Papers

Let $E$ be an elliptic curve over the rationals. In 2004, Cojocaru proved, using the Chebotarev density theorem, that the set of primes $p \leq x$ for which $m$ divides $\#E_p(\mathbb{F}_p)$ has a natural density. In 2009, Banks and Shparlinski proved an averaged version of this result over families of elliptic curves. In this article, we give a more explicit analysis of these densities. In particular, we show that, for Serre curves, the density of primes $p$ for which $m \mid \#E_p(\mathbb{F}_p)$ is approximately $1/\varphi(m)$, and is always greater than $1/m$ for every $m \geq 2$. Thus, the orders $\#E_p(\mathbb{F}_p)$ exhibit a bias toward divisibility by $m$. Finally, based on Jones' method, we prove that the average of the individual $m$-divisibility densities coincides with the average density proposed by Banks and Shparlinski.

Ten years ago, in a paper \cite{60}, a brief historical survey of the research activity in the Sector “Supersymmetry” at the Bogoliubov Laboratory of Theoretical Physics (BLTP) for more than 50 years of its existence has been given. Here, in commemoration of the 70th jubilee of Joint Institute for Nuclear Research, we review some recent sound advancements in this area. Specifically, we consider the issues of constructing the superfield quantum effective actions in $6D, {\cal N}=(1,0)$ supersymmetry and off-shell unconstrained superfield formulations of ${\cal N}=2$ higher spins. In both cases, the harmonic superspace approach plays the decisive role.

The classical Weddle surface is the locus of vertices of quadric cones through six points in $\mathbb{P}^3$ in linear general position. Equivalently, it is the closure of the locus of centers of projection from which those six points map to six points on a plane conic. Motivated by this 1850 construction of T. Weddle, we introduce $d$-Weddle schemes for finite point sets $Z\subset \mathbb{P}^n$, defined by an analogous projection-to-degree-$d$ condition. Our main tool is Macaulay duality, which yields a natural multiplication map in an Artinian algebra defined by powers of linear forms. This viewpoint connects $d$-Weddle schemes to unexpected cones and interprets them as non-Lefschetz loci for these multiplication maps. Parallel to this, we give an analysis from the point of view of interpolation matrices, and we explain the connections between these approaches. For a general set $Z\subset \mathbb{P}^n$ of $\binom{d+n}{n}$ points, we show that the $d$-Weddle scheme is a hypersurface and we compute its degree. We also study general sets whose cardinalities are "near" such a binomial coefficient, where the Weddle scheme has higher codimension. Returning to sets of six points (not always in linear general position), we discuss special configurations in which the appropriate Weddle scheme is reducible, or even nonreduced.

For two vectors $x,y\in [b]^k$, consider the bipartite graph with two copies of $[b]$ in which $i$ on the left is joined to $j$ on the right if $(x_t,y_t)=(i,j)$ for some coordinate $t$. We study the largest size of a family $C\subseteq [b]^k$ such that, for every two distinct $x,y\in C$, this bipartite graph contains a cycle. We give a natural construction for such families and conjecture that it is optimal whenever $k$ is large relative to $b$. We prove an LYM-type upper bound that is asymptotically tight with respect to this construction, and is exact when $k$ is large and divisible by $b$. We then refine the argument using a circular ordering, obtaining the sharp full-support bound when $k\equiv -1\pmod b$. In the case $b=3$, we prove the exact general result when $k\equiv -1\pmod 3$ and $k$ is sufficiently large. The problem is motivated by the Daniely–Shalev-Shwartz dimension and the pseudocube formulation of a higher-alphabet Sauer-Shelah-Perles lemma.

Stabilizer codes provide exact lattice realizations of bosonic topological orders. In contrast, systematic stabilizer descriptions of intrinsically fermionic topological phases remain much less developed. In this work, we introduce Majorana-Pauli stabilizer codes, a class of exactly solvable fermionic lattice models whose stabilizers are built from both generalized Pauli operators and Majorana operators. As a main example, we construct an exactly solvable stabilizer realization of the fermionic toric code: an intrinsically fermionic $\mathbb Z_2$ topological order in $(2{+}1)$ dimensions, using $\mathbb Z_8$ Pauli operators coupled to Majorana modes. Within this stabilizer framework, the anyons, string operators, fusion rules, and braiding statistics all follow naturally from the stabilizer algebra. More broadly, we show that the fermionic toric code belongs to a duality web generated by anyon condensation and by gauging or ungauging bosonic or fermionic symmetries. This web connects bosonic topological orders, symmetry-enriched topological (SET) phases, and both bosonic and fermionic symmetry-protected topological (SPT) phases, all within a common stabilizer description. We extend this construction to all Abelian fermionic topological orders with gapped boundaries and to fermionic SPT phases whose fully gauged theories are Abelian. Going beyond Majorana operators, we introduce fermionic versions of the clock and shift operators and use them to construct an exact bosonization map for $\mathbb Z_D^F$ symmetries for $D$ even. Using this, we realize a stabilizer model for a nontrivial $\mathbb Z_8^F$ fermionic SPT phase with no free-fermion analog. Altogether, these results extend the stabilizer-code paradigm to a broad class of intrinsically fermionic phases bridging fermionic quantum many-body physics to quantum error correction.

We introduce a new Calderón-type problem for fractional powers of Schrödinger operators, with local boundary conditions. The associated Dirichlet-to-Neumann operator maps Dirichlet data to Neumann data on the boundary. We show that this operator determines the Taylor series of the potential at the boundary. In particular, analytic potentials are uniquely determined. These are the first results for a fractional Calderón problem with sources and measurements on the boundary. Our proof builds on recent advances for pseudodifferential boundary value problems to compute the complete symbol of the Dirichlet-to-Neumann operator.

We prove a single algebraic mixed coincidence identity that unifies a broad swath of information-theoretic variational results. For any family of priors $\{\pi_i\}$ and real exponents $\{ \alpha_i \}$, the log of the mixed count $E_{x\sim\nu}\!\left[\prod_{i=1}^W \pi_i^{\alpha_i}(x)\right]$ is simultaneously a Boltzmann coincidence weight, an exponential-family normalizer, a maximum-entropy value, and a KL-barycenter optimum. The identity yields a unified derivation of classical cornerstones of information theory: concentration of empirical distributions (Sanov-type decompositions and Gibbs conditioning), hypothesis-testing error exponents (Chernoff information and its multi-way analogue), change-of-measure inequalities (Donsker-Varadhan and PAC-Bayes), and laws governing rare-pattern coincidences (Erdos-Renyi run-length, iterative guesswork, rate-distortion, and birthday thresholds). Each is recovered as a specialization of the same algebraic equality. It strictly generalizes the classical Renyi entropy and divergence variational formulas (one and two priors respectively) to a $W$-prior simplex, and holds for unnormalized and continuum-indexed priors. Among its consequences are an exact multi-prior PAC-Bayes penalty that subtracts an explicit "coincidence bonus" from the usual single-prior posterior penalty, and the asymptotic MAP error exponent for $W$-ary hypothesis testing as an edge-restricted simplex optimum. We demonstrate the calculus at scale on two large alphabets encoding richly modeled sequential languages: on language-model next-token predictives where we recover contrastive decoding, and on human genomic regulatory sequence where it separates correlated from diverse prior families along a sliding-window trace.

Recovering governing Ordinary Differential Equations (ODEs) from data is a central challenge in modeling dynamical systems across scientific domains. Existing approaches cast discovery as a static inference problem over fixed datasets, assuming that the observed trajectories are sufficiently informative. However, dynamical systems evolve over large state spaces, and limited data can make multiple equations observationally indistinguishable, leading to identifiability gaps and the recovery of incorrect governing equations. To address this, we introduce LLM-ACES, or LLM-guided Active Closed-loop Equation Search, a closed-loop framework that jointly optimizes symbolic hypothesis construction and adaptive data acquisition. In LLM-ACES, a large language model (LLM) proposes operator priors that partition the large search space into distinct regions, within which candidate equations are fit to the observed data. The disagreement among these candidates guides the acquisition of informative trajectories, creating a feedback loop that iteratively refines both the hypothesis space and the discovered dynamics. On 122 ODE systems spanning ODEBench and ODEBase, LLM-ACES achieves the lowest median NMSE, outperforming state-of-the-art baselines by several orders of magnitude while achieving a high symbolic accuracy of 46.2% and 52.4%, respectively. Our analysis further shows that LLM-ACES is sample-efficient, achieving better performance with one-tenth the data. Furthermore, LLM-ACES's feedback-driven data acquisition makes it robust to noise and recovers the correct symbolic structure, while baselines introduce spurious terms that fit the data locally but obscure the true governing relationships.

The Goodman-Strauss theorem states that for “almost every” substitution $\tau$, the family of substitution tilings is sofic, that is, it can be defined by local matching rules for some decoration of tiles. The conditions on the substitution that guarantee the soficity are quite complicated in the statement of the theorem. In this paper we propose a version of the Goodman-Strauss theorem with very simple conditions on the substitution: the family of substitution tilings must have finite local complexity (FLC), that is, the number of crowns that appear in $\tau$-supertiles is finite. Like the original theorem, our theorem provides matching rules for all known substitution tilings. We also prove a similar theorem for the family of hierarchical tilings associated with the given substitution. A tiling is called $\tau$-hierarchical if it has a composition under $\tau$, such that this composition also has a composition, and so on, infinitely many times. Every substitution tiling is hierarchical, but the converse is not always true.

High-dimensional partial differential equations (PDEs) with unknown coefficients arise widely in scientific machine learning, including continuous-time reinforcement learning, yet solving them efficiently in a data-driven way remains challenging. Existing deep learning solvers often rely on repeated automatic differentiation to evaluate differential operators, which can cause instability and amplify derivative errors in high dimensions, while probabilistic methods based on stochastic representations require explicit knowledge of the data-generating dynamics and therefore do not apply to black-box environments. We introduce two types of simulators as data-generating mechanisms, and take a “representing-then-learning" approach that learns the solutions and their derivatives under settings where the underlying PDE operators are accessible only through simulations and pointwise evaluations. Our representation of derivatives relies on the zeroth-order derivative (ZOD) estimators derived from perturbed Monte Carlo trajectories. This fully model-free approach generates targets for the gradient and Hessian networks using only function evaluations. We provide a statistical learning analysis of the proposed approach, including a bias–variance tradeoff for ZODs. Assuming a standard contraction property of the underlying operator, we establish a non-asymptotic error bound that decomposes the total error into discretization error, approximation error, statistical error, and ZOD bias. Crucially, we derive the sample complexity of the learned representations in (weighted) Sobolev space, characterizing the error up to second-order derivatives. Numerical experiments illustrate the competitive performance of the method in moderate and high dimensions.

Model Predictive Control (MPC) is widely used in industrial and robotic systems for enforcing constraints and embedding domain knowledge through finite-horizon optimization-based planning. However, despite these strengths, an MPC scheme typically does not yield optimal policies for sequential decision-making problems formulated as Markov Decision Processes (MDPs). Recent combinations of MPC with Reinforcement Learning (RL) alleviate this issue by treating MPC as a parameterized model of the optimal policy of an MDP and adjusting its parameters using data. While these approaches typically consider classical MDPs, many real-world problems include future information–such as forecasts, prices, or reference trajectories–at decision time, which must be included in the MDP state for optimal decision-making. Current MPC-RL approaches do not directly account for this augmented-state structure, raising the question of how to incorporate future information into MPC to obtain an optimal policy. This work establishes the structural requirements under which a parameterized MPC can exactly represent the optimal value functions and policy of an MDP with future information. We further demonstrate that such a parameterized MPC can serve as a structured function approximator, with its parameters learned using RL. The approach is illustrated on a point-mass racing task with future reference information.

The total isotropy index of a quadratic form $\varphi$ over a field $F$ is the maximum dimension of any totally isotropic subspace of $\varphi$. If $\varphi$ is anisotropic and $\psi$ is another anisotropic quadratic form over $F$ of the same dimension, then $\varphi$ and $\psi$ are called Vishik-equivalent if, over any field extension $E/F$, their total isotropy indices are the same. In characteristic $\neq 2$, Vishik-equivalence implies similarity in all dimensions $\leq 7$ and in all odd dimensions, but there are counterexamples in all even dimensions $\geq 8$. In this paper, we construct semi-singular anisotropic quadratic forms of dimension $2^m$ for any $m\geq 3$ and defined over a suitable extension of any given field $F_0$ of characteristic $2$ that are Vishik-equivalent but not similar, thus completing the list of such examples provided earlier by the first author and Kristýna Zemková.

Optimal transport (OT) has become a central language for comparing probability measures, but exact balanced OT is often both too rigid for data with missing, created, or destroyed mass and subject to unfavorable high-dimensional sample complexity. Entropic regularization and unbalanced relaxations address these limitations in complementary ways. Entropy smooths the geometry, improves statistical behavior, and enables fast Sinkhorn-type algorithms, while unbalanced marginal penalties replace hard conservation constraints by divergence terms adapted to noisy empirical data. This paper studies the sample complexity of entropic unbalanced OT at the level of the optimal coupling, rather than only the scalar transport value. We develop a translation-invariant dual formulation, prove compactness and strong convexity properties for the intrinsic dual variables, and convert these geometric estimates into high-probability finite-sample bounds for empirical couplings. The results clarify why regularization is a practical necessity in machine learning applications: it softens the curse of dimensionality, reduces the number of samples needed for stable transport estimation, and keeps the resulting estimators compatible with scalable Sinkhorn-type solvers.

We study linear TD(0) under Markovian sampling, where data are generated along a single trajectory. We provide high-probability guarantees for a plain unprojected TD(0) algorithm with Polyak-Ruppert (PR) averaging, using a single stepsize schedule $\eta_t \propto \frac{1}{\tau_{\mathrm{mix}}\log(t)\sqrt{t}}$ that depends on the mixing time but requires no prior knowledge of the curvature parameter $\omega$. Our first result shows that such a choice of the stepsize guarantees that the TD(0) iterates are automatically and uniformly bounded with high probability, without projections and without any stability argument based on $\omega$. Building on this result, we establish a simultaneous high-probability convergence guarantee for the PR average: the same stepsize yields both a robust curvature-free $\widetilde{\mathcal{O}}\!\left(\frac{\tau_{\mathrm{mix}}}{\sqrt{T}}\right)$ rate and a fast curvature-dependent $\widetilde{\mathcal{O}}\!\left(\frac{\tau_{\mathrm{mix}}^2}{\omega T}\right)$rate, with the bound taking the minimum of the two. The core technical ingredient is a Poisson-equation toolkit for geometrically mixing Markov chains, which decomposes Markov noise into a martingale term plus a controlled remainder and enables a new self-bounding inductive argument for pathwise stability.

The aim of this short note is to show that the Sylow theorem (respectively Hall theorem) for finite skew braces proved by Truman in arXiv:2606.18414 is a direct consequence of the Sylow structure (resp. Hall structure) of a finite trifactorised group. A Cauchy theorem for finite skew braces naturally emerges.

Let $b\ge2$ be an integer, let $p\in\mathbb{Q}[x]$ be nonzero, and let $S\subseteq\mathbb{N}$ be infinite. We prove that if $\sum_{n\in S}p(n)b^{-n}$ is rational, then, for every fixed $0<\theta\le1$, there is a constant $c>0$ such that $|S\cap(N,N+W]|\ge cW$ for all sufficiently large $N$ and every $N^\theta\le W\le N$. Thus rationality forces positive lower density; if $S=\{a_1<a_2<\cdots\}$ and $d=\deg p$, it also forces $a_{j+1}-a_j\le d\log_ba_j+O(1)$. In the binary linear case, the result implies the irrationality conjectured by Erdős whenever $a_j/j\to\infty$. The proof turns rationality into an integer carry orbit of polynomial height. Repeated gap words lock the orbit onto rational polynomial graphs, while the normalized highest Newton coefficient evolves by an expanding affine map. Denominator preservation, polynomial sampling, and an interior–exterior count then rule out sparse polynomial-scale windows.

Collective behavior arises when locally interacting units produce coordinated global organization, from synchronization in dynamical systems to task-relevant information flow on graphs. The central challenge is not only to explain how collective behavior emerges, but to design local interaction rules that can produce desired global organization and generalize across graphs, dynamics and tasks.To address this challenge, we introduce the Swarm-Inspired Emergent Synchronizer (SIES), a graph-dynamical framework that learns generalizable local-interaction laws for controllable collective organization. Each node is an agent-like dynamical unit with a state and task cue, and signed source-target-conditioned attention acts as an adaptive coupling term inside an explicit evolution model. Therefore, SIES combines an explicit dynamical engine with local agent intelligence, similar to biological swarms. For synchronization control, SIES learns a generalizable coupling operator that produces prescribed synchronization patterns for CDSs across untrained network scales, target phase relations, and intrinsic node dynamics without retraining. The learned operator also reaches gait-related modes faster than three oscillator baselines and generalizes synchronization-driven locomotion to simulated multi-legged robots of different scales and a physical hexapod after leg disablement. For graph representation learning, SIES applies the same signed interaction principle to message passing and achieves the highest performance among the compared methods on heterophilous node-classification benchmarks. Together, these results position SIES as a generalizable and learnable graph-dynamical interaction framework with promise for synchronization control, adaptive robot coordination, and heterophilous graph representation learning.

We demonstrate the different possible structures for holomorphic hulls for embeddings of compact real 3-manifolds $M \hookrightarrow \mathbb{C}^3$ along the set of complex tangents $\gamma$. Using our previous work [1], we can construct embeddings with any prescribed link and 2-plane field along it, as well as a prescription of angles at each point that determines the Bishop invariant. We show that elliptic points ($\gamma < \frac{1}{2}$) produce analytic discs filling a Levi-flat hypersurface, hyperbolic points ($\gamma > \frac{1}{2}$) add no local hull structure, and parabolic points ($\gamma = \frac{1}{2}$) may develop a "delicate" Jöricke `onion' structure. We illustrate these phenomena with explicit examples of parabolic knots and links with mixed components.

We consider the adjacency matrix of the directed Erd{\H o}s-Rényi graph. As long as the expected degree is larger than the logarithm of the number of vertices, the graph is connected, we show that all eigenvectors are completely delocalized. Below this critical scale, we prove eigenvector delocalization if the corresponding eigenvalue is away from zero. This contrasts the undirected or Hermitian setting, where large eigenvalues have localized eigenvectors [arXiv:2005.14180]. Our results also hold for sparse random matrices with independent entries, which can be viewed as weighted Erd{\H o}s-Rényi digraphs.

We resolve Erdős Problems #593 and #1177. Problem #593 asks which finite triple systems occur in every uncountably chromatic triple system; the answer is exactly the class generated from private-vertex expansions of finite bipartite graphs by finite disjoint unions and one-point amalgamations. Equivalently, after isolated vertices are removed, a finite triple system is obligatory precisely when it is linear, every hyperedge-node of its Levi graph has an incident bridge, and every Berge cycle is even. The proof uses an exact bridge-trace theorem for complete-rank one-apex sequence lifts. We also prove that, for every uncountable cardinal kappa, there is a linear triple system of chromatic number exactly kappa, with at most 2^{2^mu} vertices when kappa=mu^+. These two ingredients give a class-valued exact avoidance-spectrum dichotomy for every finite forbidden triple system. As a consequence, Erdős Problem #1177 has truth values yes, no, and yes. All results of this paper have been formally verified in Lean-4.

The classical $M/M/c$ queue is the canonical Markovian model for a single-node queue with Poisson arrivals and homogeneous independent servers. We identify its natural counterpart when the servers are dependent. For homogeneous Markovian labelled service-completion mechanisms (retaining the identities of the completing servers), consistency under restriction to subsets of busy servers is equivalent to an independent Poisson shock representation, yielding homogeneous Marshall–Olkin multivariate exponential residual-service laws. This characterization identifies the $M/M_D/c$ queue as the canonical Markovian extension of $M/M/c$ to dependent homogeneous servers, with a Markovian queue-length process. For heterogeneous servers, augmenting boundary states with busy-server identities yields a Markovian queue-state process, as in the classical independent-server setting. For arbitrary $c$ and fixed idle-set assignment, we derive an explicit matrix-analytic stationary solution and departure and customer-delay laws. Waiting time conditional on delay is exponential, and the stationary sojourn distribution has a phase-type representation with at most $c+1$ phases. A three-server comparison with identical bivariate residual-service laws demonstrates how higher-order dependence can change waiting-time behavior.

We study the last iterate of the stochastic subgradient method for one-dimensional convex Lipschitz objectives. For a fixed horizon $n$, we consider the standard fixed stepsizes $\eta =\Theta(1/\sqrt n)$. We prove that, for such stepsize policies, under additive i.i.d. subgradient noise with uniformly bounded variance, the last iterate features an optimization error of order $1/\sqrt n$, thereby removing the extra $(\log n)$ factor present in existing generic bounds. On the other hand, we show that without the i.i.d. assumption, the optimization error can be of order $(\log n)/\sqrt n$. Thus, under the uniformly bounded variance assumption alone, the last iterate of SsGM is suboptimal even in dimension one, resolving negatively an open problem posed in Koren and Segal, COLT, 2020.

Let $g(r)$ be the minimum number of edges in an $r$-uniform intersecting hypergraph with cover number $r$. Erdős and Lovász proved the lower bound $g(r)\ge 8r/3-3$. We first give a completely elementary proof that $g(r)\ge 3r-4$. We then build on the same approach and apply Kahn's small-codegree hypergraph edge-colouring theorem to improve this to $g(r)\ge ((41-\sqrt{19})/12-o(1))r\approx 3.053r$. In particular, this shows that $g(r)>3r$ for all sufficiently large $r$, addressing a question of Erdős.

For a partially ordered set ${\mathbb{P}} = (X,\leq)$ there exist hypergraphs where the vertices are the set of ordered tuples of either all incomparable elements of ${\mathbb{P}}$ or all the critical pairs of ${\mathbb{P}}$, and the edges are formed by the duals of either all the alternating cycles of ${\mathbb{P}}$ or all the strict alternating cycles of ${\mathbb{P}}$. The weak chromatic numbers of these hypergraphs are all equal to the order dimension of ${\mathbb{P}}$. Here are established upper bounds on the number of strict alternating cycles a poset ${\mathbb{P}}=(X,\leq)$ can have in terms of $n = |X|$, the cardinality of the groundset of ${\mathbb{P}}$, and the width $w$ of ${\mathbb{P}}$. These bounds also apply to the number of hyperedges of the associated hypergraph ${\mathcal{H}}^s(\mathbb{P})$, with incomparable pairs as vertices and strict alternating cycles dual to its hyperedges.

We prove the degree-four case of a path problem of Erdős, Herzog, and Piranian. If $f$ is monic of degree four and all zeros of $f$, counted with multiplicity, lie in the open unit disk, then two zeros from this list can be joined inside $$\{z:|f(z)|<1\}$$ by a possibly degenerate polygonal path of length less than $2$.

In this work, we address the problem of validating the reconstruction of a stochastic system from noisy data. We demonstrate the limitations of criteria based solely on the loss function or on standard metrics used for reconstructing deterministic dynamics. We also propose an exploratory approach, based on a two-step test, which allows for a general assessment of the reconstruction without relying on arbitrary error-tolerance thresholds. However, we discuss how system degeneracy and non-identifiability, together with features intrinsic to stochastic dynamics, impose certain constraints on the application of this test.

We resolve Erdős Problem 768. Let $A(x)$ count the positive integers $n\le x$ such that, for every prime $p\mid n$, there is a divisor $d>1$ of $n$ with $d\equiv 1 \pmod p$. Erdős asked whether $A(x)/x=\exp(-(c+o(1))\sqrt{\log x}\log\log x)$ for some constant $c>0$. We prove that this holds with $c=1/(2\sqrt{\log 2})$; equivalently, $\log(x/A(x))/(\sqrt{\log x}\log\log x)$ tends to $1/(2\sqrt{\log 2})$. The lower bound is obtained from primes in disjoint logarithmic intervals using a fourth-moment argument based on the multiplicative large sieve and a subset-product second moment. The upper bound uses canonical witness divisors, a deterministic compression map, an injective reconstruction theorem for its fibers, and growing divisor moments. Thus the paper determines the exact leading constant in Erdős Problem 768. The main theorem and its complete proof have been formally verified in the Lean 4 proof assistant.

The Alon–Frankl–Lovász theorem determines the chromatic number of Kneser hypergraphs; equivalently, it gives the sharp minimum size of a monochromatic matching in every edge-colouring of a complete uniform hypergraph. Its known general proofs are topological. We introduce a topology-free structural framework. It reduces every colouring of a pseudorandom $t$-graph, with only $o(n)$ loss in the largest monochromatic matching, to a colouring of $K_n^{(t)}$ whose vertex set has at most $r$ parts and whose edge colours depend only on intersection profiles. Together with a stability analysis at the critical scale, we prove an exact robust form: there exists $c=c(r,t)>0$ such that, if a $t$-graph satisfies $\delta(\mathcal G)\ge(1-c)\binom{n-1}{t-1}$, then every $r$-colouring of $\mathcal G$ contains a monochromatic matching of the exact optimal size for all sufficiently large $n$. This gives a topology-free proof of the Alon–Frankl–Lovász theorem for large $n$, a sparse random transference theorem, and the exact value predicted by Meunier's stable Kneser conjecture throughout the range covered by our robust AFL theorem. We further develop the framework for Ramsey graph tilings. For a graph $H$, let $Rt_r(H;K_n)$ be the minimum, over all $r$-edge-colourings of $K_n$, of the largest monochromatic $H$-tiling. We prove $$ Rt_r(H;K_n)=(\beta_{r,H}+o(1))n, $$ where $\beta_{r,H}$ is effectively computable from finitely many rational linear programs depending only on $H$ and $r$. An additional multipartite Ramsey argument is needed to reconstruct a consistent coloured template. This gives an effective asymptotic solution to the multicolour Ramsey-tiling problem, extending the classical two-colour theorem of Burr, Erdős and Spencer. We also determine explicit constants for several natural families.

What kind of physical object is an artificial-intelligence system: a machine in the sense a refrigerator is – dissipating free energy while holding an order imposed from outside – or a dissipative structure in the sense a convection cell is – an ordered state that exists only under throughput and loses stability past a critical drive? We argue the answer is split, as it is for the living cell and the star – the trained artifact is a machine, quenched and storable; the operating loop is a dissipative structure in the informational sense – and develop the framework in which the loop's classification becomes decidable. Modeling systems that couple inference to irreversible action as finite-capacity decoders under sustained informational driving, we characterize recoverable operation by a feasibility margin, local invertibility, and a stability diagnostic that diverges as capacity saturates. Making the feedback of uncertified output onto load explicit converts this continuous transition into a first-order one at mean-field level, sharpening in the fleet limit: lucid and collapsed states coexist in a cusp-organized bistable region with closed-form spinodals, collapse pre-empts the divergence, recovery is hysteretic, and for ungatedness alpha >= 1 load reduction alone cannot restore operation; reset restores only what is archived, making certification the operative stability lever. Cascades are subcritical branching with mean-field exponent 3/2 and a cutoff set by the grounded fraction of input. An instrumented real-workload pipeline experiment exhibits the collapse just below the spinodal computed from the measured service law, the backlog-delayed hysteretic recovery, and the cascade statistics. This supplies a statistical-mechanics account of the "metastable failures" documented in large-scale distributed systems, identifying recoverable dissipation as the stability criterion.

In this work we describe Roelcke and WAP compactifications of automorphism groups of discrete ultrahomogeneous cyclically ordered sets in the topology of pointwise convergence.