate reconstruction and exact recovery are fundamentally different tasks. A method may reveal coarse geometry or dominant regions without recovering the precise microscopic sequence defining the hidden path. We also discuss attack surfaces relevant to future cryptographic use, including linearization, lattice style recovery, dynamic programming, meet in the middle attacks, SAT and SMT formulations, approximation followed by rounding, witness collisions, and generic quantum search. This work does not claim a complete post quantum cryptosystem. It provides a formal framework for studying exact hidden path recovery as a possible foundation for future cryptographic constructions

We establish the foundations of categorical weave calculus, developing the diagrammatic calculus of weaves and braid varieties within the study of Calabi-Yau triangulated categories and cluster tilting theory. This is achieved by associating a perverse sheaf of triangulated categories to each Demazure weave. A central contribution is the construction and study of the categorical Lusztig cycles and their duals, which we show form simple-minded and silting collections in the category of global sections of such a sheaf of categories. These categorical collections are built using the diagrammatics of weaves and we study their behavior under changes of weaves. For instance, we show that they undergo tilts under weave mutations. En route, we develop the study of categorical weighted braid words, as canonical rigid filtered dg modules over derived preprojective algebras, and the categorical incarnation of the tropical Lusztig rules, as a gluing mechanism for such filtered objects. Appendix A contains homological results, providing a novel construction of simple-minded and silting collections from full exceptional collections, and characterizing when these arise from a highest weight structure on an abelian category.

Let $\mathcal{F}_5$ denote the $3$-uniform hypergraph on the vertex set $\{f_1,f_2,\dots,f_5\}$ with hyperedges $\{f_1f_2f_3,f_1f_2f_4,f_3f_4f_5\}$. Recently, Balogh, Clemen and Luo determined the Turán number of a one-vertex blow-up of $\mathcal{F}_5$, more specifically, they blow up the vertex $f_5$ to $t$ vertices, the resulting hypergraph is denoted by $\mathcal{F}_5(f_5;t)$. They show that for infinitely many $t$, $\mathcal{F}_5(f_5;t)$ has exponentially many extremal constructions and positive Turán density. In this paper, we determine the exact Turán number of the hypergraph obtained by blowing up $f_3$ of $\mathcal{F}_5$ to $t$ vertices and show that it also has exponentially many extremal constructions. We also give a general upper bound and lower bound of the Turán number of every blow-up of $\mathcal{F}_5$. For some special blow-ups of $\mathcal{F}_5$, for example, $t$-disjoint copies of $\mathcal{F}_5$, we determine the exact Turán number. We construct a hypergraph $\mathcal{F}_{sim}(t)$ which is a subgraph of a blow-up of $\mathcal{F}_5$, and is contained in the hypergraph obtained by adding any new hyperedge to the Turán hypergraph (the balanced complete $3$-partite hypergraph), but its extremal construction is not the Turán hypergraph. We also determine the exact Turán number of $\mathcal{F}_{sim}(t)$.

A well-known result of Burr, Erdős and Spencer [Transactions of the American Mathematical Society, 1975] determines the $2$-colour Ramsey number for any sufficiently large collection of vertex-disjoint copies of a fixed graph $H$ without isolated vertices. A focus of this paper is to give analogous results for the corresponding $r$-colour Ramsey problem. More precisely we determine, up to an additive constant, this Ramsey number in the case when $r=3$ and also in the case when the chromatic number of $H$ is at least $r$. In these cases our results depend on a parameter which, roughly speaking, describes the corresponding class of potential extremal colourings of the complete graph. Our proofs rely on our notion of $(H,r)$-gadgets, which is crucial to obtain results that are best-possible up to additive constant terms. We exploit this notion using linear programming and the Poincaré-Miranda theorem. We also determine, up to a linear error term, the corresponding $r$-colour Ramsey number in the case when $H$ is a complete bipartite graph. Here, the corresponding class of potential extremal examples exhibits a connection with the well-known clique-edge-covering problem. We also prove random versions of some of our results. In particular, we prove a random version of the Burr-Erdős-Spencer theorem, thereby generalising the random Ramsey theorem of Rödl and Ruciński [Journal of the American Mathematical Society, 1995]. Our proofs make use of coloured versions of the (sparse) regularity lemma and the KLR conjecture for random graphs.

We describe an efficient method for computing the Ehrhart polynomial of Gelfand–Tsetlin polytopes arising from Kostka coefficients. The key idea is to exploit Ehrhart–Macdonald reciprocity: evaluating the Ehrhart polynomial at negative integers reduces to counting strict Gelfand–Tsetlin patterns, which are often zero or very small for low dilations. Combined with an adaptive strategy that chooses the cheapest evaluation point (positive or negative) at each step, this yields substantial practical speedups compared to general-purpose polytope software. We benchmark against $\mathtt{OSCAR}$/$\mathtt{polymake}$, and illustrate the broader applicability of the method through order polytopes and permutation posets. The implementation is available in the Rust kostka package, with related optimizations also incorporated in the new lrcalc-rs replacement for lrcalc.

We develop a general theory of Cartesian and non-Cartesian polynomials on products of complex spaces $\mathbb{C}^{n_1} \times \cdots \times \mathbb{C}^{n_k}$. We prove that, for any fixed degree $d \ge 2$, a (Zariski) generic polynomial is non-Cartesian in a broad range of dimensions, establishing that Cartesian structure is highly exceptional. We further introduce effective sufficient criteria for a polynomial to be non-Cartesian. Moreover, we show that being (non)-Catersian can be decided algorithmically via Gröbner basis methods and quantitative forms of Hilbert's Nullstellensatz. As an application, we connect the non-Cartesian condition to incidence geometry, obtaining sharp intersection bounds and constructing extremal configurations that demonstrate the optimality of these estimates.

The weak $k$-metric dimension of a graph is roughly understood as the cardinality of a smallest set of vertices $S$ of the graph with the property of uniquely recognizing all the vertices of the graph throughout summations of differences of distances to the vertices of $S$. The weak $k$-metric dimension of the direct product of two isomorphic complete graphs is considered in this work. Specifically, the value of such parameter is computed for almost all possibilities of these products and a bound is provided in the remaining case.

An $(r, s)$-${\textit set}$ in ${\rm PG}(n, q)$ is a set of points, say $\mathcal X$, such that each $s$-dimensional projective subspace contains at most $r$ points of $\mathcal X$. We investigate $(n, n-2)$-sets and $(n-2, n-3)$-sets in ${\rm PG}(n, q)$, $n \le 6$. We show that the trivial upper bounds on $(n, n-2)$-sets in ${\rm PG}(n, q)$, $4 \le n \le 6$, $(4, 3)$-sets in ${\rm PG}(6, q)$ and $(3, 2)$-sets in ${\rm PG}(5, q)$ are essentially sharp. A $(3, 2)$-set in ${\rm PG}(13, q)$ of size $\frac{q^6-1}{q-1}$ is also constructed.

Using divisibility relations between the generators of a square-free monomial ideal $I$, we describe divisibility relations between the generators of the second power $I^2$. We then employ discrete Morse theory to produce a cellular free resolution of $I^2$ which is minimal for specific ideals that are extremal with respect to a given divisibility relation. In particular, we provide sharp bounds on the projective dimension of $I^2$ when the generators of $I$ satisfy at least one divisibility relation.

We provide a complete combinatorial and asymptotic analysis of positive linear systems of equations in one catalytic variable that appear in several combinatorial problems such as in lattice path counting or stack-sortable permutation counting. We show that the corresponding generating functions satisfy a positive polynomial system of equations (which is associated to a context-free grammar). Furthermore we prove a universal asymptotic behaviour.

A problem of Erdős asks for extremal conditions forcing edge-disjoint cycles with a prescribed nested structure. In the geometric version, the nesting is required to be noncrossing with respect to the cyclic orders. Fernández, Kim, Kim and Liu proved that constant average degree forces two such cycles. We prove a polylogarithmic bound for the natural multi-layer version: for every fixed $k\ge 3$, every sufficiently large $n$-vertex graph with at least \[ C_k n(\log n)^{k-1}(\log\log n)^{k-3} \] edges contains $k$ pairwise edge-disjoint nested cycles without geometric crossings. The proof combines the robust sublinear expander framework of Alon, Bucić, Sauermann, Zakharov and Zamir with a controlled wrapping lemma that permits the layers to be built successively with controlled length.

Let $\mathrm{PG}(n-1,q)$ denote the $(n-1)$-dimensional projective space over $\mathbb{F}_q$. We investigate the intersection of two Desarguesian $(h-1)$-spreads of $\mathrm{PG}(kh-1,q)$ and show that it is determined by a subgeometry over a suitable extension field. Our approach combines a characterization of subsets of points of $\mathrm{PG}(k-1,q^h)$ closed under $q$-order subgeometries with a matrix model for Desarguesian spreads based on Moore matrices. This leads naturally to the notion of generalized Segre varieties $\mathcal S^r_{kr-1,h-1}(q)$ and a geometric description of their maximal subspaces. As a main application, we prove that if two distinct Desarguesian $(h-1)$-spreads of $\mathrm{PG}(kh-1,q)$ contain a common pseudo-arc of size $k+1$, then their intersection is precisely the system $\mathcal R^r_{h,q}$ of $(h-1)$-dimensional subspaces of $\mathcal S^r_{kr-1,h-1}(q)$, for some proper divisor $r$ of $h$.

High dimensional expanders simultaneously satisfying spectral and combinatorial (coboundary) expansion have recently played a major role in breakthroughs in PCP and coding theory, but the only known construction of such complexes is extremely involved, requiring deep algebraic number theory. In this work, we give an extremely simple combinatorial construction of a sub-polynomial degree complex based on projections of the flags complex (subspace chains) that is (i) a local spectral expander, (ii) a coboundary expander, and (iii) a swap coboundary expander. As a corollary, we also give the first near-linear size combinatorial hypergraphs with good agreement tests in the '1%' regime, and a simple PCP construction with near-linear size.

The Ramsey number $\mathrm{R}(G_1,G_2)$ is the smallest integer $N$ such that any red-blue coloring of the edges of the complete graph $K_N$ contains either a red copy of $G_1$ or a blue copy of $G_2$. In 2022, the third author and others gave lower and upper bounds of the Ramsey number $\mathrm{R}(W_n,W_n)$, where $W_n$ is the wheel graph with $n$ vertices. In this paper, we improve their bounds by showing that $3n-2\leq \mathrm{R}(W_n,W_n)\leq 6n-6$ for even $n\geq 8$ and $2n\leq \mathrm{R}(W_n,W_n)\leq \frac{9n-7}{2}$ for odd $n\geq 7$. Furthermore, we give recursive bounds for the $k$-colored Ramsey number for $W_n$.

We consider the following problem: Let $H$ and $F$ be two graphs on $k$ vertices and assume $F \neq H$. We say that $H$ and $F$ are incomparable if neither $F$ nor $H$ contains the other. Let $H$ be a graph on $k$ vertices and let $G$ be a graph on at least $k$ vertices. Then $G$ is said to be $H$-exact if any induced subgraph of $G$ on $k$ vertices is either isomorphic to $H$ or incomparable with $H$. Exact($H$) is the family of all graphs $G$ which are $H$-exact. We pose the following problem: For a graph $H$ on $k$ vertices, determine or estimate $f(H) = \max \{n: \exists G \in \text{Exact}(H), |V (G)| = n\}$. Among the many results obtained in this paper the following are representatives concerning trees and matchings: 1. For a tree on $k \geq 3$ vertices, $ (k - 1)(\left \lceil \frac{k}{2} \right \rceil -1 ) \leq f(T) \leq ( k-1)^2$. 2. For $k \geq 4$, $f(K_{1,k-1}) = (k-1)(k-2)$. 3. For $k \geq 5$, $f(P_k) = \frac{(k-1)^2}{2}$ if $k$ is odd and $f(P_k) = \frac{(k-1)(k-2)}{2}+1$ if $k$ is even. 4. $f(nK_2) = 3n$ for $n = 2, 3$ and $f(nK_2) = 4n - 4$ for $n \geq 4$.

In the context of geometric measure theory, Llorente-Winter introduced the (average) fractal Euler number as a notion of the Euler characteristic for fractals embedded in Euclidean space. However, the class of fractals to which it is applicable remains very limited. In the present paper, we modify this notion by applying perspectives of persistent homology and partly the theory of magnitude, which have recently come from applied topology and category theory. We then demonstrate concrete calculation of our average ph-fractal Euler number for some classically well-known fractals, especially the Cantor dust and Menger sponge which are excluded from Llorente-Winter's approach.

Guo, Li, Shangguan, Tamo, and Wootters proposed a hypergraph analogue of the Nash–Williams–Tutte theorem, asserting that every $k$-weakly-partition-connected hypergraph admits a $k$-distinguishable tree assignment. We identify a sharp edge-count obstruction to the literal statement. A full tree assignment has $\rho(H)=\sum_{e\in E(H)}(|e|-1)$ labelled graph edges, whereas an ordered decomposition into $k$ spanning trees has exactly $k(t-1)$ edges. Since weak partition connectivity implies only $\rho(H)\ge k(t-1)$, every strict inequality rules out a full decomposition. In particular, for all $t\ge2$, $k\ge1$, and $q\ge1$, the hypergraph consisting of $k+q$ labelled copies of the full hyperedge is $k$-weakly-partition-connected but admits no $k$-distinguishable full tree assignment. We therefore isolate the critical regime $\rho(H)=k(t-1)$ and prove that every tree assignment of a critical $k$-weakly-partition-connected hypergraph admits a full ordered decomposition into $k$ spanning trees. Consequently, the critical conjecture reduces to the existence of some tree assignment having a decomposition whose signature fiber is a singleton. We establish this property for layer-contained certificates, without a star hypothesis or a rank restriction in interior layers. These certificates also imply weak partition connectivity by a quotient-rank argument and are stable under one-vertex sums. Finally, we derive a decomposition-indexed generalized-Laplace expansion for the relevant intersection-matrix minor, together with an exact formula for its collected monomial coefficients; we also give the required row-and-column perfect-shuffle normalization, identify criticality with the square full-assignment row count, and separate the critical conjecture from the additional overfull pruning problem.

We study the problem of sampling weighted partial triangulations of a convex polygon with $n+2$ sides. We consider the distribution $\pi_{n,\lambda}$ under which each partial triangulation $\sigma$ is assigned probability proportional to $\lambda^{|\sigma|}$, where $\lambda>0$ is a model parameter and $|\sigma| \in \{0,\dots,n-1\}$ denotes the number of diagonals in $\sigma$. This model belongs to a broad class of weighted geometric partition problems that include lattice triangulations and dyadic tilings, and is closely related to several classical combinatorial structures, including the full triangulations of a convex polygon and the associated Catalan structures. Our main result is a simple exact sampling algorithm for $\pi_{n,\lambda}$ with expected running time $O\big((\min\{n,n\sqrt{\lambda}\}+1)\log n\big)$, which is optimal up to the logarithmic factor.

The stability number of a forbidden family measures how many different structures are needed to approximate all near-extremal constructions avoiding it. An infinite stability number means that no finite list of structures suffices. We construct a simple explicit $3$-graph whose stability number is infinite. This extends the infinite-stability phenomenon for finite forbidden families, established by Hou–Li–Liu–Mubayi–Zhang, to the single-forbidden setting, and further develops the single-$3$-graph direction of Balogh–Clemen–Luo, in which exponentially many exact extremal constructions coexist with stability.

Let $q$ be an odd prime power, let $n\ge 2$, and let $V\subsetneq \mathbb F_{q^n}$ be a proper $\mathbb F_q$-vector subspace. Given a nonzero quadratic form $Q(X,Y)\in \mathbb F_{q^n}[X,Y]$, we consider the graph $\Gamma(Q,V)$ that naturally arises from the condition $Q(X,Y)\in V$. We determine all quadratic forms $Q$ for which $\Gamma(Q,V)$ is undirected for every $V$. Besides the case $Q(x,y)=XY$, studied earlier by the second author, this essentially leads to the forms $X^2\pm Y^2$ and the family $Q_b(X, Y):=X^2+bXY+Y^2, b\ne 0$. We then study connectedness and clique number for the corresponding graphs. Our results reveal a clear contrast between these cases. The graphs $\Gamma(X^2\pm Y^2, V)$ are well structured, disconnected and their clique number can be as large as $\# V$. On the other hand, the family $Q_b$ seems to yield less structured graphs: the graphs are connected (in fact, of diameter $2$) if $\# V\ge q^{3n/4}$ and, in many cases, their clique number is $o(\# V)$. Our proofs are mainly based on character sums, while requiring a few algebraic and combinatorial ideas. We end the paper with some open problems and remarks, including a short discussion of the complementary case where $q$ is even.

A Golomb ruler of order~$n$ is an integer set $\{a_0<a_1<\cdots<a_{n-1}\}$ whose $\binom{n}{2}$ pairwise differences $a_j-a_i$ ($i<j$) are all distinct. The optimal Golomb ruler problem asks for $\mathrm{OGR}(n)=\min\{a_{n-1}-a_0\}$ and is a classical combinatorial benchmark. The values $\mathrm{OGR}(2),\dots,\mathrm{OGR}(28)$ are settled through distributed volunteer search (the Distributed.net OGR project); $\mathrm{OGR}(29)$ is in active computation, with verification expected in late 2026 or early 2027. The recent upper bound $\mathrm{OGR}(29)\le 757$ of Lee, Park, and Kim (arXiv:2510.0122, October~2025) tightens the search window. This note describes a compact CNF encoding of the decision problem $\mathrm{GR}(n,L)$ (“is there a Golomb ruler of order~$n$ with length exactly~$L$?”) with $O(n^2 L)$ clauses, together with a small-case verification sweep that emits machine-checkable LRAT certificates of optimality for $\mathrm{OGR}(n)$ at $n\le 12$. We then give closed-form encoding-size estimates for $\mathrm{OGR}(29, L=757)$ and propose a cube-and-conquer decomposition aimed at a Mallob-style parallel run on commodity multi-core hardware. The certificate pipeline (CaDiCaL with –lrat=true, a structural sanity-check, then formal validation by drat-trim or cake\_lpr) is end-to-end. A formally verified optimality proof for any single $\mathrm{OGR}(n)$ value beyond the trivial $n\le 5$ would be a first in the field.

Given a positive integer $k$ and graph $G$, the $\mathbb{Z}_k$-Ramsey number $R(G,\mathbb{Z}_k)$ is the least $N$ (if it exists) such that every coloring $f:E(K_N)\rightarrow \mathbb{Z}_k$ contains a copy $G'$ of $G$ such that $\sum_{e\in E(G')}f(e)=0$. Motivated by a question of Caro and Mifsud, we study the $\mathbb{Z}_k$-Ramsey number of graphs with a sufficiently large 2-packing, i.e. a set of vertices $S\subseteq V(G)$ such that $N[u]\cap N[v]=\emptyset$ for all distinct $u,v\in S$. In particular, we prove that $R(G,\mathbb{Z}_p)\leq n+6p-9$ for all $n$-vertex graphs $G$ and all primes $p$ such that $p$ divides $e(G)$, the minimum degree of $G$ is at least $1$, and there exists a $2$-packing of $G$ with size $p-1$. This upper bound improves depending on vertex degrees in the $2$-packing, with equality in certain cases. The result also implies an upper bound of the form $R(G,\mathbb{Z}_p)\leq n+C$ for $n$-vertex graphs $G$ of bounded maximum degree.

Rank-metric codes are subspaces of matrices over finite fields endowed with the rank metric and admit a natural tensorial representation. The tensor rank provides a measure of the minimal size of a decomposition of a code into rank-one tensors. Kruskal showed that the tensor rank of a rank-metric code of dimension $k$ and minimum rank distance $d$ is at least $k + d - 1$, and codes meeting this bound with equality are called minimal tensor rank (MTR) codes. It is known from algebraic complexity theory that the existence of an MTR code implies the existence of a maximum distance separable (MDS) code. In this work, we establish new results relating the tensor rank of a rank-metric code to the parameters of associated linear codes in the Hamming metric and introduce the notion of tensor rank defect. We then develop new constructions of rank-metric codes with small tensor rank defect using algebraic geometry (AG) codes.

This paper introduces four types of subdivision products for simple directed graphs extending those from the undirected case, in particular, the subdivision-vertex join, subdivision-arc join, subdivision-vertex corona and subdivision-arc corona. Structural and spectral properties of these constructions are analyzed, with a focus on adjacency, Laplacian and signless Laplacian spectra.

The Erdős-Ginzburg-Ziv theorem states that every sequence of 2n - 1 integers contains a subsequence of length n whose sum is divisible by n. Choi, Kang, and Lim gave a simple deterministic O(n log n) algorithm for finding such a subsequence, and Leung recently improved this to O(n log log log n). We give a deterministic linear-time algorithm. The core is a linear-time algorithm for the following prime target subset-sum problem: given p - 1 nonzero residues in Z_p and a target residue, find a subset with the prescribed sum. Our algorithm maintains a compact arithmetic-progression representation of reachable sums. When two progressions intersect, a bounded Frobenius interval in their sum allows them to be merged into one longer progression, with enough growth to pay for the update. When the representation either contains a full progression or covers all nonzero residues, the target residue is recovered constructively. The standard multiplicative reduction then extends the prime algorithm to arbitrary moduli.

For $w$ in the symmetric group $S_n$, let $\widetilde C_w$ be the corresponding modified, signless Kazhdan–Lusztig basis element of the type-$A$ Hecke algebra $H_n(q)$. An extension [Ann. Comb. 25, no. 3 (2021) pp. 757–787] of a result of Deodhar [Geom. Dedicata 36, (1990) pp. 95–119] implies that any factorization of the form \begin{equation*} \widetilde C_w = \frac1{f(q)} \widetilde C_{v^{(1)}} \cdots \widetilde C_{v^{(r)}}, \end{equation*} with $v^{(1)},\dotsc,v^{(r)}$ maximal elements of parabolic subgroups of $S_n$ and $f(q) \in \mathbb N[q]$ depending on these, provides cancellation-free combinatorial interpretations of the polynomials $\{P_{v,w}(q) \,|\, v \in S_n \}$ appearing in the expansion $\sum_v P_{v,w}(q) T_v$ of $\widetilde C_w$ in terms of the natural basis $\{ T_v \,|\, v \in S_n \}$ of $H_n(q)$. While the set of permutations $w \in S_n$ admitting such a factorization of $\widetilde C_w$ has not yet been characterized, we apply a result of Gaetz – Gao [Adv. Math. 457 (2024) Paper No. 109941] to describe a set admitting no such factorization.

Phylogenetic networks and, more generally, directed acyclic graphs (DAGs) represent hierarchical structure beyond trees, for instance in the presence of reticulate evolutionary events such as hybridization or horizontal gene transfer. A central question is which parts of such graphs are essential with respect to leaf-observable information, and which parts can be removed without changing this information. Resolving this question can lead to principled simplification methods for phylogenetic networks, such as the recent normalization approach of Francis et al. In this paper, we study this question from three related perspectives: clusters displayed by a DAG $G$, least common ancestors (LCAs) of subsets of its leaf set, and visibility, a path-based property of vertices. We first introduce an LCA-based simplification procedure called $i$-regularization. For a DAG $G$ and $i\geq 1$, the DAG $\reg_i(G)$ retains precisely those vertices that occur as unique LCAs of leaf subsets of size at most $i$, removes the remaining non-leaf vertices by a graph-editing operation $\ominus$, and then deletes shortcuts. We show that $\reg_i(G)$ admits a Hasse-diagram characterization in terms of the corresponding lca-clusters. We then compare LCA-based regularization with normalization. Using the same $\ominus$-operator, we describe the cover construction underlying normalization, identify visible vertices that are nevertheless removed, and characterize when regularization and normalization coincide. Together, these results provide a unified framework for cluster-based, LCA-based, and visibility-based simplifications of DAGs and phylogenetic networks.

Ballantine–Beck–Feigon–Maurischat introduced the subsum polynomial \[ \operatorname{sp}(\lambda,x):=\prod_i (1+x^{\lambda_i}) \] attached to an integer partition $\lambda$, and studied rational functions obtained by summing reciprocals of these polynomials over natural classes of partitions. They posed ten conjectures which naturally divide into coprimality and divisibility questions, special-value and recurrence formulas, and coefficient-shape problems. We prove all of the conjectures in the first two families: the ordinary and binary coprimality/divisibility conjectures, and the odd and ternary special-value/recurrence conjectures. AxiomProver autonomously produced Lean/mathlib formalizations and machine-checkable proofs of these six conjectures, and also discovered a counterexample to the statement as printed; the corrected form remains open.

An infinite structure has the finite length property (over a given field) if, for each of its finite powers, chains of equivariant subspaces in the corresponding free vector space are bounded in length. Prior work showed that the countable pure set and the countable dense linear order without endpoints have this property. We generalise these results to (a) any structure approximated by finite substructures with few orbits, provided the field is of characteristic zero, and (b) any Fraïssé limit with free amalgamation in a finite vocabulary consisting of unary and binary relations, possibly expanded with a generic total order. As a special case, we deduce the finite length property of the Rado graph using both methods. We also describe some connections with function spaces, weighted register automata, and orbit-finite systems of linear equations.

Agustín-Aquino solved, in terms of the table of marks of $\Aff(\mathbb{Z}/2k\mathbb{Z})$, the problem of enumerating the classes of bicolour self-complementary and rigid patterns in $\mathbb{Z}/2k\mathbb{Z}$ (also known as strong dichotomy classes). In particular, the rigid pattern-inventory polynomial appeared, for odd $k$, to yield the number of strong classes with negative sign when evaluated in $-1$, and it was conjectured that this is true for $k$ a power of an odd prime. Here we prove the conjecture is true for $k$ odd in general.