Papers

The Stranded Cellular Automata (SCA) model is a grid of cells such that each cell can contain 0, 1, or 2 strands, together with two cellular automata that control when and how strands turn and cross. It was developed to study patterns occurring in fiber arts. We define a notion of what it means for a braid, in the sense of an element of a braid group, to be represented by an SCA pattern, and provide several algorithms to determine when a braid has an SCA representation with certain additional properties.

For the mass-critical generalized Korteweg-de Vries equation, \begin{equation*} \partial_{t}u+\partial_{x}\left( \partial_{x}^{2}u+u^{5}\right)=0,\quad (t,x)\in [0,\infty)\times \mathbb{R}, \end{equation*} we prove the existence of three-soliton solutions with logarithmic relative distance and with the choice of signs $(+,-,-)$. The choice of the number three and the signs of solitons are related to the solvability of the ODE system generated by the nonlinear interactions between the three solitons and some non-localized profiles. In particular, these special behaviors are due to strong interactions between the three solitons. That is, the dynamics of each soliton is perturbed at leading order by the presence of other solitons.

Let \(G\) be a graph with \(m\) edges and adjacency spectral radius\(\rho(G)\), and let \(N(C_{2k+1},G)\) denote the number of copies of \(C_{2k+1}\) in \(G\). For each fixed integer \(k\ge 2\), define \( g_k(m):=\frac{k-1+\sqrt{4m-k^2+1}}{2}. \) Li, Zhai and Shu [European J. Combin., 2024] determined the spectral extremal threshold for odd cycles by proving that, for all sufficiently large \(m\), every \(C_{2k+1}\)-free graph \(G\) with \(m\) edges satisfies \(\rho(G)\le g_k(m)\). We establish the asymptotically sharp supersaturation counterpart of their result. More precisely, for every fixed integer \(k\ge 2\), we prove that \[ \inf_{\substack{e(G)=m\\ \rho(G)>g_k(m)}} \frac{N(C_{2k+1},G)}{m^k} = \frac{\lceil k^2/2\rceil (k-1)!}{(k+1)^k}+o(1) \qquad\text{as }m\to\infty. \] Thus every \(m\)-edge graph whose spectral radius exceeds the \(C_{2k+1}\)-free threshold contains at least \[ \Big( \frac{\lceil k^2/2\rceil (k-1)!}{(k+1)^k}-o(1) \Big)m^k \] copies of \(C_{2k+1}\), and the leading constant is asymptotically best possible. In particular, taking \(k=2\), we obtain if \( \rho(G)>\frac{1+\sqrt{4m-3}}{2}\) then \( N(C_5,G)\ge \left(\frac{2}{9}-o(1)\right)m^2, \) with the constant \(2/9\) being asymptotically optimal. This answers a question of Chen, Li and Tang concerning the existence and the largest possible value of a constant \(C>0\) for which the same spectral condition guarantees at least \(Cm^2\) copies of \(C_5\). More generally, our result resolves a recent problem of Li, Lin, Liu and Zhang on spectral supersaturation for odd cycles. The proof combines spectral stability and resolvent analysis with estimates for odd spectral moments and a careful treatment of non-injective closed walks.

Let $A, B \in \mathbb{Z}[x_1, \dots, x_n]$ be multivariate polynomials with integer coefficients and let $G = \gcd(A, B)$. We present an algorithm for computing $G$ whose expected bit complexity is asymptotically linear in all fundamental parameters: the number of variables $n$, the term count $T = \max\{\|A\|_0, \|B\|_0, \|G\|_0\}$, the total degree $D$, and the logarithmic coefficient sizes $\log\Hi$ and $\log\Ho$, where $\Hi$ bounds the coefficients of the inputs and $\Ho$ bounds those of the GCD. The bit complexity is characterized by the clean bound \[ \widetilde{O}\bigl( n \cdot T \cdot D \cdot \log\Hi \cdot \log\Ho \bigr). \] To our knowledge, this is the first sparse GCD algorithm over the integers that achieves linear complexity in all these parameters simultaneously. The integer algorithm is built upon a new field GCD algorithm. For $A, B \in \K[x_1, \dots, x_n]$ over a field $\K$ with $\operatorname{char}(\K) = 0$ or $\operatorname{char}(\K) > \deg G$, we give the first algorithm that computes $G = \gcd(A,B)$ with expected \[ \widetilde{O}\bigl( n \cdot T \cdot D \bigr) \] field operations, which is both input- and output-sensitive. The key technical contribution behind both algorithms is a derivative-aided separated Hensel lifting technique introduced in this paper. By introducing an auxiliary variable and leveraging derivative information, our scheme extracts all partial exponents via a single $z^2$-lift per variable, achieving constant sequential depth $O(1)$. This stands in sharp contrast to classical Hensel lifting, which requires $O(D)$ sequential lifting steps and suffers from representation densification in the sparse setting. The field algorithm is then extended to the integer case through modular reduction and rational reconstruction.

Cheong proved that if $S$ is a simple Lie group and $P$ is a parabolic subgroup, then $S/P$ admits a unique equivariant compactification of the unipotent radical $N$ of $P$, up to isomorphism, provided that $S/P$ is not isomorphic to a projective space. We generalize this result to a smooth projective horospherical variety $X$ with Picard number 1. Let $G = \mathrm{Aut}(X)$, and let $H$ be the isotropy subgroup at a point in the open $G$-orbit. We describe a unipotent subgroup $N$ of $H$, which is not necessarily the unipotent radical of $H$ when $X$ is not homogeneous. Moreover, we prove that $X$ admits a unique equivariant compactification of $N$, up to isomorphism.

We evaluate moment constants for the times during which Brownian motion admits uniform approximation by an unrestricted affine function or by a line through the origin. The first four moments are rational linear combinations of $1$ and $\zeta(2j+1)/\pi^{2j}$, $1\le j\le4$; the means are $40\zeta(3)/\pi^2-4/3$ and $21\zeta(3)/\pi^2$, respectively. Slope integration and a common contour argument give the unrestricted moments, while time inversion relates the anchored law to the reciprocal of Kingman's absorption time. For both models we also give exponential-integral series for the standardized absolute third central moments, with explicit truncation bounds.

Short-recurrence Krylov methods that are equivalent in exact arithmetic often diverge in floating-point arithmetic. To illustrate this, we provide an exactly representable two-cycle for steepest descent with a recursively updated residual. A complementary convergence theorem gives a sufficient condition under which the stored residual decreases geometrically. We then show that a second positive definite family yields different outcomes for a Hestenes–Stiefel Conjugate Gradient (CG) implementation and a direct Lanczos–Galerkin approach. CG finds the exact solution after four updates, whereas the two-step projected system becomes inconsistent. Under stated perturbation and transfer hypotheses, a common spectral enclosure gives comparable convergence bounds. For a dense, left-to-right evaluation order, we prove a sufficient precision bound for a prescribed backward error within $n$ updates for an $n\times n$ matrix, together with a computable stopping test and specified exponent-range assumptions. We use polynomial estimates and numerical experiments to examine the trade-off between working precision and the number of CG iterations needed to achieve a prescribed backward error. We also compare results on inexact matrix-vector products and preconditioning with the frameworks of Paige and Greenbaum. For block Lanczos, we analyze Householder orthogonalization and singular-value truncation as the block size changes. Under componentwise and normwise error bounds, the computed coefficients satisfy a controlled local recurrence and an exact block Lanczos relation for a nearby symmetric problem in a larger space. A block form of Paige's identity bounds the overlap with a Ritz vector along its residual coordinate direction. An inter-block recurrence describes the evolution of overlap, and a Gram-matrix argument gives a count of additional nearby Ritz values.

We extend the wavelet and tent space characterization of $h^1(\mathbb{R}^n)$ to $0<p\le 1$ via a local dyadic tent space $t_p$, and derive the unconditional $L^2$-convergence of the resulting atomic decomposition, with an exact error identity refining Ding, Jiang and Zhu (2019).

We prove the conjecture of Borowiecki and Broere that every finite simple planar graph admits a generalized total coloring with four colors: a proper vertex coloring with four colors together with an edge coloring with four colors in which each edge color class is a forest and no edge receives the color of either endpoint. In fact every proper four-coloring of such a graph extends to an edge coloring of the required kind. The proof combines a component-count inequality for planar graphs whose vertices are partitioned into four independent sets, obtained by completing to a triangulation, with the matroid partition theorem applied to four graphic matroids. A Lean 4 formalization relative to six named background assumptions is described below.

In once-reinforced random walk, introduced by Davis (1990), every edge starts with weight one and is assigned weight $\beta\geq1$ after its first crossing. At each step the walker chooses an incident edge with probability proportional to its weight. For sufficiently large reinforcement $\beta$, the expected range in the first $n$ steps was predicted by Ordemann et al. (2000) and independently by Beffara (2011) to have order $n^{d/(d+1)}$ on $\mathbb{Z}^d$. We prove the corresponding lower bound in every dimension $d\geq2$ and for every reinforcement parameter $\beta$.

We construct a map between simply connected rational spaces of finite rational type which admits a homotopy retraction and satisfies $msecat(p)=2<secat(p)=3$. The target is the rationalization of a simply connected twelve-dimensional CW complex with eleven cells, and is rationally hyperbolic. This disproves Carrasquel–Vera's conjecture for retractive maps.

We give a counterexample to Wormald's conjecture. A 16-vertex cubic graph, the disjoint union of $K_{3,3}$ with a 10-vertex bridged graph built from two subdivided copies of $K_4$, has no partition of its edges into two isomorphic spanning linear forests. The obstruction is parity: any such partition would force every monochromatic component to have even order while requiring a five-vertex side of a bridge to be partitioned into such components. Adjoining copies of $K_4$ gives counterexamples in every order $16+4t$. The counterexample is disconnected and leaves the connected case unresolved. The full family is verified in Lean without custom axioms.

A general $4\times3\times3$ tensor defines three determinantal varieties in $\mathbb{P}^3$ and $\mathbb{P}^2$ and $\mathbb{P}^2$ where it drops rank: a cubic surface and two length 6 subschemes. We study the Geometric Invariant Theory (GIT) quotient of $\mathbb{P}(\mathbb{C}^4\otimes\mathbb{C}^3\otimes\mathbb{C}^3)$ by $\operatorname{SL}(4)\times\operatorname{SL}(3)\times\operatorname{SL}(3)$, a moduli space of linear determinantal representations of cubic surfaces. We give a geometric description of the (semi)stable locus and some of the geometry of this GIT quotient. This GIT quotient admits a surjective rational map to the GIT moduli space of cubic surfaces, undefined at only one point.

We prove a version of genus-0 mirror symmetry for toric Fano surfaces at the level of big quantum cohomology. The mirror Landau-Ginzburg model $(\check{X}, W_{\infty})$ of such a surface $X$ can be defined in terms of counts of Maslov index two tropical or holomorphic discs with bulk insertions. While these mirror LG models are not well-defined, the Jacobian ring $\operatorname{\mathsf{Jac}}(W_{\infty})$ of the superpotential $W_{\infty}$ is well-defined. We show that $\operatorname{\mathsf{Jac}}(W_{\infty})$ is isomorphic over the Novikov base to the big quantum cohomology of $X$ using tropical methods, providing a purely algebraic analogue of the cobordism argument used in works of Fukaya-Oh-Ohta-Ono.

Let $S^k_d(n)$ denote the maximum number of regular $(k-1)$-simplices spanned by $n$ points in $\mathbb{R}^d$. For all fixed $r\geq k\geq3$, the exact value of $S^k_{2r}(n)$ was recently determined by Dumitrescu and the authors for all sufficiently large $n$ when $k=3$, and conditionally on an optimization problem when $k\geq4$. In this paper we establish a sharp estimate in odd dimensions by proving that \[S^k_{2r+1}(n)=\binom{r}{k}\left(\frac{n}{r}\right)^k+\Theta(n^{k-2/3})\] for all fixed $r\geq k\geq3$. This can be viewed as a generalization of the result by Erdős and Pach on unit distances in odd dimensions. We also establish a structural characterization of nearly extremal point sets. A notable difference from the even-dimensional case is that nearly extremal point sets in odd dimensions admit two distinct configurations. The proof leverages techniques from hypergraph Turán theory and linear algebra, with additional geometric arguments.

In this paper, we introduce and investigate partially antiinvariant submanifolds of Kenmotsu manifolds. The integrability of the involved distributions is examined, and conditions ensuring that the induced foliations are totally geodesic are obtained. We further explore the geometric properties of these distributions when the associated structure tensors N and P are parallel. By making use of the curvature tensor of Kenmotsu manifolds and the properties of Kenmotsu space forms with constant psi sectional curvature, we derive various geometric characterizations of partially anti-invariant submanifolds. In addition, we characterize totally umbilical partially anti-invariant submanifolds and establish an inequality for partially anti invariant submanifolds of Kenmotsu space forms, including the characterization of the equality case.

We propose a general approach to enharmonicism within syntactic music theory; that is, we formalize several abstract structures of enharmonicism without any acoustic considerations. The structures of concern to this paper are constructed from diatonics, chromatics, and enharmonics, and we seek to present a general enharmonic theory that builds upon Hook's enharmonic theory published in "Enharmonic Systems: A Theory of Key Signatures, Enharmonic Equivalence and Diatonicism" (2007). We also propose that the two criteria of reflection and maximal evenness can be used to determine whether an arbitrary enharmonic system is musically practical. Moreover, we argue that preferential treatment of the standard enharmonic system (SES) can be motivated from an abstract mathematical perspective divorced from pitch consideration and conventional acoustic constructions of SES.

We study a class of interacting diffusion particle systems with memory and their mean-field limit using coupling methods. We first establish quantitative propagation of chaos estimates for this class of models. Our first main result is a uniform-in-time propagation of chaos estimate, under a suitable contraction condition. We also discuss conditions for existence of stationary solutions and show that long-time convergence to a non-stationary measure may also hold. This requires an asymptotic behaviour for the memory interaction and a memory loss condition. As a consequence, our second main result is that the particle system converges asymptotically to the same equilibrium up to the propagation of chaos error, which shows that the large-particle limit and the large-time limit are exchangeable.

For the power mean $P_t$, $t<1$, let \[ \Lambda_N(t)=\sup_{x_k>0} \frac{\sum_{n=1}^N P_t(x_1,\ldots,x_n)}{\sum_{n=1}^N x_n} \] be its finite Hardy constant. Negative powers, the geometric mean, and the usual positive-exponent Hardy inequality correspond respectively to $t<0$, $t=0$, and $0<t<1$. With the analytic parameter $q=t/(1-t)\in(-1,\infty)$, and writing $\lambda_N(q)=\Lambda_N(q/(1+q))$, we derive one exact scalar shooting map for all three regimes; its apparent singularity at $q=0$ is removable and gives the finite Carleman problem exactly. The map has a common terminal condition and its limiting vector field has a quadratic critical bottleneck at \[ y_*(q)=1+q,\qquad \Lambda_*(q)=(1+q)^{(1+q)/q}. \] We prove that there is a unique real-analytic phase $\kappa(q)$ for which the finite-defect expansion has zero cubic coefficient and, uniformly for $q$ in compact subsets of $(-1,\infty)$, for every fixed $L\ge2$, \[ \Lambda_*(q)-\lambda_N(q) =\sum_{j=2}^{L}\frac{A_j(q)}{(\log N+\kappa(q))^j} +O\!\left((\log N)^{-L-1}\right). \] In this phase-normalized scale, \[ A_2=2\pi^2(1+q)\Lambda_*,\qquad A_3=0,\qquad A_4=-\frac{\pi^2}{3}(2q^2+5q+5)A_2. \] Thus the classical Carleman and positive-Hardy leading corrections are projections of a single analytic family. At the Carleman parameter we obtain $A_2(0)=2e\pi^2$ and $A_4(0)=-(10/3)e\pi^4$; the latter is also recovered by a direct local residue calculation. The phase itself contains the global discrete defect. After phase normalization, the first coefficient involving a nonconstant discrete response is $A_5$, at order $(\log N+\kappa(q))^{-5}$.

We prove some new identities for homogeneous symmetric polynomials when evaluated at roots of unity. Some of these formulas are applied to obtain new insights onto cyclotomic polynomials.

Motivated by a question of Corvaja and Zannier, we investigate whether the projections on the real torus of three real algebraic plane curves admit infinitely many triple intersections. We analyze the case of three lines in detail, obtaining a complete classification. For higher-degree curves, we derive several finiteness results using Siegel's and Levin's theorems on integral points on curves. On the other hand, we construct explicit examples of higher-degree curves with infinitely many triple intersections.

This book develops algebraic and combinatorial methods for studying discrete structures. It brings together graph theory, Boolean functions, Fourier analysis on finite groups, coding theory, perfect colorings and perfect codes, association schemes, Latin squares, and related topics. A central theme is the interaction between different representations of the same object: combinatorial, algebraic, spectral, and coding-theoretic. The main mathematical object studied in this book is a perfect coloring of a graph, or, equivalently, an equitable partition of a graph. The book is intended for advanced undergraduate and graduate students in mathematics and computer science, as well as for researchers in discrete mathematics, combinatorics, coding theory, and related fields.

Let $K\subset S^3$ be a non-trivial knot and let $\mathrm{IS}(K)$ be the simplicial complex whose vertices are the ambient isotopy classes of incompressible Seifert surfaces in the exterior $E(K)$, a finite set of distinct vertices spanning a simplex exactly when its classes admit simultaneously pairwise disjoint representatives. Kakimizu proved that $\mathrm{IS}(K)$ is connected; whether it is contractible was asked by Przytycki and Schultens, who also identified the obstruction, namely that the projection used in the connectedness proof is not known to be well defined on isotopy classes. We prove that $\mathrm{IS}(K)$ is contractible. We also prove contractibility of every genus truncation $\mathrm{IS}_\ell(K)$ with $\ell\ge g(K)$ and every non-empty lexicographic complexity sublevel. The argument factors through an unconditional combinatorial theorem: every non-empty connected flag exchange complex is contractible, where an exchange complex carries a complexity function subject to two axioms which require only that an exchanging vertex exist, never that a coherent selection rule be supplied. The combinatorial proof uses hereditary descending links and transfinite induction. We compare its attachment step with existing Morse criteria, including a formulation applicable to every countable exchange complex, and distinguish the given exchange order from a dismantling order. The geometric argument also applies to links satisfying a linking condition that forces every spanning surface to be connected. For the geometric input we prove fixed-boundary area attainment among smooth neat embeddings, using smooth convex replacement domains, area-controlled disc cleaning and boundary-preserving smoothing. All area complexities use this smooth competing class. No interpretation of an undefined piecewise smooth existence class is needed.

Let $G$ be a finite abelian group of order $N$, $S \subseteq G \setminus \{0\}$ a symmetric connection set, and $K$ a field with $char(K) \nmid N$. The spectral algebra $\mathscr{A_K}(Cay(G,S)) = K[A]$ generated by the adjacency matrix of the Cayley graph is proved to decompose, via the character-orbit decomposition, as a semisimple product of field extensions of $K$, one factor for each $Gal(\overline{K}/K)$-orbit of the eigenvalues $\lambda_\chi = \sum_{s \in S} \chi(s)$. The proof uses the abelian discrete Fourier transform to diagonalise $A$, the Galois action on the character group $\widehat{G}$ to partition eigenvalues into orbits, and the Chinese Remainder Theorem to convert the squarefree minimal polynomial into a Wedderburn product. The dimension of $\mathscr{A_K}(Cay(G,S))$ equals the number of distinct eigenvalues, the idempotent count is $2^r$ where $r$ is the orbit number, and primitive idempotents are computed explicitly via the Bezout algorithm in $K[x]$. Over $\mathbb Q$, every Wedderburn summand is a real subfield of the cyclotomic field $\mathbb Q(\zeta_N)$. New results include: a tensor-product comparison for Cartesian products of Cayley graphs; a systematic analysis of the spectral algebra for elementary abelian groups $(\mathbb Z/p)^k$ (rational for $p \le 3$, requiring real cyclotomic extensions for $p \ge 5$); and a worked orbit analysis for non-cyclic groups including $\mathbb Z/6 \times \mathbb Z/2$ and $\mathbb Z/5 \times \mathbb Z/2$. The cyclic case recovers the companion result $\mathscr{A}_{\mathbb{Q}}(C_n) \cong \prod_{d \mid n} \mathbb Q(\zeta_d)^+$; the Hamming cube gives $\mathscr{A}_\mathbb{Q}(\mathbb Q_k) \cong \mathbb{Q}^{k+1}$. The characteristic-$p$ case is also treated.

Fix $r\ge2$ and let $D_k$ be the natural density of the $n$ such that none of $n+1,\dots,n+k$ is $r$-free. We prove that $\log(1/D_k)=\frac{r-1}{\zeta(r)}k\log k+\frac{r}{\zeta(r)}k\log\log k-C_rk+o(k)$, where $C_r=\frac{1}{\zeta(r)}((r-1)\log\zeta(r)+r-1+r\log\frac{\pi}{r\sin(\pi/r)})$; for $r=2$ this is $C_2=\frac{6}{\pi^2}(1+\log\frac{\pi^4}{24})=1.4595513\dots$. This determines Mirsky's constant $\alpha(h)$, the density of the $n$ with $s_{n+1}-s_n=h$ in the squarefree numbers, to the same precision. Previously the only known estimate was $\log\alpha(h)\le-\frac54 h\log\log h+O(h)$; ours improves this by a factor of order $\log h/\log\log h$, and is two-sided. The upper bound rests on the observation that the primes $p\le z$ with $\prod_{p\le z}p^r\le\sqrt k$ sieve a window of length $k$ with no freedom of alignment, so that at least $(1/\zeta(r)+o(1))k$ positions must be covered by $r$-th powers of distinct primes exceeding $k^{1/r}$. The resulting sum over injections is $m!e_m$ with $m\sim k/\zeta(r)$, evaluated by a saddle point. For the lower bound the Chung-Erdos inequality recovers the full contribution of $m!e_m$ rather than merely its largest term, which is what makes the linear coefficient explicit. As a consequence the first-moment threshold for gaps between $r$-free numbers is $\frac{\zeta(r)}{r-1}\frac{\log x}{\log\log x}(1-\frac{(1+o(1))\log\log\log x}{(r-1)\log\log x})$; for $r=2$ the constant is $\pi^2/6$, that of Erdos's 1951 lower bound. We also prove that every covering system of length $k$ has modulus at least $\exp((\frac{r}{\zeta(r)}+o(1))k\log k)$, so the Chinese remainder construction yields only $\zeta(r)/r$, half of $\pi^2/6$ when $r=2$. Erdos's bound is thus equivalent to the assertion that the first run of $k$ consecutive non-$r$-free integers occurs at the first-moment threshold, not at the scale set by the minimal modulus of a covering system.

We present an explicit infinite equational basis for the six-element additively idempotent semiring $TR_6$ and prove that $TR_6$ is nonfinitely based. We also give a complete description of the subvariety lattice of the variety generated by $TR_6$, showing that it forms a four-element chain. Our results demonstrate that the variety generated by $TR_6$ is a limit variety: it is itself nonfinitely based, yet all of its proper subvarieties are finitely based. This provides a new limit variety of additively idempotent semirings, distinct from all previously known ones. In fact, $\mathsf{V}(TR_6)$ is the first explicit limit subvariety of the variety generated by the max-plus algebra $\mathbf{N}$. Moreover, $TR_6$ is not strongly nonfinitely based: it belongs to a finitely based variety generated by a finite additively idempotent semiring. Together with the six-element additively idempotent semiring $SR_6$, these are the first two finite additively idempotent semirings that are nonfinitely based but not strongly nonfinitely based. Finally, we study the variety generated by $SR_6$ and $TR_6$, showing that it is nonfinitely based and has exactly nine subvarieties, four of which are nonfinitely based and the remaining five are finitely based.

We show that, relative to the existence of an inaccessible cardinal, it is consistent that for an inaccessible cardinal $\kappa$, there is a non-trivial automorphism of $\mathcal{P}(\kappa)/\mathrm{fin}$ whose restriction to $\mathcal{P}(\beta)/\mathrm{fin}$ is induced by a bijection on $\beta$ for every $\beta<\kappa$. This answers questions posed by Larson–McKenney, Shelah–Steprāns, and Veličković.

We give a short, conceptual proof that prime graphs have at most two transitive orientations, a much-quoted result of Gallai. Our proof uses chains, introduced by Chudnovsky, Kim, Oum, and Seymour, which provide a transparent characterization of primality. Transitivity induces a forcing relation on edges; using chains, we show that any two edges of a prime graph are equivalent under this relation, and thus any transitive orientation is unique up to reversal.

Let $R=\mathbb{Z}/(2^n-1)\mathbb{Z}$, where $n\geq 3$, and let $S_w\subseteq R$ be the residues whose canonical $n$-digit binary expansion has Hamming weight $w$. We obtain, in particular, an asymptotic formula for the additive energy of $S_w$ \[ E(S_w)=\frac{\left|S_w\right|^4}{|R|}+ \mathcal{O}\left(|R|^3 n^{-3} \right), \] which holds uniformly in $w$. The error term is optimal in order, with a matching lower bound for $w=\lfloor n/2+\sqrt{n} \rfloor$. It follows that triple sums of arbitrary unit dilates have asymptotically uniform representation counts when $\prod_{j=1}^{3} \left|S_{w_j} \right| /\left(|R| n^{-3/5}\right)^3\to\infty$, and that double sums have asymptotically full support when $\left|S_{w_1}\right| \left|S_{w_2} \right|/\left(|R| n^{-3/4}\right)^2\to\infty$. In the proof, modular collisions are represented using a cyclic binary carry automaton; this appears to be a novel approach in this area of problems.

An expository essay containing some new results. A learner holds a prior belief over Boolean maps that answer a finite set of $Q$ questions, and receives answers one by one. Each answer carries surprisal and can clear predictive uncertainty about both the question asked and questions still unasked. We quantify this effect by the leverage: table entropy cleared per bit of surprisal received. Taking expectations over the truth prior and the question order, define the aggregate leverage as the ratio of expected uncertainty cleared to expected surprisal received. It equals one for independent answers and can exceed one for correlated answers. At finite size, this ratio is determined exactly by a single sequence: the mean entropy $G_\ell$ of the answers to $\ell$ questions. As the number of input bits grows at fixed asked fraction $t=\ell/Q$, a limiting increment profile $\gamma(t)$ determines the macroscopic learning curve. With $\eta_0$ the limiting initial entropy per question, the leverage becomes $L(t)= \frac{\eta_0-(1-t)\gamma(t)} {\int_0^t\gamma(x),dx}$. For exchangeable priors, de Finetti's representation gives a constant bulk profile $\gamma(t)$: deduction is confined to a boundary layer at $t=0$, and the leverage is forced to a hyperbolic form, as surprisal grows linearly. By contrast, we construct a simplicity prior with nontrivial bulk learning by grading Boolean maps by their polynomial degree over $\mathbb{F}_2$ and allocating weight across degree classes through a CDF $F$. Reed-Muller capacity then yields $\gamma(t)=1-F(t)$. This realizes any nonincreasing profile taking values in $[0,1]$, together with the corresponding macroscopic leverage curve.