Papers

We characterize tree-child phylogenetic networks in terms of their mu-representations. First, we give a structural characterization of tree-child networks by means of ordered tree-path decompositions. We then translate this decomposition into a set of purely vectorial conditions on finite subsets M in N^n. We prove that such a set M is the mu-representation of a tree-child phylogenetic network if and only if it is tree-child mu-compatible. This provides a feasibility criterion for tree-child mu-representations which can be used as a basis for reconstruction and further algorithmic applications. Note that this paper presents results arising from ongoing research on tree-child networks and that the results will be further developed and placed into proper context in subsequent versions.

We introduce posets of type $\mathbb{D}$, a family of posets described by an admissible Dynkin quiver of type $D_n$ together with a compatible set of extra arrows (alien arrows), and show that their socle-projective representation category is of finite representation type. Continuing, in the same spirit, a program initiated for posets of type $\mathbb{A}$ [R. Schiffler, R-J. Serna, A geometric realization of socle-projective categories for posets of type $\mathbb{A}$, J. Pure Appl. Algebra 224 (2020), no. 12, 106436], we build on an existing geometric model for cluster-tilted algebras of type $D_n$, whose geometric combinatorics is more intricate than that of type $\mathbb{A}$. Our main result establishes a $\Bbbk$-linear categorical equivalence $\Theta$ between a full subcategory $(\mathcal C/T)_F$ of the arc category of a punctured $(n+3)$-gon, whose objects are certain $sp$-arcs, and the category of finitely generated socle-projective $\Bbbk\mathscr{P}$-modules, for $\mathscr{P}$ a poset of type $\mathbb{D}$. As a consequence, when the set of alien arrows is empty, we conclude that the cluster subalgebra generated by the $sp$-arcs coincides with the full cluster algebra.

This article is the first of three articles whose goal is to generalize the Cartier transform of Ogus and Vologodsky to the logarithmic setting. In this article, we generalize a local version, due to Shiho, of the Cartier transform to log smooth schemes. More precisely, let $k$ be a perfect field of positive characteristic and equip $\operatorname{Spec}k$ with the trivial logarithmic structure. For a log smooth morphism of logarithmic schemes $X \rightarrow S,$ where $S$ is log flat and locally of finite type over $\operatorname{Spec}k,$ we obtain, under the assumption that the exact relative Frobenius lifts over the Witt vectors of $k$ to a morphism between log smooth schemes over $S,$ a fully faithful functor from the category of quasi-coherent modules on the base change $X'=X\times_{S,F_S}S$ of $X$ by the Frobenius $F_S$ of $S,$ equipped with a quasi-nilpotent Higgs field, to the category of quasi-coherent modules on $X$ equipped with a quasi-nilpotent integrable connection. For this, we prove a log flat descent theorem for morphisms, based on previous work by Kato.

Let $\mathbb{P}$ be the set of primes. Suppose that $P_{i}$, $i=1,2,\ldots,9$ are nine subsets of $\mathbb{P}$ with relative lower density $\underline{d}_{\mathbb{P}}(P_{i})=\mu_{i}$. When $\sum_{i=1}^{9} \mu_{i}^{3}>8$, we prove that every sufficiently large odd integer can be written as $n=\sum_{i=1}^{9} p_{i}^{3}$ with $p_{i} \in P_{i}$, $i=1,2,\ldots,9$.

We study nonnegative entire spacelike graphs in Lorentz–Minkowski space satisfying $\sigma_k(A[u])=u^p$. For $2\le k<n$ and $p\ge k$, pointwise strict spacelikeness and closed $k$-admissibility yield universal bounds for the height and Lorentz factor. Their proof combines block coercivity in the full Gårding cone, a Lorentzian cutoff, and hyperbolic-cap comparison. For $2\le k<n/2$ and $k\le p\le k(n+2)/(n-2k)$, every such solution vanishes, without symmetry, decay, finite-energy, curvature-pinching, or global Hessian assumptions. In the subcritical range we use a trace-free Newton tensor, a weighted divergence identity, and a finite descent. At the critical endpoint, the divergence identity retains an exact nonnegative defect and a positive quartic term. Direct estimates cover $2k<n\le4k+2$. For $n\ge4k+3$, we use compactness, concentration of the $k$-Hessian measure at the first crossing, a single-pole Pohozaev argument, core counting, a recursive defect estimate, and Souplet-type radius selection. A geometric corollary gives a rigidity result for complete spacelike immersions under the corresponding curvature equation.

The Lie algebra of multiple zeta values is realized as the space $\operatorname{ARI}^{\operatorname{pol}}_{\underline{al}\ast \underline{il}}$ of alternal moulds whose swap is alternil up to a constant mould, equipped with the $ari$ bracket. In this paper, we study the larger space $\operatorname{BARI}_{\underline{il},swap}^{\operatorname{pol}}$ of alternil, swap-invariant bimoulds, which is conjecturally the Lie algebra for multiple $q$-zeta values. We propose an explicit formula for a Lie bracket $uri$ on this space. Moreover, we prove that the Lie algebra $\operatorname{ARI}^{\operatorname{pol}}_{\underline{al}\ast \underline{il}}$ embeds into $\operatorname{BARI}_{\underline{il},swap}^{\operatorname{pol}}$, with the $ari$ bracket translating directly into the $uri$ bracket. Finally, we examine the associated-depth graded of $\operatorname{BARI}_{\underline{il},swap}^{\operatorname{pol}}$, extending the known depth-graded setup for $\operatorname{ARI}_{\underline{al}\ast \underline{il}}^{\operatorname{pol}}$.

Let $M$ denote a module over a local Noetherian ring $(R,\mathfrak{m})$ with residue field $\Bbbk$ and its injective hull $E_R$. Let $D_R(\cdot) = \operatorname{Hom}_R(\cdot,E_R)$ denote the Matlis duality functor. Then $M$ is called Matlis reflexive whenever the injection $M \to D_R^2(M)$ is an isomorphism. The following results are proved: (1) For an ideal of $R$ a reflexive module is quasi-complete, generalizing part of Belshoff's results (see \cite{Br}). (2) We improve Zöschinger's numerical criterion for Matlis reflexivity (see \cite{Zh2}). (3) As shown by Enochs (see \cite{Ee}) for a prime ideal and the injective hull $E_R(R/\mathfrak{p})$ it follows that $D_R(E_R(R/\mathfrak{p}))$ is the completion of a free $R_{\mathfrak{p}}$-module of rank $\tau_{\mathfrak{p}}$, the Enochs number of $\mathfrak{p}$. We characterize when $\tau_{\mathfrak{p}} = 1$, extending author's result (see \cite{Sp16}).

We introduce an analogue of Gröbner bases, or equivalently, Bergman's diamond lemma, for linear (super)monoidal categories. It gives a systematic way to study monoidal ideals and to prove basis theorems. The central notion is that of a monoidal Gröbner system and is based on higher linear rewriting theory. We then apply the theory to categories of affine Brauer type: linear (super)monoidal categories that have the same hom-basis as the affine Brauer category, but possibly distinct composition and tensor product. Our main result is a criterion for a category to be of affine Brauer type, reducing the question to an explicit list of local computations, which we partially implement in the computer algebra system FORM. This gives new combinatorial proofs of the basis theorems for the affine Brauer category, the nil-Brauer category and the affine VW supercategory, and yields new examples. In particular, we construct the odd nil-Brauer category, a conjectural supercategorification of the split $\imath$quantum group of rank one, and the quantized affine VW supercategory, whose non-affine part is the quantized periplectic Brauer category and which we expect to satisfy a higher quantum Schur–Weyl duality. Finally, we classify the categories of affine Brauer type admitting a sufficiently simple presentation.

We describe the ideals of an idealization $D\ltimes M$ by triples $(I,N,\varphi)$, where $I$ is an ideal of $D$, $N$ is a submodule of $M$ with $IM\subseteq N$, and $\varphi\in\operatorname{Hom}_D(I,M/N)$. For a domain $D$ and torsion-free $M$, the nonzero proper ideals of $D\ltimes M$ with nonzero annihilator are exactly $0\ltimes N$. As an application, for $R=\mathbb R[[t]]\ltimes\mathbb R[[t]]$, the nonzero proper annihilating ideals form a countable set, while the set of all nonzero proper ideals has cardinality $2^{\aleph_0}$. This answers negatively a question of Behboodi and Rakeei.

We prove various properties of the universal valued Abelian groups $\mathbb{G}_r(N)$ constructed by Niemiec, for $r\in\{1,\infty\}$ and $N\in\{0,2,3,\ldots\}$. First we show that the completion of $\Gamma_0=\bigoplus_{n\in\mathbb{N}}\mathbb{Q}/\mathbb{Z}$ (for $N=0$) or $\Gamma_N=\bigoplus_{n\in\mathbb{N}}\mathbb{Z}/N\mathbb{Z}$ (for $N\geq 2$) with respect to a generic invariant metric, bounded by $1$ when $r=1$, is isometrically group-isomorphic to $\mathbb{G}_r(N)$, recovering a result of Doucha in the case when $r=\infty $ and $N=0$. This confirms an expectation of Doucha for the group $\Gamma_N$. We combine this genericity result with a criterion of Melleray and Tsankov concerning the extreme amenability of the generic completion of a countable group to obtain the extreme amenability of $\mathbb{G}_r(N)$. We then establish the strictly stronger properties that these groups are Lévy and strongly exotic. We conclude by showing that $\mathbb{G}_r(0)$ is also monothetic, from which we obtain the existence of a monothetic group structure on the Urysohn sphere, giving a bounded version of a result by Cameron and Vershik and answering a question of Niemiec.

The maximum independent set (MIS) problem is a fundamental NP-hard combinatorial optimization problem with applications in scheduling, resource allocation, and network analysis. Exact solvers can provide high-quality solutions or optimality certificates, but their computational cost grows rapidly with graph size, while hand-crafted heuristics improve scalability at the expense of guarantees. Learning-based methods offer an alternative by exploiting structural patterns across graph instances, yet directly predicting independent sets can make global coordination difficult on large graphs. We instead use learning to guide multilevel graph coarsening while retaining combinatorial search for final decision making. Our Dual-GNN Multilevel Coarsening framework uses a Partition GNN to score candidate contractions and a Representative GNN to select top-k local independent-set states for each final cluster. Experiments on Erdős–Rényi graphs with up to 2,000 vertices demonstrate a favorable quality–runtime trade-off. On 500-vertex instances with certified optima, our method achieves an average independent-set size of 19.20, corresponding to 99.5% of the optimal value of 19.30, while reducing the mean wall-clock time from 643.57 seconds for exact solving to 3.41 seconds, yielding an approximately 189$\times$ speedup. On larger graphs with 1,000 and 2,000 vertices, our method achieves the best mean solution quality among all evaluated methods. Moreover, although trained only on Erdős–Rényi graphs with edge probability $p=0.35$, the learned coarsening policy generalizes effectively across both unseen graph densities and structurally different graph families.

In this paper we present a formalism of categorical representation theory for affine group schemes of finite type over fields (both in the weak and strong settings) which applies to arbitrary base fields. This is based on the development of an appropriate $\infty$-categorical setting for the study of representations and Harish-Chandra (bi)modules of such group schemes, which constitute the building blocks and basic examples of such structures. We also study categorical traces in this context, and show in particular that the categorical trace of the identity morphism, resp.~Frobenius morphism, on the appropriate $\infty$-category of representations of a connected reductive algebraic group $G$ over an algebraically closed field $k$ of positive characteristic identifies (under suitable assumptions) with the $\infty$-category of Ind-coherent sheaves on the adjoint quotient $G/G$, resp.~of the fixed points $G^{\mathrm{F}}$ of the Frobenius.

Variational autoencoders (VAEs) offer an efficient approach to amortized Bayesian inference for inverse problems, but posterior accuracy can depend strongly on the choice of variational regularization, particularly when the inverse problem contains weakly identified parameter directions. This study investigates three objectives: a reverse Kullback–Leibler formulation (VAE-KL), an asymmetric Jensen–Shannon formulation (VAE-JS), and a Jensen–Shannon–Wasserstein formulation (VAE-JSWA), which replaces the reverse Kullback–Leibler regularizer with the squared 2-Wasserstein distance while retaining forward-Kullback–Leibler posterior supervision. A full-covariance Gaussian encoder and a pre-trained physics-based surrogate are used for amortized posterior inference. A local linear–Gaussian analysis in the generalized Fisher basis is developed to characterize the variance-dependent gradients of the three objectives. The formulations are first evaluated using linear–Gaussian benchmarks with known posterior solutions and subsequently tested on nonlinear physics-based inverse problems, including an inverse problem governed by a linear ODE and two PDE-constrained problems. VAE-KL performs slightly better than the other formulations in the well-conditioned benchmark, where all three approaches yield comparable posterior approximations, whereas VAE-JSWA provides substantially lower posterior errors in the strongly ill-conditioned benchmark. The nonlinear physics-based problems exhibit a similar conditioning-dependent trend, with JS-based formulations providing greater benefit as posterior ill-conditioning increases. These results indicate that posterior conditioning is an important factor in selecting variational objectives and motivate geometry-adaptive variational inference for Bayesian inverse problems.

The geodesic orbit property is useful and interesting in Riemannian geometry. It implies homogeneity and has important classes of Riemannian manifolds as special cases, such as weakly symmetric Riemannian manifolds and naturally reductive Riemannian manifolds. The corresponding results for indefinite metric manifolds are much more delicate than in Riemannian signature, but in the last few years important corresponding structural results were proved for geodesic orbit Lorentz and trans-Lorentz manifolds. %Here we carry out a major step in the structural analysis of geodesic orbit Lorentz nilmanifolds. Here we study pseudo-Riemannian geodesic orbit nilmanifolds of metric index three. Those are the geodesic orbit pseudo-Riemannian manifolds $M = G/H$ of signature $(n-3,3)$ such that a nilpotent analytic subgroup of $G$ is transitive on $M$. Suppose that there is a reductive decomposition $\g = \h \oplus \n$ (vector space direct sum) with $\n$ nilpotent. When the metric is nondegenerate on $[\n,\n]$ we show that $\n$ is abelian, $2$-step or $4$-step nilpotent. In contrast to the Riemannian, Lorentzian, and trans-Lorentz cases, the two-step conclusion therefore fails, but $3$-step nilpotency remains impossible. The $4$-step case is confined to a Lorentzian derived algebra with an orthogonal complement of index two, and its structure forces an invariant totally isotropic two-plane in that complement. An explicit nine-dimensional example of signature $(6,3)$ proves sharpness. When the metric is degenerate on $[\n,\n]$ we prove the existence of an invariant isotropic subspace that centralizes its orthogonal complement, yielding a double-extension reduction to a geodesic orbit metric nilpotent Lie algebra of strictly smaller index. An eight-dimensional example shows that this relative centrality need not imply centrality in the full Lie algebra.

We define product subcomplexes and intersection complexes for compact nonpositively curved square complexes and CAT(0) square complexes. Product subcomplexes are defined as equivalence classes of local isometries from products of graphs with embedded coordinate fibers. Using their factorizations, we give a common definition of the intersection complex, extending the constructions in S. Oh's paper "Quasi-isometry invariants of weakly special square complexes" (Topology and its Applications 307 (2022)). Cubical automorphisms induce actions on these complexes. We define morphisms by comparing compatible elevations of core-map labels and their factorizations along face chains. For a compact two-sided weakly special square complex, we prove that its intersection complex is canonically isomorphic to the deck quotient of the intersection complex of its universal cover. Under a simplicity hypothesis, this recovers the image-defined reduced intersection complex in that paper, and product-base groups are the full simplex stabilizers. We also prove that quasi-isometries between universal covers of compact two-sided weakly special square complexes induce isomorphisms of their intersection complexes in this sense, and give finite counterexamples explaining why images alone do not determine the quotient in general.

For every even prime power $q>2$, we explicitly construct a rank five string C-group representation of the Chevalley simple group $G_2(q)$ using its natural seven-dimensional matrix representation. We also give a census for the string C-group representations of $G_2(q)$ of ranks four and above for every prime power $3 \le q \le 11$; providing computational evidence that rank five string C-group representations of $G_2(q)$ are rare and that this construction is exceptional among the currently known examples.

We study the edge spectrum of two-dimensional lattice Hamiltonians and magnetic Schrödinger operators in a mobility gap, an energy interval in which the bulk spectrum is potentially localized rather than necessarily absent. Under suitable localization and boundary assumptions, we prove that a constant nonzero bulk topological index implies that the entire interval belongs to the absolutely continuous spectrum of the corresponding half-space operator. The absolutely continuous spectral multiplicity is bounded below by the absolute value of the index at almost every energy in the interval. The first version of the auxiliary-space cut construction in Section 2.3 was generated using ChatGPT 6.

As large language models (LLMs) are increasingly used to automate digital interactions, users can leverage LLM-generated text as cover for covert communication within seemingly benign conversations. Existing LLM steganography, however, is predominantly white-box, requiring the sender and receiver to share the cover statistics, typically through access to the model weights and prompt. Black-box schemes remove this requirement by allowing the receiver to operate solely on the generated text, but current approaches rely on fixed-length, open-loop watermarking techniques that suffer from high decoding error rates under variable-length token generation. We recast black-box LLM steganography as a sequential communication problem with causal, noiseless feedback: every generated token is observed by both parties and can guide subsequent embedding. Based on this perspective, we introduce Burnashev Adaptive Posterior Matching (BAM), a feedback-coding scheme that combines posterior matching with a decode-and-confirm phase. The design is inspired by classical information-theoretic feedback-coding principles, while its security is established through a cryptographic reduction proof. Across three open-weight language models, we demonstrate that BAM attains 0-0.1% empirical message error on an 8-bit payload in around 50 tokens, across 1000 trials, versus 10-17% for the strongest black-box baseline at comparable length. Building on the proposed steganography algorithm, we demonstrate the feasibility of an end-to-end communication protocol that achieves high communication rates across multiple conversational settings.

We determine the most and least likely shapes for an instance of the critical beta-splitting tree via a connection to data compression and Huffman's minimum redundancy codes. This allows us to answer combinatorial questions about the distribution of clades posed by Aldous and Janson, stated as problem 7 in arXiv:2303.02529.

We prove the remaining positive characteristic cases of the Borisov-Alexeev-Borisov conjecture for geometrically integral $\varepsilon$-klt del Pezzo surfaces over arbitrary fields. Bernasconi and Martin established boundedness outside characteristics $2$, $3$, and $5$, and reduced the remaining cases to a uniform bound for the irregularity. We prove this bound, thereby completing the boundedness theorem in all positive characteristics.

We establish a spectral weak-to-strong stability criterion and a sharp Sobolev dichotomy for Maxwell–Ohm evolution with prescribed velocity on $\T^d$, $d=2,3$. Uniform spatial spectral tightness, together with weak convergence of each fixed spatial Fourier mode and strong convergence of the electromagnetic initial states, yields strong convergence of the electromagnetic fields, weak convergence of the currents, and distributional convergence of the Lorentz forces. In particular, weak convergence of the velocity coefficients in $L^2_tH^s_x$ suffices when $s>d/2$. At the critical index $s=d/2$, this conclusion fails sharply. We construct smooth divergence-free velocities converging strongly to zero in the critical Sobolev space, together with electromagnetic initial states converging strongly to zero, while the corresponding terminal fields remain of order one and the currents stay bounded. In two dimensions the construction is based on a moving logarithmic core, whereas in three dimensions it uses a Fourier-capacity core built from Leray-projected lattice modes and a polarized six-component Maxwell packet with exact magnetic divergence constraint. In both dimensions, the Lorentz forces converge to an explicit nonzero measure supported on a single ray with fixed unit direction.

We construct global finite-energy weak solutions for a 1D barotropic Navier–Stokes–Maxwell system with displacement current and the algebraic Ohm law. The result holds for every \(\gamma>1\) and arbitrary finite-energy initial data, allowing vacuum and requiring only \(L^2\) initial electromagnetic fields. A key ingredient is a weak-to-strong compactness principle for the Maxwell–Ohm subsystem: weak convergence of the velocity coefficients in \(L^2_tH^1_x\), together with strong convergence of the initial fields, yields strong electromagnetic trajectories in \(C_tL^2_x\). This permits identification of the weak current and the distributional Lorentz force without strong convergence of the velocity. The electromagnetic closure is combined with the classical artificial-viscosity and artificial-pressure construction for compressible flow. A density-primitive effective-flux argument identifies the physical pressure and eliminates the physical and artificial pressure-concentration defects.

Let $\mathbf{C}=\partial E\subset\mathbb{R}^{n+1}$ be an area-minimizing hypercone. The cone may have nonisolated singularities. We prove that there is a constant $c_{\mathbf{C}}>0$ such that \[ \frac{\operatorname{Per}(F;B_R)-\operatorname{Per}(E;B_R)}{R^n} \geq c_{\mathbf{C}} \left(\frac{|F\triangle E|}{R^{n+1}}\right)^2 \] for every $R>0$ and every set $F$ of locally finite perimeter with $F\triangle E\Subset B_R$. This extends the unweighted quantitative inequality from regular area-minimizing hypercones to general area-minimizing hypercones. We follow the calibration argument in the author's earlier work on regular cones. The main change is the construction of the vector fields. The pointwise asymptotic estimates for positive Jacobi fields used in the regular case do not directly apply here. We use the minimal foliations constructed by Zhihan Wang on the two sides of the cone. On each leaf, we integrate projection kernels with a positive Jacobi field as the weight. The kernel construction gives the required divergence bound. Wang's weak Harnack inequality and growth estimates give the integral bounds needed to prove linear growth of the vector fields.

Skew-symmetric matrix Schubert varieties are determinantal varieties obtained by intersecting matrix Schubert varieties with the space of skew-symmetric matrices. They are closely related to the orbit closures of the symplectic group action on the flag variety, and their torus-equivariant K-classes are the symplectic Grothendieck polynomials. We compute the Castelnuovo-Mumford regularity of skew-symmetric matrix Schubert varieties by giving a combinatorial formula for the degree of symplectic Grothendieck polynomials. In addition, we characterize the highest-degree homogeneous component of a symplectic Grothendieck polynomial and compute the maximal Castelnuovo-Mumford regularity of skew-symmetric matrix Schubert varieties.

We study fixed point theorems for actions of lattices of semisimple groups. They are deduced from vanishing results for the group cohomology of $L^p$-representations. We show that for lattices in simple groups of higher rank, the cohomology with $L^p$-coefficients vanishes below the rank whenever there are no invariant vectors. As a corollary of the vanishing for $L^1$-coefficients, we obtain that every action on an acyclic simplicial complex of dimension lower than the rank has a finite orbit. This in particular proves a conjecture by Farb. The $L^p$-vanishing below the rank proves a conjecture by Gromov regarding $L^p$-cohomology of symmetric spaces.

Multiparameter eigenvalue problems arise in boundary value problems, stability analysis, and delay-differential equations. Despite their importance, existing methods either require solving extremely large global problems or rely on local iterative techniques that only recover a few eigenvalues at a time. In this work we develop the first contour method for analytic multiparameter eigenvalue problems. The key theoretical ingredient is a new residue formula for multivariate matrix-valued analytic functions, extending Beyn's Keldysh-based residue theorem for meromorphic operator functions to several complex variables. Using this, we derive a multidimensional analogue of Beyn's contour method that computes all the eigenvalues of a multiparameter eigenvalue problem contained in a prescribed region of $\mathbb{C}^d$. The resulting algorithm targets eigenvalues in a region without constructing preposterously large matrices and is embarrassingly parallelizable. Numerical experiments demonstrate that we can now successfully solve large multiparameter problems arising in applications where existing approaches struggle.

We show that for any number field $K$ and any rational map $f$ in $K(x)$ that is not conjugate to a power, (signed) Chebyshev or Lattès map, then $K^{\mathrm{ab}}$ has the strong Bogomolov property for the canonical height of $f$. We also classify the pairs $(f,\alpha)$ whose backward orbit contains infinitely many abelian points. This settles the Andrews–Petsche conjecture to rational maps over a number field and to infinite abelian subsets of backward orbits. The authors were led to the main idea of the proof in conversation with Astra. The key insight consists of applying the equidistribution results \cite{Yua08}, followed by a classification of $(f,f)$-preperiodic curves \cite{Pak23,Pak20, Bea25} followed by an additional application of equidistribution on a parametrized preperiodic curve.

We prove a formula for Reidemeister numbers of virtually polycyclic groups that generalises known formulas for finite groups and torsion-free virtually polycyclic groups. Based on this formula, we introduce a new invariant which we call the tamed Reidemeister number, and prove that its associated zeta functions are rational on virtually polycyclic groups. We also show that this implies rationality of (ordinary) Reidemeister zeta functions, as well as rationality of Nielsen zeta functions on compact infra-solvmanifolds.

Let $X$ be uniform on $\{-1,1\}^n$, let $Y$ be obtained by passing its coordinates independently through a binary symmetric channel with crossover probability $p$, and let $g:\{-1,1\}^n\to\{0,1\}$ be a Boolean function. We give a computer-assisted proof of the Courtade–Kumar conjecture $I(g(X);Y)\le1-H_2(p)$, where $H_2$ is binary entropy, with equality attained by dictator functions. The present work builds on the differential-equation method, itself a limiting form of the auxiliary-receiver approach in network information theory using a continuum of degraded receivers. The proof proceeds from a local inequality to a dimension-independent bound on entropy production. Differentiation along the Boolean noise semigroup expresses entropy production as an average of edge costs. The key estimate is therefore an unrestricted Bellman inequality with two mean constraints and two entropy constraints, allowing arbitrary couplings of the edge variables. This paper and its supplement provide the proofs and computational verification records. The document is lengthy because it is designed to be entirely self-contained, deriving all proofs from first principles and reproducing the proofs of cited results. We also give a self-contained expository note explaining the reduction to a low-dimensional inequality and the ideas behind the key lower bounds. The entire proof, including all numerical certificates, has been formally verified in Lean end-to-end, and is available online.

We construct global solutions of the defocusing cubic half-wave equation on the torus whose $H^s$ norms grow exponentially for every $1/2<s<3/2$. Such solutions exist at every positive mass. To the best of our knowledge, this provides the first example of a genuine infinite cascade for a defocusing fractional nonlinear Schrödinger equation on the torus. In particular, our result resolves an open question posed by Gérard and Grellier concerning the existence of unbounded Sobolev trajectories for the defocusing cubic half-wave equation on the torus.