Papers

We develop a Ramsey theory for leaf-generated subsets of finite products of trees. Our starting point is a Theorem of Furstenberg and Weiss which states that for every $k\geq 1$ and $\alpha>0$, if $A$ is a subset of the leaves of $T_n$, where $T_n$ is the complete binary tree of height $n$, with size $|A|\geq 2^{\alpha n}$, then for $n$ sufficiently large the ancestor closed sub-tree $T_A\subset T_n$ generated by $A$ must contain a copy of $T_k$ such that (1) all vertices in the same level of $T_k$ are mapped into vertices at the same level of $T_A$, (2) if a non-leaf vertex $x\in T_k$ is mapped into a vertex $y \in T_A$, then the two children of $x$ are mapped into descendants of the two children of $y$, and (3) the levels of $T_A$ occupied by the copy of $T_k$ form an arithmetic progression. For the product $T_n \times T_n$ of two binary trees, we show that for every $k\geq 1$ and $\alpha > 1$, for $n$ sufficiently large, every subset $A$ of the leaves of $T_n\times T_n$ of size at least $2^{\alpha n}$ has that its ancestor closure $\Gamma_A\subset T_n \times T_n$ contains similarly structured arithmetic copies of the height $k$ four-ary tree, where the four children of every non-leaf vertex in the source tree are required to map below the four distinct diagonal children of the image vertex, where a diagonal child of a vertex $(x,y) \in T_n \times T_n$ is obtained by moving one level down each in each component. Our results generalise to product of $d$-many finite $b$-ary trees with a sharp critical exponent of \[ \alpha_{\text{crit}}(d,b) =d-1 + \log_b(b-1).\] Geometrically, our results imply that any set $E\subset [0,1)^d$ of upper Minkowski dimension greater than $\alpha_{\text{crit}}(d,b)$ contains arithmetic $b$-adic branching patterns of arbitrary finite order.

We prove that every $(2K_2,K_4)$-free graph is recolorable. Equivalently, for every such graph $G$ and every $\ell\geq \chi(G)+1$, the reconfiguration graph of proper $\ell$-colorings of $G$, in which two colorings are adjacent if they differ on exactly one vertex, is connected. This resolves the final remaining open case in the classification of recolorable $(F_1,F_2)$-free graphs when $F_1$ and $F_2$ have at most four vertices.

Motivated by a word encoding of tree-child networks, we introduce trident tableaux, which are Young tableaux with a unique three-cell column satisfying certain conditions on successors. We construct a bijection between trident tableaux with $n+1$ columns and two-wall tableaux with $n$ columns, that is, two-row fillings with two designated columns in which vertical order is not imposed. The bijection matches the three-part decompositions of the two classes and shows that each class has cardinality $n(n+1)C_n/2$, where $C_n$ is the $n$-th Catalan number. In particular, it gives a trident-tableau interpretation of a shifted form of OEIS A002457. As a consequence, we obtain an exact enumeration of tree-child networks with one reticulation node that contain a trident, and show that their proportion among all tree-child networks with one reticulation node tends to $1/8$ as the number of leaves tends to infinity.

We prove Laugesen and Siudeja's conjecture on Neumann eigenvalues of isosceles triangles. After multiplication by the squared diameter, the first positive symmetric eigenvalue increases strictly with the apex angle $\theta$, while the first antisymmetric eigenvalue decreases strictly up to the equilateral triangle and increases strictly thereafter. \rev{The proof combines a slice-average estimate of directional energies with the explicit equilateral eigenfunction.} A quadratic correction across slices also gives a two-term expansion for every fixed positive spectral index $k$ as $\theta\downarrow0$, with relative correction $\theta^2/6$ and remainder $O_k(\theta^4)$.

We establish moment and mixed-tail inequalities for fixed-order decoupled homogeneous chaoses generated by independent, centered, sparse bounded random variables. Our bounds apply to arbitrary real rectangular coefficient tensors and describe the fluctuation scales through weighted slice and partition norms, with a Bennett-type logarithmic improvement in the largest-entry term. As an application, we derive guarantees for sparse Khatri–Rao embeddings that explicitly account for sparsity and input geometry.

The key part of the current paper is the computation of boundary and Eisenstein cohomology of $GL_4({\mathbb Z})$ with coefficient in any highest weight representations. The method we develop let us compute in an alternative way the cohomology of $SL_3({\mathbb Z})$ and of $GL_3({\mathbb Z})$ with coefficients in any highest weight representation. This is done in a simpler, faster and in a more structured way compared to \cite{BHHM}. We state a duality for the boundary cohomology of $GL_m({\mathbb Z})$ of the type of Serre's duality, where the dualizing sheaf is a power of the determinant representation. We refine this duality to a duality on the level of the spectral sequence for the boundary cohomology $E_\infty^{p,q}$. We state it as a conjecture. However, all the computations, 30 different families of representations, satisfy this conjecture. We compute the Eisenstein cohomology of $GL_4({\mathbb Z})$ with coefficients in the symmetric powers and their twist by the determinant representation. For several other representations, we compute the Eisenstein cohomology, based a few conjectures. Based on those conjectured, one can compute the Eisenstein cohomology in most of the cases. They will be included in the next version of the paper.

We generalize the Bondal–Orlov Reconstruction Theorem to smooth, proper, tame algebraic stacks with generically trivial stabilizer, whose canonical bundles are nowhere torsion: They do not become trivial after pullback along any finite morphism from a projective curve. Our main tools are coherent Tannaka duality as proved by Lurie and by Hall and Rydh, and Grothendieck duality for proper tame stacks as developed by Hall and Priver. We additionally use the notion of nowhere torsion line bundle on a scheme to prove a version of the Bondal–Orlov Reconstruction Theorem for finite type, separated, Gorenstein schemes which are not necessarily proper, which generalizes work of Ballard, Favero, Ito, and Matsui.

Large language models are increasingly used as inexpensive judges to evaluate outputs, label data, and assess whether a system meets a desired quality standard. Yet using AI judgments for formal statistical inference is fundamentally different from simply treating them as ground-truth labels: AI evaluations can be biased or noisy, and rigorous hypothesis testing requires explicit control of type-I and type-II errors. We study how to use AI judgments, together with selective human verification, to conduct a valid hypothesis test at minimum cost. We consider a population of items with hidden binary labels. After choosing a fixed pool of items, the decision maker can selectively query AI, send an item directly to a human, escalate an AI-scored item to a human after observing the AI report, or stop once sufficient evidence has accumulated. We derive an information-theoretic lower bound that captures the minimum cost of achieving prescribed testing errors and characterizes the value of AI information and human verification through a report-dependent information frontier. Motivated by this characterization, we develop SCALE, a sequential cost-aware policy that combines selective AI scoring with adaptive human escalation. SCALE is valid at finite sample sizes and matches the lower bound to first order as the target error probabilities vanish. We further extend the framework to an unknown AI-output model using paired AI-human pilot data. Numerically, SCALE approaches Human-only or AI-only testing when one source clearly dominates, while achieving its largest savings when inexpensive AI judgments and selective human verification are both valuable.

This article focuses on defining the Euler class group of stably free module and analyzes the condition for existence of unimodular element in corresponding projective module.This is an improvisation of work by Banerjee and Das, On a question of Nori:Obstructions,improvements and appliaction. If A is a smooth affine algebra over certain affaine algebras.

How many equilateral points can $\ell_p^n$ contain beyond $n+1$? For every fixed real $p>4$, we construct equilateral sets for which the number of points beyond $n+1$ grows at least linearly with $n$. For every fixed dimension, we prove that the maximum is $n+1$ at all sufficiently large finite $p$, with an explicit threshold. The least dimension admitting any fixed number of extra points is $p^{1+o(1)}$ as $p\to\infty$. The constructions and finite bounds are formalised in Lean. Thus $p=4$ marks the boundary: $n+1$ is exact in every dimension for $2\leq p\leq4$, while at every fixed finite $p>4$ the excess grows at least linearly with dimension.

Jiao and Ren proved that the varieties generated by the matrix semirings $\M_n(S_7)$ form an ascending chain of nonfinitely based varieties, and asked whether consecutive members are equal for all $n\geq2$. We give a negative answer by exhibiting an identity in five variables that holds in $\M_2(S_7)$ and fails in $\M_3(S_7)$. Consequently, $\V(\M_2(S_7))$ is a proper subvariety of $\V(\M_n(S_7))$ for every $n\geq3$. The proof uses the cubic-power property established by Jiao and Ren and a description of products equal to the constant matrix $[a]_n$. The identity forces two distinct indices to exhaust the index set in dimension two; an additional index yields an explicit counterexample in dimension three.

We study complex rank-two pinor fields on the flat Möbius band and the relation between their deck equivariance and Dirac boundary realizations. With explicit Clifford conventions, the two lifts for each Pin type are represented by $R=c\sigma_1$, where $c=\pm1$ for Pin+ and $c=\pm i$ for Pin-. The relevant longitudinal monodromy is $cJ$, with $J\phi(w)=\sigma_1\phi(-w)$, rather than the fiber matrix alone. Its eigenspaces give shifts $0,1/2$ in the positive case and $1/4,3/4$ in the negative case. We identify the resulting geometric Sobolev spaces and construct self-adjoint elliptic realizations of $D=-i(\sigma_1\partial_x+\sigma_2\partial_w)$ with a local boundary condition defined by the longitudinal tangent field. An exact graph estimate gives compact resolvent and Fredholm mappings on nonnegative Sobolev levels. The spectrum consists of a single unpaired transverse branch and paired branches whose longitudinal shifts alternate with transverse parity. Consequently, the kernel and spectral symmetry depend on the lift and boundary condition; the two negative Pin lifts have eta values $1/2$ and $-1/2$ in the stated convention. We also give a secular equation and an exact kernel criterion for the full translation-invariant family of local self-adjoint elliptic boundary lines.

We prove that the Hilbert matrix operator $H:A^2_{\ln^{\alpha}} \to A^2$ is bounded when $\alpha\geq 2$ and that $H$ does not map $A^2_{\ln^{\alpha}} $ into $A^2$ when $\alpha<2$, solving an open problem posed by B. Karapetrović and M. Jevtić. The boundedness when $\alpha=2$ is shown as a corollary to the more general statement: the Hilbert matrix $H:\ell_v^p \to \ell_w^p$ is bounded, where $v(n) = (\ln n)^p / n$ and $w(n) = 1/n$.

Alpöge constructed a complex structure on $S^6$ through a torus fibration with exactly three special fibres. We prove that every member of the associated admissible period family is Oka and that the family projection over each relatively compact parameter disc is an Oka map. The Oka property survives the deletion or simultaneous blow-up of any finite set of distinct points in any member. Blowing up a single point therefore produces an Oka complex manifold diffeomorphic, but not biholomorphic, to $\mathbb{P}^3$. The proofs combine explicit cyclic coverings with analytic extension and the invariance of the Oka property under holomorphic coverings.

We study an optimal control problem for the two-dimensional Navier-Stokes equations projected onto the tangent space of the $\mathbb{H}$-unit sphere manifold $\mathcal{M} := \{\boldsymbol{u} \in \mathbb{H} : \|\boldsymbol{u}\|_{\mathbb{H}}=1\}$. This geometric projection yields a constrained evolution system of finite codimension, ensuring that the state remains on $\mathcal{M}$ for all times. Using a Faedo-Galerkin approximation technique, we establish the global existence and uniqueness of strong solutions to the constrained Navier-Stokes equations with external controls $\boldsymbol{U} \in \mathrm{L}^2(0,T;\mathbb{H})$, together with uniform energy estimates. The associated optimal control problem, minimizing a quadratic cost functional subject to the manifold constraint, is shown to admit at least one optimal pair $(\boldsymbol{u}^*,\boldsymbol{U}^*)$. We derive the first-order necessary optimality conditions by formulating an optimal control system whose state-control pair $(\boldsymbol{u}^*,\boldsymbol{U}^*)$ evolves on the tangent bundle $T\mathcal{M}$ of the constraint manifold $\mathcal{M}$. In this setting, the control satisfies $\boldsymbol{U}^*(t)\in T_{\boldsymbol{u}^*(t)}\mathcal{M}$ for all $t\in[0,T]$, ensuring that the dynamics remain confined to the tangent bundle $T\mathcal{M}$. Furthermore, we establish the Fréchet differentiability of the control-to-state mapping $\boldsymbol{U}\mapsto \boldsymbol{u}_{\boldsymbol{U}}$ in $\mathrm{C}([0,T];\mathbb{H})\cap\mathrm{L}^2(0,T;\mathbb{V})$, with the corresponding linearized state characterized as the unique weak solution of a linearized constrained Navier-Stokes system preserving the tangency condition. These results provide a rigorous analytical foundation for the optimal control of fluid flows constrained to nonlinear manifolds.

We study the Kuramoto–Sivashinsky (KS) equation in one, two, and three spatial dimensions subject to a codimension-one constraint. The constraint preserves the \(L^2\)-norm of the solution and consequently provides a uniform a priori bound that plays a central role in the higher-order energy estimates. In one spatial dimension, we establish global well-posedness by means of a Fourier–Galerkin approximation and compactness arguments. In two and three dimensions, we first prove the existence and uniqueness of local strong solutions using analytic semigroup theory and the Banach fixed-point theorem. In two dimensions, the preserved \(L^2\)-norm, combined with the Agmon inequality and suitable higher-order energy estimates, allows us to control the nonlinear terms and obtain global-in-time a priori bounds. Consequently, we establish the existence and uniqueness of global strong solutions for arbitrary sufficiently regular initial data, without any smallness assumption. This provides a global well-posedness result for the constrained two-dimensional KS equation, in contrast with the corresponding unconstrained problem, for which global well-posedness of strong solutions remains open in general. In three dimensions, although the constraint yields a uniform \(L^2\)-bound, the available interpolation and Agmon estimates are insufficient to close the higher-order energy estimate. This identifies a genuine dimension-dependent obstruction to extending the global regularity argument to three dimensions.

We obtain restrictions on the geometry and topology of minimal equatorial fillings in $\mathbb{S}^3$ of the lowest possible Morse index.

We establish a randomized first-order lower bound for finding an $\epsilon$-stationary point, $\|\nabla f(x)\|\le \epsilon$, of a nonconvex function with initial gap at most $\Delta$, $L_1$-Lipschitz gradient, and $L_2$-Lipschitz Hessian. Each oracle call returns the exact function value and gradient. Let $Q_{\mathrm{rand},\mathrm{FO}}^{\infty}(\epsilon;\Delta,L_1,L_2)$ denote the minimax number of calls, maximized over finite dimensions, for arbitrary adaptive randomized algorithms with per-instance success probability at least $2/3$. In the regime $\epsilon\lesssim L_1^2/L_2$ and $\Delta L_2^{1/2}\epsilon^{-3/2}\gtrsim 1$, we prove $ Q_{\mathrm{rand},\mathrm{FO}}^{\infty}(\epsilon;\Delta,L_1,L_2) \ge c\,\Delta L_1^{1/2}L_2^{1/4}\epsilon^{-7/4}, $ where $c>0$ is an absolute constant. This matches the deterministic restarted accelerated-gradient upper bound of Li and Lin (2023) and extends the sharp deterministic lower bound of Zhou (2026) to unrestricted randomized algorithms. Thus, randomization does not improve the high-dimensional worst-case query rate. The construction keeps quadratic curvature visible while revealing the forcing directions sequentially. Large eigenspaces limit preprocessing, and a scheduled-disclosure coupling makes each fresh direction incur a separate cost.

In this paper, we construct primitive KMS-symmetric quantum Markov semigroups for which the $p$-logarithmic Sobolev constant vanishes at $p=1/2$ and a correlated graph-based example for which the $p$-logarithmic Sobolev constant tends to $-\infty$ as $p\to 0^+$. These examples obstruct a positive uniform reverse-hypercontractivity rate on $(0,1)$. By contrast, we show that every primitive KMS-symmetric quantum Markov semigroup satisfies quantum reverse hypercontractivity for $1/2< p\leq q < 1$ with explicit sufficient times. For each fixed $\delta>0$, a sufficiently strong dephasing perturbation satisfies a uniform positive logarithmic Sobolev inequality on $[\delta, 1]$ and hence quantum reverse hypercontractivity for $\delta< p\leq q<1$ with fixed $0<\delta<1/2$. Furthermore, we establish reverse hypercontractivity for graph-based KMS-symmetric quantum Markov semigroups for $\delta< p\leq q< 1$ with fixed $0<\delta<1$ for connected graphs when $n \geq 3$.

We construct a smooth solution of the focusing cubic nonlinear Schrödinger equation on a square two-dimensional torus which is global in both time directions and concentrates at one point as time tends to positive infinity. Its mass is the mass of the Euclidean ground state and its energy is zero. If $\lambda(t)$ denotes its concentration scale and $L$ the period, then $L/\lambda(t)=2\log t+\frac{11}{2}\log\log t+O(1)$. The solution approaches a phased, periodized ground state with an error tending to zero in every fixed Sobolev space. In particular, its $H^s$ norm is asymptotic to an explicit constant times $(\log t)^s$ for every $s>0$. We also prove uniqueness, up to constant phase and time translation, within the quantitative single-profile asymptotic class used in the construction. The mechanism is the exponentially small interaction of the profile with its periodic images. Exact mass and energy constraints select the concentrating branch, while a positive energy on the moving constrained space controls the terminal-value problem at every fixed regularity order.

In the setting of the previously reported degree-seven counterexample of ani, we present an explicit monic polynomial with real rational coefficients and a direct proof by certified separators. The polynomial has seven simple zeros in the open unit disk. Every continuous path joining distinct zeros in its closed unit sublevel set has image of one-dimensional Hausdorff measure greater than $2+10^{-8}$. Five piecewise-linear graphs exclude twenty of the twenty-one zero pairs; two short forbidden segments force a quantitative detour for the remaining pair. Exact rational Bernstein certificates verify the graph inequalities on forty-four terminal intervals and their unbounded tails. The proof does not require critical-point counting or analytic inverse branches. We separately record the general continuum and dilation arguments that transfer path-length obstructions to Hausdorff measure and to closed sublevel sets. Those transfers are not asserted as new principles. The focus is the explicit real-coefficient construction, its separation certificates, and its verified numerical margin, rather than a first negative answer to the length-two question or a disproof of every larger universal bound.

Let $b_1,\dots,b_d\ge2$ be fixed integers, and let $k$ be their multiplicative rank. We study representations $n=a+b_1^{u_1}\cdots b_d^{u_d}$, where $a$ belongs to a set $\mathcal{A}$ of positive integers and $u_1,\dots,u_d$ are positive integers, counting distinct tuples of exponents separately. Under density and correlation assumptions on $\mathcal{A}$, we obtain a lower bound for the number of integers with many such representations, in terms of $d$ and $k$. In particular, when $\mathcal{A}$ is the set of primes or the set of positive integers representable as a sum of two squares, a positive proportion of the integers $n\le x$ have at least $c_1(\log x)^{d-1}$ or $c_1(\log x)^{d-1/2}$ representations, respectively, for some $c_1>0$ and all sufficiently large $x$. No multiplicative independence or coprimality assumptions on the bases are required.

We study the Kadison–Kastler distance and the interior angle between masas in finite-dimensional matrix algebras. In \(\mathbb{M}_2(\mathbb{C})\), we obtain an explicit formula for the Kadison–Kastler distance between two masas and show that every value in \([0,1]\) is attained by \(\mathrm{d}_{\textrm{KK}}(\Delta,u\Delta u^*)\). We further prove that the distance attains its maximal value precisely when the relative unitary is a Hadamard unitary. In addition, for every $v$ in the groupoid normaliser of $\Delta$, the Kadison–Kastler distance between \(\Delta\) and \(v\Delta v^*\) assumes either its minimal or maximal possible value. We also establish the identity \[ \mathrm{d}_{\textrm{KK}}(\mathcal{A},\mathcal{B})=\sin\alpha(\mathcal{A},\mathcal{B}) \] for any two intermediate subalgebras \(\mathbb{C}\subsetneq\mathcal{A},\mathcal{B}\subsetneq\mathbb{M}_2(\mathbb{C})\). Most notably, for any unitaries \(u,v\in\mathbb{M}_2(\mathbb{C})\), we establish the equivalence of maximal Kadison–Kastler distance, maximal interior angle, maximal Connes–Størmer modified entropy, and minimal Popa probabilistic index: \[ \mathrm{d}_{\textrm{KK}}(u\Delta u^*,v\Delta v^*)=1 \iff h(u\Delta u^*,v\Delta v^*)=\log 2 \iff \alpha(u\Delta u^*,v\Delta v^*)=\frac{\pi}{2} \iff \lambda(u\Delta u^*,v\Delta v^*)=\frac{1}{2}. \] For general \(\mathbb{M}_n(\mathbb{C})\), we prove a unitary invariance property for the interior angle and derive an explicit formula for the angle between arbitrary masas of the form \(u\Delta^{(n)}u^*\) and \(v\Delta^{(n)}v^*\). As a consequence, the interior angle is \(\frac{\pi}{2}\) precisely when the relative unitary \(u^*v\) is a Hadamard unitary. Finally, we show that every value in \(\left[0,\frac{\pi}{2}\right]\) is attained as the interior angle between a pair of masas in \(\mathbb{M}_n(\mathbb{C})\).

We examine the nonlinear initial value problem $\dot{y} = C(1 - y^\alpha)$ modeling polytropic gas permeation. For rational $\alpha$, we derive the exact primitive via cyclotomic partial fraction decomposition, establishing its real trigonometric representation. By synthesizing Ritt's theorem with Rosenlicht's differential algebraic residue criterion, we prove the exact step response $y(t)$ is an elementary function if and only if $\alpha \in \{1, 2\}$. Admitting the Lambert $W$ function extends closed-form solvability strictly to $\alpha \in \{-1, 1/2\}$. Finally, the asymptotic limit $\alpha \to 0$ converges to a singular dynamics solved by the logarithmic integral. These results constitute a complete classification of all explicit closed-form solutions admitted by this system.

For a group $G$ and subsets $S,T \subset G$ we consider a family of mirror di-Cayley graphs $MX(G;S,T)$ and mirror di-Cayley sums graphs $MX^+(G;S,T)$, namely those with $T=\{e\}, S$ or $S \cup \{e\}$. We refer to them indistinctly by $MX^*(G;S,T)$. We can think of $MX^*(G;S,T)$ as two copies of the Cayley (sum) graph $X^*(G,S)$ joined by edges determined by the connection set $T$. Recently, in the work Isospectral Cayley graphs with even and odd spectrum, we study the spectrum of these graphs and several isospectrality problems. Here, we compute the energy and spectral $k$-moments of $MX^*(G;S,T)$ in terms of those of the underlying Cayley graphs $X(G,S)$. Then, we study energetic problems like hypo-, order-, and hyper-energeticity of these graphs. Finally, we give conditions for the existence of equienergetic pairs of non-isomorphic MDCGs.

We address Questions~8.1 and~8.4 of Coron–Ferroni–Li \cite{coron2025chow}. For the first part of Question~8.1, which asks whether every rank-uniform EL-shellable poset admits a rank-uniform EL-labeling, we construct a counterexample of rank $4$ and prove that this rank is minimal. For the second part of Question~8.1, concerning whether every rank-uniform EL-shellable poset is UMEL-shellable, we construct a counterexample of rank $3$. We also develop a general criterion for constructing UMEL-shellable posets of rank $3$. As an application, we give an affirmative answer to Question~8.4 of Coron–Ferroni–Li.

Adaptive arm selection changes the distribution of the observations collected by a bandit algorithm, but it need not change their limiting empirical spectrum. We study Gaussian bandit designs in which the dimension and the number of observations grow proportionally. A quantitative coupling theorem compares the design generated by any causal selection rule with an independent Gaussian design. If the logarithm of the number of available arms is sublinear in the dimension, the empirical spectral distribution converges to the Marchenko-Pastur law, uniformly over the selection rule. Consequently, Gaussian Bayesian bandits have policy-independent first-order limits for posterior mean-square uncertainty, squared posterior covariance, and information acquisition. For linear-score selection, we obtain the exact conditional arm distribution and show that two-arm selection produces an exactly Wishart Gram matrix in every dimension, despite its nonzero conditional mean. For a fixed selection direction, we identify an explicit eigenvalue and eigenvector transition governed by the second moment of a Gaussian maximum. A counterexample shows that a direction's overlap with a reference signal does not determine this transition. Finally, an exponentially large arm pool permits a different bulk limit, establishing the order-sharpness of the arm-growth condition. These results distinguish global spectral stability from directional effects in adaptive bandit data.

Let $\mathcal I$ be an admissible ideal on $\mathbb N$. Zhou, Liu, Liu, and Lin asked whether every regular space with a $\sigma$-hereditarily closure-preserving $\mathcal I$-$sn$-network is $\mathcal I$-$sn$-metrizable. The answer is affirmative for every admissible ideal. The proof rests on a local countability theorem. If $y_n\to_{\mathcal I}x$ and $\{n\mid y_n\ne x\}\in\mathcal I^+$, then every point-discrete family of $\mathcal I$-sequential neighborhoods of $x$ is countable. In a $T_1$ space, this applies in particular to every HCP family. If no $\mathcal I$-convergent sequence to $x$ has $\mathcal I$-positive support away from $x$, then $\{x\}$ is an $\mathcal I$-sequential neighborhood. It follows that every space with a $\sigma$-point-discrete $\mathcal I$-$sn$-network is $\mathcal I$-$snf$-countable. This conclusion, together with the characterization theorems of Ge and of Zhou, Liu, Liu, and Lin, settles Problem 5.2. The countability bound cannot in general be strengthened to finiteness. A countable Rudin–Keisler antichain of free ultrafilters gives an admissible ideal for which local HCP-finiteness fails. Every $P^{-}$ ideal has local HCP-finiteness. Preservation theorems for Fubini products, heterogeneous Fubini sums under cross-row Katětov absorption, and suitable increasing unions extend local HCP-finiteness beyond the $P^{-}$ class. They apply to the finite Fubini powers of $\mathrm{Fin}$, Katětov's first limit-stage ideal $\mathrm{Fin}^{\omega}$ and its finite successors, the tree-derived rank-$\omega$ ideal $\mathcal H_{<\omega}$, and the inductive-limit ideal $\mathrm{Fin}_{\omega}$. Every listed example other than $\mathrm{Fin}$ fails $P^{-}$.

A polyhex is an edge-connected set of n cells of the regular hexagonal tiling, where each cell has area one. We prove that the convex hull of a polyhex has area at most (1/6)*ceiling(n^2 + 14n/3), and we show that some polyhex reaches this bound for every n. This proves a conjecture of Kurz from 2008, which asked for the weaker bound (1/6)*floor(n^2 + 14n/3 + 1). The two bounds differ exactly when 3 divides n. We checked the upper bound in the Lean 4 proof assistant with the Mathlib library. We also report a computation over all polyhexes with at most 12 cells, which shows that for these sizes only one shape reaches the maximum, up to rotation and reflection.

Let $\lambda\succ\gamma\succ\mu$ be equal-degree nonnegative integer exponent vectors with at most $n$ parts, let $J_\eta^{(n)}$ be the labeled symmetric orbit sum associated with $\eta$, and set \begin{equation*} P_n=J_\lambda^{(n)}+J_\mu^{(n)}-2J_\gamma^{(n)}. \end{equation*} For every $n\ge4$, we prove that global nonnegativity of $P_n$ on the positive orthant is equivalent to nonnegativity on the fixed two-variable section \begin{equation*} (x,y,1,\ldots,1),\qquad x,y>0. \end{equation*} Thus the positivity of this family is determined by a fixed two-dimensional section, independently of both degree and ambient dimension. In the injective hierarchy $C_k$ introduced below, the hypothesis is $C_2\ge0$, global positivity is equivalent to $C_{n-1}\ge0$, and the main step is the stable order-raising implication \begin{equation*} C_k\ge0\Longrightarrow C_{k+1}\ge0, \qquad 2\le k\le n-2. \end{equation*} For $n\ge5$, this rank-two hypothesis is strictly weaker than positivity on the normalized full collision wall, which corresponds to $C_{n-2}\ge0$. The result complements the preceding three-variable theorem Collision Positivity for Three-Variable Symmetric Monomial Inequalities: A Complete Boundary Criterion, which proves the exceptional rank-one step $C_1\Rightarrow C_2$. The proof here is logically independent: the local three-label ingredients needed in higher rank are established directly. The main tools are a Green second-compound reserve, fixed-union exchange on adjacent subset ranks, a rank-three zero/one/two-mark contraction, and terminal boundary recombination. The theorem extends to nonnegative rational exponents by clearing denominators.