Papers

We develop a rigorous mathematical framework for statistical mechanics models on hypergraphs, and give conditions for lifting the classical connection between Potts model partition functions and the Tutte polynomial from graphs to hypergraphs. We define hypergraphical models and their associated partition functions, and extend these to special classes of models induced by families of interaction functions. For boolean interaction families, whose interaction functions map to $\{0,1\}$, we show that the partition function is determined by a combinatorial rank function, and we establish sufficient conditions for a hypergraph deletion-contraction recurrence and for when the rank function defines a polymatroid. We illustrate the theory by applying it to three hypergraph interaction families: Parity Ising, Delta Potts, and And Ising. The induced hypergraphical models are not isomorphic to each other, but the first two reduce to the same graphical Ising models. We identify three polymatroids naturally associated with hypergraphs for these models: respectively, the binary matroid of the incidence matrix over $\mathbb{F}_2$, the hypergraphical polymatroid, and the boolean polymatroid. For graphs, the first two reduce to the classical graphical matroid, and their partition functions recover the multivariate Tutte polynomial. The Tutte polynomial thus admits at least two distinct generalizations for hypergraphs, both satisfying a deletion-contraction recurrence: the Tutte polynomial of the binary matroid of the hypergraph incidence matrix, and a multivariate version of the Poincaré polynomial of the hypergraphical polymatroid. The partition functions of And Ising models are likewise multivariate versions of the Poincaré polynomial, here of the boolean polymatroid. These examples illustrate the much greater range of hypergraphical models and underscore the need for the unifying theory.

We study the $2$-cops-move variant of the game of cops and robber, in which at most two cops may move in each turn. Let $c_2(G)$ denote the corresponding cop number. We prove that every graph embeddable on the torus, Klein bottle, or projective plane satisfies $c_2(G)\le 3$, extending a recent result of González Hermosillo de la Maza and Mohar for planar graphs.

We prove quantitative stable regularity lemmas for binary real-valued functions, extending the work of Malliaris and Shelah for stable graphs. The statements of our results are modeled after non-quantitative theorems for stable functions due to Chavarria, Conant, and Pillay. One of the key tools in our quantitative proof is an "analytic symmetry lemma", which gives a function-theoretic analogue of the fact that a pair of good sets in a graph has density close to 0 or 1. We also develop a function-theoretic treatment of Malliaris and Shelah's random sampling method for refining partitions consisting of good sets into equipartitions.

We prove function-theoretic analogues of a quantitative result of Hodges on extracting the order property from a sufficiently large 2-tree coded in a binary relation. Similar analogues for functions were previously obtained by Daskalakis and Golowich and by Anderson and Benedikt. These results are from statistical learning theory, where 2-trees are captured by sequential fat-shattering dimension, and the order property is controlled by various notions of "thresholds". Our first main result (Theorem 1.11) focuses on extracting a less restrictive kind of threshold from a tree, and yields significantly better bounds compared to what can be obtained from earlier results focusing on more restrictive versions. Part of the motivation for Theorem 1.11 lies in a companion paper, where this theorem is used to obtain efficient bounds in quantitative regularity lemmas for "stable functions". Here will use Theorem 1.11 to reprove a result of Anderson and Benedikt in a stronger form and with improved bounds. We also use Theorem 1.11 to prove an at most double-exponential bound on dual sequential fat-shattering, which resolves an open problem. In our second main result (Theorem 1.14), we give a new proof of a result of Daskalakis and Golowich on extracting "tight thresholds" from large sequential fat-shattering dimension, with improved bounds. This resolves another open problem related to correcting the proof of a result claimed by Jung, Kim, and Tewari.

Let $\mathcal{F}\subseteq\binom{[n]}k$, and let $\partial_{k\to\ell}\mathcal{F}$ be the family of all $\ell$-sets containing a member of $\mathcal{F}$. Writing $\mu_k(\mathcal{F})=|\mathcal{F}|/\binom nk$, $p=k/n$, and $q=\ell/n$, we prove $$ \mu_\ell(\partial_{k\to\ell}\mathcal{F})\geq \begin{cases}\mu_k(\mathcal{F})^{\frac{\log q}{\log p}}, &0\leq \mu_k(\mathcal{F})\leq p^2,\\ q\left[1-\left(1-\frac{\mu_k(\mathcal{F})}{p}\right)^{ \frac{\log(1-q)}{\log(1-p)}} \ \ \right], &p^2\leq \mu_k(\mathcal{F})\leq p,\\ 1-(1-\mu_k(\mathcal{F}))^{\frac{\log(1-q)}{\log(1-p)}}, &p\leq \mu_k(\mathcal{F})\leq1. \end{cases} $$ This yields a dimension-free closed-form lower bound for the finite Kruskal–Katona profile and each branch is asymptotically sharp. As the main application, we settle both the uniform and biased product conjectures of Frankl and Tokushige. If $r\geq2$, $0\leq k_i\leq(r-1)n/r$, and $\mathcal{F}_i\subseteq\binom{[n]}{k_i}$ are $r$-cross-intersecting, then $$ \prod_{i=1}^r\mu_{k_i}(\mathcal{F}_i)\leq\prod_{i=1}^r\frac{k_i}{n}. $$ In biased product setting, we prove that for $r$-cross-intersecting families $\mathcal{A}_i\subseteq 2^{[n]}$ and $0\leq p_i\leq(r-1)/r$, $\prod_{i=1}^r\mu_{p_i}(\mathcal{A}_i)\leq\prod_{i=1}^rp_i.$ We further determine all equality cases in both settings.

We study tight Hamilton cycles in linearly quasirandom $3$-graphs. An $n$-vertex $3$-graph $H$ is $(p,\mu)$-dense if $e_H(X,Y,Z)\ge p|X||Y||Z|-\mu n^3$ for all $X,Y,Z\subseteq V(H)$. Araújo, Piga and Schacht asked whether the conditions $p,\alpha>1/4$ and $\delta_2(H)\ge\alpha n$ force a tight Hamilton cycle. We give a negative answer to this question. More generally, we determine the asymptotically sharp minimum-codegree threshold for the existence of a tight Hamilton cycle for every density $p\in(0,1)$. The resulting threshold is a discontinuous piecewise-defined function with four distinct regimes, and matching constructions show that every piece is best possible. The proof combines the absorption method and a fixed-length connecting lemma above density $1/3$ with a canonical-component Hamilton framework at and below density $1/3$.

A restricted-budget version of the random graph process, introduced by Frieze, Krivelevich, and Michaeli in 2025, studies the construction of structures by an online player who can purchase only a limited number of random edges. In this paper, we transfer this framework from random graphs to random subsets of integers, focusing on the construction of $k$-term arithmetic progressions. A player, Builder, is presented with a sequence of $t$ integers drawn uniformly at random from $[n]$. As the elements are revealed one by one, Builder must immediately and irrevocably decide whether to select the current integer, subject to a maximum budget of $b$ selected elements in total. We establish the optimal thresholds for this process, proving that for $t = \omega(n^{1-2/k})$, a budget of $b = \Theta((n/t)^{\frac{k-2}{2}})$ is both necessary and sufficient for Builder to successfully construct a $k$-term arithmetic progression with high probability.

We give a recursive construction of an LKTS(7q + 2) from an LKTS(q + 2) and an auxiliary permutation structure on an abelian group of order q, which we call a cubic orthomorphism. We prove that cubic orthomorphisms exist whenever q is a product of prime powers congruent to 1 modulo 6. The construction yields, in particular, LKTSs of orders 93, 177, 219, 261, and 303.

The Shannon capacity $\Theta(G)$ of a graph $G$ quantifies the maximum rate at which information can be transmitted with zero error over a noisy channel. It is lower bounded by $\alpha(G^d)^{1/d}$ for any $d$, where $\alpha(G^d)$ is the independence number of the $d$-th strong product of $G$. We construct independent sets of size $134753$ in $C_7^{10}$, $21909$ in $C_{11}^{6}$, $62530$ in $C_{13}^{6}$, and $8076974$ in $C_{15}^{8}$, improving the best known lower bounds for the Shannon capacity of these graphs to $\Theta(C_7)\geq 134753^{1/10}>3.258020$, $\Theta(C_{11})\geq 21909^{1/6}>5.289773$, $\Theta(C_{13})\geq 62530^{1/6}>6.300109$, and $\Theta(C_{15})\geq 8076974^{1/8}>7.301399$. We also improve the best known lower bounds on the independence numbers of several individual strong products of odd cycles that do not improve the Shannon capacity lower bound. The constructions were discovered through iterative interactions with a Large Language Model (LLM), illustrating the potential of LLMs for finding explicit combinatorial constructions.

We identify and study the interval Garside groups arising from the restriction of the absolute order on a Coxeter group to a lattice $[1,w]_T$, where $w$ is an involution. Those involutions $w$ for which $[1,w]_T$ is a lattice were previously classified by the second author; every such involution lies in the center of the parabolic subgroup generated by $[1,w]_T$. Except in type $B_n$, the obtained groups are isomorphic to (decomposable) right-angled Artin groups. We also investigate the situation for some finite complex reflection groups, mostly in rank two, taking for $w$ a (not necessarily involutive) central element.

Chromatic symmetric functions are well-studied symmetric functions in algebraic combinatorics that generalize chromatic polynomials of graphs. In 1995, Stanley introduced these symmetric functions and conjectured that they are Schur positive for claw-free graphs. We give examples of a line graphs, which are thus claw-free, whose chromatic symmetric function have a negative coefficient in its Schur expansion. We also give a counterexample to the 2018 conjecture of Monical that Schur positive chromatic symmetric functions have saturated Newton polytope when expanded in any finite number of variables. Both of these examples were found using ChatGPT-5.6 Sol Pro.

McKay proved that the spectra of all polynomial functions of the adjacency matrix $A$ and the diagonal degree matrix $D$ determine a tree. We prove a principal version of this theorem. Let $\mathcal A(G)=\langle I,A_G,D_G\rangle$ and let $M_G=\mathcal A(G)\mathbf1$ be the cyclic module generated by the all-ones vector. For connected graphs the ideal $\mathcal A(G)J\mathcal A(G)$, where $J=\mathbf1\mathbf1^T$, acts on $M_G$ as the full endomorphism algebra. We show that every forest satisfies $M_G=U_G$, the automorphism-orbit module, and that the induced algebra on the orbit quotient of a tree is a full matrix algebra. It follows that the scalar moments $\mathbf1^Tw(A_T,D_T)\mathbf1$ determine every tree. For general graphs these moments are degree-decorated caterpillar homomorphism counts. The resulting moment-rigidity class lies inside the amenable, compact, refinable hierarchy of color refinement, and its first small-order failures are ten-vertex integral switchings invisible to $M_G$.

A (3, 6)-fullerene is a cubic planar graph whose faces all have 3 or 6 sides. We give an exact enumeration of (3, 6)-fullerenes with V vertices and each of five possible symmetry types: *332, 332, 2*2, *222, and 222, in orbifold notation. We use this enumeration, together with some constructions, to resolve three conjectures about the existence of (3, 6)-fullerenes with 2*2$, *222, and 222 symmetry for given numbers of vertices.

Fat minors are the metric analog of graph minors that are tailored to the analysis of metric (edge-weighted) graphs and, more generally, metric spaces having a suitable notion of shortest paths. Despite a large interest in this notion, not much is known about the structure of metric graphs excluding a fixed fat minor. We prove that if a metric graph $G$ excludes a fixed graph $H$ as a $\delta$-fat minor, for some $\delta>0$, then $G$ enjoys the metric analog of flatness (aka uniform quasi-wideness) - a structural property from the field of Sparsity. In essence, our flatness result says that for any $\alpha\geq \beta$ large enough compared to $\delta$, in every large enough set $A$ in $G$ one can find a sizable subset $B$ that becomes $\alpha$-scattered after removing a bounded number of balls of radius $\beta$. We call this property drill-flatness. Notably, the proof only relies on excluding shallow fat minors: every branch set has radius at most $2\alpha$. As a corollary, we prove that metric graphs that exclude a fixed $\delta$-fat minor have bounded $\varepsilon$-scatter dimension if we consider only $\varepsilon$-scatters at distances large enough compared to $\delta$. By combining this with the results of Abbasi et al. [FOCS 2023], we infer that the $k$-Center problem on instances excluding $H$ as a $\delta$-fat minor admits an approximation algorithm that finds a solution of cost at most $(1+\varepsilon)\cdot\mathsf{OPT}+{\cal O}(\delta/\varepsilon^2)$ in time ${\cal O}_{H,\varepsilon}(n^{{\cal O}(1)})$. This is one of the first algorithmic results for general fat-minor-free metrics. We also study drill-flatness in hereditary classes of (unweighted) graphs, where we obtain a characterization equating drill-flatness with excluding shallow induced minors. This is an induced analog of the equivalence between flatness and nowhere denseness - one of central results of Sparsity.

We construct a new reduction process which takes a $(132,213)$-avoiding permutation to a shorter one that is cyclic if and only if the original was. Iterating it determines whether a given $(132,213)$-avoiding permutation is cyclic. Reversing it gives four moves that build every cyclic $(132,213)$-avoiding permutation, uniquely, from $1$ if $n$ is odd, and $21$ if $n$ is even. Our main application is the first non-trivial lower bound for the growth rate of $\mathcal{C}_n(132,213)$, the cyclic permutations of length $n$ avoiding $132$ and $213$. We also give several other consequences of the reduction, including a bijection between the odd and even size classes and an exact enumeration for those permutations with a restricted number of layers.

We prove Reiner's conjecture for higher Bruhat orders in corank 3: the facial intervals of B(n,n-3) are precisely the spherical intervals, and all other intervals are contractible.

We prove Borsuk-Ulam-type results governed by selection structures. Selection structures extend the matroidal framework for colorful theorems in discrete geometry and include non-matroidal examples such as chessboard complexes. Motivated by Frick and Wellner's Radon-type strengthening of Fan's theorem and its colorful variants, we prove selection-structure analogues whose conclusions are determined by Radon partitions. We also prove a prime-power selection-structure covering version of Volovikov's theorem, governed by Tverberg partitions. We include applications to fair partitions, including selection-structure versions of the ham sandwich and necklace splitting theorems.

The target-free clique conjecture asserts that the supports of stable fixed points of a nondegenerate combinatorial threshold-linear network (CTLN) are exactly its target-free cliques: bidirected cliques for which no outside vertex receives an edge from every clique vertex. We give an explicit six-neuron counterexample. For every sufficiently small $\varepsilon>0$, the CTLN defined by one fixed graph at $\delta=29\varepsilon/25$ is nondegenerate and has a stable fixed point with nonclique full support. Its values of $q=\delta(1-\varepsilon)/\varepsilon$ tend to $29/25$. In the complementary direction, for any CTLN on $n\geq3$ vertices, we prove that in the parameter range \[ q\geq n-2-\frac{n-3}{2}\varepsilon, \] no nonclique support can satisfy both the fixed-point positivity and linear stability conditions. Consequently, throughout this range, every nondegenerate CTLN has exactly its target-free cliques as supports of stable fixed points. In particular, this holds when $\varepsilon\leq\delta/(\delta+n-2)$.

We prove a weighted sum formula for double Eisenstein series. Its corresponding identity for the generating series of multiple divisor sums was conjectured by the author in his master's thesis. The double Eisenstein series identity follows from the restricted double-shuffle relations proved by the author and Tasaka, while the proof of the divisor-sum identity is combinatorial and uses generating series.

This short note provides two examples of aperiodic frieze patterns of the plane, supported on the rhombic Penrose tiling and the Godrèche–Lançon–Billard tiling. That is, we provide a decoration of their vertices with positive integers which satisfy the diamond rule, in analogy to the usual (in)finite frieze patterns as defined by Conway and Coxeter.

Let $\mathrm{IMS}_n(m)$ count the $n\times n$ nonnegative integer matrices whose row sums, column sums, main-diagonal sum, and antidiagonal sum are all $m$. We determine the Ehrhart series $F_n(q)=\sum_{m\geq0}\mathrm{IMS}_n(m)q^m$ as reduced rational functions for $n=7$ and $n=8$. Their numerator–denominator degrees are respectively $(366,373)$ and $(540,548)$. Both numerators have positive integer coefficients, are palindromic and strictly unimodal. The proofs share one finite architecture: a signed SimpCone decomposition is evaluated by quotient characters over finite fields, a certified common denominator and Ehrhart reciprocity reduce the rational identity to finitely many coefficients, an explicit counting bound lifts modular congruences to integer equalities, and exact gcd computations prove reducedness. For order eight, a face-index pole certificate gives a degree-$598$ common denominator without enumerating the full face lattice, leaving $296$ independent coefficients in degrees $0$ through $295$. Once the candidate rational function is known, the first six production primes certify this finite prefix by the same bounded-coefficient argument used for order seven.

We study the critical-exponent, or equivalently entropy, spectrum arising from free type-preserving actions on the biregular tree $\mathcal T_{r+1,s+1}$. For an action with a nonempty finite quotient core, the critical exponent agrees with both the volume entropy of the universal cover of the core and the topological entropy of the associated non-backtracking edge shift. The unrestricted spectrum is $[0,\frac12\log(rs)]$, whereas the finitely generated spectrum is a countable dense subset of this interval obtained by taking logarithms of Hashimoto spectral radii. We stratify this finite-state spectrum by the circuit rank of the quotient core. At each fixed rank, finitely many typed kernels parametrize all values. Moreover, every positive entropy value has only finitely many kernel–length realizations, up to type-preserving isomorphism, and every positive accumulation point of a fixed-rank spectrum belongs to a lower-rank stratum. At rank two, the corresponding exponential rates admit a complete inverse classification in terms of three explicit polynomial families, together with a finite exact membership test for algebraic-integer inputs.

Delcourt and Postle reduced the Linear Hadwiger Conjecture to coloring $K_t$-minor-free graphs on $O(t\log^4 t)$ vertices. An important theorem in their proof process asserts that every sufficiently dense $K_t$-minor-free graph contains a small, highly connected subgraph. In this paper, we show that such a subgraph can be chosen to be smaller. More precisely, there exists an integer constant $C\geq 1$ such that, for all integers $t\geq 3$ and $k\geq t$, every $K_t$-minor-free graph $G$ with $d(G)\geq Ck$ contains a nonempty $k$-connected subgraph $H$ satisfying $v(H)\leq C^2t\log t$. Thus the structural bound improves from $O(t\log^3 t)$ to $O(t\log t)$. We also give a probabilistic construction showing that the $t\log t$ bound on $v(H)$ is best possible up to a constant factor. Consequently, the graphs occurring in the reduction have order $O(t\log^2 t)$ rather than $O(t\log^4 t)$.

We study new higher-dimensional analogs of graph chordality and review the existing ones. Our main results for simplicial complexes are: (1) $\Delta$ skeleton-E-chordal $\Rightarrow$ $\Delta^\vee$ vertex-decomposable $\Rightarrow$ $\Delta$ skeleton-clique-chordal. Moreover, for subflag complexes, $\Delta$ skeleton-E-chordal $\Longleftrightarrow$ $\Delta^\vee$ vertex-decomposable. (For $d=1$ this boils down to “$G$ chordal $\Longleftrightarrow$ $G^\vee$ vertex-decomposable”, a result closely related to Fröberg's theorem.) (2) For subflag complexes, $\Delta$ is skeleton-E-chordal $\Longleftrightarrow$ it splits as $\Delta = \Delta_1 \cup \Delta_2$, with each $\Delta_i$ a skeleton-E-chordal induced subcomplex of $\Delta$, and with $\Delta_1 \cap \Delta_2$ a complex whose $1$-skeleton is a clique. (This generalizes “$G$ chordal $\Longleftrightarrow$ $G$ splits as a union of chordal graphs that intersect in a common clique”). (3) $\Delta$ skeleton-E-chordal $\Longleftrightarrow$ every nonempty induced subcomplex of $\Delta$ has a skeleton-E-simplicial vertex. (Generalizes “$G$ chordal $\Leftrightarrow$ every nonempty induced subgraph has a simplicial vertex”.) (4) $\Delta$ underclosed $\Rightarrow$ $\Delta$ skeleton-weakly-chordal and weakly-closed. (Generalizes “$G$ interval $\Rightarrow$ $G$ chordal and co-comparability”.) (5) All pure E-chordal complexes are vertex-chordal; all pure mid-chordal complexes are weakly-vertex-chordal; all pure very-weakly-chordal complexes are weakly-ridge-chordal. (This expands Bigdeli, Yazdan-Pour and Zaare-Nahandi's work on ridge-chordality.)

We develop an explicit geometric and algebraic description of shadows in the affine Coxeter complex of type $\tilde{A}_2$, by introducing a coordinate system based on a decomposition of the complex into tunnels and channels. We also provide an algorithm for converting arbitrary reduced words into coordinates. Using this framework, we identify geometric symmetries of shadows and show that they are governed by the underlying channel structure. This allows us to derive explicit, piecewise formulas for the cardinality of shadows in $\tilde{A}_2$, depending on the parity of the coordinates. Furthermore, we establish a simple criterion for shadow membership via a counting function that detects admissible positions within channels. These results provide a concrete and computationally effective description of shadows in affine type $\tilde{A}_2$, bridging the gap between combinatorial definitions and geometric realisations.

The root lattice $A_n$, equipped with its graph distance (equivalently, one half of the ambient $\ell_1$ metric), is isometric to $\mathbb{Z}^n$ with the asymmetric Manhattan metric. We study two extremal problems in this space – the isodiametric problem, i.e., determining the maximum anticode cardinality, and the (non)existence of linear diameter-perfect codes, i.e., lattice tilings by optimal anticodes – and solve them in dimension $3$. We show that, for every integer $D\ge 0$, the largest cardinality of a diameter-$D$ subset of $A_3$ is $\binom{D+3}{3}+(D+1)\lfloor D^2/4\rfloor$, and this value is attained by the balanced difference of two discrete simplices. We then prove an integrality-refined simplex-packing obstruction: a sublattice of $\mathbb{Z}^n$ of asymmetric Manhattan distance greater than $D$ induces a lattice packing by $(D+1)\Delta_n$ in $\mathbb{R}^n$. Combining this observation with the exact lattice-packing density of the tetrahedron yields a complete classification in dimension $3$: lattice diameter-perfect codes in $A_3$ exist precisely for $D=1$ and $D=2$. We also give the equivalent statement for perfect $B_h$ sets of cardinality four. Finally, we formulate a conjecture regarding optimal anticodes in arbitrary dimension, and restate it as an intersection problem for uniform multisets.

We establish new lower bounds for the strength and partition rank of the determinant. For every prime $p$, we prove the exact identity \[ \operatorname{str}(\mathrm{det}_p)=p. \] A weak monotonicity argument, combined with a bound for gaps between consecutive primes, then gives $\operatorname{str}(\mathrm{det}_n)\ge (1-o(1))n^{0.475}$ for sufficiently large $n$. Since the Birch rank of $\mathrm{det}_n$ is always $4$, this gives the first explicit family showing that the dependence on the degree in bounds for strength in terms of Birch rank is unavoidable. Viewing $\mathrm{det}_n$ as an $n$-linear form in its columns, we also prove that its partition rank is at least the largest prime not exceeding $n$. Consequently, \[ n-n^{0.525}\le \operatorname{prk}(\mathrm{det}_n)\le n \] for all sufficiently large $n$, and hence the partition rank of the determinant is $n-o(n)$. The proof introduces an intersection-theoretic method for lower-bounding strength: a short strength decomposition produces a nowhere-vanishing section of a split vector bundle on the complement of the determinantal hypersurface, while a nonzero top Chern class in the Chow ring of $\mathrm{PGL}_n$ obstructs such a section.

Let $D$ be an oriented graph (a digraph with no directed 2-cycles) with maximum degree $\Delta\ge 1$, equipped with nonnegative arc weights of total weight $w(D)$, and let $\mathrm{fas}_w(D)$ denote the minimum weight of a feedback arc set of $D$. Alon (2002) proved $\mathrm{fas}_w(D)\le(\frac{1}{2}-\frac{1}{16\sqrt{2\Delta}})w(D)$. We determine the optimal constant: \[\mathrm{fas}_w(D)\le(\frac{1}{2}-\frac{\sqrt{2}}{6\sqrt{\Delta}})w(D).\] In fact, we show a stronger result: $\mathrm{fas}_w(D)\le\frac{1}{2}w(D)-\frac{\sqrt{2}}{12}\sum_v w_2(v)$, where $w_2(v)$ is the $\ell_2$-norm of the weights of the arcs incident with $v$. Both bounds are attained by the unit-weight directed triangle, so the constant $\sqrt{2}/6$ is best possible (already among unweighted oriented graphs). The proof combines the vertex-peeling scheme of Berger and Shor with a continuous random-ordering analysis: realizing the random order by independent uniform labels renders the expected local imbalance at each vertex exactly an integrated Khintchine-type functional, and the theorem reduces to the sharp evaluation \[\inf_{\|a\|_2=1}\int_0^1 \mathbb{E}|\sum_j a_j B_j(q)|\,dq = \frac{\sqrt{2}}{6},\] where the $B_j(q)$ are i.i.d. Bernoulli$(q)$ random variables, which we prove via Fourier analysis. The proof also yields a randomized, near-linear-time algorithm attaining the bounds in expectation.

Engel, de Gaay Fortman, and Schreieder attach to each prime $\ell$ a minor-closed class $\mathcal{M}_\ell$ of regular matroids, whose excluded minors govern the failure of the integral Hodge conjecture for curve classes on very general principally polarized abelian varieties. For $\ell=2$ the class is the cographic matroids, with excluded minors $M(K_5)$ and $M(K_{3,3})$ by Tutte's theorem; for $\ell=3$ the only excluded minor explicitly identified so far is $M(K_{3,5})$, and the general characterization is [EGFS, Problem 8.8]. We exhibit five new excluded minors for $\mathcal{M}_3$, of ranks 8, 9, 9, 9, and 10; none contains $M(K_{3,5})$ or any of the other four as a minor, so the excluded-minor list of Problem 8.8 has at least six members, with excluded minors at every rank from 7 through 10. Combined with $M(K_7) \notin \mathcal{M}_3$, established by the same authors, and the minor-closedness of $\mathcal{M}_3$, some minor of $M(K_7)$ of rank between 4 and 6 is a further excluded minor, so the list has at least seven members, six of them explicitly identified. The list is structurally diverse: two bipartite apex constructions with automorphism group $S_4$, and three non-bipartite, non-planar rank-9 graphs with no such apex structure. Every verdict is certified by a finite $\mathbb{F}_3$-linear-algebra computation with explicit machine-checkable witnesses, cross-checked by independent implementations; the certificates further show that the five matroids lie outside the larger class $\widetilde{\mathcal{M}}_3$. We also record what are, to our knowledge, the first $\ell=5$ data on this corank-8 slice: $M(K_{3,5})$ and the rank-8 and rank-10 minors lie in $\mathcal{M}_5$ by explicit certificates, and the radical distance $d(M(K_{3,5}))=6$ is determined exactly.

For every $l\geq1$, we construct explicit closed-form coordinate formulas for the local energy functions on the tensor products $B_l\otimes B_l$ of level-$l$ perfect crystals in classical affine types. A single finite-level max-linear formula covers all seven types: its two branches coincide in type $A_n^{(1)}$, yielding a cyclic maximum of partial sums, whereas in the remaining six types each branch is the maximum of finitely many explicit piecewise-linear expressions in barred coordinates, with type-dependent boundary data. We verify the defining local-energy recursion directly on the finite crystals and derive equivalent recursive forms, allowing the energy to be evaluated without applying the combinatorial $R$-matrix. Substitution into the KMN path character formula gives explicit positive coordinate path sums for the characters of all level-$l$ integrable highest weight modules. After principal specialization, the path exponent can be rewritten as a weighted sum of a position-independent adjacent-pair statistic; comparison with the specialized Weyl–Kac character formula yields a uniform family of level-$l$ Rogers–Ramanujan-type identities equating these sums with explicit infinite products. For a representative low-rank case at level two in each family, we display the complete adjacent-pair degree matrix and list the resulting identities.