Papers

We determine the critical probability for Bernoulli bond percolation on essentially any finite graph. Namely, letting $\lambda(G)$ denote the spectral radius (maximum eigenvalue) of $G$, we prove that the critical probability is at $1/\lambda(G)$: above this probability there is typically a component of order $\Omega(\lambda(G))$, whereas below it all components are of order at most $O(\sqrt{|G|})$. These results in particular confirm a conjecture of Krivelevich and Samotij about percolation on graphs of a given average degree, and vastly extend theorems of Bollobás, Borgs, Chayes, and Riordan, who proved analogous results but only for dense graphs. Our theorems are optimal in many regimes, and also demonstrate that percolation has an unexpectedly subtle behaviour on graphs whose spectral radius is roughly the square root of their maximum degree.

Let $\pi : M \to B$ be an elliptic fibration over $B = D^2$ or $B = S^2$ with $n$ nodal fibers over $\Delta \subseteq B$. We study the universal liftable braids for $\pi$: those braids that admit a fiber-preserving lift to $M$ for all choices of coordinates on $(B,\Delta)$. When $B = S^2$, we show that nontrivial universal braids do not exist by proving a Zariski-density theorem on the $\mathrm{SL}_2$-character variety for $(S^2,\Delta)$. When $B = D^2$ we classify when the subgroup of universal braids has finite index in the braid group $B_n = \mathrm{Mod}(D^2,\Delta)$, and relate these examples to Coxeter's factor groups of braid groups, which in turn are related to the platonic solids. Finally, we generalize the results derived in the finite-index cases by considering a canonical family of branched covers of the base $B$ associated to any elliptic fibration. The generalization naturally connects the universal braids to the integral Burau representation reduced modulo 3.

The goal of this paper is to prove an upper bound for the $L^4$ norm of a trigonometric polynomial whose spectrum is a nontrivial strictly monotone polynomial with integer coefficients of degree three and higher, via its $L^2$ norm. Our condition on the coefficients of the trigonometric polynomial in question is that they form a complex sequence whose modulus is decreasing. Second, we obtain a similar result in the case where the spectrum is formed by perfect squares, under a more strict condition on the coefficients. Our results strengthen and complement those by S. Bochkarev and A. Córdoba. We also give an answer to a conjecture of Eceizabarrena and Da Rocha and determine the sharp order of growth of the $L^4$ norm of the trigonometric polynomial in this conjecture.

We consider the spherical Borel Eisenstein series induced from the trivial representation for a split semisimple linear algebraic group over a number field. We prove that its regularization at the special point corresponding to half the weighted marking of a distinguished coadjoint nilpotent orbit in the Langlands dual Lie algebra is nonzero and square-integrable. Our proof follows the philosophy of Kazhdan and Okounkov. We give a geometric interpretation of Langlands' square-integrability criterion in this setting and, using the equivariant integration formula, prove that the regularization satisfies this criterion. As an immediate consequence, we obtain a simple and uniform proof of Arthur's unitarity conjecture, without case-by-case analysis or machine computation.

The Erdős Matching Conjecture is governed by two competing ways of excluding $s+1$ disjoint edges: one may concentrate all edges on fewer than $k(s+1)$ vertices, or force every edge to meet a fixed $s$-set. We determine a near-optimal range in which the second construction is extremal. For every fixed $k\ge2$, there is $s_0(k)$ such that, whenever $s\ge s_0(k)$ and $n\ge(k+1)s$, every $\mathcal{F}\subseteq\binom{[n]}k$ with $\nu(\mathcal{F})\le s$ satisfies \[ |\mathcal{F}|\le\binom nk-\binom{n-s}k, \] with equality only for the family of all $k$-sets meeting a fixed $s$-set. This improves the best previous general linear coefficient from $(5k-2)/3$ to $k+1$. In particular, the parameterized form of our argument further lowers the coefficient to $k+0.6$ for $k\ge5$. Since the two conjectured constructions exchange asymptotic dominance at $n=(\rho_k+o(1))s$ for a coefficient $\rho_k\in(k,k+1)$, our range lies less than one unit above the unavoidable barrier. We also prove a stability theorem showing that cover families are the only near-extremal configurations throughout this range. A key ingredient in our proof is a probabilistic rigidity statement which forces near-extremal fractional covers to be almost integral.

As a close relative of the Jacobian conjecture, the Hessian conjecture in dimension $n$ states that the local Legendre transform of a polynomial solution to the Monge–Ampère equation $\det(\operatorname{Hess}(\phi))=\pm1$ is also a polynomial solution. The general Hessian conjecture is false for $n\geq5$, while in Riemannian signature it follows from the Jörgens–Calabi–Pogorelov theorem. We study the four-dimensional Hessian conjecture in Lorentzian signature. For a polynomial potential $\phi$ in four real variables whose Hessian matrix has index $1$ and determinant $-1$, we define a constant pivot for $\phi$ to be a nonzero constant vector $\xi$ such that the second directional derivative $D_\xi^2\phi$ is constant. We then prove that the gradient mapping of every potential admitting a pivot is a polynomial automorphism, and that a pivot always exists when $\phi$ decomposes into homogeneous pieces as $\phi=\phi_d+\phi_{d-1}+\phi_2+\phi_1+\phi_0$ with $d\geq4$. More generally, we prove the same conclusion when $$\phi=\phi_d+\cdots+\phi_{d-k}+\phi_2+\phi_1+\phi_0,$$ where $k\geq0$ and $d\geq4k+3$. Then, we associate with each potential a linear system of quadrics, called the Hesse system, and a canonical homomorphism $\mu_\phi$. We prove that the existence of a pivot is equivalent to $\operatorname{rank}(\mu_\phi)\leq55$. As an application, we prove that the gradient mapping is a polynomial automorphism whenever the Hesse system has complex dimension at most 4. We also show that if ${\det(\operatorname{Hess}(\phi-\phi_2))\equiv0}$, then the potential $\phi$ admits a pivot. After that, we then give an analytic degeneracy criterion for $\operatorname{Hess}(\phi-\phi_2)$. Finally, we prove the Hessian conjecture in this setting for every polynomial potential of degree at most five.

We prove a global-in-time uniform bound for solutions to thermo-diffusive systems of reaction-diffusion equations with $L^\infty$ initial conditions.

We show that any measurable, translation and $\mathrm{SO}(n)$-invariant valuation on polytopes in $\mathbb{R}^n$ is a linear combination of the intrinsic volumes, which extends Hadwiger's classical characterization of rigid motion invariant continuous valuations on convex bodies. This result is based on a novel regularity result for translation invariant, measurable, $1$-homogeneous, and simple valuations. As a consequence, we obtain a similar characterization of the Steiner point map on polytopes.

Using the theory of complex spinorial forms, we prove that an odd-dimensional Riemannian manifold admits a pure spin-c Killing spinor with an imaginary Killing function $i\mu$ if and only if it is an exact $\mu$-Kenmotsu manifold, thereby obtaining an extension of a recent result by the first named author that, under the purity assumption, does not require simple connectivity or completeness. We then reinterpret $\mu$-Kenmotsu manifolds in terms of metric connections with vectorial torsion and give a second proof of this characterization, combining the theory of complex spinorial forms with the theory of metric connections with torsion. Finally, we describe the global structure of exact $\mu$-Kenmotsu manifolds by means of Morse-Bott theory.

Let $G$ be an almost simple group whose socle is an alternating, classical, or sporadic group, and let $H$ be a non-parabolic maximal subgroup of $G$. We prove that any maximal subgroup $M$ of $H$ can be generated by at most $7$ elements, and that this bound is sharp when the socle of $G$ is alternating or classical; this improves the bound of $12$ due to Burness, Liebeck and Shalev. When the socle is sporadic, at most $5$ generators suffice, and this is again best possible. The proof relies upon a detailed structural analysis of $H$ and $M$, especially when $H$ is a subgroup of a wreath product. In particular, we determine the chief factors of $M$ and subsequently bound its number of generators using the theory of crowns. We also correct the classification of maximal subgroups $H$ of almost simple groups requiring more than three generators, established by Lucchini, Marion and Tracey. Their bound of five generators remains valid, but their classification is missing several pairs $(G,H)$, including cases in which $G$ has socle $\mathrm{PSU}_n(q)$.

Let $X_1,X_2, \ldots $ be independent uniform random points in a bounded region $A \subset {\bf R}^d$ having a smooth boundary, $d \geq 1$. Let $B \subset A$ be compact. The _coverage threshold_ of $B$, $R_n$, is the smallest $r$ such that $B$ is covered by the balls of radius $r$ centred on $X_1,\ldots,X_n$. The _maximal spacing_ $\tilde{R}_n$ is the volume of the largest ball contained in $A \setminus \{X_1,\ldots,X_n\}$. We investigate the asymptotic frequency of _record times_ in the sequence $(R_n)$, that is times $n$ for which $R_n < R_{n-1}$. Let $N_m$ denote the number of records in the sequence $(R_n)$ up to time $m$, and let $\nu_m$ be the time at which the $m$th record value of the sequence $(R_n)$ occurs. For $B \subset A^o$, we show that almost surely, $N_n \sim \frac12 (\log n)^2$ and $\nu_n^{1/\sqrt{n}}\to \exp \big(\sqrt{2}\: \big)$ as $n \to \infty$, and likewise for $\tilde{N}_n$ and $\tilde{\nu}_n$, defined analogously in terms of $(\tilde{R}_n)$. But if $B=A$ and $d \geq 3$, then $N_n \sim \frac12 (1- \frac{1}{d}) (\log n)^2$ and $\nu_n^{1/\sqrt{n}}\to \exp \big( \sqrt{2d/(d-1)} \: \big)$. We also discuss the generalization (for fixed $k \in {\bf N}$) to $k$-coverage thresholds, maximal $k$-spacings and non-uniformly distributed points $X_i$ in $A$.

We show that the existence of a growth gap for infinite-index subgroups of a given finitely genrated group can depend on the finite generating set. More precisely, for any irreducible lattice $\Lambda$ in a higher rank semisimple Lie group $G$ with Kazhdan's property (T), the group $\Lambda \times \Lambda$ admits one finite symmetric generating set with a growth gap and another without a growth gap. We also prove that the growth gap can be made arbitrarily small. In contrast, for a non-elementary hyperbolic group the existence of a growth gap is independent of the finite generating set.

We construct nanopteron solutions to the differential equation \[ \epsilon^2u^{(4)}+u''-u+u^2 =0, \] which is a singular perturbation of the traveling wave problem $u''-u+u^2=0$ for the Korteweg–de Vries (KdV) equation. These nanopteron solutions are the superposition of the sech$^2$-type KdV traveling wave profile, a small localized error, and a periodic "ripple." The ripple is parametrized by its amplitude and its phase shift. In the process of constructing the nanopteron, we preselect one of these parameters and then solve for the other, and we provide detailed commentary on how the amplitude and phase shift solution processes differ. Our work revisits several prior constructions of nanopteron solutions for this singularly perturbed KdV equation in light of modern developments of nanopterons for lattice differential equations. In the context of these contemporary methods, the simple structure of the singularly perturbed KdV equation allows it to serve as an accessible template for nanopteron constructions in more complicated problems.

Calderbank-Shor-Steane (CSS) codes, constructed from nested classical codes $C_2 \subseteq C_1$, are typically optimized for good code parameters. However, practical quantum computing equally demands fault-tolerant logical gates. In this work, we characterize nested pairs $(C_1, C_2)$ whose resulting CSS codes realize a target logical diagonal gate via transversal physical $Z$-rotations. In doing so, we recover a result of Camps-Moreno et al. that CSS codes can realize only logical single-qubit $Z$-rotations and multi-qubit controlled $Z$-rotations via transversal physical $Z$-rotations. Building on our characterization, we develop the ”appending construction”, that takes as input an $[[n',k']]$ CSS code $Q'$ and a target logical $Z$-rotation (single-qubit or multi-controlled) $U_L$, and extends $Q'$ by systematically appending $n''$ physical qubits to obtain an $[[n,k]]$ CSS code $Q$ with $n = n'+n''$ and $k=k'$. The target logical gate $U_L$ is realized in $Q$ by applying a well-chosen physical transversal $Z$-rotation to the $n''$ appended physical qubits. Moreover, any logical gate realized via transversal physical $Z$-rotations in the input code $Q'$ remains transversally realizable in the extended code $Q$. The CSS code $Q$ may incur a loss in minimum distance, but the loss can be controlled through the parameter choices made in the construction. By repeatedly applying the appending construction, we can extend any CSS code $Q'$ to obtain a CSS code $Q$ that supports fault-tolerant implementations of multiple desired logical $Z$-rotations. The cost to be paid for this is the increased physical qubit overhead as the number of target logical gates grows. To illustrate our methodology, we construct CSS code families that transversally realize addressable logical single-qubit and multi-controlled $Z$-rotations.

Among $n+1$ equiprobable equal-energy signals in $\R^n$ under additive white Gaussian noise with maximum-likelihood decoding, which arrangement maximizes the probability of correct decoding? The question is Shannon's, recorded by Rice in 1950. Mulgund proved in 2026 that the regular-simplex value bounds the correct-decoding probability of every signal set at every signal-to-noise ratio, leaving open whether the simplex is the only maximizer. This paper determines the equality cases in a form stronger than uniqueness. A signal set other than a regular simplex falls strictly below the bound at every positive signal-to-noise ratio. Hence a code meeting the bound at one positive operating point is already a regular simplex, up to vertex relabeling and an orthogonal map. In probabilistic form, among the correlation matrices that signal sets induce, any matrix other than the identity gives a lower-orthant probability strictly above its independent counterpart at every finite threshold, leaving no room for a nontrivial equality. No code of ambient dimension below $n$ attains the bound. Under an energy budget $E$ with unrestricted blocklength the optimal codebook is uniquely the regular simplex of circumradius $\sqrt{E}$. Every optimal codeword therefore exhausts its allowance. Equality in the Simplex Mean Width Conjecture likewise occurs only at the regular simplex. The proof strengthens the first self-convolution step of Mulgund's argument with Royen's correlation theorem. The single-parameter rigidity is machine-checked in Lean 4.

Let $G$ be a nontrivial, finite abelian group. Then $\nu (G)$ is the smallest integer $\ell$ such that every zero-sum free sequence $T$ over $G$ of length at least $\ell$ has the following property: all nonzero elements of $G$ that do not occur as a subsequence sum of $T$ lie in a proper coset of some subgroup of $G$. We study the invariant $\nu (G)$, which was introduced in Zero-Sum Theory in the 1960s.

The oriented diameter of a connected bridgeless graph $G$, denoted by $\overrightarrow{\operatorname{diam}}(G)$, is the minimum diameter among all strong orientations of $G$. We study the oriented diameter of planar triangulations, and show that $\overrightarrow{\operatorname{diam}}(G)\leq \frac{2n+44}{5}$ for any $n$-vertex planar triangulation $G$. This improves the leading constant in the previous best general upper bound $\lceil \frac{n}{2}\rceil$, due to Ge, Liu, and Wang, from $1/2$ to $2/5$. We also prove that every $n$-vertex $4$-connected planar triangulation satisfies $\overrightarrow{\operatorname{diam}}(G)\leq \frac{n+17}{3}$.

In this paper, we establish interior $C^2$ estimates for admissible semiconvex solutions to the general Hessian quotient equation $\frac{\sigma_k}{\sigma_l}(D^2u)=f(x,u),$ for the cases $l=k-1$ and $l=k-2$, where $f$ is a positive $C^2$ function. Such estimates are known to fail in general for $k-l\geq 3$, even for convex solutions, as shown by counterexamples due to Lu \cite{LuGeneral}. The main ingredient is a quantitative concavity inequality for the Hessian quotient operator under the semiconvex condition. Our result provides a unified argument to such general Hessian quotient equations for $2\leq k\leq n-1$ in arbitrary dimensions.

Let $g$ be analytic in the unit disc and consider the generalized Hilbert operator $$ \mathcal{H}_g(f)(z)=\int_0^1 f(t)g'(tz)\, dt. $$ The boundedness of $\mathcal H_g$ on $H^p$ is characterized by the mean Lipschitz condition $g\in\Lambda\left(p,\frac{1}{p}\right)$ when $1<p\leq2$, while the problem remains open for $2<p<\infty$. It has been recently proved that the condition $g\in\Lambda\left(p,\frac{1}{p}\right)$ does not imply the boundedness of $\mathcal H_g$ on $H^p$, $2<p<\infty$ \cite{GuoTang2026}. We show that this condition is far from sufficient in the latter range: for every $2<p<\infty$, there exists a function $g\in\Lambda\left(p,\frac{1}{p}\right)$ such that $\mathcal H_g$ is not bounded even from $H^p$ into $H^1$. The main ingredient is an exact characterization of the boundedness of $\mathcal{H}_g:H^p\to H^2$ for all $1\leq p\leq\infty$. In particular, when $2<p<\infty$, this mapping is bounded if and only if $g'$ belongs to a certain mixed-norm space. For lacunary symbols, the same mixed-norm condition also characterizes the boundedness of $\mathcal H_g$ on $H^p$, and hence gives a complete solution of the open problem within this class of symbols. We also show that, for $1\leq q\leq\infty$, boundedness of $\mathcal H_g:H^1\to H^q$ is characterized by the condition $g'\in H^q$. We also characterize compactness of $\mathcal H_g$ in the aforementioned cases.

We give a counterexample to the convergence conjecture in Remark 12 of [Bolte & Pauwels, 2021] for mini-batch stochastic approximation with definable potentials. The construction uses two convex piecewise-affine, hence semialgebraic, summands on $\mathbb{R}$. We choose a deterministic nonincreasing block stepsize sequence satisfying $\alpha_k = o(1/\log k)$ and an admissible minimum-norm selection from each aggregate batch field. On successive blocks, the iterates form lazy reflected random walks on nested dyadic lattices. An explicit endpoint-cover-time estimate, Markov's inequality, and the first Borel-Cantelli lemma imply that almost surely every sufficiently late block's iterates visit their entire lattice. Consequently, the iterates remain in $[-1,1]$ but do not converge, and their accumulation set is exactly $[-1,1]$, on which the averaged objective is constant. Finally, the construction has $\sum_k \alpha_k^2 =\infty$. Both Chat-GPT 5.6 (Sol) and Gemini Pro 3.1 (DeepThink) were used in the development and drafting of this result.

We show that tree tensor network varieties, including tensor train varieties, are general Markov models associated to spaced trees. This allows us to prove that the prime ideals of these varieties are generated by minors of matrix flattenings. In the case of tensor train varieties, we discuss whether these minors form a Gröbner basis and provide a combinatorial method to compute the degree for order $3$ tensor trains.

This paper investigates the estimation of the regression operator in function-on-function regression models. While traditional research has predominantly focused on linear models or their immediate nonlinear extensions, we propose a neural operator approach to accommodate general regression operators under mild smoothness assumptions. Operator learning has emerged as an active area of machine learning, particularly for solving physical models governed by partial differential equations. Using this paradigm, our methodology introduces the separable neural operator, a neural-operator architecture that represents the regression operator through input-dependent coefficient functions and output-dependent basis functions. Beyond adapting this architecture to the regression operator estimation problem, we establish the consistency of the estimator under relatively mild smoothness and sampling conditions, allowing functional data to be observed on dense, possibly irregular, discrete grids. We also apply the proposed approach to the BGC Argo data and demonstrate its potential for oceanographic research.

The Jacobian Conjecture is a known unsolved problem and it is the problem number 16 of the list ”Mathematical Problems for the Next Century”, made by Stephen Smale, in 1998. The problem asks whether or not the Jacobian matrix of a polynomial mapping $F:\mathbb{C}^n\to\mathbb{C}^n$ at every point being invertible implies that $F$ is an automorphism. The case $n = 1$ is trivially true, while the case $n\geq 3$ has been recently proven to be false by a counter-example provided by Levent Alpöge, and the case $n = 2$ is still an open problem. In this paper, we show that, for all $n \geq 1$, there exists a non-empty Zariski dense open set $U$ such that, for all $F \in U$, if the Jacobian matrix of $F$ is invertible, then $F$ is an automorpshim.

We construct a metric on $S^2 \times S^2$ with positive sectional curvature. Starting from the standard metric on $S^2 \times S^2$, we first perform a Cheeger deformation. The resulting metric has nonnegative sectional curvature. We refer to it as a Cheeger-Müter metric. We then consider a suitable third order perturbation of this Cheeger-Müter metric and show that the perturbed metrics have positive sectional curvature. The proof requires various calculations, some of which have been carried out with the help of MATHEMATICA. The MATHEMATICA code is attached to this submission.

The empirical success of diffusion models in generative modelling has motivated theoretical work, including quantitative error bounds and qualitative analyses that characterise the different phases of denoising. We bring these two areas together by studying the adaptivity of diffusion models to the structured geometry of multimodal high-dimensional data that consists of multiple clusters in $\mathbb{R}^D$, each with its own low-dimensional structure, and inter-cluster separation depending on $D$. We employ $K$-mixture Gaussian distributions as a canonical framework to capture this geometry and establish two theoretical results. First, we interpret denoising as a dynamical Bayesian classifier: the mixture score is a posterior-weighted average of cluster-wise scores, and we show that, with high probability, the posterior class probabilities concentrate on a single cluster once the signal-to-noise ratio reaches the scale $\Theta (\log (KD)/D)$. Second, by separately analysing the denoising process in its mixing and cluster-commitment phases, we prove that the KL error bound depends linearly on the maximum intrinsic dimension of a cluster, up to a logarithmic factor, even when $K$ grows polynomially with $D$. This improves on ambient-dimensional bounds and extends existing low-dimensional adaptivity analyses to multimodal distributions with heterogeneous, approximately low-rank covariances.

An infinite jointly exchangeable adjacency matrix is a mixture of graphon models, and its graph is almost surely empty or dense; classical matrix exchangeability therefore excludes sparse nonempty graphs. We study a point-process alternative, graph processes, whose vertices are identified with the atoms of a locally finite point process on a Polish latent space and whose edges are drawn conditionally independently from a symmetric connection kernel $W$. For a Poisson graph process with diffuse $\sigma$-finite intensity, we show that its edge measure is jointly exchangeable as a random measure if and only if $W$ is essentially constant. Non-constant kernels therefore require a different exchangeability notion. We introduce bounded exchangeability, under which every nontrivial bounded restriction determines an infinite jointly exchangeable adjacency matrix that is unique in distribution, with an associated local graphon representation. We then study threshold models on homogeneous Riemannian manifolds, in which Poisson-distributed vertices are connected whenever their intrinsic distance is at most a fixed threshold. On every infinite-volume manifold in this class, the model is sparse, its empirical degree distribution converges to a Poisson law, and its limiting clustering coefficient depends on an additional geometric parameter that can vary while the limiting degree law is held fixed.

Noise-activated switching between coexisting stable states is a fundamental mechanism underlying stochastic dynamics in systems ranging from chemical reactions to neural networks. While this phenomenon is well understood for stationary attractors, it remains largely unexplored for limit cycles, whose periodic motion cannot be described by a static potential landscape. Here we experimentally demonstrate activated switching between two coexisting limit-cycle attractors in a driven nonlinear system of coupled resonators. Specifically, we introduce controlled fluctuations to directly observe the rare stochastic transitions between two limit cycles and measure their dependence on noise intensity and driving strength. The measured switching rates are well described by a large-deviation theory, which replaces the conventional activation barrier by the action along the most probable transition path. Our results extend the concept of activated dynamics from stationary to limit-cycle attractors and establish a framework for modeling stochastic transitions between limit cycles in driven-dissipative systems.

We state and prove the Theorem: for primes $p < 10^n$ with base-$10$ expansion $p = \sum_{k=0}^{n(p)-1} d_k(p) 10^k$, the positional digit probabilities $P_n(d \mid k)$ satisfy \[ \lim_{n \to \infty} P_n(d \mid k) = \begin{cases} 1/10, & k \ge 1,\ d \in \{0,\dots,9\}, \\[4pt] 1/9, & k = \mathrm{lead},\ d \in \{1,\dots,9\}. \end{cases} \] The limiting behavior splits cleanly into two distinct mechanisms: an arithmetic regime for interior digits and an Archimedean regime for the leading digit. For fixed interior positions ($k \ge 1$), digit extraction modulo $10^{k+1}$ reduces the problem to prime counts in reduced residue classes, where uniform distribution follows from Siegel–Walfisz (with Bombieri–Vinogradov allowing $k$ to grow with $n$). For the leading digit, the $1/9$ limit is not a Benford-type scale invariance, but arises from the local near-constancy of the prime density $1/\log t$ within single decades combined with a Toeplitz-type error averaging. Explicit classical and conditional error bounds are recorded for both regimes.

We study optimal online algorithms for embedding a permutation $\pi$ of $[k]$ into an iid stream of uniform $[0,1]$ random variables. This problem is a broad generalization of the classical online monotone subsequence selection problem, recovered in the special case $\pi=\mathrm{Id}_k$. Our first contribution is an efficiently solvable dynamic program for the optimal embedding time of any $k$-permutation $\pi$. This dynamic program also yields an explicit optimal online embedding algorithm. We then investigate the asymptotic scaling of the optimal embedding time for uniformly random target permutations, as well as the extremal problem of identifying the permutations with largest expected online embedding time. Our second main result shows that, to first order, random permutations are strictly faster to embed than monotone permutations, which in turn are strictly faster to embed than the extremal permutations. This separation stands in sharp contrast to prevailing conjectures and heuristics in the offline theory of permutation embeddings.

We present Eureka, a task-conditioned Meta-Agent architecture that compiles long-horizon tasks into dynamic obligation graphs with explicit acceptance semantics. During execution, Eureka forms Macro-Agents with specialized state, memory, operators, tools, verifiers, and local topology via receding-horizon planning, architecture promotion, and minimal-sufficient compilation. When bottlenecks recur, cost-benefit-gated evolution updates the local architecture under constraints. Theoretically, we establish results on regret, planning invalidation, amortization, subtree interfaces, serializability, and verification. Experimentally, Eureka completes 170/170 recursive tasks and generates 3,948 certificates with no false acceptances. Active context compresses median input from 9,490 to 4,005 tokens; incremental processing avoids 65.38% recomputation across 12,000 tasks; 16,000 concurrent executions serialize consistently. The same Meta-Agent instantiates a Theory-Discovery Agent and a Math/Conjecture Agent. The former yields structural results in quantum-process and spacetime theory. The latter identifies bottlenecks in Riemann Hypothesis research and advances a positivity certificate for Suzuki's localized Weil quadratic form to 0 < a <= 69/200 = 0.345, reaching ~99.55% of (log 2)/2. These results suggest that scientific-agent capability depends not only on the base model but on whether an architecture can be formed to match the task's cognitive structure.