Papers

Computer-assisted proofs in dynamics establish results about nonlinear systems by rigorous numerical computation. Their correctness rests on a trusted base of interval-arithmetic libraries and analytic estimates checked by hand. We formalize in Lean~4 a framework for the radii polynomial method, which certifies an exact solution near a numerical approximation by verifying four norm bounds and the resulting polynomial inequality. Weighted coefficient algebras provide the common setting for polynomial equations and initial value problems in Taylor and Chebyshev series. Their universal properties construct the bounded operators and the evaluation maps, and the universal property of the free commutative algebra makes polynomial substitution commute with evaluation. Finite/tail reductions turn the four norm bounds into finite rational inequalities, which are checked in Lean. The radii theorem then yields an exact coefficient solution, and realization theorems carry it to a solution of the original equation. The worked examples are a square-root branch given by a convergent power series and polynomial initial value problems, among them the Lorenz system, for which the library proves existence, uniqueness within the trajectory ball, and analyticity of the function-level solution.

Mobile embodied AI networks (MEAN) enable embodied agents to perceive, reason, communicate, and act in wireless environments. In such networks, agent mobility can improve channel conditions, while semantic compression can reduce transmission payloads. However, movement consumes energy, and stronger compression incurs additional computational cost. This paper studies joint movement, semantic compression, and transmit power design for an uplink MEAN system. We formulate a max-min energy efficiency (EE) problem by jointly optimizing transmit power, movement distance, and semantic compression ratio under controllable power constraints. The problem is non-convex due to the coupled signal-to-interference-plus-noise ratio (SINR), mobility-dependent channel gains, and fractional EE objective. To solve it, we propose an alternating optimization (AO)-Dinkelbach algorithm, where the fractional objective is handled by the Dinkelbach transformation, transmit power is updated via successive convex approximation (SCA), and movement distance is updated by coordinate-wise grid search. Simulation results show that the proposed scheme outperforms no-mobility and no-compression baselines, demonstrating the benefit of jointly exploiting mobility control, semantic compression, and power allocation in MEAN.

Using the equivalence of sphere order representations of posets with Minkowski space embeddings, we show that the Boolean lattice $\mathcal{B}_6$ is not a sphere order. This is the first explicit finite example of a poset that is not a sphere order.

We extend the classification of manifolds with positive isotropic curvature to all dimensions $n \geq 5$. This result also completes the classification of uniformly positive isotropic curvature on complete non-compact $\kappa$-noncollapsed ancient solutions and gradient shrinking Ricci solitons. The proof involves a construction of pinching cones in all dimensions, refining and extending to all dimensions the results of the first author. We have used MATHEMATICA to find viable parameters for the cone families. The final presentation is self-contained and can be verified manually. No stage of this work involved LLMs.

We develop a finite element method (FEM) for partial differential equation (PDE) solver based on the entanglement-basis representation introduced in our companion work. By lifting non-linear finite-element equations into an augmented coefficient space, the governing PDE together with boundary, initial, and inter-element constraints can be expressed through a unified quadratic residual minimization. Although this augmented space grows exponentially with the number of elements, its tensor-product structure allows it to be represented efficiently using tensor networks. Using the matrix product state (MPS) as a concrete example, we show that density matrix renormalization group (DMRG) sweeps enable element-by-element optimization without explicitly constructing the full augmented space. For bounded bond dimension, the resulting computational cost scales polynomially with the number of finite elements. We extend the framework to time-dependent problems through implicit temporal discretization and demonstrate convergence under both mesh and polynomial refinement using diffusion equations.

Recent work has shown that large language models (LLMs) can be vulnerable to jailbreak attacks in which harmful intent is obscured through composition with benign tasks. A harmful request refused in isolation may elicit a different response when embedded within a larger, seemingly benign query. We study these compositional intent-hiding jailbreaks from an information-theoretic perspective. Our formulation associates each task with an estimated probability of being judged harmful: the average over the full task collection defines the prior probability of harmful intent, while the average over a selected bundle containing the target defines the posterior. Selecting auxiliary tasks so that these averages agree, which we call prior-posterior matching, leaves the estimated intent unchanged even though the harmful target remains in the bundle. We study two settings that differ in whether query construction is part of the optimization. In the query-independent setting, tasks are selected without regard to how they will be expressed in the final query. We show that exact prior-posterior matching under a bundle-size constraint is computationally hard, derive an optimal water-filling solution for fractional weights, and characterize the smallest bundle satisfying a prescribed safety threshold. In the query-dependent setting, task selection and query construction are considered jointly, and intent concealment and target preservation are evaluated on the resulting query. We evaluate jailbreak effectiveness and preservation of the target behavior across bundle sizes, query generators, and several open-source models. These results show that compositional queries can elicit target behaviors beyond the direct-request baseline under the evaluated search budgets, while revealing a trade-off: as bundle size increases, response-level target preservation tends to decrease for several models.

The union-closed sets conjecture asks whether every finite union-closed family containing a nonempty set has an element contained in at least half of its members. Let $c_{\mathrm{UC}}$ denote the largest universal lower bound on this proportion. The best previously proved lower bound is due to Liu and is above $0.3823455$; his own numerical optimization suggested a stronger threshold $c_{\mathrm L}>0.382709087918735$. The main argument of this paper is to use different protocols (suitable families of probability measures) within Liu's framework. In particular, we combine the independent protocol with Liu's Example 5 to prove the conjectured bound $c_{\mathrm{UC}}\ge c_{\mathrm L}$. We then vary the parameter in Example 5 and optimize the choice of the two coefficients in the combination to obtain $c_{\mathrm{UC}}\ge 0.3828852549667978$. We also show that the inequality used in this proof cannot give a threshold larger than $0.3828852599667\ldots$.

Unital quantum channels are the noncommutative analogues of doubly stochastic matrices. In 1959, Marcus and Ree showed that the squared Frobenius norm of a doubly stochastic matrix is at most its maximal diagonal sum. We prove a quantum version of this inequality for mixed unitary channels on $\mathbb{M}_d$ and give an example showing that it fails for general unital channels. As in the classical case, we call the channels attaining equality as Erdős channels. We show that the trace-preserving conditional expectation onto any unital $C^*$-subalgebra of $\mathbb{M}_d$ is an Erdős channel. It is also locally isolated among channels of the same Choi rank. Next, we introduce quantum RCDS channels. These are the noncommutative analogue of doubly stochastic matrices whose supported permutations all have the same diagonal sum. We prove a structure theorem for them and use it to characterize the quantum RCDS Erdős channels. We give an explicit form, up to unitary equivalence, for the Erdős channels of Choi rank $2$, and further show that there are uncountably many unitary equivalence classes of Erdős channels unlike the classical case. We also characterize the Erdős channels among mixed Weyl-unitary channels of any Choi rank. For qubits, we prove that there are exactly four unitary equivalence classes of Erdős channels and they are represented by the identity channel and uniform averages of Pauli channels. Finally, restricting to the diagonal channels, our results recover the Marcus–Ree inequality, the structure of RCDS doubly stochastic matrices, and the characterization of Erdős.

This study considers the problem of portfolio choice, where we recommend a portfolio to an investor to maximize the expected utility of their wealth. Our goal is to construct an asymptotically optimal portfolio choice rule in terms of expected utility regret, the difference between the expected utility of an oracle investor and that achieved by a portfolio chosen from data. We propose the Expected Utility Regret (EUR) rule, which jointly selects a portfolio class and estimates its weights. In a regular parametric return model, a single EUR rule attains both the minimax and the Bayes lower bounds, including their leading constants, without using the prior distribution that defines the Bayes criterion. We then derive the mean–variance and risk-parity portfolios as special cases of this framework. Under smooth increasing and concave utility, the EUR rule and the sample mean–variance portfolio attain the same leading expected regret when expected excess returns approach zero sufficiently fast. When the returns divided by their volatilities have a joint distribution that does not depend on the order of the assets, the EUR rule and the risk-parity portfolio coincide.

We prove that the maximal cp-rank $p_n$ satisfies $p_n\ge n(n-3)/2$ for odd $n\ge5$ and $p_n\ge n(n-3)/2-1$ for even $n\ge6$. Together with the known upper bound $p_n\le\binom{n+1}{2}-4$, this gives $p_n=n^2/2+O(n)$.

We prove the Koumandos–Ruscheweyh conjecture for every $0<\rho\leq1$. If $\nu(\rho)$ is the unique root in $(0,1]$ of $\int_0^{(1+\rho)\pi}t^{\mu-1}\sin(t-\rho\pi)\,\mathrm{d}t=0$, then $(1-z)^\rho s_n^\mu(z)\prec((1+z)/(1-z))^\rho$ for every $n\geq0$, $0<\mu\leq\nu(\rho)$ and $z\in\mathbb{D}$, where $s_n^\mu(z)=\sum_{k=0}^n(\mu)_kz^k/k!$. The parameter $\nu(\rho)$ is optimal. The proof combines an all-parameter gamma-coefficient comparison with an exact beta integral and a binomial variance estimate, reducing the infinitely many degrees to finitely many continuous interval inequalities. The remaining inequalities are certified by 256-bit ball arithmetic; rational covers, source code and alternative-formula verifiers are provided. The weak positive-real-part conjecture follows as a sharp corollary. We also derive sharp consequences for starlike functions and Gegenbauer polynomial sections: a full-parameter convolution subordination, the optimal starlike order for uniform partial-sum sectors, and the optimal Gegenbauer parameter and sector angle. The necessary bounds and angular sharpness are obtained from explicit kernels and interior scaling limits.

Let $P$ be a Markov operator on a general state space $(S,\Sigma)$ with invariant probability $m$, assumed ergodic. When $P$ is normal in $L_2(m)$, we prove the necessity of uniform ergodicity, under some spectral conditions (satisfied when $P$ is symmetric), for the annealed CLT to hold for every centered $ f \in L_2(m)$. We also obtain results without assuming normality under some Ritt-type conditions.

For edge-disjoint graphs meeting along a finite boundary, we construct relative shellings of NBC faces with prescribed boundary connectivity. The construction adds paths in spanning forests to the restriction faces of auxiliary rooted-forest complexes. When every edge of the first graph precedes every edge of the second, these relative shellings combine into a shelling with consecutive connectivity blocks. The resulting nonnegative decomposition of the $h$-polynomial realizes the chromatic specialization of the Negami splitting identity. We also give path formulas for gluing graphic Orlik–Terao Gröbner bases across boundaries of size two or three, and a virtual-clique elimination formula for arbitrary boundaries. An exponential output lower bound at treewidth two distinguishes explicit basis enumeration from coefficient computation at fixed treewidth. Finally, state polytopes and their Cayley lifts describe an obstruction to linear block orders in the original coordinates and a lifted realization of these orders.

We study deformations of foliated nonsingular varieties defined by locally free subsheaves of tangent sheaves and locally free subshseaves of cotangent sheaves in the language of functors of Artin rings. We identify first-order deformations and obstructions.

Choose a uniformly random ordering of the vertices of a finite tree and run the usual greedy maximal-independent-set algorithm. We compare the resulting law on maximal independent sets with the uniform law. We prove that exact uniformity occurs only for the one-vertex tree and the single edge. The proof is structural: a diameter endpoint exposes a pendant star, and the remaining one-pendant-leaf case is resolved by a strict injection between exact permutation fibres obtained by swapping the pendant leaf with its support vertex. Exact uniformity is therefore rigid, but it can be approached closely. For an explicit mixed-spider family $T_{k,l}$ we count the maximal independent sets and compute the exact probability of every output. With $l=2^k-k$ the total-variation bias is positive and satisfies [ b(T_{k,2^k-k})=O!\left(\frac{\sqrt{k}}{4^k}\right) =O!\left(\frac{\sqrt{\log n_k}}{n_k^2}\right), \qquad n_k=2^k+k+1. ] The theorem package has also been formalised in Lean and registered with Palomar. These records document machine-checked formal verification and the checked axiom boundary; they are not peer review or a certificate of novelty.

An edge open packing is a set of edges whose endpoints induce a disjoint union of stars. We study the problem of finding a maximum edge open packing in AT-free, chordal, and convex bipartite graphs. For AT-free graphs, we use the oriented star-conflict graph $A_G$, introduced by Das and Santra, for which $\rho_e^o(G)=\alpha(A_G)$, and prove that $A_G$ is AT-free whenever $G$ is AT-free. Consequently, a known maximum independent set algorithm for AT-free graphs yields an $O(n^2+m^4)$-time algorithm for Maximum Edge Open Packing, where $n$ and $m$ denote the numbers of vertices and edges of $G$, respectively. For chordal graphs, we develop an $O(n^4)$-time dynamic programming algorithm over a nice tree decomposition derived from a clique tree, exploiting the structural fact that the endpoint set of an edge open packing intersects every clique in at most two vertices. Finally, for convex bipartite graphs, we obtain an $O(n^4)$-time dynamic programming algorithm based on two boundary indices that separate consecutive star components. This result extends the previously known algorithm for biconvex bipartite graphs and, in particular, provides an improved explicit running-time bound for that class.

Let $G$ be an $n$-vertex connected nonregular graph with maximum degree $\Delta$ that attains the maximum spectral radius. For $3\le\Delta\le n-2$, Liu and Li (2008) conjectured that $G$ has the degree sequence $(\Delta,\ldots,\Delta,\delta)$ with $\delta<\Delta$ as large as possible. In this paper, we study the dense case where $\Delta=n-k$ with $k\ge4$ fixed and $n$ sufficiently large. We prove that \[ d(G)= \begin{cases} (\Delta,\ldots,\Delta, 1), & \text{if $n$ is even and $k$ is odd};\\ (\Delta,\ldots,\Delta,\Delta-1), & \text{if $n$ is odd and $k$ is even};\\ (\Delta,\ldots,\Delta,2), & \text{if $n$ and $k$ have the same parity}. \end{cases} \]

A prime labeling of a graph labels its vertices bijectively with the integers from one to the number of vertices, so that adjacent vertices receive coprime labels. I prove the Entringer-Tout conjecture that every tree admits a prime labeling, extending the result of Haxell, Pikhurko, and Taraz from sufficiently large trees to all orders. My proof combines structural decompositions of trees, arithmetic organization of the labels, and matching arguments. I reduce the labeling problem to a structured bipartite matching problem by separating a small part of the tree, distributing labels according to their divisibility properties, and applying Hall's theorem to complete the labeling. I handle the finite and intermediate ranges using exact computer-assisted certificates whose validity is justified mathematically, while the remaining range is covered by analytic estimates. Together, these ingredients yield a proof for trees of every order.

For a positive integer $n$, let $S(n)=\{\sigma(d): d\mid n\}$ be ordered by divisibility, and let $a(n)$ be its width. We study how much of the divisor lattice of $n$ survives under the map $d\mapsto \sigma(d)$. For every fixed prime-exponent pattern, the largest possible value of $a(n)$ is the width of a corresponding product of divisibility posets, and this maximum is attained infinitely often. In the squarefree case with $\omega(n)=r$, every width from $1$ to $\binom{r}{\lfloor r/2\rfloor}$ occurs, again infinitely often. For repeated exponent patterns, Engel's limit theory reduces the asymptotic width problem to optimal representation variance. We show that for the divisibility poset $D_m=(\{1,\ldots,m\},\mid)$, the prime directions $2$ and $3$ already determine the exact variance. The structural reason is the critical dilation $4$, lying strictly between $3$ and the next prime $5$. This yields an exact finite formula, the limit $V(D_m)\to 1.0367934201598\ldots$, and asymptotics for repeated exponent patterns. The complementary minimum-width problem remains largely open.

A binary word is a shuffle square if its positions can be partitioned into two identical subsequences. A finite word over $\{1,2\}$ is smooth if repeated finite run-length differentiation remains binary until the empty word is reached. This paper proves that every smooth binary word in which both letters have even multiplicity is a shuffle square once its length is at least $18$. There are exactly $34$ exceptions, all of length at most $16$. The infinite part of the proof is a parity-return atom induction: a smooth even-Parikh word factors uniquely into smooth parity atoms, and every such atom has length at most $8$. A finite, independently checkable certificate handles lengths $18$ through $40$. As a consequence, a nonempty prefix of the classical Kolakoski sequence is a shuffle square if and only if both symbols occur an even number of times and its length is not $4$ or $8$. There are infinitely many such prefixes. In fact, the entire infinite Kolakoski sequence can be partitioned into two identical infinite subsequences with monochromatic blocks of uniformly bounded length.

We give explicit positive factorizations for a genus-19 Lefschetz fibration over the torus and a family of genus-4 Lefschetz fibrations over a genus-2 surface. All these examples have only three nodes, and their total spaces are symplectic 4-manifolds of general type on the Bogomolov-Miyaoka-Yau line. In the course of deriving the monodromy relation in genus 19, we obtain several interesting identities, including a factorization of a Dehn twist into a product of three order-3 mapping classes. We show that the total space of this fibration is topologically s-cobordant to the Cartwright-Steger surface, and we determine its real quotient up to homeomorphism. The genus-4 family consists of variations of the Bauer-Catanese fibrations and contains 65 isomorphism classes of Lefschetz fibrations on 45 diffeomorphism types of compact complex ball quotients. We describe their topology and that of the real quotients of the three original surfaces. The handle monodromies in our main factorizations are pseudo-Anosov, and we show more generally that for holomorphic genus-4 Lefschetz fibrations on the BMY line, this holds for every choice of handle generators.

Full-parameter fine-tuning of large language models (LLMs) incurs substantial optimizer state memory overhead, limiting the model sizes that fit on modern GPUs. Existing approaches either compress optimizer state, abandon first-order gradients, or change the update geometry while retaining dense state. The recently introduced Muon optimizer reduces optimizer memory through matrix-valued updates. Still, its geometry differs from AdamW and can lead to performance degradation when fine-tuning AdamW-pretrained models. To reduce optimizer memory without sacrificing accuracy or computational efficiency in LLM fine-tuning, we propose Ternary Absolute-max Column-wise One-sparse optimizer, or TACO, which follows Muon's operator-norm steepest-descent view but takes the geometric route further. TACO computes the exact steepest-descent direction under a dimension-normalized $1\to1$ operator norm by selecting the sign of the largest magnitude entry in each column of two-dimensional weight matrices. This retains first-order gradients while making optimizer state memory nearly negligible. Our practical TACO optimizer maintains only a small set of low precision gradient components per column, reducing persistent optimizer state by $174\times$ relative to AdamW8bit (from 27.7 GB to 0.16 GB) and peak training memory by $2.9\times$ (from 80.6 GB to 27.5 GB) on OPT-13B, while achieving comparable accuracy and runtime. TACO further enables full-parameter fine-tuning of 30-32B-parameter models on a single 80 GB H100 GPU across multiple model families and tasks.

Step-size selection remains a central challenge in large-scale neural network optimization; conservative steps slow convergence, while aggressive steps can destabilize it. We combine Zero-and-First-Order optimization~(ZFO) and propose a lightweight framework that decouples direction selection from step-size. ZFO uses a trusted first-order optimizer to determine the direction and performs zeroth-order evaluations only along this one-dimensional subspace to choose how far to move. Using the current {gradient information} and two additional objective function evaluations, ZFO instances construct a local model of the objective function along the proposed direction and select a curvature-aware step within a bounded search interval. This yields an adaptive step-selection mechanism that costs less than a full line search. We provide theoretical guarantees to show that shared-sample evaluations produce reliable finite-difference curvature estimates, that the induced local model selects a near-optimal step along the search interval, and that ZFO converges to a neighborhood of a stationary point. Across the evaluated settings, language models and datasets, ZFO frequently improves optimization and final performance relative to fixed-step first-order baselines, with the magnitude and preferred local model depending on the objective. Our code is publicly available at: https://github.com/nizswan/Zeroth-First-Order-Framework.

On broad classes of linear systems, the shortest experiments are almost as good as the best possible ones. For $n$ states and $m$ inputs, the shortest input sequences that support robust data-driven stabilization of every controllable plant have $mn+1$ steps with exact states and $m(n+1)$ with noisy states. We show that, when the spectral radius is bounded and the spectrum is well separated near the unit circle, these sequences tolerate a fixed fraction of the error level achievable by any experiment, even one designed with full plant knowledge and allowed to use any finite duration. This constant-factor comparison can fail for slowly actuated systems. For $A=I+hG$ with controllability depth $\nu\ge2$, short experiments lose a factor of order $h^{\nu-1}$, and duration of order $1/h$ is both necessary and sufficient to recover a fixed fraction of the optimal tolerance.

In 1993, Ruzsa showed that for every $k \geq 2$, there exists a constant $C$ such that every subset $A \subseteq [N]$ of size at least $C N^{1/k}$ contains $2k$ distinct elements $a_1, \ldots, a_k, b_1, \ldots, b_k \in A$ such that $a_1 + \cdots + a_k = b_1 + \cdots + b_k$. We strengthen this result by proving that the elements $a_1, \ldots, a_k, b_1, \ldots, b_k$ can be chosen to have the additional property that $\{a_1, \ldots, a_k, b_1, \ldots, b_k\}$ has $2^{2k}-1$ distinct subset sums, with the only coincidence being that $\{a_1, \ldots, a_k\}$ and $\{b_1, \ldots, b_k\}$ have the same sum. Our proof also applies to any finite Abelian group of odd order $N$, and it provides a corresponding supersaturation result: that whenever $|A| \geq CN^{1/k}$, there are at least $\Omega(|A|^{2k}/N)$ choices for $a_1, \ldots, a_k, b_1, \ldots, b_k \in A$ satisfying these properties. We prove a slightly weaker statement for Abelian groups of even order. We also apply our methods to the vector space setting and prove the following $\mathbb F_q$-analogue of the Bondy-Simonovits Theorem on the extremal number of even cycles in graphs: Any rank-$n$, simple, $\mathbb F_q$-representable matroid with no circuit of size exactly $2k$ has size at most $C q^{n/k}$ for some constant $C$ depending only on $q$ and $k$. When $q=2$, this is best possible up to the constant $C$ for all $k \geq 2$. Our methods also apply to the original graph setting and give a new proof of the Bondy-Simonovits Theorem and its supersaturation version.

We show that the canonical embeddings of the Jiang–Su algebra $\mathcal Z$ into the universal unital free product $\mathcal Z \ast_{\mathbb C}\mathcal Z$ are not approximately unitarily equivalent. This answers one of the $99$ problems in the list of Schafhauser, Tikuisis, and White. By contrast, we prove that any two such embeddings into a simple, unital, monotracial $\mathrm{C}^\ast$-algebra with strict comparison by the unique trace are strongly asymptotically unitarily equivalent.

The Ahlfors–Weill construction explicitly extends a univalent function $f$ with Schwarzian norm less than $1/2$ to a quasiconformal homeomorphism of the Riemann sphere, an extension that can be understood geometrically through Epstein surfaces. For asymptotically conformal curves, the uniformization map can be extended to a larger disk, and Loewner theory can be used to verify that this extension is quasiconformal. In this paper, we construct a local quasireflection with such an extension. Using this local quasireflection, we modify classical estimates of Ahlfors and show that domains bounded by asymptotically conformal curves have asymptotically similar hyperbolic and quasihyperbolic metrics. Furthermore, by combining these estimates with a geometric reinterpretation of the extension through Epstein surfaces, we show that the Epstein-Poincaré surfaces of the complementary domains are arbitrarily close to one another near the curve.

We introduce a method for studying state preparation complexity in dense quantum $p$-spin Hamiltonians on $n$ qubits, going beyond bounds based only on circuit lightcones. The key input is the class's effective profile complexity, which is derived from the metric entropy of its Pauli profiles. These profiles record expectations of all Pauli operators supported on exactly $p$ qubits. Classes with uniformly bounded quadratic effective profile complexity remain separated from the ground-state energy by a positive multiple of $\sqrt n$ for sufficiently large fixed $p$. At subquadratic effective profile complexity, the class cannot outperform a suitable benchmark class at leading order, with product states providing a universal benchmark. The proof combines an adaptation of a nonsymmetric quantum de Finetti theorem of Berta et al. (arXiv:1810.12197) with Gaussian process entropy bounds. Applying this framework, we show that attaining near-ground-state energy requires $\Omega(n^2/\log n)$ one- and two-qubit gates, even with arbitrary discardable ancillas. We also obtain depth-width tradeoffs, entanglement-depth and matrix product state bond-dimension lower bounds, and obstructions for both orientations at every fixed level of Parham's magic hierarchy (arXiv:2504.19966), with total circuit width $O(n)$. In first-level reverse magic, a shallow circuit is followed by an unrestricted Clifford circuit. The latter can spread local observables across the system, preventing a direct application of small-lightcone bounds. For this first-level class, our bounds also allow arbitrarily many clean ancillas at fixed shallow-circuit depth. A sharper benchmark shows that Clifford+$T$ circuits with $o(n)$ $T$-gates have no leading-order energy advantage over product stabilizer states, even with unrestricted Clifford operations and arbitrary discardable ancillas.

What is the maximum number of rainbow triangles in an edge-colored graph with $m$ edges and $r$ colors? Using entropic techniques, we prove an upper bound of $C_rm^{3/2}$ rainbow triangles with $C_r=\sqrt{\frac{2(r-2)}{9r}}$; this constant is best possible whenever there exists an affine plane of order $r-1$. We also show that constructions attaining at least $(C_r-\varepsilon_r)m^{3/2}$ rainbow triangles must exhibit an affine plane structure, which further improves the upper bound if no such affine plane exists. We also consider the problem of counting properly edge-colored cliques of larger sizes. Surprisingly, if the number $r$ of colors is odd, this count is instead maximized by blowups of a properly edge-colored $K_{r+1}$.

We establish the uniqueness of the soliton vector field of a complete shrinking gradient Kähler-Ricci soliton up to pushforward by biholomorphism.