Papers

We construct quintic surfaces in the three-dimensional projective space $\mathbb P^3$ with $18$ ordinary cusps. Our starting point is the Barth–Rams description of quintics containing a $3$-divisible set of $12$ cusps. A specialization in which the two contact cubics are singular along two skew lines produces a family with $16$ cusps, and examples with $18$ cusps can be found over small finite fields. Our main construction is based on quintics admitting two Barth–Rams decompositions. The corresponding sets of $12$ cusps meet in $7$ points, and we prove that the locus of quintics admitting two such decompositions contains a $6$-dimensional component in the moduli space whose general member has $17$ cusps. This makes it possible to find members with $18$ cusps efficiently over finite fields. We lift one of these surfaces to characteristic zero using Newton–Hensel lifting and LLL reconstruction, obtaining a quintic over a number field of degree $22$. We verify that this surface has $18$ ordinary cusps and no other singularities.

We construct phase transitions that proliferate condensable anyons in general 2+1d topological orders, including non-abelian ones. The central tool that provides a systematic approach to this question is the Symmetry Topological Field Theory (SymTFT). For a given topological order, we identify the relevant symmetry from the transparent lines generated by the condensable anyons, and thereby realize the topological order in terms of a 3+1d SymTFT sandwich. The proliferation phase transition is realized by coupling scalar fields to the anyons purely on the symmetry boundary of the SymTFT. We illustrate the construction for abelian theories, as well as non-abelian ones, $D(S_3)$ and $SU(2)_k$ Chern-Simons theories, and extend it to anomalous anyons.

We find an explicit rank-one convex non-quasiconvex integrand in $\mathbb{R}^{2\times 4}$: to falsify the quasiconvexity inequality, we exhibit a map $\mathbb{T}^4\to \mathbb{R}^2$ with $12$ non-zero Fourier modes. In fact, this map is obtained from a scalar potential, so we also find a rank one convex integrand in $\mathbb{R}^{4\times 4}_\text{sym}$ which is not quasiconvex. These examples are obtained by transpositions and restrictions of Grabovsky's example of a rank-one convex, non-quasiconvex integrand in $\mathbb{R}^{8 \times 2}.$ We also modify Šverák's example to construct a rank-one convex non-quasiconvex integrand in $\mathbb{R}^{3\times 3}_\text{sym}$.

We compute explicit defining equations for eight fake quadrics arising as $\mathbb Z/2\times\mathbb Z/4$-covers of two singular $\mathbb Z/2$-Godeaux surfaces obtained in earlier work of the second author. Starting from explicit equations for the universal covers of the Godeaux surfaces, we reconstruct the relevant character eigenspaces and determine the homogeneous ideals of the bicanonical models of the eight fake quadrics in $\mathbb P^8$. All eight models are defined over $\mathbb Q$. We prove that the surfaces are pairwise non-isomorphic and rigid. Combined with the non-product result established in the earlier work, this gives the first explicit projective models of fake quadrics which are not isogenous to a product of curves.

The min-plus process is a stochastic coagulation-annihilation-type process on the binary tree, of interest in mathematics, physics, and computer science as a tractable instance of max-type recursive distributional equations. We carry out large Monte Carlo simulations at effective tree depths up to $N=60$ that provide finite-depth corroboration of the Beta(2,1) stretched-exponential limit for its root value $X_{N}$ at $p=1/2$, on the asymmetric $\sqrt{N}$ side of the random-homogeneous-systems classification recently introduced by Chen, Duquesne, and Shi and by Morfe. Off criticality, our simulations confirm the sub-critical closed form $\mathbb{P}(X_{\infty}=1)=(1-2p)/(1-p)$ within Monte Carlo error and document a super-critical mean growth exceeding the elementary $(2p)^{N}$ lower bound at the depths we reach. For a Bernoulli($q$)-initial-condition variant, we identify an elementary closed-form identity at $p=1/2$ that pins down the order parameter $\mathbb{P}(X_{N}=0)=q$ exactly, locates the absorbing-state phase transition at $p_{c}=1/2$ in the operator-mixing probability rather than in the initial-zero density, and shows that the conditional law on positives deforms substantially with $q$. Our simulations use a level-wise recursion and an FFT-based precomputed leaf table which reduce the effective simulation depth while preserving the recursive tree law and may be useful for the simulation of related recursive equations on large trees.

Let $(p,q)$ be a Jacobian pair. We show that the real Jacobian conjecture holds if the degree of $p$ is 7 and the highest degree homogenous part is of the form $\alpha x^7 + \beta x^6y$ for $\alpha^2+\beta^2 \neq 0$. We then show that there are no atypical Jacobian pairs such that $\mathrm{deg} \, p =7$ and $\mathrm{deg}\, q$ is even and coprime with 7.

Prediction markets have become a prominent way of aggregating beliefs about binary future events, and their price processes are often interpreted as evolving win probabilities, or “win-martingales.” Motivated by this perspective and recent work on Aldous' “most exciting game,” we study when a decision maker should stop observing a win-martingale and make a decision about the outcome. In particular, we allow the true outcome to be revealed at a fixed finite horizon, as in a sports game or election. Under a general terminal loss and running cost, we reduce the Bayes risk to an optimal stopping problem for the win probability process. When the win-martingale is a diffusion and its volatility separates into a deterministic time factor and a state-dependent factor, a deterministic time change transforms the problem into one for a time-homogeneous diffusion with a generally time-inhomogeneous running cost. Under explicit structural assumptions, we obtain a complete free-boundary characterization of the solution to the stopping problem in both finite and infinite horizons without discounting. We prove smooth-fit and $C^1$ regularity of the value function, $C^1$ regularity of the optimal stopping boundaries before the horizon, and derive a nonlinear integral equation that characterizes the boundaries uniquely. Taken together, these results yield a common decision-theoretic framework and solution theory for a broad class of posterior dynamics that includes the Aldous, Bass, and binary sequential-inference martingales as special cases. Our analysis requires no temporal monotonicity of the running cost and therefore accommodates highly nonmonotone stopping boundaries. In particular, we exhibit an example in which the optimal boundary has no limit as the calendar-time horizon is approached.

We study the quasisymmetric classification of limit sets of Kleinian groups and obtain a characterization of those geometrically finite limit sets that are quasisymmetrically universal. This allows us to obtain a quasi-isometric classification of finitely generated Kleinian groups. We also obtain a classification of virtually topologically rigid geometrically finite Kleinian groups.

Many nonlinear models across physics, chemistry, and biology exhibit multiple solutions for the same parameters, and capturing this entire solution set is essential for understanding pattern-forming systems. Yet existing learned surrogates are fundamentally single-valued: neural operators map each parameter to a single output, and physics-informed neural networks converge to one branch. We develop PatternFormer (PF), a large language model-based framework for learning the multiple solutions of nonlinear partial differential equations. By transforming unordered coexisting solutions into canonical sequences, PF produces structured solution sets in a single autoregressive pass, terminating automatically for finite families and enforcing physical residual constraints for unbounded ones. On nonlinear elliptic problems it recovers all solution branches in one inference step; on Gray–Scott it generates coexisting Turing patterns, including physically valid states absent from the reference data and beyond training. PF can also be sequentially fine-tuned across multistable systems, toward general foundation models for solution landscapes.

We focus on the finite basis problem for flat semirings of the form $S(W)$, where $W$ is an arbitrary set of nonempty words. We prove that $S(W)$ generates a Cross variety (and hence is finitely based) whenever every word in $W$ has length at most $3$, whereas it is nonfinitely based whenever there exists $k \geq 3$ such that $W$ is $x^{k+2}$-free but not $x^{k+1}$-free. In particular, if $W_k$ denotes the set of all words of length $k$, then $S(W_k)$ is finitely based if and only if $k \leq 3$. Moreover, $S(W)$ is nonfinitely based whenever $W$ is finite and not $x^4$-free. These results provide a partial answer to an open problem raised by Jackson et al.~(J Algebra 611: 211–245, 2022).

We establish a sharp version of Grothendieck's theorem for Bessel sequences. Precisely, given a Bessel sequence $\{ x_j \}_{j\in\mathbb{N}}$ with Bessel bound $1$ in a Hilbert space, we show that there exists functions $\{ f_j \}_{j\in\mathbb{N}}$ belonging to the unit ball of $L^\infty([0,1])$ such that for all $j,k \in \mathbb{N}$ one has $$ \langle x_j,x_k\rangle = \int_0^1 f_j(x)\overline{f_k(x)}\,dx.$$ As an application, we give an affirmative answer to an extension problem of Olevskii: if $E \subset [0,1]$ is a Lebesgue measurable set such that $[0,1]\setminus E$ has positive measure, then every Bessel sequence in $L^2(E)$ with Bessel bound $1$ extends to an orthonormal system in $L^2([0,1])$ that is bounded by the (optimal) constant $\lambda([0,1]\setminus E)^{-1/2}$ on $[0,1]\setminus E$. A formalization of our main result in Lean 4 accompanies the paper.

While the solution to the $1+1$ dimensional stochastic heat equation with multiplicative noise is closely related to the exponential of a Brownian motion, the two-dimensional picture exhibits an additional weak-to-strong disorder transition. In [CSZ '23], the critical two-dimensional stochastic heat flow (SHF) was constructed as the scaling limit of the partition function of $2+1$ dimensional directed polymers under the logarithmic intermediate-disorder scaling at criticality. The SHF is a random measure and, like many naturally occurring random measures, it is expected to exhibit rich intermittency. [CSZ '25] established that it is almost surely singular with respect to the Lebesgue measure. More recently, [GT '26] showed that the logarithm of the SHF averaged over small balls is asymptotically Gaussian, with both its mean and variance diverging as the ball radius tends to zero. In this paper we prove a sharp result quantifying the singularity of the support of the SHF as well as its intermittency. In particular, we show that, almost surely, for all small $\varepsilon>0$, up to a vanishing error, all the mass of the point-to-plane SHF in any domain is concentrated on ${1}/{\big(\varepsilon^2\log^{1/2+o(1)}(1/\varepsilon)\big)}$ balls of radius $\varepsilon$, each containing $\varepsilon^2{\log^{1/2+o(1)}(1/\varepsilon)}$ mass, thus precisely establishing its logarithmic fractal behavior. A key ingredient in the proof is a refined large-deviations theory, which allows access to conditional distributions, by taking advantage of the Gaussian-like behavior of the SHF at quasi-critical scales. A further useful observation that features prominently is that conditioning a Brownian motion on its endpoint being unusually large essentially induces a shift in the mean of its increments, and consequently, at small enough scales, their distributions do not alter significantly.

We prove the conjecture of Huh, Matherne, Mészáros, and St.~Dizier that the normalization of every skew Schur polynomial in finitely many variables is Lorentzian. We first realize every nonzero skew Schur polynomial in finitely many variables as a specialization of a Schubert polynomial and prove that it is dually Lorentzian. The dual Jacobi–Trudi identity then identifies its normalization with the finite dual of a skew Schur polynomial obtained by rectangular complementation. As a consequence, skew Kostka numbers satisfy log-concavity inequalities along the root directions.

This paper develops an analytical framework to evaluate the feasibility and performance of satellite infrastructure sharing among multiple low Earth orbit (LEO) satellite operators. Motivated by the growing demand for universal connectivity under limited satellite resources, the proposed model captures uncoordinated deployments where independently operated constellations coexist without predefined orbital agreements. To describe such heterogeneous configurations, the spherical Cox point process is employed to jointly generate orbital structures and satellites. Then, each satellite is further assigned a random communication range, reflecting variations in coverage capability across operators. The overall coverage region is modeled through a spherical Cox-Boolean model that captures the spatial overlap of individual satellite spherical footprints on Earth. Using the proposed framework, the feasibility and benefits of satellite infrastructure sharing are mathematically analyzed, and closed-form expressions are derived for key performance metrics such as the connection probability, connection number, and downlink signal characteristics including the nearest serving distance, total received signal power, and the signal-to-interference ratio (SIR) distribution in the interference-limited regime. The analytical results, validated through system-level simulations, provide a tractable characterization of how orbital geometry governs coverage, connectivity, and interference, and reveal the inherent trade-offs induced by coverage overlap in heterogeneous satellite constellations.

A nonlinear model of a memristor based on charged mobile vacancies is considered taking into account the electrostatic interaction between them. This interaction significantly affects the stationary (limiting) vacancy distributions formed under the action of the electric current flowing through the memristor, for which analytical expressions are obtained in this work. Between the regions with reduced and increased vacancy concentrations, an intermediate electrically neutral region is formed due to the electrostatic interaction. Interestingly, the limiting resistances in the “on” and “off” states of such a memristor do not depend on the strength of the electrostatic interaction, at least in the leading first order in the vacancy concentration.

In this paper we prove the Corona theorem for slice hyperholomorphic functions in a general way that applies to functions with values in a Clifford algebra $\mathbb R_n$ for any $n\geq 2$. This general proof is based on a matrix version of Corona's theorem and then exploiting some symmetries of the matrices that allow to reconstruct the values of the functions. The crucial idea is to tie together the algebra of the functions, the Bezout identities and the Carleson-type conditions. The results we obtain rest on a new way of writing the slice product which improves the standard approach and which holds in general dimension.

Let $\pi$ and $\pi'$ be unitary cuspidal automorphic representations of $\mathrm{GL}(n)$ and $\mathrm{GL}(n')$ over a number field $F$. Let $\mathfrak{C}_{\pi}$ be the analytic conductor of $\pi$. We develop a new approach to zero-free regions for $L$-functions via lower bounds for power sums, proving for all $\varepsilon>0$ the existence of ineffective constants $c=c_{n,F,\varepsilon}>0$ and $c'=c'_{n,F,\pi',\varepsilon}>0$ such that the standard $L$-function $L(s,\pi)$ satisfies \[ |L(\sigma+it,\pi)|\geq c(\mathfrak{C}_{\pi}(|t|+3))^{-\varepsilon},\qquad \sigma\geq 1-c(\mathfrak{C}_{\pi}(|t|+3))^{-\varepsilon} \] and the Rankin-Selberg $L$-function $L(s,\pi\times\pi')$ satisfies \[ |L(\sigma+it,\pi\times\pi')|\geq c'(\mathfrak{C}_{\pi}(|t|+3))^{-\varepsilon},\qquad \sigma\geq 1-c'(\mathfrak{C}_{\pi}(|t|+3))^{-\varepsilon}. \] Applications include improvements to the prime number theorems for these $L$-functions and new generalizations of the Brauer-Siegel theorem.

We develop a general principle for passing from the analysis of partial differential operators on the whole space to boundary value problems on the half-space. The reduction separates the boundary problem into restricted invertibility of the interior operator and a complementing problem formulated entirely on the trace space. For a broad class of operators admitting suitable factorizations, the restricted invertibility is governed by the whole-space theory of the factors, while the complementing problem determines both well-posedness and the admissible combinations of interior and boundary data. In this way, boundary value theory is reduced to whole-space operator theory together with an explicit analysis on the trace space. We realize this principle for time-periodic partial differential equations and systems of mixed order, developing the required Newton-polygon function spaces, interpolation theory, and $L^p$-trace theory. For mixed-order differential operators under general boundary conditions, well-posedness and the admissible data space are determined by an explicit complemented boundary matrix. This yields new $L^p$-well-posedness results for Cahn–Hilliard equations with dynamic boundary conditions and for Cahn–Hilliard–Gurtin systems. Most notably, we obtain the first maximal $L^p$-regularity result for a genuinely mixed-order interior system coupled to dynamic boundary conditions carrying a distinct mixed-order structure.

Let $\mathcal{P}_{N}$ be the $(N+1)$-dimensional Hilbert space of analytic polynomials of degree at most $N$. This is the natural environment to define $SU(2)$ (Bloch) coherent states. Let $Q_{\rho }$ be the Husimi function of a density operator $\rho $ on $% \mathcal{P}_{N}$. We prove an isospectral version of Lieb-Solovej inequality: if $\rho ^{\downarrow }$ is obtained by placing the eigenvalues of $\rho $ in decreasing order along the monomial basis, then \begin{equation*} \int_{\mathbb{C}}\Phi (Q_{\rho }(z))\,dm(z)\leq \int_{\mathbb{C}}\Phi (Q_{\rho ^{\downarrow }}(z))\,dm(z) \end{equation*}% for every convex function $\Phi $ on $[0,1]$. Applying the corresponding reversed inequality to the concave function $\Phi (t)=-t\log t$ gives the Wehrl entropy. In the process it is show that the output state of $\rho $ under Lieb-Solovej's channel is majorized by the output of the state $\rho ^{\downarrow }$. As an application, among all measurable subsets of the sphere having a fixed area, spherical caps maximize every partial sum of the eigenvalues of Toeplitz operators with symbol $1_{\Omega}$. Equivalently, caps maximize all Ky Fan norms, leading to the sharp inequalities, previously known for $r=1$: \begin{equation*} \sum_{j=1}^{r}\lambda_{j}(\Omega) \leq r-\sum_{k=0}^{r-1}(r-k)\binom{N+1}{k} m(\Omega)^{k}\bigl(1-m(\Omega)\bigr)^{N+1-k}. \end{equation*} This implies isoperimetric inequalities for all Schatten sums of $T_{\Omega}$, obtained without using the spherical isoperimetric inequality.

Let $\widehat R$ be the Pearson sample correlation matrix formed from $n$ independent Gaussian observations in $p$ dimensions, and write $m=n-1\ge p$. Under the null correlation $R=I_p$, the classical independent beta product, exact cumulants, and full Fourier inversion yield, along every sequence $p\to\infty$ with $m\ge p$, a uniform first Edgeworth expansion for $\log\det\widehat R$, centered by its exact mean and scaled by its exact standard deviation. The expansion identifies the exact finite dimensional skewness correction and gives the sharp Kolmogorov equivalent $A_{m,p}/\{6\sqrt{2\pi}V_{m,p}^{3/2}\}$, where $V_{m,p}$ is the exact variance and $A_{m,p}$ is the absolute third cumulant. This equivalent unifies the square, fixed gap, growing gap, proportional, and dilute regimes; in the square regime the error has order $(\log p)^{-3/2}$ with an exact constant. For every positive definite population correlation matrix $R$, we prove a uniform finite sample Berry-Esseen bound that explicitly tracks population dependence. All theoretical results have exact or proved equivalent Lean 4 formulations whose declarations and dependencies are kernel checked.

Let $K$ be a global function field with full constant field $\mathbb{F}_q$, let $T$ be a finite set of finite places, and let $\infty$ be a distinguished place. We study the maximal pro-$\ell$ extension of $K$ which is unramified outside $T$ and in which $\infty$ splits completely. Using factorisation results for $\ell$-primary idele class characters, we prove that its Galois group is multiplicator-free if $\ell \nmid q - 1$ or $T \neq \varnothing$. In particular, if $T \neq \varnothing$, the conclusion holds for every prime $\ell$.

Let $R$ be a $d$-dimensional Noetherian local ring with maximal ideal $m_R$ and let $I$ be an ideal of $R$. The epsilon multiplicity $\epsilon(I)$ of $I$ is defined by Ulrich and Validashti in \cite{UV} as a limsup of normalized lengths. We show in this paper that the epsilon multiplicity is always a limit of these normalized lengths. Specifically, $$ \epsilon(I)= \lim_{n\rightarrow\infty} \frac{\ell_R((I^n)^{\rm sat}/I^n)}{n^d/d!}. $$ We prove that the epsilon multiplicity satisfies an associativity formula, analogous to that of the ordinary multiplicity. We prove these results by comparing the epsilon multiplicity to a suitable limsup of normalized Amao multiplicities, the Amao epsilon multiplicity $a\epsilon(I)$. We show that the epsilon multiplicity is always equal to the Amao epsilon multiplicity, which is itself a limit.

The $n$-dimensional hypercube $Q_n$ is the graph with vertex set $\{0,1\}^n$ in which two vertices are adjacent if they differ in exactly one coordinate. For a graph $H$, let $\operatorname{ex}(Q_n,H)$ be the maximum number of edges in an $H$-free subgraph of $Q_n$. The hypercube Turán density of $H$ is defined by $\pi_{\square}(H)=\lim_{n\rightarrow\infty}\operatorname{ex}(Q_n,H)/|E(Q_n)|$. In this note, we prove \[ \frac{1}{8} \leq \pi_{\square}(C_{10}) \leq 0.36577. \] For the upper bound, we prove $\pi_{\square}(C_{10}) \leq \pi_{\square}(C_6)$, which, together with a result of Baber, gives the stated upper bound. For the lower bound, we prove that $\operatorname{ex}(Q_n,C_{10}) > |E(Q_n)|/8$ for every $n \geq 2$.

The dissipative dynamics of a pair of wedge disclinations with opposite Frank angles confined to a circular domain is investigated under the assumption of radially symmetric motion. Two distinct dynamical regimes are identified: a diverging-disclination regime and an annihilating-dipole regime. The stationary points associated with both regimes are determined and fully characterized in terms of their stability, and quantitative estimates for the characteristic evolution times in their vicinity are derived. In the annihilating-dipole regime, it is shown that, after a suitable time rescaling, the resulting law of motion for the dipole coincides with that of an edge dislocation.

The goal of this paper is to investigate the traveling wave solutions with one-sided far field conditions for the one-dimensional hyperbolic Keller-Segel equations with quorum sensitivity. The traveling wave solutions are piece-wise smooth in the moving coordinate, and satisfy the entropy inequality at the discontinuity.

We develop an intrinsic calculus for smooth functions of paths of arbitrarily low regularity on smooth manifolds. The regularity of paths is defined in terms of a $p$-th variation tensor along a sequence of partitions, for an arbitrary integer $p$; this tensor is constructed as a local symmetric tensor measure along the path. We define pathwise integrals of closed one-forms along paths with finite $p$-th variation and derive a change of variable formula for smooth functions of such paths. For $p=2$, our results give a manifold version of H. Föllmer's pathwise Itô calculus. Our construction only requires an affine connection on the manifold and may be viewed as a higher-order analogue of L. Schwartz's second-order differential geometry. The connection provides a splitting of higher-order tangent vectors into symmetric tensor components and leads to a geometric transfer principle: the change-of-variable formula defines an intrinsic, connection-independent functional of the reduced $p$-jet of the test function, whose canonical highest-order component is determined by the $p$-th variation tensor. Although our results are purely geometric, they apply to manifold-valued stochastic processes with highly irregular paths and yield a higher-order Itô-type calculus for such processes. We illustrate this calculus for exponential lifts of fractional Brownian motions to Riemannian manifolds and Lie groups.

Classical rate-distortion theory characterizes the fundamental limits of lossy compression under fidelity constraints, but minimizing distortion often yields perceptually unsatisfying reconstructions - blurry images, over-smoothed textures, and unnatural artifacts. This has motivated a growing body of work on compression with realism constraints, which require reconstructions to be statistically indistinguishable from natural signals, giving rise to the three-way rate-distortion-perception (RDP) trade-off. This paper provides an accessible overview of this emerging area and reveals deep connections to another fundamental problem: distributed coordination under rate-limited communication. Under strong distribution matching formulations, both problems lead to nearly identical information-theoretic characterizations, both require common randomness (CR) for optimal performance, and both rely on similar analytical tools such as the soft covering lemma. Beyond a unifying perspective, we survey recent developments in formalizing realism, including batched critics and algorithmic realism, and propose to transfer such paradigms to coordination - illustrating how the connection continues to generate new problems.

We study the quantum geometry of doubly degenerate energy levels in a four-dimensional parameter space. We find that the scalar quantum metric $g$ and the Berry curvature $F$ obey $(\operatorname{tr} g)^2/16\geq\sqrt{\det g}\geq |2\Tr(F\wedge F)-\Tr F\wedge \Tr F|/24$. The first inequality characterizes the anisotropy in the metric. The second determinant inequality measures the self-duality of the traceless $SU(2)$ part of the curvature under Hodge star operation and the algebraic closedness of the inter-level polarization amplitudes under $SU(2)$ rotations in the doubly degenerate levels. The saturation of the determinant bound imposes a quaternionic Cauchy-Riemann equation, analogous to the complex analyticity imposed by ideal-band conditions in two-dimensional Chern insulators. As examples, four-band Dirac Hamiltonians automatically saturate the determinant bound and possess a topological zero in $\operatorname{tr}(F\wedge F)$. We compare the differences between Kramers degeneracy and ordinary $U(2)$ degeneracy. In addition to the non-Abelian geometric bound, the latter also obeys an independent first-Chern bound.

A rigorous framework for consistent study of junctions in a coordinate independent manner is presented. This is achieved by extending a theory of distributions in curved spacetimes with torsion and combining it with covariant formalism. In this way, one can derive general conditions on the differentiability of the joined manifold. Given a specific theory of gravity, in particular the Einstein-Cartan-Sciama-Kibble one, we evaluate the conditions for obtaining a smooth junction. These conditions can be successfully applied independently of the choice of coordinates used to describe the metric, thereby, evading the drawbacks of coordinate dependent Israel-Darmois framework.

The Hajnal–Szemerédi theorem states that every graph with $\Delta(G)\le k-1$ admits an equitable $k$-coloring. Using a reduction implicit in Aboulker, Oijid, Petit, Rocton, and Simon (which itself builds on the literature surrounding feedback arc sets), we prove that this theorem is equivalent to each of four digraph analogues guaranteeing equitable acyclic $k$-colorings under various degree conditions. A minor modification of the proof method in makes the construction algorithmic, so the equitable acyclic coloring guaranteed by these statements can be found in polynomial time. The same reduction shows that the Ore-type Hajnal–Szemerédi theorem is equivalent to its four digraph analogues. These equivalences settle the conjecture of Treglown that every $n$-vertex digraph with $r\mid n$ and $\delta^{\max}(D)\ge(1-\frac{1}{r})n$ contains a $T_r$-factor, answer a question of Molla and Treglown by determining the exact Ore-type threshold for $T_r$-factors, and recover the directed Hajnal–Szemerédi theorems of Czygrinow, DeBiasio, Kierstead, and Molla as immediate corollaries of this reduction. We also discuss related problems for oriented graphs which do not follow immediately from the reduction mentioned above.