Papers

Let $N\ge 2$ and $1<p<N$. Let $H$ be a possibly non-reversible Minkowski norm on $\mathbb{R}^N$, with $H\in C^2(\mathbb{R}^N\setminus\{0\})$ and $D^2(H^2/2)$ uniformly positive definite away from the origin. We classify all nonnegative local weak solutions $u\in W^{1,p}_{\rm loc}(\mathbb{R}^N)$ of \[ -\operatorname{div}\big(H(\nabla u)^{p-1}\nabla H(\nabla u)\big)=u^{p^*-1},\qquad p^*=\frac{Np}{N-p}, \] without any finite-energy, integrability, or asymptotic assumption. Every nonzero solution is an anisotropic Sobolev bubble. This settles, for the full range $1<p<N$, the anisotropic analogue of Problem A of Antonini–Ciraolo–Gatti, whose Euclidean case was settled by Zhang, and removes the restriction on $p$ in the non-energy anisotropic classification of Chen–Wu–Yan–Yang. We also prove a scale-invariant Schoen-type Harnack inequality for local solutions. The proof follows the strategy of Zhang for the Euclidean $p$-Laplacian, combining rigidity of bounded blow-up profiles and a first-contact argument with the integral bootstrap of Ciraolo and Gatti. The anisotropic analysis requires two additional ingredients. First, we establish a Bochner identity for the conjugated stress tensor in distributional form on the noncritical set, using only $C^2$ regularity of $H$. Second, we prove a two-term expansion at isolated singularities of Finsler $p$-harmonic functions, including the gradient control needed by the Pohozaev invariant. For non-reversible anisotropies the argument is formulated with the reflected norm. A Morrey-type estimate proved directly for the relevant Leray–Lions operator replaces the potential-theoretic input used in the Euclidean argument.

We give an exposition of some of the ideas leading up to Jacob Tsimerman's groundbreaking work developing the theory of o-minimal definability in arithmetic and algebraic geometry.

Let $X$ be a connected finite-type CW-complex with fundamental group $G$. We show that the profinite completion $\widehat{G}$ determines the cohomology jump loci $\mathcal{V}^q_s(X,\mathbb{C})$ under two hypotheses: that the loci are finite unions of torsion-translated subtori, as for smooth quasi-projective varieties, and that $\widehat{G}$ determines the Betti numbers of the finite cyclic covers of $X$ in degrees $\le q$, which holds unconditionally for $q=1$, and in all degrees when $X$ is aspherical and $G$ is good in the sense of Serre. When the isomorphism of completions is compatible with an identification of the abelianizations, the loci correspond exactly; in general, they correspond up to an isogeny. Applied to finite covers, this shows that the tropical bounds for the Bieri–Neumann–Strebel–Renz invariants are profinite invariants, and recovers the profinite invariance of the BNS invariant of Kähler groups due to Hughes, Llosa Isenrich, Py, Stover, and Vidussi. We also show that $\widehat{G}$ determines the graded abelian groups $\mathrm{gr}_r(G/W(G))$, torsion included, for every verbal subgroup $W(G)$; the cases $W(G)=1$ and $W(G)=G''$ give the lower central series quotients and the Chen groups. For hyperplane arrangements, it follows that no arithmetic Zariski pair is distinguished by any of these invariants, while two known lattice-isomorphic pairs show, respectively, that the profinite completion of an arrangement group is not combinatorially determined, and that it does not determine the group.

In this paper, we establish the weak type $(1,1)$ boundedness of noncommutative singular integral operators with rough kernels. To deal with the rough kernels, we develop some orthogonality of the kernels themselves in the pseudo-localization arguments for the good functions and use the microlocal decomposition and $TT^*$ arguments for the bad functions.

We study the zero distribution of the sum of the first $n$ Gegenbauer polynomials \[ S_n^{(\alpha)}(z)=\sum_{k=0}^n C_{k}^{(\alpha)}(z). \] We prove that for $\alpha>1$, the zeros of $S_n^{(\alpha)}(z)$ lie on the interval $(-1,1)$ for all large $n$.

Z-Sigil is a public-key cryptosystem in which the plaintext selects successive keys from a fixed module-lattice family. Messages are length-prefixed, zero-padded and divided into 32-byte blocks. Each public vector is a shared public matrix applied to a small secret vector, plus a small error. Key indices label torsion points of a flat Kähler torus; the secret family forms a section of a key bundle over them. A public nonce initializes a hash state that selects each block's key and bit mask. The sender updates the state with the plaintext block; the receiver does so after recovering it. The stream and nonce determine a discrete walk along which decryption reads the secret section. We specify the algorithms, prove correctness under an explicit noise condition and bound decoding failure for messages chosen after the public key. Under stated decisional Module-LWE assumptions, we establish IND-CPA confidentiality for the chain without modelling the state hash as a random oracle. The reduction covers quantum adversaries under quantum hardness assumptions, with classical keys, messages and ciphertexts; no concrete security level is established. With independent uniform selectors, a restricted direct-decryption model quantifies reduced fragment recovery under partial key exposure, without improving full-message recovery probability over an independent-block baseline. Known-plaintext and candidate-message attacks, and parallel candidate-table decryption, delimit this result. Neither a universal sequential lower bound nor authentication or chosen-ciphertext security is established. Replacing an earlier scalar-exposing proposal, we give an augmented-lattice interpretation, integral-transport obstructions and a noise budget for research on curved, nontrivial bundles. A byte-level specification, pseudocode, test vectors and numerical checks support verification.

We investigate Miller's bias conjecture for general one-parameter families of elliptic curves over $\mathbb{Q}$ with non-constant $j$-invariants. Utilizing methods developed by Michel, we view the second moment as the sum of distinct cohomological components: a main $p^2$ term, a $p^{3/2}$ term arising from the first cohomology over the projective line, and terms of order $p$ or lower corresponding to singular fibers. By assuming the generalized Sato-Tate conjecture, we employ representation and motive theory to demonstrate that the coefficient of the $p^{3/2}$ term averages to zero in the limit. We then independently analyze the lower-order contributions without assuming Sato-Tate. By applying the Chebotarev density theorem and analyzing Galois representations, we show that the contributions to the order $p$ term from additive singular fibers also average to zero. Consequently, the only remaining average contribution to the $p$ term is $-B_pp$, where $B_p$ denotes the number of “bad” points where the discriminant vanishes. Because $B_p$ averages to a strictly positive value, the overall coefficient of the $p$ term averages to a strictly negative value in the limit, thereby establishing the bias conjecture for these families.

We study multi-armed bandits (MAB) with multiple optimal arms, motivated by the fact that many practical decision making problems admit multiple correct answers. For $K$-armed bandits with $A$ optimal arms, we first provide a sharper analysis of previous sub-sampling algorithms (De Heide et al., 2021; Zhu and Nowak, 2020), establishing a $\tilde{O}\Big(\frac{K-A}{\sqrt{KA}}\sqrt{T} \Big)$ minimax regret, where $T$ is the total number of interactions and $\tilde O(\cdot)$ drops all constant and logarithmic factors, improving the previous $\tilde{O}(\sqrt{KT/A})$ regret. We then provide a matching lower bound up to logarithmic factors, indicating that our established rate is nearly minimax-optimal. We further show that the knowledge of $A$ up to $\tilde{O}(1)$ factors is necessary to achieve near-optimal regret, as near-optimal algorithms for one number of optimal arms must incur substantially larger regret than optimal regret for a smaller number. Overall, our results provide a comprehensive minimax characterization of $K$-armed bandits with $A$ over the entire range of $1 \leq A \leq K-1$.

In this paper, we investigate two related questions concerning the Boolean polynomial semiring in two variables. First, we study prime ideals in $\mathbb{B}[x,y]$, beginning with a complete classification of those generated by linear polynomials. Second, motivated by the limitations of ideals in capturing geometric phenomena over semirings, we introduce a notion of projective closure via congruences. We prove that this projective closure coincides with the topological closure in the projective space of prime congruences.

Let $M(N)$ be the number of similarity classes of positive rational perfect cuboids with ordered edges $a,b,c$, space diagonal $g$, and squarefree invariant $N=\operatorname{sf}(abcg/2)$. We prove that $M(N)$ is finite for every positive squarefree $N$, without any restriction on the rank of the associated congruent-number elliptic curve. More precisely, $M(N)\le C^{1+r_N}$ for an absolute constant $C>1$, where $r_N$ is the rank of $y^2=x^3-N^2x$. Using the established Paulsen-West rank obstruction and Smith's results on quadratic twists, we then prove \[ \lim_{X\to\infty}\frac1X \sum_{\substack{N\le X\\N\text{ positive squarefree}}}M(N)^q=0 \qquad\text{for every fixed }q>0. \] Thus the result counts cuboid classes with their full multiplicities at each $N$. A quantitative decay estimate is also obtained. The geometric step excludes every positive-dimensional real abelian coset through the positive part of the compatibility surface; quantitative Mordell-Lang then gives the required bound. The argument is unconditional and uses the cuboid correspondence and rank obstruction as prior results.

Nonlinear parabolic partial differential equations frequently occur in the modeling of applied phenomena. They often exhibit complicated dynamics which is organized by their associated equilibrium solutions. As a function of the involved model parameters, the equilibria can be detected using a bifurcation analysis, which in turn is centered around a deeper understanding of bifurcation points. In many applied systems establishing these organizing centers rigorously is difficult to achieve using traditional mathematical techniques. In this paper, we propose a theoretical framework for computer-assisted proofs of symmetry-induced transcritical and pitchfork bifurcations in nonlinear problems. This s accomplished by reformulating the existence question via a zero finding problem for an associated extended nonlinear system. The approach is general and allows for a wide variety of underlying symmetries. We demonstrate how these results can be applied in the setting of parabolic systems and higher-order equations, on both one- and two-dimensional domains.

Xiao, Qiu, and Huang (AAMAS 2023) and independently Lonc (personal communication) asked whether indivisible chores located at the vertices of a tree can always be allocated to $n$ agents in connected bundles so that the cost of every agent is at most its connected maximin share; for goods, this is a theorem of Bouveret, Cechlárová, Elkind, Igarashi, and Peters. We answer the question affirmatively, even for monotone costs. The answer follows from a combinatorial theorem: if $\mathcal P_1,\ldots,\mathcal P_k$ are partitions of the vertex set of a finite tree, each into at most $k$ connected parts, then the vertex set can be split into disjoint sets $B_1,\ldots,B_k$, some possibly empty, such that each nonempty $B_i$ is connected and contained in a part of $\mathcal P_i$. Equivalently, if each of $k$ colours occurs on at most $k-1$ edges of a tree, then the vertices can be partitioned into connected sets labelled by distinct colours, none containing an edge of its own colour; in particular, one can choose for every colour a component of the forest obtained by deleting that colour so that the chosen components cover the tree. The bound is already best possible for paths, and for additive costs, the theorem is equivalent to the fair-division statement. The proof reduces the problem to inward partitions of oriented trees, which we obtain from the colourful KKM theorem on a simplex of edge weights, using a leaf-elimination labelling that remains compatible when weights vanish. We also give an algorithm running in time $k^{O(k)}$ plus polynomial time.

The Gaussian curvature of the graphs of cubic polynomials $f(z)=(z-a)(z-b)(z-c)$, with triangle of zeros $\triangle=\triangle(a,b,c)$, is analyzed. We introduce the composed Pythagorean mean $\mathtt{B}_{\triangle}$, a combination of the three classical Pythagorean means, which plays a central role in the description of the critical points of the Gaussian curvature. When $\triangle$ is not equilateral, the curvature has exactly five critical points: a global maximum at the barycenter, two local minima, and two saddle points, forming a rhombus whose vertices lie on the axes of the Siebeck-Marden ellipse of the triangle. We also initiate a comparative study of $\mathtt{B}_{\triangle}$ with the skew of the triangle by describing a family of polynomials whose zeros determine a specific family of isosceles triangles. The cubic case provides a concrete model strongly suggesting that the critical geometry of the Gaussian curvature of the graph can reveal both analytic and algebraic properties of the underlying holomorphic function.

High-frequency athlete-tracking data can be summarized as quantile cubes (Thomas and Hannig, 2026): counts of time spent in bins of velocity, acceleration, and movement angle. In our application, each athlete-match session has 100 such bins, with counts ranging from under a hundred to several thousand. Whether a bin's negative binomial distribution is well approximated by a normal distribution depends on its shape parameter, the product of its mean and dispersion, rather than on the count alone. We develop a hybrid generalized fiducial framework that uses exact negative binomial structural equations where the estimated shape is small and moment-matched normal approximations where it is large. This approach provides uncertainty quantification for hundreds of regression and dispersion parameters without specifying a prior distribution. Only the normal components contribute to the fiducial Jacobian, which has a closed-form block-diagonal structure, and we prove a Bernstein-von Mises theorem for the correctly specified hybrid model. Simulations show near-nominal coverage, agreement with a flat-prior Bayesian analysis, and the importance of partitioning by estimated shape rather than by the mean or observed responses. In 216 sessions from 17 professional women's soccer athletes, the estimated position effects concentrate at the velocity extremes, while the estimated age effects follow an acceleration gradient. However, an athlete-level permutation check indicates that these patterns are not distinguishable from those obtained under athlete-label permutations.

In this note we present a direct relation between the geometrical definition of a quasi-Einstein metric and Horava gravity: the solutions of the large-distance effective action of Horava gravity that are static and orthogonal are quasi-Einstein metrics. The converse is also true if the function over the quasi-Einstein manifold satisfies a given equation.

Let $B_\infty$ and $P_\infty$ be the braid group and the pure braid group on infinitely many strands. We show that the inclusion $C^*_r(P_\infty)\subseteq C^*_r(B_\infty)$ is $\mathcal{Z}$-stable and selfless. Hence every intermediate inclusion and every intermediate $C^*$-algebra is $\mathcal{Z}$-stable and selfless. The intermediate $C^*$-algebras are natural examples of simple nonnuclear $\mathcal{Z}$-stable $C^*$-algebras, and they are singly generated. The proof of $\mathcal{Z}$-stability uses a simple relative centralizer criterion. We also show that $C^*_r(\mathbb{F}_\infty)\subseteq C^*_r(B_\infty)$ is a selfless inclusion, where $\mathbb{F}_\infty$ is the free group of braids in which only the first string moves. All results hold for reduced twisted group $C^*$-algebras. Selflessness of $C^*_r(B_\infty)$ itself already follows from work of Ozawa. On the other hand, $\mathcal{Z}$-stability fails for all twisted group $C^*$-algebras of groups that are not inner amenable. A variant shows that it fails for all twisted group $C^*$-algebras of $B_n$ with $n$ finite, and for the twisted crossed products from Artin's representation of $B_n$. Finally the method applies to other increasing unions of groups, such as $\mathrm{SL}_\infty(\mathbb{Z})$, whose reduced group $C^*$-algebra is $\mathcal{Z}$-stable.

We consider the multi-spinor Seiberg-Witten equations on a 4-manifold in a general setting, where we allow the auxiliary bundle to be non-trivial. We show that when the metric is Kähler and the auxiliary connection is unobstructed anti-self-dual Yang-Mills, the moduli space of solutions is compact for all parameters in some open neighborhood of the given metric and auxiliary connection. We use this fact to compute a count of solutions on some examples of Kähler surfaces. These computations have a surprising purely algebro-geometric corollary which may be of independent interest.

We establish instantaneous $H^2$ norm explosion for the two-dimensional Boussinesq equations around the linearly stable background profile $-x_2$. While previous ill-posedness results for this system relied on $L^\infty$-type spaces, this work provides the first strong ill-posedness result in the energy-critical space, where stable background stratification acts to suppress deformation growth. We adapt an idea from our previous work on the IPM equation \cite{BCMZ2025} and select an initial density perturbation vanishing near the origin to neutralize the stable stratification. Crucially, unlike the IPM setting, the evolution is here driven by the non-linear coupling between density and vorticity. Since the initial vorticity vanishes identically, the instantaneous loss of regulari\-ty is driven by density-vorticity feedback - a mechanism fundamentally distinct from both $H^1$ Euler ill-posedness of Bourgain & Li and the IPM evolution. Through an iterative scheme yielding a sequence of approximations with strong deformation at the origin, we construct a strong solution $(\rho (t), \omega (t))$ originating from data with arbitrarily small $H^2 \times (H^1\cap \dot H^{-1})$ norm, which immediately exhibits $H^2 \times H^1$ norm explosion for all $t > 0$ while maintaining uniqueness within the energy class.

These notes introduce the martingale dimension of a strongly local regular Dirichlet form and the analytic dimension of a Lipschitz differentiability space. We explain why these dimensions coincide under two-sided Gaussian heat kernel estimates and discuss their relationships with other notions of dimension. We also introduce the energy image density property, its recent resolution, and its applications to martingale dimension. Finally, we show that spaces carrying diffusions with two-sided sub-Gaussian heat kernel estimates of walk dimension greater than two are not Lipschitz differentiability spaces.

We give a cohomological criterion for the automorphism group scheme of a hypersurface in a smooth Deligne-Mumford stack to be discrete in tame characteristics. The criterion converts Bott-type vanishing on the ambient stack into vanishing of infinitesimal automorphisms of every quasismooth hypersurface in a linear system. We apply it to toric orbifolds, obtain uniform finiteness results for quasismooth weighted-projective hypersurfaces in tame characteristic and hypersurfaces in products of projective spaces, and identify the precise exceptional windows in which the method fails.

c-Polynomials are polynomials whose (non-zero) coefficients are all equal to 1. Taking a theorem by Carlitz and Moser as a motivation, we prove a structure theorem for the factorisations of the c-polynomial $(x^n-1)/(x-1)$ into c-irreducible factors by relating c-torisations into c-polynomials to joint ordered factorisations arising from the integer n. The question of how many different joint ordered factorisations there are leads to the precise chromatic polynomial for path graphs, counting the ways of path graph colourings with $m$ colours under the condition that all colours are used. Moreover, we give similar counting formulae for joint ordered factorisations under the constraints that all factors are primes, or that all factors are square-free.

Zeroth-order optimization methods are essential for solving black-box problems where gradient information is unavailable or expensive to compute. This paper presents POEM-ES, a novel parameter-free stochastic zeroth-order algorithm that extends the recent POEM method by integrating subspace preconditioning with ellipsoidal randomized sampling. In contrast to traditional zeroth-order approaches that rely on isotropic random directions, POEM-ES performs anisotropic sampling guided by a fixed structural symmetric positive semi-definite (SPSD) preconditioner $\hat{\Sigma}$ that encodes the underlying low-dimensional geometry. Under a standard structural spectral normalization where $\lambda_{\max}(\hat{\Sigma}) = 1$, we introduce the use of the empirical effective dimension $d^* = \operatorname{tr}(\hat{\Sigma})$, which reflects the intrinsic dimensionality of the problem and guides both the sampling and randomized smoothing parameter schedules. In practice, such a preconditioner can be effectively obtained via pilot sampling, historical trajectories, or domain-specific expert knowledge. We prove that POEM-ES achieves a dimension-reduced convergence rate under low-rank structural assumptions, requiring only $\tilde{\mathcal{O}}\left( \frac{\left( r^2 \kappa(\hat{\Sigma}) + d^* \right) L^2 D_{\mathcal{X}}^2}{\varepsilon^2} \right)$ stochastic zeroth-order oracle queries. The method remains fully parameter-free and demonstrates significant improvements over the original POEM in problems with low-rank structure where $d^* \ll d$. Numerical experiments on hinge-loss binary classification tasks using LibSVM datasets confirm the practical superiority of the proposed approach.

Let $p\geq 5$ be a prime, and let $f$ be a $p$-ordinary cuspidal newform of even weight $k = 2r \geq 2$ and level $N$, with $p \nmid N$ and trivial nebentype. In this article, we prove Kato's Iwasawa main conjecture for $f$, formulated in terms of the $p$-adic $L$-function of $f$, under the sole assumptions that the associated Galois representation of $f$ satisfies the big-image hypothesis and that its residual representation is absolutely irreducible. This generalises the main result of Burungale–Castella–Skinner, which treats elliptic curves over $\mathbb{Q}$ with good ordinary reduction at $p$. Our proof follows their general strategy. A key new ingredient is the construction, over an imaginary quadratic field in which $p$ splits, of a zeta element attached to $f$, together with a proof of its explicit reciprocity laws, generalising the corresponding work of Burungale–Skinner–Tian–Wan. These results provide the higher-weight analogue of a crucial input in the argument of Burungale–Castella–Skinner and allow their method to be carried out for higher-weight modular forms.

We determine the intrinsic mirror of an orbifold cubic surface with three nodes and interpret the resulting geometry through mirror symmetry between the $PGL(2,\mathbb{C})$ and $SL(2,\mathbb{C})$ character varieties of the once-punctured torus. We further construct a quantization of the mirror using higher genus Gromov-Witten theory and show that it recovers the $A_1$ spherical double affine Hecke algebra.

We study a two-parameter deformation of scalar curvature associated with the Gauduchon connections on compact Hermitian manifolds. A central identity expresses its dependence on the connection parameter through the squared norm of the trace-free part of the torsion of the Chern connection. We establish existence criteria for metrics of constant deformed scalar curvature and analyze the global dependence of the deformed Yamabe invariant on the connection parameter, including when the critical threshold is reached. We also study a prescribed-curvature problem in positive conformal classes. Applications include attainment of the critical threshold for an explicit Hopf example in every complex dimension at least two and on non-locally conformally Kähler Iwasawa products.

Lagrangians of the moduli space of Higgs bundles are a fundamental piece of its study under the lens of mirror symmetry. We construct a new class of Lagrangians defined via a zero 'velocity condition' imposed on the natural action of the non-zero complex numbers on the moduli space, motivated by a certain 'infinitesimal conormal principle'. These are Lagrangian subvarieties analogous to the upward flows considered by [CW19] and [HH22], but which fiber over the $\mathbb{C}^*$-fixed point subvarieties. We then prove a formula for the number of intersection points of this Lagrangian with the generic fiber of the Hitchin map, and build a vector bundle over the dual moduli space which we conjecture to correspond to its mirror hyperholomorphic bundle.

We study the local geometry, singularity classification, and wavefront dynamics of complex evolutoids for regular holomorphic curves in $\mathbb{C}^2$ equipped with the standard complex bilinear metric. By integrating the generating family of rotated complex affine lines, we derive an explicit parametrization for the 1-parameter family of complex wavefronts $\Gamma^{\theta, w}$ and establish that their singular cusp locus sweeps out the complex evolutoid $E^\theta$. Using a square-root-free formulation, we classify the local behavior of $E^\theta$, proving that evolutoids remain regular and tangent to the curve at isotropic points ($q=0$) while escaping to infinity at complex inflections ($\kappa=0$). Finally, we analyze the three-dimensional complex discriminant surface $\mathcal{D}_F \subset \mathbb{C}^3$ formed by the evolving family $E^\theta$. We prove that $A_2$ cusps are versally unfolded by spatial parameters, whereas $A_3$ swallowtail singularities are holomorphically versally unfolded by $(x_1, x_2, \theta)$ if and only if the complex non-degeneracy condition $\kappa^4 + \kappa_s^2 \neq 0$ holds. Crucially, we demonstrate that simple isotropic points drive the breakdown of this versality condition, inducing degenerate, non-transverse sections of the complex swallowtail surface.

In this paper we study Euclidean triangles associated with Diophantine $D(4)$-triples. For a $D(4)$-triple $\{a,b,c\}$, $a<b<c$, we prove that the lengths $r=\sqrt{ab+4}$, $s=\sqrt{ac+4}$ and $t=\sqrt{bc+4}$ associated with this $D(4)$-triple form a non-degenerate triangle if and only if the triple is regular and $r>b-a$. For a fixed shortest side $r$, we describe all triangles associated with $D(4)$-triples in terms of the factorisations of $r^2-4$, give a divisor criterion for their number, and determine their angle type. We also prove that two similar triangles associated with $D(4)$-triples are necessarily congruent. Finally, for the family of $D(4)$-triples $\{r-2,r+2,4r\}$, we obtain a Pell-equation characterisation of the Heronian triangles associated with these $D(4)$-triples.

We show that every UCP extension problem for a finite dimensional operator system can be reduced to three dimensions after finite matrix amplification. The set of extensions is preserved by an affine bijection. The matrix size is of order $\sqrt{d}$, where $d$ is the dimension of the original system. This also gives a three dimensional (smallest) counterexample to Arveson's hyperrigidity conjecture.

For ordered sets $X$ and $Y$, let $Y^X$ denote the ordered set of order-preserving maps from $X$ to $Y$, where $f\le g$ in $Y^X$ if $f(x)\le g(x)$ for all $x\in X$. Let $P$ and $Q$ be finite ordered sets such that $P^P\cong Q^Q$. It is proven that $P\cong Q$ if $P$ or $Q$ has height at most $1$ or if $P$ and $Q$ are ordered sets of the following form: replace each element of an ordered set with a three-element antichain. The latter is an elaboration of a proof of Tim Campion.