Papers

We prove quantitative polynomial Szemerédi-type theorems involving polynomial progressions with shift parameter restricted to the set of shifted primes $\mathbb{P}-1$. The types of configurations covered are distinct degree progressions and progressions involving integer multiples of a fixed polynomial. For nonlinear configurations of length at least three, these results provide the first quantitative versions of such theorems. In the linear case, our results improve on work by the last two authors. Our density bounds are strongest in the case of distinct degree polynomials, where they give polylogarithmic bounds, of the same shape as recent bounds by Shao and Wang with integer shifts. The proofs combine recent quantitative results for polynomial configurations in the integers with quantitative Gowers uniformity bounds of the primes. For multiples of a fixed polynomial, we adapt a comparison argument of Altman and Sawhney to obtain uniformity over the polynomial families produced by the $W$-trick. For distinct degree progressions, we establish a comparison between prime-weighted and unweighted polynomial counts that is uniform throughout the density increment argument and accounts for a possible Siegel zero.

We study the initial run of equal adjacent parts in an integer composition. For a composition alpha, let P(alpha) be the common value of the parts in the initial run, let K(alpha) be the number of parts in that run, and set S(alpha)=P(alpha)K(alpha). We obtain exact finite formulas for the joint parameters (P,K,S) and the corresponding limiting laws. In the unrestricted model, each fixed event (P,K)=(p,k) reaches its limiting probability once n is above an explicit threshold. The limiting distribution of the total size is P(S=s)=2^{-s} sum_{d|s}(1-2^{-d}). Pointing a unit cell in the initial run gives M_1(z)=((1-z)/(1-2z)) sum_{m>=1} sigma_1(m) z^m. For every r>=1, the r-th power-moment generating function has coefficient function A_r(n)=sum_{j=1}^r (-1)^{j+1} binom(r,j) n^{r-j} sigma_j(n), so higher moments involve explicit finite combinations of generalized divisor sums. We also give factorial-moment formulas for the initial-run length and extend the construction to compositions with parts in a prescribed set A, where the corresponding formulas involve restricted divisor sums and the dominant root of the part generating function. Copy

We prove the cyclic-product conjecture of Holmes, Holroyd and Ramírez: among all circular arrangements of a decreasing positive sequence, the circular-symmetric “greedy” arrangement maximises every consecutive cyclic product sum $P_r$.

Let $r\ge 3$ and $k\ge 2r+1$ be fixed integers. We determine, for all sufficiently large $n$, the maximum adjacency-tensor spectral radius of an $n$-vertex $r$-uniform hypergraph containing no Berge cycle of length at least $k$. Write $s=\left\lfloor\frac{k-1}{2}\right\rfloor$. If $k=2s+1$ is odd, the unique extremal hypergraph consists of all $r$-sets containing at most one vertex outside a fixed $s$-set. If $k=2s+2$ is even, one additionally includes all $r$-sets containing a fixed pair outside the $s$-set together with $r-2$ vertices inside it. Consequently, the maximum spectral radius is given as \[ \operatorname{spex}_r(n,k) = \left[ \binom{s}{r-1} \left(\frac{r-1}{s}\right)^{(r-1)/r} +o(1) \right]n^{(r-1)/r}. \]

Let $G$ be a finite connected simple graph, let $uv$ be a nonedge, and let $s$ be a starting vertex. We study the exact change in the expected cover time of simple random walk when $uv$ is inserted. A killed Green matrix update, combined with target-set inclusion-exclusion, gives an exact formula using only the original graph. The response can have either sign. Our main result is a nonlocality theorem. For every radius $r\geq 1$, we construct two marked configurations whose ambient-degree-labelled radius-$r$ neighbourhoods at $s,u,v$ are isomorphic but whose fixed-start cover-time responses have opposite signs. The construction also matches the three marked degrees, the marked distance, and the effective resistance $R_{uv}$. The two graphs have different orders; equal-order nonlocality remains open. We complement this result with a conductance interpolation theorem and an occupation interpretation of its high-conductance coefficient. After contracting $u$ and $v$, that coefficient is a positive multiple of the expected pre-cover occupation of the contracted vertex, and its zero case is classified exactly. A path with two pendant insertion endpoints is solved for all starting vertices and shows a sharp change of sign across the start set.

For any $m, s \in \mathbb{N}$, we study the number $N_{m\times s,q}(\kappa, \beta)$ of solutions $(x_1,\ldots,x_s) \in (\mathbb{F}_q)^s$ of the monic system of diagonal equations $$ X_{1}^{k_i} + \cdots + X_{s}^{k_i}= \beta_i, \qquad (1\le i \le m), $$ with $\kappa=(k_1,\ldots,k_m) \in \mathbb{N}^m$ and $\beta=(\beta_1,\ldots,\beta_m) \in (\mathbb{F}_q)^m$. We show that this number can be obtained in terms of some data of diagonal GP-graphs $\Gamma(\kappa,q)$. This is a new family of graphs that we introduce here, $\Gamma(\kappa,q)$, with $\kappa = (k_1,\ldots,k_m) \in \mathbb{N}^{m}$, is the directed graph with vertex set the finite field $\mathbb{F}_q$ and there is an arc from $u$ to $v$ if and only if $v-u \in R_{\kappa} = \{ (x^{k_1},\ldots,x^{k_m}) : x \in \mathbb{F}_{q}^*\}$. In particular, we give three different expressions for $N_{m\times s,q}(\kappa, \beta)$: one in terms of walks, another in terms of adjacency matrices of $\Gamma(\kappa,q)$ and the last one in terms of the spectrum of $\Gamma(\kappa,q)$. Finally, we explicitly derive combinatorial formulas for the number of solutions $N_{m}(s,q) = N_{m\times s,q}(\kappa_\ell, 0)$ of monic homogeneous systems of diagonal equations of the form $$ X_1^{q^{\ell_i}+1} + \cdots + X_s^{q^{\ell_i}+1} = 0 \qquad (1\le i \le m),$$ with $\kappa_\ell=(\ell_1,\ldots,\ell_m)=(1,3,\ldots,2m-1)$ and $m\ge 2$, via the known spectrum of Hermitian-form graphs, which can be viewed as diagonal GP-graphs. For any $m,s \in \mathbb{N}$, we give general summation and recursive formulas for $N_m(s,q) \in \mathbb{Z}[q]$. For the small cases $N_{1}(s,q)$, $N_{2}(s,q)$ and $N_{m}(s,q)$, with $1\le s \le 5$, we give explicit expressions.

We establish a general framework for developing fast sampling and counting algorithms for stoquastic spin systems at high temperature. Our framework is based on a rapidly mixing Markov chain for polymer models and a subcritical percolation process for sampling individual polymers. We apply our framework to obtain fast algorithms for approximating the partition function and sampling from the thermal distribution of (1) general stoquastic spin systems, (2) ferromagnetic Heisenberg models, and (3) antiferromagnetic Heisenberg models on bipartite graphs. For the Heisenberg models, we obtain an improved bound on the inverse temperature by using their respective cycle and loop representations.

We prove that every finite point set of size at least $10^{11055931}$ has four collinear points or six points that pairwise see each other. This resolves the first open case of the big-line-big-clique conjecture of Kára, Pór, and Wood.

The likelihood of a finite simple undirected graph $G$ on $n$ vertices is the probability that the uniform sequential attachment process, which at each step joins a new vertex to a uniformly random subset of uniformly random size of the vertices already present, outputs a graph isomorphic to $G$. Dervovic, Mocherla and Severini conjectured that the likelihood is minimised by the balanced complete bipartite graph. We prove that, among complete bipartite graphs of a given order, the balanced one uniquely minimises the likelihood. Exact computation shows that it also minimises over all graphs for every order from $6$ through $14$, and that the first counterexample occurs at $n=15$. The blow-up of the five cycle by independent sets of size three, equivalently the circulant on fifteen vertices with connection set $\{1,4,6\}$, has likelihood $0.20128\ldots$ times that of $K_{7,8}$, and it is again triangle-free. We show that the failure is not sporadic by proving that the likelihood of the balanced complete bipartite graph is $2^{-(1/2-1/(8\ln 2)+o(1))n^2}$, whereas the minimum over all graphs of order $n$ is $2^{-(1/2+o(1))n^2}$, so the conjectured minimiser exceeds the minimum by a factor exponential in $n^2$. We also determine the Shannon entropy of the process to leading order, namely $n^2/(4\ln 2)$ bits, which shows that the conjectured minimiser is in fact more likely than a typical output of the process. The proofs rest on a vertex deletion recurrence which evaluates the likelihood in time $O(n\,2^n)$ and which closes on the blow-ups of any fixed base graph.

The twisted Foulkes character polynomial is an algebraically defined polynomial attached to an integer partition. We determine precisely how much combinatorial information this polynomial encodes by completely classifying all pairs of partitions that give rise to the same polynomial. Our main result shows that equality of twisted Foulkes character polynomials admits a purely combinatorial characterization in terms of two explicit local operations on partitions.

Henning and Yeo conjectured an upper bound on the identifying vertex cover number of a graph in terms of its order, size, and maximum degree. A two-parameter family $H_{t,r}$ of connected diameter-two graphs disproves the bound for every maximum degree at least four; after denominators are cleared, its margin is exactly $-(t-1)(r-1)$. The complement relation $\tau_D=n-\rho$ exposes the mechanism: diameter-two fibres admit at most one packing vertex, while degree deficit accumulates under tree gluing with controlled port loads. Writing $A_\Delta$ for the supremal additive gap at maximum degree exactly $\Delta$, an exact transfer formula gives $A_\Delta=+\infty$ for every $\Delta\ge 4$, using Petersen fibres in degrees four and five and the original $H_{t,r}$ blocks in higher degrees. If $c_\Delta$ denotes the corresponding supremal gap per vertex, rooted rook-graph fibres match a universal square-graph packing bound to first order. Consequently, $c_\Delta\sim 1/\Delta$, equivalently $\Delta c_\Delta\to 1$ as $\Delta\to\infty$.

First formulated by Amarilli, Monet, and Suciu (arXiv:2401.16210, 2024), the Non-Cancelling Intersections (NCI) conjecture is an open problem in combinatorics stating that any set union can be constructively built from its algebraically non-cancelling intersections using only disjoint unions and subset complements. In the same paper, two orthogonal possible strengthenings are proposed: using only left-linear trees, and using non-trivial intersections only positively or only negatively depending on the sign of their Möbius value. Here we show that using only left-linear trees, the conjecture is false (independent of the other strengthening). Our argument is non-constructive. We prove the existence of a counterexample, though it is of immense size.

In this paper, we examine the structure of strongly regular graphs with parameters $\lambda = 1$ and $\mu = 2$. In particular, we provide a complete classification of induced subgraphs of order seven and determine their relative frequencies. These findings contribute to a finer understanding of the local structure of such graphs and may be useful in related combinatorial and algebraic investigations.

We present here a generalized construction of a finite solvable group whose prime character degree graph has the shape and structure of the bowtie graph. As with the original bowtie, the graphs obtained by this generalized construction, under certain restrictions, cannot be realized by the usual method of taking direct products of smaller graphs. Within the condition of $n=1$, we show how this recovers the original bowtie graph, which has five vertices. We also provide examples and explicit choices of primes which generate graphs with more vertices.

The connective constant $\mu(G)$ of a graph $G$ is the exponential growth rate of the number of self-avoiding walks starting at a given vertex. We prove upper and lower bounds for the connective constants of Cayley graphs $G_\omega$ of a general Grigorchuk group encoded by a sequence $\omega\in\{0,1,2\}^{\Bbb N}$. In particular, $\mu(G_\omega) > \phi$ for any such Cayley graph (subject to a simple condition on $\omega$), where $\phi:= \frac12(1+\sqrt 5)$ is the golden mean. This extends earlier work of the author and Zhongyang Li in "Cubic graphs and the golden mean”, Discrete Math. 343 (2020), article 111638, where it was conjectured that $\mu(G)\ge\phi$ for all infinite, vertex-transitive, cubic graphs. The current work includes an analysis of the proportions of appearances of given label-sequences in the orbital Schreier graphs of general Grigorchuk groups.

In a $k$-uniform hypergraph, the minimum $d$-degree for some $0\le d\le k-1$ is the minimum number of edges containing any given $d$-set of vertices. An extension of the classical Dirac theorem guarantees that whenever the minimum $d$-degree of a $k$-uniform $n$-vertex hypergraph, $k\mid n$, is larger than a certain Dirac threshold, it contains at least one perfect matching. Moreover, it has been known for some time, due to Kwan, Safavi, and Wang, that for $d\ge k/2$ such hypergraphs contain not only one, but “many” perfect matchings, that is, at least as many as are expected in a random hypergraph with the same edge density. However, it has also been known that such a result could not be hoped for in general, as it already fails for $(d,k)=(1,3)$. In this paper we introduce new notions of the counting thresholds and approximate counting thresholds, above which a hypergraph is guaranteed to have at least this many perfect matchings. We show that these thresholds are well-defined and nontrivial for all $d,k,n$, that they are asymptotically related, and finally, we derive improved upper bounds by reducing to cases with smaller $d$ and $k$.

In this paper we study generalised total colourings of graphs where the colour classes formed by vertices and edges, respectively, induce forests, while incident edges/vertices receive distinct colours. In [M. Borowiecki and I. Broere, Hamiltonicity and Generalised Total Colourings of Planar Graphs, Discussiones Mathematicae Graph Theory 36 (2016) 243–257] it was conjectured that for planar graphs, four colours suffice for this type of colouring. We confirm this conjecture for two infinite families of planar graphs.

A set of points $C\subset \mathbb{R}^n$ is canonically Ramsey if there is some larger set of points $S\subset \mathbb{R}^{n'}$ such that any colouring of $S$ contains either a monochromatic copy of $C$ or a rainbow copy of $C$. Mao, Ozeki, and Wang introduced this notion, showing that the 30-60-90 triangle is canonically Ramsey. Since then, many other configurations have been shown to be canonically Ramsey. The author showed that cuboids are canonically Ramsey. Ge, Shu, Xu, and Yu later showed that all simplices are canonically Ramsey, after which the author showed that all products of simplices are canonically Ramsey, a class which, together with its closure under taking subsets, includes all previously known canonically Ramsey sets. We prove that regular polygons with a prime number of sides are canonically Ramsey—the first known sets outside this class.

The kernel of a random symmetric p-adic matrix is a random abelian group, equipped with a symmetric pairing. If we consider not only the matrix but also its top-left corners, we get a process valued in isomorphism classes of abelian groups, equipped with such a pairing. We show that when the matrix is Haar random, this process is a Markov chain, generated by an operator that we explicitly describe. We will also prove that this operator is reversible with respect to a Cohen-Lenstra type measure.

The unfair $0$–$1$ polynomial conjecture asks whether a factorization \[C(x)=A(x)B(x),\] with $A$ and $B$ monic and having nonnegative real coefficients, and $C$ a polynomial with all the coefficients 0 and 1, must already be a factorization into $0$–$1$ polynomials. Let $k$ be odd and $0<a<1$. We study the possibility that \[1+a x^2+x^k\] divides a $0$–$1$ polynomial with a nonzero cofactor having nonnegative real coefficients. Ghidelli settled the first nontrivial case $k=5$. We prove that no such factorization exists for any odd $k\ge 341$. The intermediate cases (5 < k < 341) are treated in the companion paper.

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.

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.

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 $\overrightarrow{\operatorname{diam}}(G)\leq \frac{n+17}{3}$.

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.

A set S of vertices of a graph is odd independent if it is independent and every vertex outside S has either zero or an odd number of neighbors in S. The largest size of such a set is the odd independence number alpha_od. Caro, Petrusevski, Skrekovski and Tuza [2] conjectured that alpha_od = 1 for every finite Queen graph. They also asked whether the infinite Queen graph has alpha_od = 1 or alpha_od = infinity. We prove that alpha_od = 1 in both cases. In particular, in the case of an infinite board we prove that alpha_od = 1 holds on the quarter plane and on the whole plane.

We say that a matrix over a finite field $\mathbb{F}_q$ is positive definite if it is symmetric and each of its leading principal minors is a nonzero square in $\mathbb{F}_q$. In previous work of the authors [J. Algebra, 2025], the entrywise positivity preservers on $M_n(\mathbb{F}_q)$ were classified in every case except when $n=2$, $q\equiv1\pmod4$, and $q$ is not a square. We settle this remaining case, thereby completing the classification of entrywise positivity preservers over every finite field and in every dimension $n\ge2$. Our proof is based on a novel idempotent reduction that not only resolves the remaining case but also yields a self-contained proof of the complete classification, while avoiding several technical results used in the earlier arguments. As a further application of the same reduction, we classify the entrywise preservers of strongly nonsingular matrices, i.e., matrices whose leading principal minors are all nonzero. We also prove a more general theorem in odd characteristic: for every prescribed sign pattern of nonzero leading principal minors of matrices of a fixed dimension $n\ge2$, the entrywise preservers are precisely the positive scalar multiples of field automorphisms. Thus, in odd characteristic, preserving any nonzero leading-principal-minor sign pattern surprisingly forces the preservation of every such sign pattern.

Norin and Turcotte's asymptotically sharp bound for graph burning [J. Combin. Theory Ser. B 168 (2024), 208–235] led them to an exact random-cover conjecture for finite metric trees. Let $U[0,r]$ be the uniform probability measure on $[0,r]$. They conjectured that every finite metric tree $T$ of length $L\ge2r$ admits a probability measure on $0$-good ball covers whose expected radius measure is at most $(L/r)U[0,r]$. We prove the conjecture for every finite metric tree. We recast the bootstrapping calculation of Norin and Turcotte as a zero-error replacement certificate. The resulting local scale reduction, together with a three-piece decomposition and a macro-recursion, produces a fractional marked-ball cover with the exact radius budget. We then pass from the fractional cover to random finite covers by a compact rounding argument. For metric-tree balls, Tamir's balancedness theorem and standard balanced-matrix ideality provide the finite-dimensional integrality input. We also prove an arbitrary-budget duality criterion. If $0<R\le L$ and $\beta$ is a finite positive Borel measure on $[0,R]$, then $\beta$ dominates the expected radius measure of a random $0$-good cover if and only if $\sigma(T)\le\int_{[0,R]}\max_{v\in T}\sigma(B_T(v,s))\,d\beta(s)$ for every finite positive Borel measure $\sigma$ on $T$; it is enough to test finite atomic measures. We use this criterion to extend the uniform range to every $r\le L-\operatorname{diam}(T)/2$, determine the exact range for equal-arm metric stars, and derive deterministic bounds, interval rigidity, and a diameter-defect stability estimate.

For a graph $G$, let $h(G)$ be the minimum cardinality of a vertex set meeting every maximum independent set of $G$. We establish two complementary reduction principles for the Bollobás–Erdős–Tuza conjecture: the conjecture for arbitrary graphs is equivalent to its restriction to regular graphs of any fixed positive linear degree, and, within every hereditary graph class, a uniform sublinear bound is equivalent to a sublinear bound on graphs of every fixed positive linear vertex connectivity. We prove the sharp general estimate \[ h(G)\le \left\lfloor\frac{|V(G)|}{2\alpha(G)+\delta(G)-|V(G)|}\right\rfloor \] whenever the denominator is positive, with equality for balanced complete multipartite graphs. Consequently, every $3$-colorable graph of order $n$ with $\kappa(G)\ge\rho n$ and $\rho>1/3$ has a hitting set of size at most $\lfloor(\rho-1/3)^{-1}\rfloor$; direct use of a $3$-coloring improves this to $6$ when $\kappa(G)>4n/9$ and to the sharp bound $3$ when $\kappa(G)>n/2$. For dense regular graphs with independence ratio greater than $1/4$, we obtain a logarithmic bound, while constructions with linear degree and linear independence number show that $h(G)=\Omega(\sqrt n)$ can still occur. We also prove a logarithmic bound for near-regular $3$-colorable graphs and exhibit a critical family at connectivity $n/3$ that explains the limitations of the degree-surplus and degree-ratio methods.

We develop additive-energy estimates and weighted Young inequalities for finite sets on bounded-degree real algebraic varieties. For an irreducible $m$-dimensional variety $V$, let $\sigma(V)=2m-\dim\overline{V-V}^{\mathrm{Zar}}$ and $\alpha(V)=\max{2,1+\frac{2\sigma(V)}{m}}$. For every $a\in[\alpha(V),3)$ we define a finite-degree translation-partition flag parameter $\Lambda_{a,R}(X;V)$ and prove $E(X)\ll \Lambda_{a,R}(X;V)^{3-a}|X|^{a+\varepsilon}$. This recovers the line-concentration theorem of Jing and Wu for algebraic surfaces in $\mathbb{R}^3$. For codimension-two quadratic threefolds ${(u,Q_1(u),Q_2(u))\in\mathbb{R}^3}\subset\mathbb{R}^5$ with positive-definite $Q_1$ and simple generalized spectrum, we prove the sharp estimate $E(X)\ll_{\varepsilon}|X|^{2+\varepsilon}$ without a flag loss. Hereditary versions of these estimates imply weighted $L^4$ restriction bounds and off-diagonal Young inequalities; at the near-diagonal threshold the sharp region is $1\le p,q\le 2$ and $p^{-1}+q^{-1}\ge 1$. We also prove a sharp turning-complexity extension of the Cushman-Demeter-Wu theorem: $J_3(P)\ll_{\varepsilon}\kappa(P)^2|P|^{3+\varepsilon}$, with matching examples at every power scale.