Papers

The repeated assembly of the sparse Jacobian matrix constitutes a primary computational bottleneck in the Newton-Raphson power-flow analysis. Through an algebraic equivariance analysis of the Jacobian matrix, this paper proposes a direct sparse Jacobian assembly method based on symbolic permutation. By pre-applying the permutation to the admittance matrix during a one-time symbolic phase, the runtime bus-type projection overhead is entirely eliminated. Consequently, complex Jacobian differentiation and global matrix assembly fuse into a single operation, yielding a single-pass, branchless algorithm that streams values directly into contiguous memory. The framework is realized in the open-source RustPower solver. Benchmarks on IEEE test systems and PEGASE 9241 demonstrate that the proposed assembly is 24.4-30.1x faster than the MATPOWER- style baseline and 6.0-7.8x faster than the Schafer-Braun optimized assembly. End-to-end, the grid-level solve time is 11.7x-154.3x faster than pandapower 3.5.4 with Numba, while leaving numerical results and convergence behavior unchanged. Compared with the native C++ LightSim2Grid 1.0.0, grid-level performance is comparable on IEEE 39 and 1.3x-1.8x faster on IEEE 118 and the PEGASE systems. The method extends directly to general problems where a sparse Jacobian inherits its structure from a static network operator, including transient stability analysis, EMT simulation, and OPF Hessian assembly.

A tropical quantum field theory is an exactly solvable deformation limit of an ordinary QFT, governed by a non-linear partial differential equation. We use the integrable two-dimensional sine-Gordon theory as a testbench to compare the exact features of an undeformed model with its tropical counterpart. We classify scale-invariant tropical QFTs in $D=2$ and leverage functional renormalization group methods to demonstrate that all scale-invariant theories with periodic effective actions are infrared fixed points of flows originating from sine-Gordon Lagrangians. For the tropical sine-Gordon model, we compute the $\beta$-functions exactly, solve the renormalized theory both via a convergent expansion and numerically for all admissible couplings, and illustrate the complete renormalization group flow from the Gaussian to the IR fixed point. Finally, we study the large-order behaviour of the loop expansion and relate it to the Hamilton-Jacobi equation, whose solutions we interpret as tropical instantons.

Nested integrals $\int f\left(\int g(\boldsymbol{y},\boldsymbol{x})\mathrm{d}{}\boldsymbol{x}\right)\mathrm{d}{}\boldsymbol{y}$ where $\boldsymbol{y}\in[0,1]^{d_1}$ and $\boldsymbol{x}\in[0,1]^{d_2}$ frequently arise in Bayesian experimental design, medical decision making, and computational finance. Statistical estimators for nested integrals are more computationally demanding than for standard integrals. Recent studies demonstrate that the median randomized quasi-Monte Carlo (mRQMC) method achieves super-polynomial convergence rates for analytic integrands. We extend mRQMC methods to nested integrations and propose a novel double-loop mRQMC estimator with super-polynomial convergence rate if both the inner and outer integrands are analytic. Established estimators for nested integrals, such as double-loop or multilevel Monte Carlo, are by contrast limited to algebraic convergence rates. However, expected information gain (EIG) applications from experimental design typically do not fulfill this condition due to a lack of regularity in the outer integrand. We address this through a theoretically-grounded importance sampling regularization scheme. Although super-polynomial convergence rates remain achievable for the inner integral, only the algebraic rate $\alpha+3/2-\epsilon$ for any $\epsilon>0$ can be theoretically guaranteed for the outer integral in the EIG, where $\alpha\in\mathbb{N}\cup\{0\}$ denotes the order of the mixed derivatives of the regularized outer integrand. Our approach shows substantial improvements over double-loop randomized quasi-Monte Carlo methods based on the sample mean in moderately high-dimensional settings for a simple polynomial example. Moreover, we demonstrate improved convergence behavior of the median-based estimator for the more challenging setting of EIG estimation.

We study how interpolation affects local steps and parallel sampling in monotone variational inequalities (VIs), with $m$ workers, $K$ oracle calls per worker per round, and $N$ rounds. Under the strong growth condition (SGC), we determine the minimax rate for affine VIs, attained by local extragradient with worker averaging. For nonlinear monotone VIs and convex-concave saddle problems with Lipschitz operators satisfying SGC, we improve the stochastic terms of bounded-variance expected-gap guarantees from $(KN)^{-1/2}$ to $(KN)^{-1}$ with local steps and from $(mKN)^{-1/2}$ to $(mKN)^{-1}$ with minibatches. Under strong monotonicity, we improve the noise-dependent condition-number scaling of existing extragradient bounds from quadratic to linear. The resulting monotone-gap and strongly monotone accuracy bounds match lower bounds up to logarithmic factors in $m$. Under generalized smoothness and SGC, adaptive steps give polynomial dependence on the initial radius. For affine VIs with globally bounded oracle variance, we also attain the minimax rate: the deterministic rate over $KN$ sequential calls plus the statistical rate over all $mKN$ samples. For composite VIs with smooth convex sample interpolation and an exactly evaluated monotone Lipschitz component, accelerated mirror-prox sliding achieves an $N^{-2}$ gradient term, a $(KN)^{-1}$ monotone term, and a $(mKN)^{-1}$ sampling term, without assuming SGC.

A proper coloring of vertices of a graph $G$ is called $t$-frugal if no color appears more than $t$ times in the neighborhood of any vertex of $G$. While $1$-frugal coloring coincides with $2$-distance coloring, we study $2$-frugal colorings of planar graphs with maximum degree $4$ and $5$, respectively. We prove that every planar graph with maximum degree $4$ admits a $2$-frugal coloring with $7$ colors, while a planar graph with maximum degree $5$ admits a $2$-frugal coloring with $10$ colors. The results present variations of parts of the famous Wegner's conjecture, which for planar graphs with maximum degree $\Delta\in\{4,\ldots,7\}$ states that their $2$-distance chromatic number is at most $\Delta+5$.

We compute the first homology of the unordered configuration space of $n$ open unit-squares in the $w$ by $h$ rectangle for all positive integers $n$, $w$, and $h$. Letting $w$ and $h$ vary, these square configuration spaces form a bi-filtration of the unordered configuration space of $n$ points in the plane, so their fundamental groups approximate the $n$-strand braid group. Using a pair of spectral sequences, we prove that these square and point configuration spaces have naturally isomorphic first homology groups if $w,h\ge 3$ and $n\le wh-3$. In the $h=2$ case, we show that the first homology groups of these square configuration spaces are free of rank $n-1$ for $n\le 2w-2$, whereas they are free of rank $w-1$ for $n=2w-1$.

The Tu-Deng Conjecture, made in 2011, asserts that for $k\geq2$ and $1\leq t<2^k-1$, the set \[ S_{t,k}=\left\{(a,b)\in\{0,1,\ldots,2^k-2\}^2: a+b\equiv t\pmod{2^k-1},\quad \wt(a)+\wt(b)<k\right\} \] satisfies \[ |S_{t,k}|\leq 2^{k-1}. \] Proofs of the conjecture were given independently by the author and by Liu, Luo and Xie in arXiv in 2026. Let $r=\wt(t)$, and let $z=k-r$ be the number of zeros in the $k$-bit expansion of $t$. We prove that equality holds in the Tu-Deng Conjecture if and only if every cyclic distance between consecutive zeros is at least $z$. We give several other simple necessary and sufficient conditions, including a forbidden-pattern condition, a packing condition for the zero positions, a composition condition for the gaps, and equivalent conditions involving the coefficients of the polynomial $H_W(u,v)$ in the proof of the Tu-Deng conjecture. We also count the equality words having a prescribed number of zeros.

The paper studies $l^p$-wedge sums of metric spaces and provides a two-sided estimate for the pointed Gromov-Hausdorff distance between them. Using an upper bound for the Gromov-Hausdorff distance, the existence of an isometric embedding of $l^\infty$ into the Gromov-Hausdorff space of all non-empty metric spaces is proved.

We give a reduction from a smooth-domain theorem excluding interior critical points of first positive Neumann eigenfunctions to a non-strict hot-spots conclusion on arbitrary bounded simply connected planar domains with compact Neumann resolvent and simple first positive eigenvalue. The main analytic step is spectral convergence under exhaustion by conformal images of concentric disks. A reflection extension, conformal invariance of Dirichlet energy, and compactness of the Sobolev embedding give a direct proof. A truncation argument justifies square integrability of the transferred extensions without any boundary regularity. The smooth-domain theorem proved by OpenAI is an external input; the approximation argument is proved here in full. The paper was generated by ChatGPT with prompts from the listed author.

For a positive integer $d$, let $\Delta_d^L$ denote the supremum of the densities of lattice sphere packings in $\mathbb R^d$. We prove that, for every sufficiently large $d$, $\Delta_d^L \gg d^2\log\log d\,2^{-d}$. Abuya-Gargava-Zhao obtained the same bound along a sparse sequence of dimensions by combining Venkatesh's cyclotomic symmetries with Klartag's stochastic ellipsoid method. We introduce a multi-cyclotomic gluing construction that allows this argument to work in every sufficiently large dimension.

Physics foundation models are an increasingly popular approach for building versatile emulators for spatiotemporal systems. They require tokenisation layers that lift physical field data into a latent representation suitable for dynamic processing. A typical choice for this layer is a convolutional encoder, which lifts field information in a locality-aware way and can be trained very efficiently. However, this choice cannot be applied to a large class of scientific computing problems where data are represented on unstructured meshes, for example simulations on domains with curved boundaries. Such data typically require graph neural network architectures, which are more expensive to train and, in the form currently used, encode information differently from their CNN counterparts, thus limiting the portability of physics emulators between these data regimes. In this work we introduce Mesh-Native Convolution (MNC), a locality-aware tokenisation layer that encodes unstructured mesh-based data in a way that is analogous to CNNs. This approach leads to models that can be trained once on mesh data and then deployed to arbitrary unstructured meshes, even if these are irregular, with minimal retraining. We show that this tokenisation approach can be used to retrofit grid-pretrained foundation models efficiently on unstructured meshes and provide theoretical results to confirm the consistency of this approach with classical CNN encoders. We demonstrate the favourable performance of this retrofitting approach, which in many cases reaches acceptable accuracy even in zero-shot deployment, and outperforms existing baselines after only a few epochs of fine-tuning, thus providing an efficient bridge between the two data regimes.

The linear arrangement of a simple graph $G = (V,E)$ is a bijective labeling $\pi$ of the vertices of $G$ to the set $\{1, \ldots, n\}$. The maximum linear arrangement ($\texttt{maxLA}$) problem is to determine whether $G$ has a labeling such that the sum of labels across $G$'s edges, $\sum_{uv \in E} |\pi(u) - \pi(v)|$, is at least a given value $k$. Minimum linear arrangement ($\texttt{minLA}$) is defined analogously. Both $\texttt{maxLA}$ and $\texttt{minLA}$ are known to be NP complete for general graphs. $\texttt{minLA}$ has shown to be NP complete for bipartite graphs by Evans and Shiloah in 1975. It has been an open problem whether the same holds for $\texttt{maxLA}$. In this paper we prove that $\texttt{maxLA}$ is NP complete for bipartite graphs.

We completely resolve Question~5.3 of Kania's paper \cite{Kania} and establish a more general realization theorem. We show that for infinite cardinals $\mu\leqslant\nu<\kappa<\lambda$, a space with \[ \pc(X)=\mu,\qquad\pcl(X)=\nu,\qquad \pcc(X)=\kappa,\qquad\pcd(X)=\lambda \] exists if and only if $\kappa\leqslant2^\mu$. The witnesses may be zero-dimensional and Tychonoff spaces, of the least possible cardinality $\lambda$. They may have the least possible number of non-isolated points, or alternatively be crowded. The main ingredient is an exact computation of the centered local $\pi$-character of star filter spaces in terms of ultrafilter character, obtained from a finite selection lemma for filter pushforwards. We also construct a single space with countable $r$-linked local $\pi$-bases for every finite $r$, centered value $\omega_1$, and any prescribed decreasing value greater than $\omega_1$.

We present a Fourier-based method for evaluating two-electron four-center Coulomb repulsion integrals over Slater-type orbitals. Spherical Fourier-Bessel coefficients of the convolution factors are precomputed and reused. Since the spherical part of the integral can be evaluated directly, there remains only a one-dimensional quadrature problem in the radial direction. For the ten reference integrals reported here, the largest absolute discrepancy is $1.7\times10^{-9}$. A preliminary resident-cache benchmark computes approximately $4.2\times10^5$ unique integrals per second on an NVIDIA V100 at the contraction stage.

We prove that for any prime $p$ and tame level $N$, the weight map from the cuspidal Hilbert eigenvariety constructed by Andreatta-Iovita-Pilloni for a totally real field $F$ to the weight space is proper in the sense of Buzzard-Calegari, under a non-vanishing hypothesis on certain Sen polynomials of Galois representations at the primes above $p$. In particular, this yields the properness of the eigenvariety away from non-positive integral Sen weights. Our approach extends the $p$-adic Hodge-theoretic argument of Diao-Liu.

We prove that a tubular group is linear if and only if it is residually finite, equivalently virtually primitive. Every such group embeds in $\SL_n(\Z[u])$. For a primitive tubular decomposition with at most $m$ edge slopes at each vertex, our representation has dimension $2\max\{2,m\}$. In particular, Gersten's group embeds in $\SL_6(\Z[u])$, and this dimension is minimal.

Reversible motion can produce a directed circulation of surviving ancestry in a \FV\ particle system. We prove that two particles are the exact threshold for universal ancestral balance in finite systems with uniform copying. For two particles, the stationary current of surviving labels vanishes for every finite irreducible reversible Markov chain and every positive killing vector. On a fixed complete three-state graph, nonzero circulation occurs for each population size at least three on an open dense set of killing vectors of full Lebesgue measure. A single killing vector produces circulation simultaneously for every population size at least three, and such vectors form a dense $G_\delta$ set of full measure. The current is divergence free and vanishes on bridges, so three sites and three particles are the smallest possible setting for circulation. We also identify the probabilistic effect of this circulation: for sufficiently rare transplants, migration followed by fixation is more likely in one direction than in the reverse direction.

We introduce the global obstruction property (GOP) for Artin algebras: for every finitely generated module of infinite projective dimension, two distinct syzygies have a common nonprojective indecomposable direct summand. We prove that GOP implies the generalized Auslander–Reiten condition, and hence the Auslander–Reiten conjecture. We give criteria for GOP to hold for pullbacks over semisimple algebras and construct an example which is not syzygy-finite. For each Artin algebra $A$, we construct an algebra $\mathcal{G}(A)$ satisfying GOP by using triangular matrix extensions. We show that $\text{fin.dim}\mathcal{G}(A)=\text{fin.dim} A+2$ and that $A$ and $\mathcal{G}(A)$ are singularly equivalent. Finally, we give a pair of derived equivalent finite-dimensional algebras for which GOP holds on only one side.

In this note we show that if a contractible Stein manifold $X$ with the density property admits a proper holomorphic embedding into $\mathbb{C}^k$, then $X \times \mathbb{C}^k$ is biholomorphic to $\mathbb{C}^{n+k}$ with $n=\dim X$. Combined with an announced result of Auroux, Bai and Pomerleano, this shows that for some $4 \le n \le 8$ there is a complex affine variety $X$ such that $X \times \mathbb{C}$ is biholomorphic to $\mathbb{C}^{n+1}$ and $X$ is not biholomorphic to $\mathbb{C}^n$. This settles the Holomorphic Cancellation Problem in the negative.

We prove that every connected, locally finite, quasi-transitive graph which is quasi-isometric to a graph excluding a countable clique as a minor is quasi-isometric to a planar graph. Consequently, every finitely generated group with such a quasi-isometry is virtually planar, meaning that some finite-index subgroup admits a planar Cayley graph. We also show that a connected, locally finite, quasi-transitive graph admitting a regular map into a bounded-degree graph excluding some finite minor admits a regular map into a bounded-degree planar graph. If the source is accessible, the target need only exclude a countable clique. Combined with work of Mackay, Spriano, and the present author, this shows that a finitely presented group admits a regular map into a bounded-degree graph excluding a countable clique if and only if it is virtually planar.

Let $\Theta\subset\mathbb{R}^n$, $n\geq 1$ be compact with upper Minkowski dimension $\beta$, and let $\gamma(x,t,\theta)$ be a family of curves which is bi-Lipschitz in the first variable and Lipschitz in the last two. We prove the maximal estimate $$\left\|\sup_{(\tau,\theta)\in[0,1)\times\Theta}\left|e^{i\tau\Delta}f(\gamma(\,\cdot\,,\tau,\theta))\right|\right\|_{L^2(B^n_1)}\lesssim_s\|f\|_{H^s(\mathbb{R}^n)},$$ for every $s>\tfrac{n(\beta+1)}{2(n+1)}$, in the range $0\le\beta\le n-1$. In the range $n-1<\beta\leq n$ we prove the threshold $s>\min\big\lbrace\frac\beta 2 + \frac 1{2(n+1)},\frac n2\big\rbrace$. The proof relies on a fractal $L^2$ estimate of Du–Zhang. We also use the temporal localization of Lee, the locally constant property, and a counting argument allowed by the Lipschitz properties of the curve.

We show how triality, the symmetry permuting the three 8-dimensional representations of $\mathrm{Spin}(8)$, can also appear as a symmetry of geometric objects such as connections and Dirac operators. Take any Riemannian metric $g$ on the manifold $M = \mathrm{SU}(3)$ and let $\nabla$ be a metric connection on $TM$. Using an explicit model of Clifford multiplication by deformed Okubo products, we combine spinorial lifts with isometries $TM \simeq S^+ \simeq S^-$ to show that $\nabla$ can be rotated into two new metric connections on $TM$. So there is a canonical $\mathbb{Z}_3$-orbit of metric connections determined by $(g, \nabla)$, with no other choices. The three connections coincide when $g$ is bi-invariant and $\nabla$ belongs to the Cartan-Schouten line, but in general they are different. The rotations of $\nabla$ preserve the norm of the curvature, so they define a new $\mathbb{Z}_3$-symmetry of the moduli space of Yang-Mills connections on $TM$. They also preserve the Chern-Simons 3-form and adjoint Yang-Mills-Higgs functionals. After choosing an orientation on $M$, the rotated connections define three self-adjoint, Dirac-type operators on the spinor bundle. We prove that these operators are natural, in the sense that they commute with the Kosmann derivatives of spinors along all invariant Killing vector fields that preserve $\nabla$. We also show that they are not isospectral, in general. This is verified by computing the simplest eigenvalue for the Levi-Civita connection and observing that it splits among the three operators as the bi-invariant metric on $M$ is deformed to one with isometry group $(\mathrm{SU}(3) \times \mathrm{SU}(2) \times \mathrm{U}(1)) / \mathbb{Z}_6$. Finally, we outline how the constructions in the paper may extend to general 8-manifolds with a $\mathrm{PSU}(3)$-structure.

Masked diffusion models (MDMs) generate discrete data by iteratively unmasking multiple tokens in parallel using a learned mask predictor. Their output carries two errors: parallelization error due to parallel sampling, and estimation error from the mask predictor learned with finite samples. Existing theory treats these errors separately, and the available statistical bounds typically scale with the number $V^L$ of length-$L$ sequences over a vocabulary of size $V$, suffering from the curse of dimensionality. In this work, we establish end-to-end statistical guarantees showing that transformer-based MDMs learn structured distributions with a sample complexity governed by their latent structure. Specifically, we model this structure through mixtures of hidden Markov trees, which capture latent dependencies within components and heterogeneity across components. When the data distribution is well approximated by such a mixture, a transformer-based MDM with parallel unmasking schedules achieves $\varepsilon$-accurate sampling in KL divergence with $\widetilde{O}(LK_{\mathrm{par}}/\varepsilon)$ training samples, where $K_{\mathrm{par}}$ is polynomial in $L$, $V$, the number of hidden states, and the number of mixture components. The transformer network and sampling procedure require no knowledge of the tree topologies or component parameters, thereby adapting to the unknown latent structure automatically. Overall, our results provide theoretical justification for the empirical effectiveness of MDMs on real-world data.

We prove asymptotic formulas for the number of $m$-full ideals of bounded norm in a given ideal class and for the number of $m$-full algebraic integers of bounded Weil height.

Large-scale tensors arising in science and engineering are often too large to form or store explicitly. Their multilinear contractions, however, can frequently be evaluated efficiently, which makes randomized Frobenius-norm estimation a practical alternative to exact computation. In this paper, we study estimation of the squared Frobenius norm of a real order-$d$ tensor from such contractions using independent real, complex, or quaternion Gaussian probes. For a rank-one tensor, one normalized sample is a product of $d-1$ independent Gamma variables. This gives exact relative-variance factors $3^{d-1}-1$, $2^{d-1}-1$, and $(3/2)^{d-1}-1$, respectively. Under the same budget of underlying real Gaussian vectors, the quaternion probe has smaller rank-one variance than four independent real probes for $d\ge3$; a real-channel representation identifies the gain as modewise pooling. For orthogonal CP tensors, the exact relative variance equals the corresponding rank-one factor divided by the participation-ratio effective rank and the number of samples. For arbitrary real tensors, we prove sharp uniform moment bounds, attained at rank one, and obtain Chebyshev guarantees. A classical block-median construction then gives two-sided exponential confidence for the modified estimator. Numerical examples test the rank-one and effective-rank formulas. An implicit infinite-series kernel tensor illustrates contraction-based norm evaluation; EEG maps provide a qualitative spatial interpolation illustration.

We give an explicit formula for the Lusztig parameters of determinantial modules over symmetric quiver Hecke algebras. The formula applies to every Bruhat pair and every fixed reduced expression of the upper Weyl group element. In finite types $A$, $D$, and $E$, we identify the parameters of the modules in Ménard's seed using Demazure weaves. We then factor prefix determinantial modules into these seed modules, with multiplicities given by Lusztig cycles. We also give a criterion for recognizing monomials in this fixed seed and an explicit formula recovering their multiplicities from Lusztig parameters.

Large language models (LLMs) are increasingly being explored to automate operations research modeling, reduce modeling time and lower the expertise barrier to optimization based decision support. However, this process often introduces inconsistencies between generated artifacts and the original problem, which we refer to as optimization modeling hallucinations. Despite progress in general-purpose hallucination mitigation, optimization modeling poses a distinct repair challenge: restoring fidelity to the original problem through coordinated model-code updates. We introduce OPT-Kraken, a training-free hallucination repair framework. To the best of our knowledge, it is the first post-hoc framework for repairing hallucinations in LLM-based optimization modeling artifacts. It extracts problem requirements and instance data, localizes errors by checking problem-model fidelity, model-code consistency, and data grounding, and selects repair granularity by the affected scope. For local repair, category rules and instance evidence are compiled into contracts specifying permitted edits, expected effects, and protected content. Experiments on established optimization modeling benchmarks show positive gains across automatic modeling frameworks and fine-tuned models. On zero-shot artifacts from multiple base models, OPT-Kraken effectively repairs optimization modeling hallucinations while preserving initially correct artifacts. These results demonstrate effective and controlled repair for LLM-based optimization modeling artifacts.

It is conjectured that in the Lorentz mirror model on $\Z^2$ all trajectories are finite. Quas showed that, almost surely, the two halves of any infinite trajectory, cut at an edge, meet at infinitely many vertices, each of which is pivotal: changing its state can make the trajectory finite. We construct plane graphs on which infinite trajectories exist, and all of them have this property. Given $p\in(0,1)$, we replace each edge of a random one-ended subtree of $\Z^2$ by a number of parallel edges, depending on $p$, that is at most logarithmic in the height of the corresponding descendant tree. The medial graph of the resulting plane multigraph is a factor of the tree, and for the uniform spanning tree it has finite vertex intensity. In the mirror model on this medial graph, with probability $p$ for mirrors along primal edges and any probability $r\in(0,1-p]$ for mirrors along dual edges, infinite trajectories exist, and the two halves of each of them meet infinitely often. The construction rests on Bernoulli percolation: every random one-ended locally finite tree can be decorated in this way such that bond percolation with parameter $p$ is critical and percolates, and every vertex of the infinite cluster has infinitely many pivotal edges. We characterize this property for bond and site percolation and for mirrors.

For $\Delta\ge2$ and $q\ge11\Delta/6$, we prove that the antiferromagnetic $q$-state Potts partition function on finite simple graphs of maximum degree at most $\Delta$ has no Fisher zeros in a graph-uniform complex neighbourhood of $[0,1]$. For $q>11\Delta/6$, the proof establishes coupling independence throughout $[0,1]$ using a soft version of Vigoda's flip dynamics. Our main tool is a separator-shell transfer theorem for the Potts model on induced-subgraph closed classes of graphs of maximum degree at most $\Delta$, with $q\ge\Delta+1$. It yields a graph-uniform zero-free neighbourhood of $[0,1]$ from Hamming coupling independence at $0$ and a uniform coupling-independence bound on each interval $[\delta,1]$, $\delta\in(0,1]$. We also obtain zero-free Lee-Yang polydiscs around the uniform field for vertex- and edge-colour fields. For Boolean Holant problems on bounded-degree graphs whose signatures come from a fixed finite family of log-concave symmetric signatures $f$ with $f(0)>0$, such as $b$-matchings, we obtain graph-uniform zero-free polytubes around every bounded box of nonnegative activities; their union is an open zero-free neighbourhood of the nonnegative orthant. An appendix summarizes further coupling-independence inputs and the zero-free regions they yield.

We construct a polar action of $\mathrm{SU}(3)$ on a closed simply connected $11$-manifold with a compact three-dimensional section that is not injectively immersed. The section is injective over the regular part and has two distinct local branches along a singular stratum. All isotropy groups are connected, and all orbits are simply connected. This gives a negative answer to the injectivity question raised by Grove and Ziller.