Papers

Complex hypergraphs (chygraphs) contain graphs, hypergraphs, multiplex and interacting networks as special cases, and the percolation threshold of all of them follows from one symbolic calculation: the spectrum of a tensor $A$ built from four matrices of first moments. That calculation stops at the threshold. Here I show that $A$ is the Jacobian, at its trivial fixed point, of a non-linear self-consistency map whose non-trivial fixed point is the giant component fraction, so threshold and order parameter are two orders of one object on one index set. Carrying the expansion one order further gives the critical amplitude $B$ in $S=B\Lambda+O(\Lambda^2)$ in closed form, and exposes a hierarchy: first moments fix the threshold, second moments the amplitude, and only the generating functions themselves the order parameter away from it. Dropping the assumption that a complex's participation in different layers is independent generalises the tensor again, replacing unconditional first moments by inclusion-biased second moments: distributions with identical marginals have different thresholds and different order parameters. Six constructions from the literature are then solved by substitution, among them AND- against OR-logic hypergraph percolation, which share a chygraph and differ in one generating function, and SIR epidemics with two levels of mixing, whose household reproduction number falls out of the tensor. A reported failure of mean-field theory on strongly clustered graphs is shown to be a failure of applying it to the wrong object: mapped onto the treelike backbone rather than the clustered graph, the same calculation reproduces the exact threshold, order parameter and critical exponents. Everything is implemented in one class, in which a chygraph is specified once by its generating functions and then returns all three, and validated against Monte Carlo simulation.

A Gray code for $S_n$ is genlex when words sharing a suffix are consecutive. We determine the genlex Gray codes for $S_n$ that use the fewest possible operations: Zaks' recursion generalises to a superfactorial family of such codes, and no code outside this family attains the minimum. Every code in the family closes into a cycle, and the cycle is invariant under a group of left translations, cyclic of order $n$ or dihedral of order $2n$ according to an explicit criterion on the operations. In the pancake graph the family reduces to a single member, the classical Zaks order; with respect to the standard parabolic chain of $W(A_{n-1})$, that order attains one extreme of total Coxeter length.

We determine by exhaustive enumeration all main-diagonal symmetric Costas arrays of orders 37 through 42. There are 4 arrays of order 37, none of order 38, 16 of order 39, 2 of order 40, 12 of order 41, and 4 of order 42. All belong to known finite-field constructions. The arrays at orders 37-40 are Lempel arrays over $\mathbb{F}_{41}$ or their corner deletions and augmentations; the twelve arrays at order 41 are the Lempel arrays over $\mathbb{F}_{43}$; and the order-42 census consists of two corner augmentations and a reverse-complement pair in the Rickard-Golomb family. Combined with the published census through order 36, these results show that no sporadic main-diagonal symmetric Costas array occurs at orders 24-42. We also develop a collision model for random involutions. The first two terms of its collision exponent are derived analytically, $n^2/18+n^{3/2}/360$, with an $O(n)$ remainder. The lower-order model and clumping correction are empirical. The six new censuses are consistent with the model's prediction of a rapidly declining expected sporadic population.

We exhibit a binary linear $[50,40]_2$ code of covering radius $2$, so $\ell_2(10,2)\le 50$, one column below the Kaikkonen–Rosendahl length $51$ that has stood since 2003 and that still seeds the $R=2$ family of Davydov–Marcugini–Pambianco (arXiv:2511.02542). The new matrix admits a $(2,0)$-partition into ten blocks, so Construction $\mathrm{QM}_2^2$ propagates it to exhaustively verified codes of lengths $815$ and $1631$ at $r=18$ and $r=20$, and to the family $n=51\cdot 2^{r/2-5}-1$ of asymptotic density $2601/2048$. The matrices, verifiers, and source are at https://github.com/wustep/maths, pin problems/covering/share/2026-08-24/ at commit 736a38f.

In this paper, we establish a gluing formula for the pseudo-determinant of the Laplacian on a simple finite graph. We achieve this by using the gluing formula for the determinant of massive Laplacian and the perturbation theory technique. In addition, we apply this gluing relation to derive a gluing formula for the number of spanning trees and rooted spanning forests on simple finite graphs.

We investigate structural and enumerative properties of binary single parity-check product codes. For each $n\geq 2$, $\operatorname{SPC}(n)$ denotes the binary single parity-check code of length $n$, consisting of all binary vectors of length $n$ having even Hamming weight. We determine the generalized Hamming weight hierarchy of the product code $\mathcal{C}_{m,n}=\operatorname{SPC}(m)\otimes\operatorname{SPC}(n)$, whose codewords can be represented as $m\times n$ binary matrices in which every row and every column has even Hamming weight. For the square product $\mathcal{C}_n =\operatorname{SPC}(n)\otimes\operatorname{SPC}(n)$, we also determine the maximum codeword weight and prove that its homogeneous weight enumerator is symmetric if and only if $n$ is even. After characterizing the dual code, we apply the MacWilliams identity in its Walsh–Hadamard formulation to derive an exact closed-form expression for the weight enumerator. By grouping the auxiliary binary vectors according to their Hamming weights, we obtain an explicit formula for each coefficient in terms of binomial coefficients and alternating convolutions. Finally, using Krawtchouk polynomials, we present an exact procedure for computing the full weight distribution without exhaustively enumerating all codewords. Numerical examples illustrate the formulas and verify the resulting computations.

For a fixed integer $n\ge 2,$ we consider the homogeneous polynomial $$ P(x_1, x_2, \ldots, x_{n+2})=\sum_{i=1}^{n} (x_2-x_1)^{i-1} x_1^{n-i} x_{i+2}. $$ We prove that, for any finite set $A$ of complex numbers, $$ \Bigl|\bigl\{P(x_1,x_2,\ldots,x_{n+2}): \, x_i\in A\bigr\}\Bigr|\gg |A|^{n}. $$ The implicit constant in $\gg$ may depend only on $n.$

We show a birthday paradox for random non-backtracking walk on regular graphs of degree at least $3$: such a walk of length $k$ has high probability of self-intersecting when $k$ is significantly greater than $\sqrt n$, where $n$ is the number of vertices of the graph. This resolves a conjecture of Noga Alon and Yuval Peres for the fixed degree case.

We study the connectedness and enumeration of reconfiguration graphs of valid triangulations of convex polygons whose vertices are cyclically colored with $j \ge 3$ colors, where every triangle has vertices of three pairwise distinct colors. For $j = 3$, we settle a conjectural expectation of Acharya, Mütze, and Verciani: we prove that the twist graph $\mathcal{H}_{3k+2}$ is connected for every $k \ge 4$, whereas $\mathcal{H}_8$ and $\mathcal{H}_{11}$ are disconnected. Using a colored root-edge decomposition that induces Cartesian products in the state space, we obtain coupled recurrences for $T(3k)$ and $T(3k+2)$. The corresponding generating functions reduce to the equation $U(x) = 1 + xU(x)^4$, and the difference between the two consecutive families is given by the Raney number $T(3k+3) - T(3k+2) = R_{4,5}(k-1)$. For $j \ge 4$, reconfiguration is performed by validity-preserving diagonal flips. We extend the root-edge decomposition to all admissible classes $N \not\equiv 1 \pmod{j}$, obtaining, for each fixed $j$, a finite algebraic system of functional equations. We further prove that the flip graph $\mathcal{G}_N^{(j)}$ is connected whenever valid triangulations exist. Thus, the root-edge decomposition provides a unified structural framework for the enumeration and reconfiguration of cyclically colored triangulations.

We study the positivity of the tropical multidegrees of a tropical variety contained in a product of real vector spaces. These multidegrees are obtained by stably intersecting the tropical variety with pullbacks of positive tropical divisors. We introduce projection-purity and facet-selectability, two conditions under which positivity is determined by the dimensions of the natural projections, and the support of the tropical multidegrees is precisely the set of lattice points of a polymatroid base polytope. This extends He's theorem for translation-admissible tropical varieties. We also show that these conditions alone do not force the corresponding tropical volume polynomial to be Lorentzian. By contrast, for the augmented Bergman fan of any polymatroid, the positive multidegrees are supported precisely on the lattice points of the polymatroid base polytope, and the tropical volume polynomial is Lorentzian for every sequence of positive tropical divisors.

In this paper, we investigate the neighbourhood convexity ($n$-convexity) on graphs, a new finite convexity space grounded in the common closed neighbourhood closure operator. Unlike standard path-based graph convexities, $n$-convexity shows a non-canonical behaviour, giving rise to compelling structural properties and being almost never hereditary. Focusing on the properties of graphs that form $n$-convex geometries, a parity distinction emerges: an $n$-convex geometry contains a star vertex if and only if the number of its vertices is odd. Every odd-order $n$-convex geometry can be uniquely constructed by attaching a star vertex to an even-order one. We introduce the concept of quasi-stars (vertices of degree $\vert{}V\vert{}-2$) and prove a reduction property that allows systematically reducing an $n$-convex geometry by removing a pair of vertices, one of which is a quasi-star. Finally, we explore the connections between $n$-convexity and $P(G)$, the neighbourhood preorder, demonstrating that $n$-convex sets are upsets of $P(G)$ and that, in star-free $n$-convex geometries, quasi-stars correspond precisely to the maximal elements of $P(G)$. We complete our study by classifying quasi-threshold and threshold $n$-convex geometries.

Let $s_2(n)$ be the binary sum-of-digits function and let $c_t$ be the natural density of the integers $n\ge0$ for which $s_2(n+t)\ge s_2(n)$. Earlier work of the author proved the universal exponential bound $$c_t-\frac12\ge 2^{-2s_2(t)-1},$$ thereby resolving Cusick's conjecture for every $t$. This estimate, however, does not reflect the true size of the smallest possible bias at a given large Hamming weight. In this paper, we determine this extremal scale sharply: $$\inf_{s_2(t)=k}\left(c_t-\frac12\right) \sim \frac{1}{2\sqrt\pi} \left(\frac{\log_2 k}{k}\right)^{3/2} \qquad(k\to\infty).$$ Thus the optimal fixed-weight gap is polynomial-logarithmic rather than exponential, with the explicit sharp leading constant $1/(2\sqrt\pi)$. The proof combines the five-cumulant Edgeworth expansion of Spiegelhofer and Wallner with a new extremal rigidity mechanism for near-extremal binary block patterns. We also prove a stability theorem for asymptotic extremizers and give a separate shadow-energy interpretation of the same constant.

A celebrated result of Bayer and Stillman from 1987 states that for a homogeneous ideal $I$ of a polynomial ring $S$, the regularities of $S/I$ and $S/\textrm{GIN}(I)$ are the same under the reverse lexicographic monomial ordering, where $\textrm{GIN}(I)$ is the generic initial ideal. If $R$ is a polynomial ring whose variables are subdivided into disjoint blocks of variables $X_1,\dots,X_c$, there is a natural multi-grading on $R$, and one can analogously define a multi-graded version of the generic initial ideal for any multi-homogeneous ideal $I$ of $R$. However, the full strength of the Bayer–Stillman Theorem fails in the multi-graded setting; there are multi-homogeneous ideals $I$ such that the regularities are not preserved after passing to the multi-graded generic initial ideal no matter the choice of monomial ordering. We prove lower bounds on the regularity of $R/I$ in terms of almost regular sequences of the multi-graded generic initial ideal of $I$ restricted to each block of variables. Again, we use the reverse lexicographic monomial ordering, but interestingly, the lower bound result requires a particular choice of ordering on the variables. As an application, we prove the optimal fractional Helly theorem for $d$-Leray simplicial complexes, a problem stemming from the work of Kim in 2017.

For an $F$-free graph $G$, a non-edge is $F$-saturating if adding it to $G$ creates a copy of $F$. We denote by $f_{p+1}(n,m)$ the minimum number of $K_{p+1}$-saturating non-edges in a $K_{p+1}$-free $n$-vertex graph with $m$ edges. Erdős and Tuza conjectured that $f_4\left(n,\mathrm{ex}(n,K_3)+ 1\right)= (1 + o(1)) \frac{n^2}{16}$. Balogh and Liu (JCTB, 2014) disproved this conjecture and determined the asymptotic value of $f_4(n,\mathrm{ex}(n,K_3)+1)$. He, Ma, Ma and Ye (JCTB, 2023) later determined $f_{p+1}(n,\mathrm{ex}(n,K_p)+1)$ asymptotically for every $p\ge 3$, and asked for the value of $f_{p+1}(n,m)$ for all $\mathrm{ex}(n,K_p)+1\le m\le \mathrm{ex}(n,K_{p+1})$ and every $p\ge 3$. In this paper, we answer their question asymptotically for all $\mathrm{ex}(n,K_p)+1\le m\le \mathrm{ex}(n,K_{p+1})$ and every $p\ge 3$. We also determine the exact value of $f_3(n,m)$ for all $0\le m\le \mathrm{ex}(n,K_3)$ by a different method.

We prove that Pyber's upper bound $2^{O(n\log(n))}$ for the number of minimally transitive subgroups of $S_n$ is best possible along the powers of every fixed prime, even when the groups are counted up to permutational isomorphism. As a byproduct, our construction shows that, along the powers of every fixed prime, the maximum order of a minimally transitive permutation group of degree $n$ is $2^{\Theta(n)}$. For completeness, we also present Pyber's previously unpublished proof of his upper bound. We further deduce that the numbers of labelled vertex-transitive graphs and digraphs of order $n$ are both $2^{\Theta(n\log(n))}$, and discuss the implications of our results for approaches to the McKay–Praeger conjecture.

As a continuation of a previous study of isolation numbers in Cartesian and lexicographic products, we investigate isolation numbers and, more generally, $\cal F$-isolation numbers in direct, strong, lexicographic, and Cartesian products of graphs. For direct products, we derive upper bounds for the $\{K_{n_1,\ldots,n_d}\}$-isolation number in terms of isolation and total domination parameters of the factors, and establish lower bounds based on open packings. We also determine exact values for several infinite families of direct products, including $\iota(P_{4\ell}\times C_{2k+1})=\ell(k+1)$. For strong products, we prove a general lower bound on $\iota(G\,\boxtimes\, H,{\cal F})$ involving the $2$-packing number and provide an upper bound on $\iota(G\,\boxtimes\, H)$. For lexicographic products, we determine the $\cal F$-isolation number in several general settings, obtaining exact formulas in terms of domination and total domination numbers of the first factor. Finally, for Cartesian products, we extend results from our previous work to arbitrary graph families $\cal F$. We introduce $\cal F$-isolation graphs and use $\cal F$-transversals to derive general upper bounds, together with corresponding lower bounds.

For a fixed tree $T$ on $t$ vertices, let $\ex(n,K_r,T)$ denote the maximum number of $r$-cliques in an $n$-vertex $T$-free graph. Write $n=\alpha(t-1)+\beta$ with $0\le\beta<t-1$. A conjecture proposed in extremal-graph form by Alon and Shikhelman~\cite{alon2016}, later stated explicitly as an exact clique-counting formula by Gerbner and Palmer~\cite{GP2026}, asserts that \[ \ex(n,K_r,T)=\alpha\binom{t-1}{r}+\binom\beta r, \] with the extremal construction given by the vertex disjoint union of $\alpha$ copies of $K_{t-1}$ and a graph with $\beta$ vertices. This paper proves this conjecture for every $n$ and for every tree $T$ whenever $C\log t\le r\le t-1$ and $t$ is sufficiently large, where $C$ is an absolute constant. This lowers the previously known general threshold $r\ge\lceil(2t-1)/3\rceil$, which is linear in $t$ to logarithmic order. The paper also proves the conjecture for every $3\le r\le t-1$ when at least $\lceil(t-3)/2\rceil$ leaves of $T$ have a common neighbor, and this extends the known results for stars and double stars.

Quadratic ball discrepancy measures how uniformly a code meets Hamming balls over all centers and radii. Perfect codes minimize this quantity among codes of the same cardinality. We prove a quantitative refinement: at the parameters of every nontrivial perfect code, excess discrepancy is at least a positive multiple of the discrepancy at the correction radius. The latter equals the squared deviation of the covering multiplicity from one, divided by the square of the code cardinality. The comparison applies to all codes of that cardinality. For one-error parameters, the coefficient is at least one, which is sharp uniformly over the parameters; explicit coefficients depending on the parameters can be much larger. For two-error parameters of length at least five, we obtain a positive coefficient against an algebraic benchmark under the sphere-packing and Lloyd integrality conditions, even when existence of a perfect code is unknown. The multiplicity defect also measures the failure of uniform ball noise to smooth a code distribution. Consequently, the same discrepancy excess gives explicit bounds on holes, overlaps, and divergence from the uniform distribution.

We introduce stable symmetric series which encode normalized conjugacy classes and their multiplication operators simultaneously for all symmetric groups. This gives a direct route from the Ivanov–Kerov algebra to shifted symmetric functions and to differential operators in $U(\mathcal W_{1+\infty})$. Using the Goulden–Jackson product, we extend the construction to Jack polynomials, recover shifted Jack eigenvalues and Pieri-type relations, and obtain explicit candidate operators in degrees three and four. Their real and quaternionic specializations to zonal polynomials are verified by Gaussian matrix integrals and exhaustive Wick enumeration.

Graph burning is a round-based process which can be viewed as a discrete one-player game that models the spread of influence throughout a network. Extending this process to hypergraphs can be done in numerous ways; two such processes that have been studied include the burning process and the lazy burning process for hypergraphs. Combinatorial designs can be thought of as hypergraphs with useful and interesting characteristics. In this paper we explore the burning and lazy burning processes in Steiner triple systems (STSs). We obtain logarithmic bounds on the burning number of an arbitrary STS and prove the existence of an STS with burning number $\rho$ for any integer $\rho \geq 3$. We observe that the concept of lazy burning is equivalent to the notion of the dimension of an STS, and we consider ramifications of this equivalence. We also show that the difference between the burning and lazy burning number of a Steiner triple system can be arbitrarily large. Finally, we consider burning and lazy burning numbers of affine and projective triple systems in detail.

For a set of $n$ alternatives, the best–worst choice polytope is the convex hull of the deterministic best–worst choice vectors induced by all linear rankings of the alternatives. The question of determining the dimension of this polytope as a function of $n$ has been raised by Doignon \cite{Doignon2023}. In this paper we answer this question by finding an explicit formula for this dimension. Our proof is motivated by a representation-theoretic interpretation of the relevant rank problem for the symmetric group, but is presented entirely in elementary linear-algebraic terms. The connection with representation theory will be explored elsewhere.

We study multipartite hypergraphs definable in $1$-semi-equational theories and prove almost-linear Zarankiewicz bounds in every fixed arity $r\geq2$. If $T$ is a $1$-semi-equational theory, then, for every formula $\varphi$ and fixed $t,r\geq2$, there is a constant $c$ such that each $K_{t,\ldots,t}$-free $r$-partite hypergraph defined by $\varphi$ on $n$ vertices has $O_{T,\varphi,t,r}\!\left( n^{r-1}(1+\log(1+n))^c \right) $ edges. Put $\alpha_k=\min\{k-1,2\}$. In the bipartite case, a Boolean combination of $m$ $(k,1)$-semi-equations has $ O_{k,t,m}\!\left( n(1+\log(1+n))^{(m-1)\alpha_k} \right) $ edges whenever it is $K_{t,t}$-free. In particular, a relation defined by one $(k,1)$-semi-equation or its negation has a linear bound. The proofs combine incidence estimates for indexed set systems with low-crossing orderings of finite $k$-wise laminar families. Consequently, no $1$-semi-equational theory locally trace-defines an infinite domain.

In this paper, we focus on left-compressed families \(\mathcal{F}(\mathcal{G})\) generated by a family of sets \(\mathcal{G}\). We establish a necessary and sufficient condition on \(\mathcal{G}\) under which \(\mathcal{F}(\mathcal{G})\) is intersecting. We then introduce a construction of maximal left-compressed intersecting families \(\mathcal{F}_G\) from a single generator by adding suitable sets not containing \(1\) to \(F(G)\). From this, we derive several estimates for the size of \(\mathcal{F}_G\).

Ellingsrud, Göttsche, and Lehn proved a remarkable universality result for tautological integrals on Hilbert schemes of points on surfaces. We prove a relative form: over a base $B$, the tautological pushforwards are governed by universal power series paired with relative $\kappa$-classes. As applications, we focus on Segre classes on families of curves and surfaces. For curve families, we determine all universal coefficients for total Segre pushforwards. These formulas answer a question of Oprea–Pandharipande, and the resulting recursions are surprisingly related to monotone Hurwitz numbers. For surface families, motivated by Marian–Oprea–Pandharipande's work on Lehn's conjecture, we obtain a codimension-one relative form of the conjecture, with explicit formulas for every universal series.

Phylogenetic trees are used in evolutionary biology to represent the evolutionary history of a collection of taxa. As we have incomplete information about any evolutionary history, recovering trees from partial information is a focus of phylogenetic combinatorics. However, in some cases the available data does not describe a single phylogenetic tree. We consider recovery of pairs of phylogenetic trees from their combined subtrees with $k$ leaves, which we call a $k$-bucket. We establish the exact cases in which these pairs of trees are recoverable from their subtrees with a single leaf removed, both when just considering the structure of the trees, and when additionally considering the set of taxa on the leaves. We also consider recovery of pairs of trees with labelled leaves from their rooted triples, and establish that they are recoverable up to a sequence of subtree swaps.

The game of Cops and Robbers is a classical pursuit–evasion game on graphs. For a graph $G$, the cop number $c(G)$ is the minimum number of cops needed to guarantee the capture of a robber on $G$. Although this parameter has been determined for several fundamental graph classes, comparatively few exact results are known for partial cubes and their subclasses. We first establish an upper bound for every finite median graph $M$ in terms of its tree-dimension, which improves Crawford and Iršič Chenoweth's bound significantly. This result refines the previous upper bound expressed in terms of a hypercube embedding dimension and can give a substantially smaller estimate. Then we investigate the cop numbers of simplex graphs—a subclass of partial cubes. For a finite graph $G$, the simplex graph $S(G)$ has the cliques of $G$, including the empty clique, as its vertices, with two cliques adjacent whenever they differ in exactly one vertex. We establish a general lower bound for $c(S(G))$ in terms of the clique number of $G$ and a general upper bound in terms of its chromatic number. Finally, as direct applications, we determine the exact values of cop numbers of some special simplex graphs—bipartite wheels, Fibonacci and Lucas cubes.

We prove the 3-Decomposition Conjecture: every finite connected cubic loopless multigraph decomposes into a spanning tree, a 2-regular subgraph, and a matching. The proof rests on a new theorem on matching complements in subcubic graphs. Let H be a finite connected bridgeless simple graph of maximum degree three, and let S be its set of degree-two vertices, with |S| = k >= 2. We show that H has a matching of size equal to its cyclomatic number, |E(H)| - |V(H)| + 1, whose deletion leaves a single tree containing all of S, together with cycles disjoint from S. The proof is by induction on k, using an alternating-path exchange that stops at the first entry into the growing tree component.

A graph $H$ is $K_{r+1}$-Turán-good if, for every sufficiently large $n$, the Turán graph $T_r(n)$ maximizes the number of copies of $H$ among all $n$-vertex $K_{r+1}$-free graphs, and it is strictly $K_{r+1}$-Turán-good when this extremal graph is unique. Morrison, Nir, Norin, Rzążewski and Wesolek proved that for every graph $H$ with at least one edge, when $r\ge 300v(H)^9$, $H$ is $K_{r+1}$-Turán-good. They asked whether the above bound could be reduced to quadratic order in $v(H)$. In this paper, we answer their question affirmatively, and give a stronger result. We prove that there is an absolute constant $C>0$ such that every graph $H$ with at least one edge is strictly $K_{r+1}$-Turán-good whenever $r\ge Cv(H)$. Our proof uses a substantially different cluster method based on polymer models. Besides proving the vertex-linear Turán-good threshold, this method appears to have further applications. As one illustration, we prove that for every graph $H$ with at least one edge, the normalized chromatic polynomial $P_H(x)/x^{v(H)}$ is strictly increasing for every real $x\ge C\Delta(H)$, which strengthens the previously best result where $x\ge C\Delta(H)^{3/2}$. This proves a conjecture of Fadnavis.

Let $F_k(G)$ be the $k$-token graph of a finite graph with nonnegative edge weights, and let $A_k$ and $D_k$ be its weighted adjacency and degree matrices. For every $-1\leq\vartheta\leq1$, we prove that $k\mapsto\lambda_{\max}(A_k+\vartheta D_k)$ is discretely concave. Complement symmetry then makes this sequence nondecreasing up to the middle level. At $\vartheta=1$ and $\vartheta=0$, this gives the signless-Laplacian and adjacency spectral-radius monotonicity conjectures of Apte, Parekh, and Sud. The spectral result follows from a finite-time theorem: for every $t\geq0$, the heat contents $\binom{n}{k}^{-1}\mathbf{1}^{*}e^{t(A_k+\vartheta D_k)}\mathbf{1}$ are log-concave in $k$. We encode all token levels in one Lorentzian polynomial. A four-variable operator symbol proves preservation by each edge heat gate, and the Lie-Trotter formula passes this preservation to the full semigroup. Large-time growth rates recover the top eigenvalues. The same construction also yields Lorentzian polynomials from top spectral projections. Finally, we show that the local symbol certifies exactly the parameter range $[-1,1]$.

Recently, a new record kissing arrangement of $841$ points in $\mathbb R^{12}$ was obtained numerically by optimization (Takhanov-Assylbekov-Yun, 2026). The configuration was released as a coordinate file, without a mathematical description of its structure. The purpose of this paper is to provide such a description. The key observation is that the geometry becomes transparent once we regard $\mathbb R^{12}\cong \mathbb H^3$ as the Cartesian product of three copies of the quaternion algebra. We first introduce a new $840$-point kissing arrangement with a certain quaternionic structure. It consists of three mutually orthogonal regular $24$-cells, supported on the three quaternionic coordinate factors $\mathbb H\times\{0\}\times\{0\}$, $\{0\}\times\mathbb H\times\{0\}$, $\{0\}\times\{0\}\times\mathbb H$, together with two $384$-point families obtained by lifting affine sets of the form $$\{(u,v,w)\in (\mathbb F_2^2)^3\mid u+v+w=\eta\},$$ to quaternionic triples (whose components belong to the binary octahedral group $2O$) and then applying suitable component-wise rotations and weightings. A characteristic feature of this construction is a pronounced asymmetry among the three quaternionic factors. For the $816$ vectors obtained after removing the third $24$-cell, most of the squared norm is concentrated in the first two quaternionic coordinates, while the third coordinate carries systematically less mass. Thus, the third four-dimensional factor contains more available space than the first two. We then show that this $840$-point configuration provides a natural structural model for the numerical $841$-point record. Finally, we introduce a notion of the general quaternionic construction in dimensions divisible by $4$, and check that record kissing arrangements in ${\mathbb R}^{4k}$, $k\leq 5$, admit a quaternionic construction.