Papers

We give counterexamples to the remaining homological conjectures of Peskine-Szpiro concerning modules of finite projective dimension over Noetherian local rings. Specifically, we show that the dimension inequality, the strong intersection conjecture, and the grade conjecture are all false in general.

This paper studies the existence of relaxed equilibria for finite-horizon continuous-time time-inconsistent mean field games. We work directly on the product space of relaxed feedback policies and population flows. The policy component is endowed with the stable topology of Young measures, while the population component is restricted to a compact convex set of Wasserstein-continuous flows with uniform moment and time-regularity bounds. For every policy-flow pair, we establish uniform Sobolev and Hölder estimates for the associated auxiliary value function and prove its stability under Young-measure convergence of policies and uniform Wasserstein convergence of population flows. We also establish continuity of the induced population-flow map. The latter requires a duality argument for the Fokker-Planck equations because Young-measure convergence yields only weak-$*$ convergence of the controlled drifts. We then construct a set-valued best-response/consistency map with nonempty compact convex values and apply the Kakutani-Fan-Glicksberg fixed-point theorem. The resulting fixed point satisfies both the equilibrium response condition of the intra-personal game and the mean field consistency condition, , thereby establishing the existence of a relaxed equilibrium

There is a diassociative algebra structure on the tensor product of an associative algebra and a perm algebra. In this paper, we elevate this conclusion to the level of bialgebra.We prove that the tensor product of an antisymmetric infinitesimal bialgebra and a quadratic perm algebra has a diassociative bialgebra structure, and this diassociative bialgebra structure is coboundary (resp. quasi-triangular, triangular, factorizable) if the original antisymmetric infinitesimal bialgebra is coboundary (resp. quasi-triangular,triangular, factorizable). As an application, we provide the close relationship between Lie bialgebras, Leibniz bialgebras, diassociative bialgebras and antisymmetric infinitesimal bialgebras, and the close relationship between symplectic Lie algebras, symplectic Leibniz algebras, symplectic associative algebras and symplectic diassociative algebras.

Classifying link types via their n-bridge positions requires navigating vast Hilden double coset classes H2n\B2n/H2n. We provide a restriction on this by proving that every Hilden double coset admits a positive braid representative whose Dehornoy minimum is explicitly bounded. The proof relies on the key observation that $\Delta\in H_{2n}$, which allows us to utilize the Garside left-greedy normal form to remove all powers of $\Delta$ from any given braid representative. We provide an explicit upper bound on its Dehornoy minimum in terms of the Garside length of the initial presentation. This bounded representative restricts the algebraic space of plat presentations, providing a potential computational tool for studying bridge isotopy classes and offering a constructive step toward algebraic approaches to the Bridge Finiteness Conjecture.

In this work, we study quantum CSS codes with transversal $T$ gates. Here, $T$ gate transversality is meant in the strongest sense; the application of physical $T$ to every physical qubit yields logical $T$ on every logical qubit, without Clifford corrections. Despite the importance of the $T$ gate in fault-tolerant quantum computation, the parameters of asymptotic families of such codes have not been improved since the work of Hastings and Haah in 2017, and Haah in 2018. In this work, we significantly broaden the achievable parameters of quantum code families with transversal $T$ gates, both expanding the regime of achievable polynomial rate and distance, and constructing such codes with constant rate and growing distance; this is the first time the latter has been achieved, even when allowing Clifford corrections after the transversal $T$ gate. These are also the first codes achieving $\gamma \to 0$ for a code with a transversal $T$ gate, where $\gamma$ is the overhead exponent of magic state distillation. To do this, we develop a framework of divisible decreasing monomial codes, punctured at a downward-closed set on the Boolean hypercube to create logical qubits. We prove a closed-form expression for the distance of such a code punctured at such a set, which may be of independent interest. We first instantiate this with an explicit construction based on weighted Reed-Muller codes, puncturing at low Hamming-weight points, and then with a randomised construction, where a small random set of points is protected from the puncturing to save quantum code distance, achieving improved parameters.

Let $A_1,\ldots,A_T$ be independent uniformly selected subsets of a finite population of size $n$, with prescribed cardinalities $m_1,\ldots,m_T$. For each population element, define its occupancy level as the number of selected subsets containing it. Let $x_t$ and $x_{\geq t}$ denote the numbers of elements with occupancy exactly $t$ and at least $t$, respectively. The 2025 work introduced a general multi-set allocation occupancy (MAO) representation for higher-order occupancy moments. The present paper establishes its joint-probabilistic interpretation and provides a rigorous unified derivation for arbitrary joint occupancy categories and moment orders. Specifically, for arbitrary $B_1,\ldots,B_\ell\subseteq\{0,1,\ldots,T\}$, we prove the exact representation $F_\ell(B_1,\ldots,B_\ell)=G_T(B_1,\ldots,B_\ell)/(n)_\ell^{T-1}$, where $G_T(B_1,\ldots,B_\ell)$ is the corresponding generalized MAO transversal sum. This identity unifies the joint factorial moments of arbitrary occupancy categories within a single exact finite-population framework. In particular, writing $B_{\geq t}=\{t,t+1,\ldots,T\}$, the factorial moments of exact- and threshold-occupancy counts are obtained as the specializations $\mathbb{E}[(x_t)_\ell]=F_\ell(\{t\},\ldots,\{t\})$ and $\mathbb{E}[(x_{\geq t})_\ell]=F_\ell(B_{\geq t},\ldots,B_{\geq t})$. Mixed factorial moments, raw moments, variances, and covariances follow from the same representation through standard transformations. The formulas are verified by exhaustive enumeration over feasible parameter ranges and by Monte Carlo simulation. The resulting theory provides a rigorous and unified finite-population foundation for exact and threshold multi-set occupancy statistics.

For a smooth projective curve over a finite field with a fixed point at infinity, we establish a universal correspondence between finite residue duality and geometric kernel functions. We prove that the finite residue Casimir tensor is realized geometrically as the intrinsic principal part of the normalized Szegő kernel for any acyclic line bundle, and equivalently that this kernel acts as a reproducing kernel for the finite residue pairing, in exact analogy with the classical Cauchy integral formula. In the polynomial case these equivalent descriptions yield a closed formula involving the rank-two Weil operator, identified with the classical divided difference, recovering a remainder identity of Hu–Ou. These results provide a geometric foundation for the study of Anderson generating functions and the Weil pairing for Drinfeld modules.

In this paper, we confirm a conjecture of Ozawa and others asserting that every finitely generated, relatively hyperbolic, exact group is bi-exact (in the sense of Ozawa) relative to its natural peripheral structure. As a consequence, every such group gives rise to a prime group von Neumann algebra. As an application, we construct a continuum family of property (T), relatively hyperbolic groups $\{G_i\}_{i\in I}$ such that, for every fixed arbitrary free, ergodic, probability measure-preserving action $G_i \curvearrowright Z_i$, the collection of associated group measure space von Neumann algebras $\{L^\infty(Z_i)\rtimes G_i\}_{i\in I}$ are pairwise non-stably $\ast$-isomorphic.

We prove that, on every connected bounded-degree graph, the infinite collision property holds in discrete time if and only if it holds in constant-speed continuous time, and the same equivalence holds for the finite collision property. The proof is based on a pointwise comparison between the Green kernels of the synchronous discrete-time pair chain and the asynchronous pair chain obtained by Poissonization. The main estimate exploits the binomial interlacing of the coordinate updates together with a local binomial estimate. We then use Martin capacities and collision zero–one laws to relate this Green-kernel comparison to infinite visits of the diagonal.

Large language model providers are compute constrained, and a common response to congestion is to degrade service. A degraded answer fails with some probability, and a failed answer either returns as a retry or departs as churn, destroying LTV on an unaccounted ledger. We model inference allocation as a newsvendor whose stockout cost is churned lifetime value, a geometric retry multiplier in which the recycled product is dissatisfaction, and a two-regime transient fluid queue whose arrival rate is made endogenous by retries. Statically, there is a regime in which a cheaper model saves energy per initiated task while consuming strictly more capacity per initiated task, so the discount inverts when capacity binds. Dynamically, a reactive throttle fired during a surge can cross an ignition threshold beyond which it manufactures more traffic than it sheds, and a release rule set below the degraded equilibrium converts a transient surge into a permanent degraded regime. With heterogeneous customers, throttling is a transportation problem in retry-inflated load whose optimal policy rations intelligence by critical ratio, and whose dual, the shadow price of intelligence, prices a marginal query by class and by hour. Under congestion, throttling is not a cost lever but a demand lever.

In this paper, we study optimal control problems for stochastic semilinear partial differential equations, which lack the maximum principle, and whose coefficients do not have bounded Frechet derivatives. We propose an approximation scheme for the corresponding optimization problem, and prove convergence of the approximating solutions on both finite and infinite time intervals.

We observe that standard derived equivalences give a counterexample to a conjecture of Kalck on global dimension for weighted projective lines. For the root stack $X=\mathbb{P}^1\langle \infty,0,1;2,3,3\rangle$ we exhibit a $13$-dimensional radical-square-zero algebra $A$ such that $$ D^b(\mathrm{coh}X)\simeq D^b(\mathrm{mod}A), \qquad \mathrm{gldim}A=4>3. $$

This paper is concerned with a structure-preserving neural network-based framework for the Cahn-Hilliard equation in mixed form. We employ generalized transferable neural networks (GTransNet) for spatial approximation and stabilized backward differentiation formulas (BDF) for temporal discretization. The resulting first- and second-order in time GTransNet-BDF schemes are shown to conserve mass and satisfy energy stability at the time-discrete level. The schemes are implemented by a collocation-based method, in which a least-squares system with constant coefficient matrix needs to be solved at each time step. The solution of this system, which determines the output-layer weights of the network, violates mass conservation due to the expected nonzero least-squares residual. To overcome this issue, we introduce a novel post-processing mass-conserving projection that enforces the mass constraint through a minimization problem, whose solution can be computed at negligible computational cost. A key advantage of the proposed method lies in its predetermined hidden layers and mesh-free nature, making the method applicable to complex domains, variable mobility, and long-time simulations. Extensive numerical experiments in two and three dimensions verify convergence, mass conservation, and energy dissipation as well as demonstrate the accuracy and robustness of the proposed GTransNet-BDF schemes.

We study a nice class of Tambara functors, called multiplicatively cohomological Tambara functors. The first result is a proof of a conjecture of Balchin, Quigley, and Spitz showing that every multiplicatively cohomological Tambara functor is also additively cohomological. Building on this, we prove Nakayama's lemma for multiplicatively cohomological Tambara functors. As an application, we identify a condition under which finitely generated projective modules over local Tambara functors are always free. This applies, in particular, to the constant $C_{p^n}$-Tambara functor $\underline{\mathbb{Z}}_p$.

For positive integers $n\ge s>r$, let $T(n,s,r)$ denote the minimum number of edges in an $r$-uniform hypergraph on $n$ vertices such that every $s$-set of vertices contains at least one edge. A simple averaging argument shows that the ratio $T(n,s,r)/\binom nr$ is non-decreasing in $n$ and we denote its limit as $n\to\infty$ by $t(s,r)$. The case $s=r+1$ has a rich history, with the previously best known asymptotic bounds for $r\to\infty$ being $1\le r\cdot t(r+1,r)\le 4.91...$ . In this paper, we present a simple probabilistic construction which shows that $(r+2)\cdot t(r+1,r)\le 4$ for every $r\ge1$. We also derandomise it and discuss applications to covering codes.

Spatial patterns, such as those in dryland vegetation models, have historically been studied in systems of reaction-diffusion systems with pattern onset via a Turing bifurcation from a spatially uniform state. More recently, spatial patterns have been considered in models that incorporate spatially extended interactions via nonlocal interaction kernels. It remains largely underexplored if and how the choice of nonlocal interaction kernel contributes to differences in pattern formation and persistence, particularly in models that contain competition and facilitation. Here, we investigate spatial patterns in a reaction-diffusion model for a single species that includes nonlocal competition and facilitation processes; Gaussian, exponential, algebraic, hat, and smooth hat kernels are considered as specific examples. Via a center manifold analysis, and using the relative spatial scale of competition to facilitation and the death rate as bifurcation parameters, we identify that the choice of kernel has impacts on the pattern forming bifurcation. Bifurcations using the Gaussian, exponential, and algebraic kernels largely follow expectations of Turing patterns, but patterns in the hat and smooth hat kernels can form even when the scale of competition is less than that of facilitation. The dynamics of patterns far from onset are investigated via numerical continuation methods. The model produces the so-called "Turing-before-Tipping" phenomenon demonstrating that the arrangement into spatial patterns is an effective resilience mechanism against harsh conditions. Again, there is a kernel-dependent dichotomy in pattern behavior. Early warning signs for population extinction are observed with the Gaussian, exponential, and algebraic kernels, but not under the hat or smooth hat cases.

The classical Borell–Brascamp–Lieb inequality for multiple functions is an integral inequality relating finitely many input functions to a single output function. In this paper, we give an extension that allows a distinct output function for each input function. More precisely, we establish the following inequality. For a weight $ \lambda = ( \lambda_1 , \dots , \lambda_m ) $, we set $ z_\lambda ( x ) := \sum_{ i = 1 }^m \lambda_i x_i $ and denote by $ M_p^\lambda $ the weighted power mean of power $ p $. In addition, the power transform $ Q_d $ is defined as the continuous extension of $ p \mapsto p / ( 1 - d p ) $. We consider integrable functions $ f_1 , \dots , f_m , g_1 , \dots , g_m \colon \mathbb{ R }^d \to \mathbb{ R }_{ \geq 0 } $ satisfying $ 0 < \| f_i \|_1 < \infty $ for every $ i = 1 , 2 , \dots , m $. Then one has \[ \operatorname*{ess\,inf}_{ x = ( x_1 , \dots , x_m ) } M_p^\lambda \left( \frac{ g_1 ( z_\lambda ( x ) ) }{ f_1 ( x_1 ) }, \dots , \frac{ g_m ( z_\lambda ( x ) ) }{ f_m ( x_m ) } \right) \leq M_{ Q_d ( p ) }^\lambda \left( \frac{ \| g_1 \|_1 }{ \| f_1 \|_1 }, \dots , \frac{ \| g_m \|_1 }{ \| f_m \|_1 } \right) \] for any $ - \infty \leq p \leq 1 / d $, where the essential infimum is taken over the set of points satisfying $ f_i ( x_i ) > 0 $ for every $ i = 1 , 2 , \dots , m $. When all output functions $ g_1 , \dots , g_m $ are equal, this recovers the classical Borell–Brascamp–Lieb inequality for multiple functions. When $ p = 0 $ and the hypothesis is imposed pointwise, it coincides with a special case of the multi-output Prékopa–Leindler inequality of Cordero-Erausquin–Maurey.

Missing labels are usually regarded as a source of information loss in classification. We study a semi-supervised setting in which the probability of label missingness depends on the observed features through posterior classification uncertainty. In this setting, the missingness indicator is not only a record of an unobserved label, but also an observable signal generated by a mechanism linked to the classifier. We develop a likelihood-based information theory for such uncertainty-dependent missing labels. Under correct specification, we derive a Fisher-information decomposition that separates a partial-labeling component from a nonnegative mechanism-curvature term. Under joint misspecification of the label model and the missingness mechanism, we obtain the corresponding Godambe–Eicker–Huber–White sensitivity and sandwich-covariance partitions. We also clarify the relevant complete-data benchmark: favorable missingness can increase information relative to ordinary fully labeled or budget-matched non-informative labeling baselines, but cannot exceed the information in the augmented experiment in which labels and mechanism indicators are both observed. For plug-in classifiers, we connect the information decomposition to margin-based excess-risk bounds. In regular two-component mixture settings this yields the parametric \(n^{-1}\) excess-risk rate, with constants determined by the nuisance-adjusted information in discriminant directions. Gaussian-mixture calculations and a medical diagnosis example illustrate how uncertainty-dependent labeling mechanisms can improve estimation and classification under a fixed labeling budget.

Using Gaussian heat-kernel estimates, we classify planar Dunkl Newton kernels in weighted Euclidean BMO and obtain the corresponding local classification for unit-ball Green kernels. For atomic Newton potentials supported on a regular reflection orbit, orbit BMO detects exactly whether the coefficients are constant along the orbit. We also prove an intrinsic Dunkl–CLMS theorem in the heat-semigroup Hardy space and derive intrinsic and Euclidean-source Newton–Wente estimates, extending the BMO–Hardy-space method of Chanillo and Li to the Dunkl setting. Further consequences include sharp local atomic estimates, same-domain criteria, a localized $L^1$-to-BMO bound, and a Brezis–Merle-type estimate.

We study the homogenization for obstacle problems of convex Hamilton-Jacobi equations in periodic media with obstacle functions depending on both macroscopic and microscopic spatial variables. Based on the methods of analyzing variational representation formulas, we establish the optimal convergence rate $O( \varepsilon )$ for the homogenization limit under the assumption that the obstacle function is Lipschitz continuous. Moreover, we show that this convergence rate may deteriorate due to the presence of the obstacle when the obstacle function is not Lipschitz continuous, by constructing an example.

In topological data analysis, in particular persistent homology analysis, extracting "optimal" representatives for homology classes is crucial for identifying geometric regions of interest. In prior work, optimality is defined in terms of minimizing length or volume. In this work, we restrict our attention to a single homology class in degree $1$ and introduce the total absolute curvature of cycles as the cost function. We show that this cost function, based on angles between edges of cycles, penalizes departures from planarity, convexity, and simple-ness of the cycle representative. We formulate the "angle-optimal homologous cycle problem", recast it as a binary quadratic optimization problem, and show the results of experiments on artificial toy data.

We prove a free probabilistic Bakry–Émery criterion for logarithmic Sobolev inequalities and hypercontractivity. Our approach relies on a noncommutative version of the carré du champ operator and its iteration, which we are able to define in the free probability setting. This allows us to formulate a positive curvature condition, which is shown to be sufficient for logarithmic Sobolev inequalities and hypercontractivity in this setting. In the setting of free Gibbs measures, we also show that this curvature condition can be expressed in terms of the positivity of the Hessian of the associated potential.

Let $\Omega\subset\mathbb C$ be a bounded John domain and set $\delta(z)=\text{dist}(z,\partial\Omega)$. For $1<p<\infty$ and $\alpha>\text{dim}_A(\partial\Omega)-2$, we establish a norm equivalence between $\int_\Omega |g|^p\delta^\alpha\,dA$ and $\int_\Omega |g'|^p\delta^{\alpha+p}\,dA$ for analytic functions $g$ on $\Omega$, with a point-evaluation term fixing the additive constant. The estimate of the derivative term is local and holds on every proper planar domain, whereas the converse follows from a distance-weighted Poincaré inequality on John domains. Taking $\alpha=mp-2$ yields the corresponding comparison between the $m$-th and $(m+1)$-st derivatives. For $m\ge2$ the boundary-dimension condition is automatic, so the only dimension-sensitive case is the comparison between $\int_\Omega |f'|^p\delta^{p-2}\,dA$ and $\int_\Omega |f''|^p\delta^{2p-2}\,dA$ when $1<p<2$. We show that this restriction is sharp within the class of quasidisks by using self-similar Rohde snowflakes. We also construct, for every $s>1$, an inward-cusp $s$-John domain on which the comparison fails, showing that the ordinary John condition cannot in general be weakened.

Modern score-based generative models have achieved remarkable empirical success in high-dimensional tasks such as image, audio, and video synthesis. These models reduce distribution learning to a sequence of regression problems that, if solved exactly on finite data, would ultimately reproduce the training samples. Their ability to generalize must therefore arise from the implicit or explicit regularization during training. In this work, we develop a generative counterpart to the theory of benign overfitting and algorithmic regularization for overparameterized neural networks in the supervised lazy-training regime. We study denoising score matching in a vector-valued reproducing kernel Hilbert space with an inner-product kernel. In the proportional high-dimensional regime $n\asymp d$, we derive exact risk trajectories under gradient flow training. These trajectories exhibit three phases governed by qualitatively distinct estimators: a spectral estimator that generalizes, a pure-noise score with localized peaks that interpolate the training objective, and an empirical Bayes estimator that memorizes the data. We then analyze how these estimators combine along the reverse-time SDE and characterize the distribution of the resulting samples. The analysis reveals familiar mechanisms from supervised learning, including kernel linearization and self-induced regularization from the nonlinear part of the kernel, but also reveals a distinct phenomenology specific to generative modeling.

We study the approximation of the number of solutions of Laurent polynomials over finite fields. For a Laurent polynomial \[f(x)=\sum_{j=1}^{s}a_jx^{u_j}\in \mathbb{F}_q[x_1^{\pm1},\ldots,x_n^{\pm1}], \] let $U$ be its augmented support matrix whose columns are $(1,u_j)$ with rank $\rho$ and $N(f) := \# \{x\in (\mathbb{F}_q^\times)^n \mid f(x)=0\}$ be its torus point count. Our first main result is a quantum algorithm that outputs $\widehat{N}(f)$ satisfying \[ |\widehat{N}(f) - N(f)| \le \varepsilon q^{n+s/2-\rho} \] with success probability $1-\delta$. Provided that $\rho$ and $\|U\|_\infty$ are bounded, the algorithm runs in both classical bit and quantum gate complexity $\mathrm{poly}(n, s, \log q, 1/\varepsilon, \log(1/\delta))$. It provides finer resolution than relative-error approximations in general settings. To the best of our knowledge, in the explicit finite-field input model considered here, no previous algorithm achieves this additive accuracy with running time polynomial in $\log q$. Van Dam (arXiv:quant-ph/0405081) conjectured the existence of such an algorithm under the assumption of an oracle reflecting the algebraic properties of the polynomial. In contrast, by exploiting a point-counting formula derived from character sums over finite fields, we develop an alternative approach that efficiently approximates the number of points without assuming the existence of such an oracle. As a second main result, we prove that the same approximation problem becomes $\#$P-hard under randomized polynomial-time Turing reductions when the support matrix $U$ varies freely as part of the input. Thus, taken together, our results clarify how the effectiveness of the quantum approach depends on the tradeoff between the accuracy scale and the support parameters of the input polynomial.

We prove that every triangle-divisible balanced tripartite graph whose vertex classes have size $n$ and whose partite minimum degree is at least $4n/5$ admits a fractional triangle decomposition. Combined with the multipartite decomposition theorem of Barber, Kühn, Lo, Osthus, and Taylor, this implies that, for every fixed $\varepsilon<1/5$ and all sufficiently large $n$, every $\varepsilon$-dense partial Latin square of order $n$ is completable, improving the previous asymptotic bound of $2/25$. The fractional decomposition theorem is finite and exact. Its proof uses a minimum-weight perfect matching to normalize an arbitrary Farkas dual weighting, followed by an explicit direct-and-two-step routing scheme whose aggregate congestion on every off-matching edge is at most one.

Finiteness and injectivity of the moduli morphisms $\mu_n\colon|nH|_{\rm{sm}}\to\mathcal{M}_{g_n}$ associated with linear systems on K3 surfaces are understood in many cases. In this paper we study their ramification. Existing vanishing and stability results already imply unramifiedness when the polarisation or the multiple is sufficiently positive, so our focus is on low-degree phenomena. In particular, for a general K3 surface of degree $2$, we prove that the primitive moduli map is quasi-finite but ramified at exactly $171$ points, identified with the jumping lines of a logarithmic tangent bundle associated with the branch sextic. On the other hand, we prove that if $X\subset\mathbb{P}^n$ is a general hypersurface of degree at least $4$, then $\rm{H}^0(Y,T_X|_Y)=0$ for every smooth hyperplane section $Y$ of $X$. In particular, the moduli map for smooth hyperplane sections of a general quartic K3 surface is unramified. We moreover show that ramification does occur for higher multiples of the primitive polarisation on both degree-$2$ and degree-$4$ K3 surfaces, and relate the question to the stability of the tangent bundle.

In 1974, C. D. Aliprantis posed several questions concerning topological completions of locally solid vector lattices. We answer all of them in the negative. We construct Hausdorff locally convex-solid vector lattices showing that, without metrizability, neither the $\sigma$-Lebesgue property nor property (B, i) need pass to the completion; a positive element of the completion need not be the limit of a decreasing sequence of upper elements; the generalized (A, 0) property need not make the canonical image a regular sublattice; and regularity of this image need not imply order density. All five counterexamples have been formalized in Lean 4 using the Banach lattice Lean library.

Let the vertices of a graph be revealed one at a time in a uniformly random order, and let the last-isolation time be the last time at which the induced graph on the revealed vertices contains an isolated vertex. For the cycle C_n, write K_n for the number of vertices still unrevealed at this time. We obtain an exact finite-n tail formula for K_n in terms of Stirling numbers of the second kind, together with a compact bivariate generating function. The enumeration comes from reversing the process. An isolated revealed vertex then corresponds to the cyclic pattern 101, but the tail event requires this pattern to be absent from every earlier prefix, not merely from the final set. This prefix condition forces the final one-blocks to be separated by zero-gaps of length at least two and forces the reveal order inside each one-block to be peakless. Since a block of size m has 2^{m-1} peakless orders, summing over block sizes produces the Stirling numbers. We also prove K_n/sqrt(n) -> R, with P(R>x)=e^{-x^2}, and, for every fixed d>=1, K_{n,d}/n^{1-1/(2d)} -> W_d, with P(W_d>x)=e^{-x^{2d}}. Uniform stretched-exponential tail bounds imply convergence of all fixed positive moments. Thus the ordinary cycle has a Rayleigh limit, while its fixed powers give a Weibull family with shape parameter 2d.

In the presence of information-sharing noise, row-stochastic distributed optimization over directed and unbalanced graphs can suffer not only from noise accumulation in gradient tracking, but also from distortion of the left eigenvector-based gradient scaling used for imbalance compensation. To address these issues, this paper proposes a noise-robust distributed optimization algorithm using only row-stochastic weights, termed Robust Xi-row (R-Xi-row). The optimization update is driven by the increment of an auxiliary cumulative variable to exclude historical tracking noise, while two decaying gains and a normalized scaling gain are used to suppress the effect of noisy eigenvector estimation and asymptotically recover the required gradient scaling almost surely. Under strongly convex and smooth objectives and appropriate decaying gains, we prove almost sure convergence to the optimal solution. For polynomially decaying gains, we further establish a near-$\mathcal{O}(k^{-1/3})$ expected convergence rate. Numerical experiments validate the theoretical results.