Riordan arrays have traditionally been studied as infinite lower-triangular matrices that encode generating-function transformations and combinatorial relationships. This note proposes a complementary direction: using Riordan arrays as structured operators in artificial intelligence and data analysis. For a data sequence with generating function $A(z)$, a Riordan array $(g(z),f(z))$ produces the transformation \[ A(z)\longmapsto g(z)A(f(z)). \] This yields a parameterized family of causal linear operators that combine convolution, substitution, and multiscale mixing. The resulting structure may support parameter-efficient neural-network layers, variable-length sequence models, interpretable recurrent systems, invertible architectures, and models for hierarchical or recursively generated data. We explain what the infinite-dimensional viewpoint does and does not provide, outline several possible architectures, identify theoretical and numerical challenges, and propose an initial experimental program. The aim is not to claim that Riordan arrays are appropriate for all data, but to identify settings in which their algebraic structure supplies a useful inductive bias.
Predictive tasks constitute subprocedures within human--AI interactions (HAIs) involving professionals and AI systems. Complementarity occurs when an HAI outperforms a benchmark defined by its constituent agents. Existing formalizations, however, focus on a single human and a single AI, leaving multi-agent interactions insufficiently characterized. We introduce a tree-based mathematical framework for multi-agent complementarity. Ordered agent-role configurations decorate the leaves of rooted planar binary trees and local aggregation rules generate protocol outputs evaluated against either a best-average or a pointwise-min benchmark. We establish four results. First, under linear pooling, every protocol tree provides a bijective coordinate system for the interior of the simplex of global leaf weights. Arbitrary trees with the same ordered leaves admit output- and complementarity-preserving reparameterizations, realized locally through path-independent Tamari rotations. However, HAI constraints can violate admissibility preservation between equivalent protocols. Second, for regression under squared loss, we characterize complementarity through residual correction and derive optimal linear-pooling weights. We also characterize protocol-indifference loci under identical local parameters. Third, selector-based protocols cannot achieve complementarity relative to the pointwise-min benchmark, independently of task and loss. Fourth, in binary classification, HAIs cannot achieve complementarity under endpoint-monotone losses and the pointwise-min benchmark, with an analogous obstruction for multiclass cross-entropy. Numerical illustrations examine the main claims. We provide a mathematical foundation for analyzing multi-agent predictive subprocedures within HAIs, establishing a first step toward understanding how interaction, aggregation, and real-world constraints affect HAI performance.
Soft set theory provides a systematic framework for parameterized decision-making by associating each parameter with a subset of a given universe, thereby enabling structured representations of uncertainty [1, 2]. Over the past decades, the theory has evolved into a broad range of extensions, including hypersoft sets, superhypersoft sets, TreeSoft sets, bipolar soft sets, and dynamic soft sets, while also establishing connections with various mathematical disciplines, particularly topology and matroid theory. This book presents a comprehensive survey of soft set theory and its principal extensions, with particular emphasis on parameter spaces, value spaces, representative mathematical constructions, and the underlying assumptions necessary for rigorous comparisons among different frameworks. It carefully distinguishes established definitions from the literature from exploratory formulations and critically examines the limitations of selected claims concerning representation and reduction. Building upon the first edition, this second edition introduces additional concepts and illustrative examples, addresses previously identified mathematical inconsistencies and typographical errors, and refines the overall structure and presentation to enhance mathematical rigor, clarity, and readability.
List replicability measures how many accurate hypotheses suffice to contain the output of a learning rule with arbitrarily high confidence. We prove that every nonempty class of $m$ binary concepts on $n$ domain points has list replicability at most $805\ln(2m)\ln(2n)$. This resolves the logarithmic-product conjecture of Blondal, Hatami, Hatami, Lalov, and Tretiak. Hanner matrices give lower bounds of the same order, for square matrices and for explicit families with unequal logarithmic aspect ratios. The upper bound is obtained by constructing nested acyclic sets in the faces of a cube. A critical-point estimate supplies highly connected regions, weighted cell attachments keep the sets compatible across faces, and a cubical carrier theorem converts them into accurate finite covers and learning rules. Appending one constant coordinate turns list replicability into an exact geometric quantity, the face width, and graph-orientation classes satisfy exact product formulas. In particular, there are explicit classes with list replicability $4k+1$ and equivariant index $3k$, so the additive gap between the two is unbounded. We also develop entropy-tilting and two-sided Gibbs constructions, give combinatorial upper bounds adapted to structured matrices, compute exact values on classical families, and study the index of Paley matrices.
Conway's 99-graph problem asks whether a strongly regular graph with parameters srg(99,14,1,2) exists. We develop two complementary lines of attack. Fixing one vertex, the conditions lambda=1 and mu=2 force its neighbourhood to be a perfect matching and determine every edge between that neighbourhood and the remaining vertices. For (99,14,1,2), the unresolved part is therefore a constrained 12-regular graph on 84 vertices. We encode this reduction in CP-SAT and validate it by recovering the unique srg(9,4,1,2). We also prove by exhaustive enumeration that no circulant graph on Z/99 satisfies more than 68.0% of the CAISc constraints, and we give a validated orbit formulation for prescribed automorphisms. We then study the partial-score search problem. Fourteen human-designed search configurations reached at most 69.43%. Separately, we supplied the scoring function to an evolutionary program-search system. It produced a degree-preserving 4-vertex-switch tabu search whose best verified artifact scores 70.73%. The generated move differs from those used in our own searches and crosses a plateau that was stable under them. A tabu search restricted to graphs with a prescribed symmetry group, first Z/33 and then its subgroup of order 11, reaches a 14-regular graph scoring 76.67%. These results do not resolve the existence problem, but they reduce the exact search space and improve the best verified partial construction found in our experiments.
The running-time distribution of a randomized algorithm is computed under independent randomness, but independence is rarely what a source guarantees. We study sources $\nu$ with $\nu[w]\le DP[w]^s$ for every finite prefix $w$ of the tape, where $P$ is an independent reference law, $0<s\le1$, and $D\ge1$; individual choices may be predictable, even determined by the past. The worst-case probability of any closed tape event $E$ is exactly $\min\{1,DC_s(E)\}$, where $C_s(E)$ is the infimum cost of covering $E$ by prefixes of varying depth. This tree max-flow/min-cut identity makes worst-case probabilities exact and optimizable. Three applications follow. In scheduling, $n$ tasks on one machine are each retried until the next random symbol lies in its success set; every schedule has the same makespan distribution under independent randomness, yet under our constraint two fixed orders miss a linear deadline with worst-case probabilities $1$ and $\exp[-\Theta(n)]$, and a static priority rule is optimal at every deadline. For the standard local walk on a $k$-CNF with an assignment satisfying a majority of each clause, we find the sharp source threshold $s_*$: below it, on some instances one admissible source forces nontermination for every clause selection; at it, with fixed $D$, the expected number of flips is at most cubic in the initial distance, against linear under independence, with matching instances at distances up to a constant fraction of the number of variables; a fixed margin above it restores linear dependence. For noisy predecessor search with prefix-bounded comparison errors, verified backtracking attains any prescribed error probability at or above a source threshold whose sharpness we leave open, with the same cubic-to-linear query bounds. Restarting on the shared tape restores exponential tails exactly when the nontermination event has a prefix cover of cost below one.
We view a $k\times n$ Latin rectangle as $n$ points in the grid $[n]^k$. Each symbol gives one point, whose coordinates are the columns it occupies. The rectangle is balanced when these points are pairwise equidistant in the $\ell_1$ metric. Its imbalance measures the distance from this ideal. We study the minimum imbalance from three sides. Constructively, we use a simulated annealing that moves symbols within rows and only penalises column clashes. Against the reference table of Voboril et al. (2025), it improves 33 of the 51 non-square values for $8\le n\le 13$, by up to 39.4%, and matches the rest. A cheap invariant, the row group, certifies that none of the 33 new rectangles is isotopic to rows of a group table, so the earlier column-swapping local search cannot reach any of them. Exactly, a computer-assisted branch and bound proves that the minimum imbalance of a $3\times 9$ rectangle is $56$. D\'iaz et al. found this value, but its optimality was open. A sharper pruning bound, derived from the identity $I=2\sum(d-t)^+$, reduces the search tree by a factor of $2.4$ to $3.3$ on the instances with $k\ge3$ that we solve. Asymptotically, the minimum imbalance is $\Theta(n^3)$ for every fixed $k$. A pigeonhole argument gives a lower constant of order $k\,4^{-k}$, and random rectangles give an upper constant of order $\sqrt k$. For $k=2$, the limit of $\min I/n^3$ exists, a permuton attains it, and it lies between $4/375$ and $0.1196$. Perfect balance is an $\ell_1$-equilateral problem, so it forces $k=\Omega(n/\log n)$ rows. For $2\times n$ rectangles, single two-minute annealing runs improve five of the seven six-hour MIP upper bounds for $14\le n\le 20$. Finally, the Lean 4 kernel checks every rectangle reported in this paper against a 74-line specification that imports only Lean's core library. We also prove the pigeonhole lower bound in Lean with Mathlib.
Let $d_1,\dots,d_m$ be directions in $\mathbb{R}^k$, where $k\ge 2$. A multiset $X$ of $n$ points of $\mathbb{R}^k$ projects to the $m$ multisets of numbers $\{\{x\cdot d_j : x\in X\}\}$, or equivalently to the $m$ sorted lists of these numbers. How many directions are needed for these projections to determine every multiset of $n$ points? Two simple bounds are known in every dimension: more than $(k-1)\log_2 n$ directions are always needed, and $(k-1)n+1$ directions in general position always suffice. In the plane much more is known: Matousek, Privetivy and Skovron showed that almost every choice of $O(\log n \log\log n)$ directions suffices. Whether anything similar holds in higher dimensions was left open by Dym, Wellershoff, Tsoukanis, Levy and Balan. We observe that two lemmas of Hair and Sahai, proved for list recovery of random linear codes, answer that question, and by tightening their argument we show that in every dimension almost every choice of $(4k-2)\ln(2n)\ln(k\log_2 2n)+k+1$ directions suffices. For $k\le 10$ this beats the linear bound once $n$ is larger than about 100: for example, 144 directions suffice for multisets of 1000 points in the plane, and an exact computation lowers this to 133. Letting some of the random blocks in their argument fail removes a factor $\log k$: almost every choice of $O(k\log n\log\log n)$ directions suffices, uniformly in $k$ and $n$. We also show that $2k-1$ directions never determine three points, so that the logarithmic lower bound is not tight; that every $m$ directions in the plane fail on some pair of multisets of size $O(1.78^m)$, improving the bound $O(1.80^m)$ of Matousek, Privetivy and Skovron; and that for each $k$, $m$ and $n$, either almost every choice of $m$ directions works or almost none does.
The Bethe-Hessian matrix, introduced by Saade, Krzakala, and Zdeborov\'a (2014), is a Hermitian matrix designed for applying spectral clustering algorithms to sparse networks. Rather than employing a non-symmetric and high-dimensional non-backtracking operator, a spectral method based on the Bethe-Hessian matrix is conjectured to also reach the Kesten-Stigum detection threshold in the sparse stochastic block model (SBM). We provide the first rigorous analysis of the Bethe-Hessian spectral method in the SBM under both the bounded expected degree and the growing degree regimes. Specifically, we demonstrate that: (i) When the expected degree $d\geq 2$, the number of negative outliers of the Bethe-Hessian matrix can consistently estimate the number of blocks above the Kesten-Stigum threshold, thus confirming a conjecture from Saade, Krzakala, and Zdeborov\'a (2014) for $d\geq 2$. (ii) For sufficiently large $d$, its eigenvectors can be used to achieve weak recovery. (iii) As $d\to\infty$, we establish the concentration of the locations of its negative outlier eigenvalues, and weak consistency can be achieved via a spectral method based on the Bethe-Hessian matrix.
Graph neural networks are often trained on small graphs and deployed on larger ones, raising the question of which graph information must be preserved for combinatorial optimization values to transfer across sizes. We establish size-transfer guarantees for normalized graph optimization values using finite collections of energy features. For affine objectives, including MAX-CUT, coloring defect, and independent-set density, differences in task value are controlled by discrepancies in normalized scalar energy probes on all bounded-degree graphs. For general Lipschitz objectives, such as those with balance penalties, an additional representation error appears; this error vanishes on hyperfinite graph families but can remain bounded away from zero on expanders. Thus, at fixed degree bound, state space, Lipschitz constant, and target accuracy, a finite energy representation independent of graph size suffices universally for affine tasks and asymptotically for general Lipschitz tasks on hyperfinite families. These results yield prediction guarantees under explicit conditions on encoder accuracy, readout stability, and representation coverage, together with statistical generalization bounds. Theoretically, we prove Shapley-Folkman bounds with sharp parameter dependence, including a dimension-free $\ell_2$ regime, and realize the resulting energy features with a fixed-weight message-passing network using random node initialization on bounded-treewidth graphs. Experiments with exact solvers and trained models test the resulting transfer guarantees.
For a known finite-dimensional model subspace, classical Gaussian regression theory guarantees that least squares eventually attains the parametric rate, with prediction error scaling as the subspace dimension divided by the sample size. Yet subspaces with the same number of unknown coefficients can require very different sample sizes to reach this regime. For subspaces of polynomials on the Boolean cube, a fundamental question is how restrictions on degree and ambient dimension jointly determine this threshold. Our main result sharply characterizes the worst-subspace minimax sample threshold over a substantial parameter range, revealing an exponential dependence on degree and precisely quantifying the joint effects of degree, ambient dimension, and subspace dimension. A small modification of our lower-bound construction reveals a strong separation: a single subspace can admit noiseless recovery with optimal sample complexity independent of degree, yet require essentially the worst-case sample size, exponential in degree, to attain the noisy parametric rate. To establish the sampling lower bound, we sharpen the Polyanskiy-Samorodnitsky uncertainty principle (J. Funct. Anal., '19) to uniformly optimal precision and extend its concentration guarantee to large subspaces in which every function concentrates most of its energy on one shared Hamming ball. The sampling upper bound follows from the sharp moment bound of Kirshner and Samorodnitsky (IEEE Trans. Inf. Theory, '21) combined with empirical-process bounds.
We show that outlier eigenvectors of the Bethe Hessian achieve weak recovery in the stochastic block model down to the Kesten-Stigum threshold, for every fixed common expected degree $d>1$. This extends previous recovery guarantees, which required $d$ to be sufficiently large. This threshold is information-theoretically optimal for the symmetric two-community model and is conjectured to mark the limit of efficient recovery in broader symmetric models. Our result matches the recovery guarantees of non-backtracking spectral methods while using Hermitian $n\times n$ matrices. To do this, we identify recursive formulas for the negative outlier eigenvalues and show that their eigenspaces have nonvanishing overlap with the corresponding planted eigenspaces. The proof constructs approximate eigenvectors from the planted labels on the limiting Poisson tree, with weights given by its Green's function, and transfers these approximations to the finite graph.
We study conformal prediction under covariate shift with a known covariate density ratio, seeking prediction sets with minimum expected length under the target distribution. For every calibration sample size, we prove that the rule combining the inverse conditional-density score with randomized weighted conformal calibration minimizes this expected length among all rules valid over the full weighted-exchangeability class. We approximate the oracle using quantile and kernel histograms, including forest-weighted estimators, and select estimators and tuning parameters by resampling to estimate calibrated target length. On a bounded response domain, we bound excess length linearly in $L^1$ density error at fixed calibration size, without requiring a uniform positive lower bound on the true density. We also bound selection-pool and resampling errors. Under explicit growth conditions, the selected procedure approaches the same oracle while the calibration size remains fixed. Experiments on 36 known-density configurations and four public regression datasets compare coverage, prediction-set length, and runtime with existing methods. In targeted low-coverage and strong-shift comparisons, inverse-density selection yields shorter mean sets in most cases and lower mean runtime than the evaluated weighted baselines, with some differences in achieved coverage.