Papers

We classify Boolean cubic forms in ten variables up to GL(10,2)-equivalence. The catalog contains all 3691560 nonzero orbits. For every orbit we provide a representative with small monomial count, the stabilizer order, and the alternating rank together with an explicit decomposition. The classification is obtained by rank-stratified enumeration. We verify completeness by the Burnside orbit count and independently by the orbit–stabilizer identity. We also provide a fast, complete GL(10,2)-invariant. By polarization, this gives the first complete classification of alternating trilinear forms in dimension 10 over GF(2).

We show that the category of lax involutive $n$-Frobenius algebras is Quillen equivalent to the category of right comodules of the $n$-Poisson cooperad. It follows in particular, that the cohomology of a parallelized $n$-manifold is naturally endowed with a homotopy involutive $n$-Frobenius structure extending the rational homotopy type of $M$, solving a long-standing question.

In 1973, Russian mathematician I.R. Shafarevitch delivered a lecture to the Göttingen Academy of Sciences on the purpose of mathematics. The conclusion he reached in his address is that the ultimate purpose of mathematics must be religious. In this talk, we will explore a possible way in which this claim can be justified by understanding the purpose that mathematics served within a person's intellectual formation according to Plato. To place Plato's view into a Christian perspective, we will then investigate the thought of St. Augustine of Hippo, the great fifth century theologian and bishop. Augustine's insight on the role that number plays in the development of reason sheds light on how knowledge of mathematics conduces to knowledge of God.

Beamfocusing is the established near-field strategy for a large array serving a single-antenna user. We consider the single-user line-of-sight MIMO link, free of multipath, in which the user, too, carries an extended aperture, and show that the focusing prescription inverts: beyond a modest Fresnel number, focusing on the user is outperformed by far-field steering. Under fully analog, unit-modulus beamforming, we derive closed-form power gains for focusing (each aperture phase-matched to the other's center) and for steering (a planar phase ramp) in the Fresnel regime, and prove that their comparison is governed by two dimensionless quantities: the link Fresnel number, the product of the two aperture lengths normalized by wavelength and link distance, and the aperture ratio, irrespective of how many elements discretize the apertures. For equal apertures the two gains cross exactly once, at the universal value 1.947; beyond it, focusing loses ten dB per decade of Fresnel number, and the advantage celebrated in the MISO literature survives only as the receive aperture vanishes. We then derive the strategy that is order-optimal at every Fresnel number, con-focusing: both apertures aim at the common point from which they subtend equal angles. It attains the rank-one eigenbound in leading constant, needs no channel knowledge, degenerates to plain steering for equal apertures, and is acquirable within one beam-refinement round with no geometry exchange between the terminals.

The ranking of fuzzy numbers has become a challenging task in fuzzy set theory due to their complex, multi-dimensional nature. While the Klir-Yuan partial order provides a natural term-wise comparison of $\alpha$-cuts, it often leaves many fuzzy numbers incomparable. To address this, various ranking methods have been developed to construct total preorders between them. However, many classical approaches suffer from significant information loss as they imply a defuzzification process. On the other hand, approaches such as admissible orders allow defining total orders, but at the expense of imposing strict algebraic rules that may contradict human intuition. In this study, we introduce a generalized sequential ordering framework to overcome these limitations. By establishing a sequence space over a totally preordered base space, we construct a flexible lexicographical structure that sequentially resolves ties. We prove that this framework yields total preorders and, under injectivity conditions, total orders. Furthermore, we analyze the compatibility of these sequential orders with the notion of admissibility. We also show that our proposed framework provides a unified mathematical umbrella that encompasses and generalizes existing ranking techniques, offering highly discriminative ordering relations for fuzzy numbers and beyond.

This paper treats variable-order time-fractional subdiffusion with discontinuous coefficients across a curved interface using $L2\!-\!1_\sigma$ time stepping on graded meshes and a symmetric interior penalty FEM on body-fitted meshes. Stability and optimal a priori error estimates in a discrete-in-time $L^2$ norm are established, yielding second-order temporal accuracy. While analysis typically assumes $\alpha_n$ at $t_{n-\sigma_n}$ lies in the range of $\alpha(t)$ on $[t_{n-1},t_n]$ and $\alpha_n\le \alpha(t_{n-\alpha_n/2})$, experiments indicate the second inequality can be relaxed or omitted, enabling straightforward selection of $\alpha_n$ from many admissible values without solving a nonlinear equation. Numerical results verify temporal rates $\min\{2,r\delta\}$, spatial order $\min\{s,k+1\}$, and robustness to superconvergent points and interface geometry.

Let $(X,\mu)$ be a measure space and let $1< p< \infty$. We study quantitative stability refinements of Minkowski's inequality \[ \| f + g\|_{p}\leq \| f\|_{p} + \| g\|_{p} \] for real-valued functions in \(L^p(X,\mu)\). We first establish a stability estimate for arbitrary real-valued functions and show that its constant is sharp. We then prove that, for nonnegative functions, the constant can be improved when \(p\geq 2\), again to its optimal value. More precisely, if \(f,g\geq 0\) and \(f,g\neq 0\), then \[ \| f + g\|_{p}\leq \| f\|_{p} + \| g\|_{p} - c_{p}\min \{\| f\|_{p},\| g\|_{p}\} \left\| \frac{f}{\| f\|_{p}} -\frac{g}{\| g\|_{p}}\right\|_{p}^{\alpha_{p}}, \] where \[ c_p = \begin{cases} \dfrac{p-1}{4}, & 1<p\leq 2,\\[6pt] \dfrac{1-2^{1-p}}{p}, & 2\leq p<\infty, \end{cases} \qquad \alpha_p = \begin{cases} 2, & 1<p\leq 2,\\ p, & 2\leq p<\infty. \end{cases} \] Both constants are best possible.

Anytime-valid confidence sequences and e-processes are built almost universally from one recipe: average exponential test statistics over a prior on the tilting scale, then invoke Ville's inequality on the resulting nonnegative supermartingale. The mixing prior sets the width of the detection boundary and is usually chosen by hand. We recast the recipe as a two-player game with information as currency. A Learner commits to the prior; Nature adaptively produces a mean-zero score process whose difficulty is priced by a cumulant-generating-function charge. The Learner's mixture wealth obeys a single pathwise Gibbs-variational identity that holds along every realized path with no expectation operator; Ville's inequality, the equalizer condition, the GROW characterization, and the saddlepoint formula are all specializations of it. Three messages organize the rest. First, the law of the iterated logarithm (LIL) is the minimax boundary of this sequential-detection game, not arbitrary combinatorial slack. Second, the optimal prior is not a design choice but the forced equalizer strategy – the unique law that makes every boundary-crossing time equally costly for Nature – and it yields the sharp first iterated-log correction in closed form, with coefficient 3/2 = 1 + 1/2 (one for the Erdős baseline, one half for the Laplace envelope around the saddle). Third, in the log-log scale chart the equalizer is exactly the Jeffreys prior on the scale-of-scales. The Erdős-Kolmogorov integral test is the criterion that selects it. The two-stage finite-time LIL proof, the Howard-Ramdas mixture and stitching constructions, and betting confidence sequences all read as instances of this equalizer principle. A companion empirical evaluation confirms the central identities and locates the Erdős threshold at the predicted value.

People's opinions can change both from their interactions with each other and from their interactions with media sources. Bounded-confidence models (BCMs) of opinion dynamics provide one framework to study such dynamics. In a BCM, the nodes of a network are agents with continuous-valued opinions, and these agents interact with each other via the edges of the network. In this paper, we extend the original Deffuant–Weisbuch (DW) BCM by incorporating influence from two media sources – one with a positive value and one with a negative value – to capture the effects of a polarized media landscape. We show both numerically and analytically that our extended DW model exhibits drifting behavior in which a large cluster of opinions shifts toward one of the media agents. We analyze how the drift trajectory and speed depend on the model parameters, and we identify conditions in which drift is promoted or suppressed. Our results provide insight into how competing media sources can influence collective opinion formation in social systems.

This article relates the theory of embedded contact homology (ECH) with the dynamics of Anosov flows. We show that in many cases the ECH capacities of a symplectic 4-manifold are infinite, including cotangent disk bundles over closed oriented surfaces of genus at least two. We prove that ECH obstructs Reeb Anosov and Hamiltonian Anosov flows, addressing the four-dimensional case of a question posed by Herman in 1998. Further, we obtain Floer-theoretic obstructions to a 3-manifold admitting any Anosov flow. As an application, we give new constraints on the existence of embedded Lagrangians of genus at least two in symplectic 4-manifolds. In an appendix, some related results in all dimensions are proved for capacities constructed from rational symplectic field theory.

Many combinatorial designs ask for equal distribution of given symbols across the entries of a matrix. The paramount examples are Latin squares, where each symbol from $\{1,\dots,n\}$ appears once per row and column of an $n\times n$ matrix. Frequency rectangles extend this to $\lambda n \times \mu n$ matrices with repeated symbols under controlled column and row frequencies. In this more general setting, we examine structural properties of pairwise reflection-symmetry, which requires that, on every pair of columns, each ordered symbol pair $(p,q)$ occurs as often as its reversal $(q,p)$. This order-balance is precisely what makes head-to-head comparisons unbiased, i.e., no symbol gains a systematic advantage from the position it occupies relative to another, a fairness demand arising for instance when scheduling tournaments or laying out comparative trials. Existence of such objects for odd $\lambda$ turns out to be remarkably more subtle than for even $\lambda$. After showing that existence holds also for sufficiently large odd $\lambda$, we initiate the search for the smallest possible value of $\lambda$ in this setting. We obtain the insight that a per-column multiplicity of $\lambda=1$ can be achieved if and only if $n$ is a power of two. We complement the existence results with a direct product construction and add several further observations on the property. Finally, we propose and evaluate a quadratically constrained integer program to computationally search for these objects. The resulting experiments reveal that many of them possess an underlying group-theoretic structure which, as we conjecture, may even be unavoidable in certain settings.

This paper presents a multi-period mixed-integer linear programming (MILP) framework for planning the transition from conventional to electric aircraft in regional aviation. The model jointly optimizes fleet acquisition, infrastructure deployment, and service allocation over time, while accounting for policy constraints such as emissions reduction targets, electric service share, and budget limits. A real-world case study based on Helijet's short-haul network in British Columbia demonstrates the applicability of the model. The results show that electrification can reduce emissions by more than 70% within five years while remaining economically viable. However, the transition is primarily limited by the capacity of the fleet and operational structure, rather than the charging infrastructure, leading to unmet demand under direct aircraft replacement. These findings emphasize the need for coordinated planning across fleet sizing, scheduling, and route prioritization to ensure a practical and efficient transition to electric aviation.

Binary classification from positive-only samples is a variant of PAC learning in which the learner receives i.i.d. samples from the positive region of an unknown target concept, but is evaluated under the original distribution (which places mass on both positive and negative regions). This model dates back to Natarajan [1987, STOC], and the characterization of improper learning is well-known – it even appears in textbooks. The characterization of proper positive-only learning, however, has long remained open. In this work, we revisit and settle this question: a concept class is properly learnable from positive-only samples if and only if it has finite VC dimension and satisfies a new combinatorial condition, which we call uniform exterior separability. Together with several separation results, this characterization reveals a surprisingly rich landscape that differs sharply from standard PAC learning: proper and improper learning are separated, randomized and deterministic proper learning are separated, there are classes for which no ERM is a learner, and finite VC dimension does not suffice even for non-uniform learning. Along the way, we introduce new combinatorial dimensions that we believe can be of broader interest in learning theory.

We analyze Bregman ADMM for nonconvex linearly constrained problems under two-sided relative smoothness, a condition that replaces the standard Lipschitz gradient assumption with a Hessian comparison relative to a Bregman kernel. This setting covers polynomial objectives arising in matrix and tensor models for which a global Lipschitz-gradient constant need not exist. We show that on an invariant open state-space domain, one iteration of Bregman ADMM defines a smooth primal–dual fixed-point map whose strict-saddle KKT points are unstable fixed points; consequently, from random initialization the iterates converge to a strict saddle with probability zero. Combined with existing first-order convergence results, this yields almost-sure second-order stationarity of limiting KKT points. We extend the analysis to a multi-block star consensus formulation for distributed optimization. The technical novelty lies in a determinant reduction with a Bregman-specific symmetrization and scaling step in the two block spectral argument, together with a null space cancellation exploiting the star graph structure in the consensus case. Numerical experiments on distributed matrix factorization illustrate the theory, and a symmetric tensor factorization example demonstrates the broader Bregman proximal splitting idea beyond the separable consensus setting.

In recent work the authors determine complete columns of symmetric-group decomposition matrices in odd prime characteristic $p$ labeled by $p$-regular partitions for which every hook of length divisible by $p$ has even arm length. In the present paper we enumerate these partitions and prove that each block of $p$-weight $w$ contains precisely \[ \binom{w+\frac{p-3}{2}}{w} \] such partitions. More generally, for any integers $d,e>1$, we study and enumerate $d$-balanced $e$-regular partitions – partitions for which every hook of length divisible by $e$ has arm length divisible by $d$. Our first main result is that the crystal (affine) reflections preserve the $d$-balanced property for all $d,e > 1$. It follows that, for fixed $d$, $e$, and $w$, the number of $d$-balanced $e$-regular partitions in a block of $e$-weight $w$ is independent of the $e$-core. We then compute this number by working in RoCK blocks, obtaining an explicit binomial formula valid for every block. We also investigate closely related odd sequences of partitions. Among others, we find the generating function of the number of odd sequences occurring in a block. Alongside their representation-theoretic relevance, we expect these results to be of independent combinatorial interest.

This paper introduces a new application of the perfectly matched layer (PML) for mitigating model top wave reflections in geophysical fluid models. Typically, a strong Laplacian or Rayleigh damping sponge layer is used near the upper boundary, but these often need many vertical levels or a high model top to be sufficiently effective. An advantage of the PML is that, at the continuous level, it is free of wave reflection at the onset of the damping layer. This enables the PML to be effective even with a thin damping layer. We derive PMLs for the linear and nonlinear versions of the Boussinesq equations, which are a simplified model for vertical dynamics in the atmosphere. In the nonlinear system, we define a novel PML that damps perturbations from a hydrostatically balanced reference state. We approximate the PML equations using the compatible finite element method for numerical experiments. First, tests with the linear Boussinesq system show that the PML is more effective than a typical sponge layer in absorbing acoustic waves near the model top. Next, tests in the nonlinear system show that i) the PML can damp acoustic waves even when they are under-resolved by the time discretisation, and ii) the PML can avoid the standing wave pattern caused by model top reflection of orographic gravity waves. We propose that the PML is worth further development and investigation as a sponge layer alternative in dynamical cores for atmospheric modelling.

Inference-time scaling is a promising paradigm to improve generative models, especially when outputs must satisfy structural constraints or optimize downstream rewards. We consider Masked Diffusion Model (MDM) and introduce MDM-VGB, a discrete diffusion sampler that augments unmasking generation with theoretically principled reward-guided remasking. Inspired by the recent success of the classical Jerrum-Sinclair backtracking Markov chain in reward-tilted generation, MDM-VGB extends the backtracking random walk from a fixed prefix tree to a masked-state graph, allowing tokens to be unmasked and remasked at arbitrary positions. The resulting sampler favors unmasking and remasking moves that lead to higher-value partial configurations, enabling both effective high-reward generation and efficient repair of low-reward samples. We prove that MDM-VGB is robust to process-verifier noise and achieves quadratic complexity, while popular test-time heuristics such as best-of-$N$ can incur exponential complexity due to error accumulation. Our theoretical findings are corroborated by strong empirical performance, particularly on popular constraint-satisfaction and scientific benchmarks such as Sudoku and QM9.

We introduce Carleson analogs of the Bourgain–Brezis–Mironescu spaces $B$ and $B_0$ by measuring mean oscillation over upper Carleson tents. For these spaces, denoted by $B_{\mathcal C}^p$ and $B_{\mathcal C,0}^p$, we prove two types of structural results. First, we show that they contain several natural classes of functions, including BMO/VMO–Carleson spaces, tent-space potential classes, and fractional Sobolev classes. Second, motivated by Zhu's structural theorem for BMO spaces induced by the Bergman metric, we establish decompositions of $B_{\mathcal C}^p$ and $B_{\mathcal C,0}^p$ into bounded-oscillation and bounded-average components. We then revisit the Bourgain–Brezis–Mironescu rigidity phenomenon in the Carleson setting. Although the direct rigidity statement fails for $B_{\mathcal C,0}^p$, we introduce a natural $B_{\mathcal C}^p$-trace and prove that the rigidity theorem survives at the level of traces.

We study the composition of classically-controlled quantum instruments–the natural quantum analogue of Markov kernels. Classically, Markov kernels compose by integrating one kernel against another. Defining this composition for quantum instruments with continuous outcomes requires an integral of quantum channel-valued functions with respect to a quantum instrument. We construct this integral in the Heisenberg picture using the Okamura-Ozawa normal extension to a von Neumann tensor product. This integral recovers the expected finite formula, preserves normal complete positivity and subunitality, and provides the multiplication for a monad governing the composition of quantum instruments. As an immediate consequence, we identify the category of quantum Markov kernels as the Kleisli category of this monad.

We consider the initial value problem (IVP) for a two-parameter family of derivative nonlinear Schrödinger equations on the torus, known as the Majda-McLaughlin-Tabak (MMT) model arising in weak wave turbulence theory. For positive derivative order, we show that the flow map is not $C^3$ at the origin. Using an enhanced energy method, we prove unconditional local well-posedness in Sobolev spaces. At the energy regularity, conservation of the Hamiltonian and a mass-type quantity yields unconditional global well-posedness.

We study commutative topological algebras naturally associated with translation-invariant reproducing kernel Hilbert spaces whose direct integral decomposition has one-dimensional fibers. Starting from the bounded algebra of translation-invariant operators, we pass to a common dense domain generated by reproducing kernels and identify the corresponding diagonalizable operators with multiplication by symbols in an intersection of weighted $L^2$-spaces. On the symbol side this gives a canonical space $\mathcal F_0$ and a maximal multiplicative subalgebra $\mathcal F_M$, which is a complete locally convex $*$-algebra. Transporting the structure back yields corresponding algebras of operators and integral kernels. We also discuss when the inclusions $L^\infty(\Omega)=\mathcal F_\infty\subset \mathcal F_M\subset \mathcal F_0$ are strict, and illustrate the results with vertical and radial operators on classical Bergman and Fock spaces.

We construct a family of continua and pointwise periodic homeomorphisms realizing arbitrary polynomial entropy values in $[0,+\infty]$. In particular, this provides examples of pointwise periodic homeomorphisms with positive polynomial entropy. This contrasts with the fact that pointwise periodic homeomorphisms on connected manifolds and local dendrites have zero polynomial entropy.

We study first-order oracle complexity under the $\alpha$-Polyak-Lojasiewicz condition $f(x)-f_\star\leq \tau\|\nabla f(x)\|^\alpha$ for $\alpha\in[1,2]$. For $\alpha<2$, we first show that global $L$-smoothness together with a global $\alpha$-Polyak-Lojasiewicz inequality forces the objective to be constant. This motivates a nontrivial model in which smoothness remains global but the inequality is required only on the initial sublevel set. On this class, we establish sharp minimax lower bounds for every $\alpha\in[1,2)$. Deterministic first-order methods require $\Omega(L\tau^{2/\alpha}\varepsilon^{-(2-\alpha)/\alpha})$ oracle calls, matching gradient descent. With unbiased stochastic gradients of conditional variance at most $\sigma^2$, randomized first-order methods require $\Omega(L\tau^{2/\alpha}\varepsilon^{-(2-\alpha)/\alpha}+L\sigma^2\tau^{4/\alpha}\varepsilon^{-(4-\alpha)/\alpha})$ calls, matching the corresponding SGD dependence when the inequality holds along the stochastic trajectory. At the classical endpoint $\alpha=2$, a separate construction yields the variance-dependent lower bound $\Omega(L\tau^2\sigma^2/\varepsilon)$ even for globally smooth objectives satisfying the Polyak-Lojasiewicz inequality globally. In contrast, the sharp variance-dependent complexity for smooth $\mu$-strongly convex objectives is $\Theta(\sigma^2/(\mu\varepsilon))$; with $\tau=(2\mu)^{-1}$, the worst-case global Polyak-Lojasiewicz class exhibits an additional factor $L\tau=L/(2\mu)$.

In this paper, we introduce and demonstrate a simple geometric algorithm to determine which critical points, both complex as well as real, contribute to the asymptotic evaluation of multiple integrals with exponential integrands of the form $e^{ikf(\boldsymbol{x})}$ over $\mathbb R^d$, for finite $d\ge 1$ and $f$ is analytic. In so doing, the algorithm removes the need to compute the flows of $-\text{Re} (i\nabla f)$ in $\mathbb C^d$ that is required to identify such relevant critical points in Picard-Lefschetz approaches to the derivation of such asymptotic expansions. By contrast, our algorithm relies on the combination of three simple features: the values of $f$ at all the critical points plotted in the complex Borel plane, the concept of adjacency between such points derived from algebraic resurgence/hyperasymptotic approaches and the new result here of a geometric "South-East" rule. The algorithm incorporates functions $f$ that remain bounded or unbounded on $\mathbb R^d$. We illustrate this new approach with both pedagogical and advanced examples, and draw conclusions as to its importance for resolving issues associated with Wick rotations and its implications for path integrals. This is a significant step towards a systematic way of identifying instanton contributions in real-time path integrals.

The Signed Particle Formulation provides a particle-based interpretation of quantum mechanics in phase space, where quantum dynamics are represented through the creation and evolution of signed particles. A central computational challenge in this framework is the evaluation of the Wigner kernel, which generally involves highly oscillatory integrals and can become computationally demanding in time-dependent simulations. This paper proposes an analytical approximation of the Wigner kernel for one-dimensional single body quantum systems by exploiting a series-based representation of the potential function. The resulting expression provides an efficient way to approximate the Wigner kernel and the associated Gamma function, which governs the particle-generation process in the Signed Particle Formulation framework. The proposed approximation is evaluated for several Gaussian-based potential profiles, including single, double, triple, and quadruple Gaussian potentials. Numerical comparisons between the approximated and directly computed Wigner kernels and Gamma functions show that the proposed method captures the main behavior of the exact quantities while significantly reducing the computational cost. These results indicate that the proposed approximation can serve as an efficient computational component for scalable Signed Particle Formulation based quantum simulations.

This article presents a unified overview of contact Hamiltonian geometry as a natural framework for the description of dissipative and non-conservative systems. Starting from the symplectic cover of a contact manifold, we clarify the structural relation between contact and symplectic dynamics and show how dissipation is geometrically encoded through the contact structure and the Reeb vector field. Following the introduction, which provides a guided overview of the subject through key references, a dedicated section illustrates the scope of the theory through applications ranging from thermodynamics, statistical mechanics, and integrable and KAM systems to field theories, quantum and Lie systems, optimal control, control theory, and economic models, where dissipation, constraints, and optimization play a central role. The subsequent sections review and adapt classical constructions of geometric mechanics, such as integrability, Hamilton–Jacobi theory, symmetries, and reduction, to the contact setting. Particular emphasis is placed on recent developments in contact reduction, Dirac structures, and constrained systems. The article also surveys emerging approaches to the geometric quantization of contact manifolds and discusses how ideas from generalized geometry provide a unifying perspective for symplectic, contact, and related frameworks.

This paper continues the study of resonance phenomena initiated in [3] for rank-one perturbations. We consider finite-rank multi-parameter perturbations $H_\alpha$ of the Laplacian on \(L^2(\mathbb{R}^3)\) and establish Breit–Wigner-type asymptotics for the spectral density of $H_\alpha$ along the resonance $\lambda(\alpha)$ near a simple embedded eigenvalue $\lambda_0$ of $H_a$ as $\alpha\to a$. We also obtain similar asymptotic behaviour for the scattering cross-section and the average time delay.

Recent progress on the structure of the quantum connection for monotone symplectic manifolds has used two approaches, which share the common feature of reducing to mod $p$ coefficients. We refine and compare those approaches. In particular, we establish a relation with quantum Steenrod operations which is stronger than that in Chen's work, leading to more precise information about the singularity at $\infty$ of the quantum connection. For the version of the connection relative to a smooth anticanonical divisor, we draw attention to the implications of the categorical mod $p$ Fontaine-Laffaille structure established by Petrov-Vaintrob-Vologodsky.

This paper studies the mean field game of mutual holding proposed by Djete and Touzi(AAP, 2024), and consider the case where the interactions among agents are described by a graphon. We adopt the formulation on the enlarged space which is modeled using the joint law of the value process and the graphon label, as in Lacker and Soret(MOR, 2023). Under suitable conditions on the graphon function, we are able to provide the explicit characterization of the optimal strategy, prove the wellposedness of associated Mckean-Vlasov SDE and establish the convergence results of the Nash equilibria. The key technique consists in a detailed analysis of the continuity property under the $\mathcal{WOP}_2$ metric, and tailor-made arguments for different graphon equilibria under different regularities of the model.

We settle the global nonlinear stability problem for the family of Kerr black holes in the full subextremal range: spacetimes evolving from initial data close to those of a subextremal Kerr black hole as solutions of the Einstein vacuum equation ${\rm Ric}(g)=0$ settle down to a nearby member of the Kerr family at the rate $\mathcal{O}(t_*^{-2-\epsilon_{\mathcal K}})$ in spatially compact regions. For the initial data, we require $\mathcal{O}(r^{-1-\epsilon_0})$-decay for $\epsilon_0>0$ – more precisely, an arbitrary but finite expansion into terms $r^{-z}(\log r)^k$ where $z>1$, $k\in\mathbb{N}_0$, plus a remainder term with $\mathcal{O}(r^{-3-\epsilon_0})$-decay. We use a generalized wave map gauge modified using gauge source terms that lie in a suitable finite-dimensional space determined by the expansion of the initial data. Like the final black hole parameters (mass and angular momentum) and the gravitational wave tail, the gauge source terms are treated as unknowns in a nonlinear (Nash-Moser) iteration scheme. We work directly with the tensorial equation and in particular do not rely on reductions to scalar equations (except insofar as a reduction to the Teukolsky equation is used in the proof of linear mode stability). This paper relies on two companion papers by the author. The first one introduces a strong form of constraint damping in the full subextremal range, which we use in our formulation of the gauge-fixed Einstein equation as a black box. The second one provides tame estimates (albeit with weak decay) for forward solutions of a general class of wave-type equations, which we show here to include the linearizations of the gauge-fixed Einstein equation arising in our nonlinear iteration scheme; these estimates are the starting point for our detailed asymptotic analysis.