Papers

The online Ramsey game for graphs $G$ and $H$ is played on the infinite complete graph $K_\mathbb{N}$. In each round, Builder chooses an edge, and Painter colors it red or blue. The online Ramsey number $\tilde{r}(G,H)$ is the smallest integer $t$ for which Builder has a strategy guaranteeing a red copy of $G$ or a blue copy of $H$ within $t$ rounds. For every fixed integer $k\ge4$, the best-known lower bounds for $\tilde{r}(K_{1,k},P_n)$ and $\tilde{r}(K_{1,k},C_n)$ are $\left(\frac{k+3}{4}+o(1)\right)n$ as $n\to\infty$. We improve the corresponding asymptotic upper bounds from $(k+o(1))n$ to $\left(\frac{2k+4}{5}+o(1)\right)n$ as $n\to\infty$.

Erdős and Gallai in 1959 proved the seminal result that every $n$-vertex graph with no cycle of length at least $2t+2$ has at most $\tfrac{2t+1}{2}(n-1)$ edges. We prove the extension that, for every sufficiently large $t$, the same quantity is also the sharp extremal bound for graphs with no $t$ consecutive even cycle lengths, resolving a conjecture of Verstraëte. Thus, at the Erdős-Gallai threshold, forcing an entire interval of even cycle lengths costs no more than forcing its longest member. More precisely, every $n$-vertex graph $G$ with $e(G)\ge \tfrac{(2t+1)(n-1)}2$ $\bullet$ either contains $t$ consecutive even cycle lengths, $\bullet$ or equality holds and $G$ is connected with every block isomorphic to $K_{2t+1}$. As consequences, for every sufficiently large even $k$ we determine the sharp edge thresholds forcing a cycle of length $0\pmod k$ or $2\pmod k$, answering questions of Bai, Grzesik, Li, and Prorok and of Gao, Li, Ma and Xie, respectively, for sufficiently large even $k$. The proof develops a stability-enhanced sublinear expander method. Its main new ingredient is a dense-case decomposition that recovers the lengths lost in the expander extraction by combining a flexible dense core with rooted cycle families in the vertices outside the core.

We study the moduli of isolated non-normal singularities by introducing the functor $\mathrm{Crimp}_S^d(X)$ of crimpings of $(X \to S)$ corank $d$. This globalizes Ishii's moduli functor of subrings of finite colength. By verifying Artin's criteria, we prove that if $X \to S$ is a separated morphism of finite presentation, then $\mathrm{Crimp}_S^d(X)$ is represented by a separated algebraic space of finite presentation over $S$, which is proper when $X \to S$ is proper. We use this to construct a semistability condition and moduli space for geometrically unibranch curves of genus zero.

Let $\mathbf{C}=\partial E\subset \mathbb{R}^{n+1}$ be a regular area-minimizing hypercone. We first prove that $\mathbf{C}$ is simultaneously strictly stable and strictly minimizing if and only if there exists $c_\mathbf{C}>0$ such that \[ \operatorname{Per}(F;B_R)-\operatorname{Per}(E;B_R) \geq c_\mathbf{C} \int_{F\mathbin\triangle E} \frac{\operatorname{dist}(x,\mathbf{C})}{|x|^2}\,dx \] for every $R>0$ and finite-perimeter set $F$ with $F\mathbin\triangle E\Subset B_R$. Thus this intrinsic distance-weighted inequality characterizes exactly the simultaneous strictness of the stability and minimizing properties. Second, without either strictness assumption, every regular area-minimizing hypercone satisfies the scale-invariant quadratic inequality \[ \frac{\operatorname{Per}(F;B_R)-\operatorname{Per}(E;B_R)}{R^n} \geq c_\mathbf{C} \left( \frac{|F\mathbin\triangle E|}{R^{n+1}} \right)^2. \] This extends the inequality previously established for area-minimizing Lawson cones. Finally, the area-minimizing assumption is unnecessary for our spectral result: for every stable regular minimal hypercone, the first Dirichlet eigenvalue $\lambda_\mathbf{C}^D(R)$ of the Jacobi operator on $\mathbf{C}\cap B_R$ is given exactly by \[ \lambda_\mathbf{C}^D(R) = \frac{j_{b_1,1}^2}{R^2}, \qquad b_1^2 = \frac{(n-2)^2}{4}+\mu_1, \] where $\mu_1$ is the first eigenvalue of the link Jacobi operator and $j_{b_1,1}$ is the first positive zero of the Bessel function $J_{b_1}$. In particular, this identifies the optimal Dirichlet spectral constant for every stable regular minimal hypercone.

For every natural number $r\geq 2$, we construct $r^{th}$ powers of bipartite graphs whose chromatic number is quadratic in their clique number, showing that the straightforward quadratic upper bound is best possible. We thereby settle an open problem posed by Chakraborty, Chandran, Jacob and Pillai [J. Graph Theory 112(3) (2026), 235-254] by establishing the sharpness of the quadratic bound for squares of bipartite graphs.

Power grid state estimation relies on sensor measurements that are increasingly vulnerable to adversarial corruption in cyberphysical environments, potentially leading to significant deviations in system observations. This motivates the need to identify critical measurement nodes whose compromise results in the most severe system-level impact. However, existing node criticality methods primarily rely on structural or steady-state analyses and do not explicitly account for adversarial effects on system behavior. To address this gap, this paper proposes an adversarially informed framework for identifying critical measurement nodes in linearized power systems. Within this framework, a structured attack generation mechanism is developed to construct stealthy and effective false data injection attacks (FDIAs) against an H-infinity resilient state estimator. Node criticality is then evaluated using coalition-based marginal contributions of compromised sensor subsets, estimated via permutation sampling over a prescribed set of admissible nodes, with the resulting importance scores mapped to the corresponding physical buses. Simulation results on the IEEE 14-bus system show that adversarially identified nodes induce larger deviations in frequency, voltage angle, and net power compared to randomly selected nodes, demonstrating the effectiveness of the proposed framework.

The zero divisor graph $\Gamma(R)$ of a finite commutative ring $R$ has as vertices the non-zero zero divisors of $R$, with an edge between two elements exactly when their product is zero. We determine the boxicity and threshold dimension of $\Gamma(R)$ for two classes of finite commutative rings: reduced rings and quotients of principal ideal domains. Our proofs use a new combinatorial gadget, the integral covering graph, that captures the structure shared by both ring families and generalizes the disjointness graph on subsets of $[n]$, where two subsets are adjacent if and only if they are disjoint. In doing so, we answer two questions recently posed by L.~Sunil Chandran and Suraj Kumar Sahoo in Boxicity of Zero Divisor Graphs, Discrete Applied Mathematics 391 (2026).

In this article, using the theory of $\delta$-geometry, we construct a canonical $z$-isocrystal $(\mathbf{H}_\delta(E), \mathfrak{f}^*)$ admitting a Hodge-Pink structure for any abelian Anderson module $E$. The Hodge-Pink structure on $\mathbf{H}_\delta(E)$ induces a natural filtration $(\mathbf{H}_\delta(E) \supset \mathbf{X}_{\mathrm{prim}}(E) \supset \{0\})$. The elements of $\mathbf{X}_{\mathrm{prim}}(E)$ are represented by primitive delta characters associated to $E$. We establish a natural morphism from $\mathbf{H}_\delta(E)$ to the associated de Rham cohomology module $\mathbf{H}^*_{\mathrm{dR}}(E)$, which is strictly compatible with the aforementioned filtration and the classical Hodge filtration $(\mathbf{H}^{*}_{\mathrm{dR}}(E)\supset {\mathrm{Lie}(E)^{*}}\supset \{0\})$ on $\mathbf{H}^*_{\mathrm{dR}}(E)$. Moreover, we show that the map induces an isomorphism between $\mathbf{X}_{\mathrm{prim}}(E)$ and $\mathrm{Lie}(E)^*$. Hence our isomorphism provides an interesting interpretation of the invariant differentials of $E$ as primitive delta characters of $E$. Furthermore, when $E$ is a Drinfeld module, we show that the constructed $z$-isocrystal $\mathbf{H}_\delta(E)$ is weakly admissible. Consequently, the positive equal characteristic analogue of the Fontaine functor associates a crystalline $z$-adic Galois representation to the $\delta$-geometric object $\mathbf{H}_\delta(E)$. In the case, when $E$ is the Carlitz module, we show that the Galois representation associated to $\mathbf{H}_\delta(E)$ is indeed the usual one coming from the Tate module.

We consider the distribution-constrained optimal stopping problem introduced by Bayraktar and Miller (Mathematical Finance, 2019) and Beiglbock et al. (PTRF, 2018). Motivated by the multi-marginal Skorokhod embedding problem (SEP) and applications in financial mathematics, we generalize the Root-type solution to the multi-marginal case. The key difficulty is that the associated stopping barriers need not be ordered, so the problem in general cannot be reduced to a sequence of one-marginal problems. First, we give a probabilistic characterization in terms of sequential optimal stopping for an auxiliary time-reversed process, in the same spirit as Cox et al. (PTRF, 2019). Then, we prove the optimality of this construction by martingale arguments for a general class of reward functions, including multi-marginal versions of all examples in Beiglbock et al. (PTRF, 2018) as special cases.

We establish a sharp phase transition for fitting random vectors by an ellipsoid. The random vectors have independent subgaussian coordinates with mean zero, variance one, and a common fourth moment, and the number of vectors is proportional to the square of the dimension. We identify an explicit satisfiability threshold such that, with high probability, a positive definite ellipsoid passes through every data point below the threshold, whereas no positive semidefinite fit exists above it. We also determine the optimal squared fitting error throughout the unsatisfiable regime. In particular, the threshold depends on the coordinate distributions only through their common fourth moment, revealing a fourth moment universality phenomenon. For standard Gaussian data the threshold is $1/4$, resolving the ellipsoid fitting conjecture.

In this paper, we propose a unified framework for accelerated gradient methods, dubbed UGM, which subsumes a wide range of accelerated and conventional gradient-type methods designed for minimizing $L$-smooth and $\mu$-strongly convex functions. We demonstrate that the iteration update of the proposed framework can be intrinsically interpreted as a hybrid combination of the heavy-ball method and vanilla gradient descent. This interpretation reveals that classical accelerated gradient methods essentially integrate a conservative gradient descent step into the fast yet unstable heavy-ball dynamics, which achieves a favorable trade-off between acceleration and stability. We further establish a unified convergence analysis using Lyapunov functions. Guided by our analysis, we develop a family of enhanced accelerated gradient algorithms that leverage the inner product relationship between gradient information and iterative variables to optimize iterative updates. Extensive numerical experiments on unconstrained quadratic optimization and logistic regression validate that the proposed algorithms achieve superior performance compared with existing baseline methods under typical structural conditions.

In several articles, the authors assume that the convex roof structure of the EoF and finite-dimensionality of subsystems $A$ and $B$ guarantee the existence of the (global) supporting affine functional for the EoF at any state of the system $AB$. This means that for any state $\rho$ of $AB$ there is a Hermitian operator $\Lambda_\rho$ on $\mathcal{H}_{AB}=\mathcal{H}_A\otimes\mathcal{H}_B$ such that $E_F(\rho)=\mathrm{Tr}\Lambda_\rho\rho$ and $E_F(\sigma)\geq\mathrm{Tr}\Lambda_\rho\sigma$ for any state $\sigma$ of $AB$. We present an explicit example showing that, when $\rho$ is degenerate, this is not true even in the simplest case when $A$ and $B$ are qubit systems. The construction is based on the fact that the existence of a supporting affine functional for the EoF at a state $\rho$ is equivalent to the Lipschitz lower semicontinuity of the EoF at this state $\rho$. We use Wootters' formula and the help of Claude Fable 5 to find a state $\rho$ of the system $AB$ for which the latter property does not hold. We also describe conditions for the existence the local and global supporting affine functionals for the EoF at a given state of both finite and infinite-dimensional bipartite quantum systems. These conditions allow us to find Lipschitz lower semicontinuity bounds for the EoF at a given finite rank state $\rho$ (i.e. inequalities of the form $\,E_F(\rho)-E_F(\sigma)\leq C_\rho\|\rho-\sigma\|_1$) with and without restrictions on the support of the state $\sigma$.

In this paper we study the behaviour of $H^*(x,y,z)$, the number of integers less than $x$ possessing a divisor in the interval $(y,z]$ of the form $p-1$, where $p$ is a prime, for all values of $y = y(x)$ and $z= z(y)$. We observe multiple phase transitions at critical values of $z$ in terms of $x$ and $y$ guided largely by the anatomy of $n$. Our results generalize a result of Ford from 2017, which corresponds to the case $z = x$.

Chaos in classical systems can be characterized by Lyapunov exponents that measure the exponential divergence of nearby trajectories, but directly extending this framework to quantum mechanics has been a persistent challenge. The wavelike nature of quantum states and the non-commutative geometry of quantum phase space obstruct a straightforward generalization of classical chaos theory. Here we develop a rigorous approach to quantum chaos by leveraging quantum optimal transport theory, which provides the missing geometric foundation for measuring distances between extended quantum distributions. We define quantum Lyapunov exponents that naturally avoid divergences encountered when naïvely generalizing classical exponents, and show that in the semiclassical limit they recover the classical global expansion rate, and hence the usual maximal Lyapunov exponent when they coincide. Our framework provides a tight connection between the divergence of classical trajectories and semiclassical phase space evolution, and additionally clarifies the role of out-of-time-order correlators as diagnostics of quantum chaos. These results establish quantum optimal transport as a unifying mathematical foundation for quantum chaos theory, providing new tools to characterize dynamical behavior across the full range of quantum dynamics from simple few-body models to complex many-body systems.

We settle the complete Crouzeix conjecture for matrices of order at most three and prove the Q-algebra case of the completely bounded Clouâtre–Ostermann–Ransford (COR) conjecture for homomorphisms into matrices of order at most three, via sharp abstract column and row estimates. Our approach also establishes the scalar COR conjecture in a stronger form, for homomorphisms with commutative range on Banach algebras with unity satisfying von Neumann's inequality. Under contractivity of the symmetrized map, this result holds on arbitrary Hilbert spaces without initial boundedness assumptions on the homomorphism or the antilinear map. For arbitrary operator algebras, we disprove the complete COR conjecture by an exact three-dimensional example with target matrix order two. We also prove the sharp complete bound for every matrix subalgebra containing the diagonal, in arbitrary matrix order and on arbitrary target Hilbert spaces. We also obtain column and row square-function inequalities with sharp norm bounds, strict scalar bounds for operators similar to normal operators, sharp complete spectral constants for scaled $q$-numerical ranges in dimensions two and three, and rigidity, stability, and representing-measure results.

On the lattice graph $\mathbb{Z}^n$, we establish the existence of sign-preserving solutions to the Tzitzéica equation. We prove the existence of positive solutions and two classes of negative solutions under different assumptions, and derive their decay estimates. The proof is based on a suitable approximation scheme, the monotone convergence theorem, and an exhaustion argument. These results extend those of Hua, Huang, and Wang (Anal. PDE, 2026) by establishing the existence of both positive and negative sign-preserving solutions together with their decay estimates.

Given a smooth projective variety $X$ of dimension $n$ and a closed subscheme $Z \subseteq X$, it is in general a difficult problem to determine whether $X \setminus Z$ is isomorphic to $\mathbf A^n$. In the case where $X$ is a projective bundle over $\mathbf P^1$ and the restriction of every irreducible component of $Z$ to every fibre is a hyperplane, we give a complete geometric characterisation in terms of the irreducible components of $Z$ and their intersections. In the appendix, we use this to obtain a coordinate-free explanation for the recent first counterexample to the Jacobian conjecture.

Block drafters propose several tokens in one forward pass, before earlier target tokens are realised. Their rejection mixes two losses: missing within-block path information and imperfect modelling of observable information. Accepted length cannot distinguish them. We separate the two with an information floor, the minimum expected rejection at a specified conditioning order; rejection above this floor is the model gap. Measurements across four domains and four open-weight targets, plus floor estimates on a frontier API target, yield three findings. First, the all-parallel floor reaches $0.286$ at the final slot on Qwen3-4B, limiting even the best proposal to $71\%$ per-slot acceptance. Second, one realised token removes $86$–$100\%$ of this floor, a locality also recovered by an independent mutual-information analysis. Third, current drafters remain far above their floors: the final-slot model gap accounts for $43$–$64\%$ of DFlash rejection and $85$–$92\%$ of DSpark's oracle-conditioned rejection. Guided by this separation of short-range conditioning from proposal quality, we replace a context-independent predecessor correction with a prefix-attention head that reads committed context conditional on the predecessor. With supporting backbone components, the resulting Qwen3-4B drafter improves mean serving accepted length by $3.93\%$ over the released DSpark checkpoint across nine tasks, without increasing the conditioning order. Code and data: https://github.com/tie-pilot-qxw/specfloor; drafter checkpoints: https://huggingface.co/TIE-Pilot/dspark-attnconv-block7-qwen3-4b.

We extend the conjecture of Erdős and Gyárfás on monochromatic path covers to the setting of monochromatic cycle covers. We prove that, for all $n$, every 2-edge-coloring of the complete graph on $n$ vertices contains a collection of at most $\lceil\sqrt{n}\rceil$ monochromatic cycles, all of the same color, that together cover all vertices. The order of the bound is best possible, and the ceiling is necessary for infinitely many $n$.

Lie algebras describe how control Hamiltonians combine in closed quantum systems. In open Markovian systems, however, dissipation introduces irreversible directions that the Lie algebra alone does not retain. A dynamical Lie wedge preserves this information as a convex cone of locally admissible generators. The classical Levi decomposition separates a finite-dimensional Lie algebra into semisimple and solvable parts. In this work, we prove a corresponding decomposition and reconstruction theorem for dynamical Lie wedges. The theorem decomposes the wedge into semisimple and solvable data, records how these components are coupled, and provides a converse reconstruction of the wedge. The framework yields four structural types of the generated algebra. We further prove that the total dissipation strength of every admissible generator depends only on its coordinate in the solvable radical. In particular, if the generated Lie algebra is semisimple, every admissible generator is Hamiltonian. Together, these results extend Lie-algebraic structural methods to irreversible Markovian control and clarify how dissipation is encoded in the generator geometry.

In this paper, we establish systematic estimates for the entropy and its density on Riemannian manifolds, focusing on the more challenging cases where the Ricci curvature changes sign or the boundary is nonconvex. For example, the refined second law of thermodynamics states that the entropy in a compact domain in $\mathbb R^n$ is increasing in time, furthermore, it is concave if the domain is convex. The concavity property is equivalent to the property that the Fisher information is decreasing, which also holds for convex domains in a Riemannian manifold with nonnegative Ricci curvature (cf. \cite{NiLei}). In view of the wide application of entropy in mathematics, information theory, physics, etc., there is certain desire in the community to extend the property to broader settings, especially to the case with nonconvex boundary (see e.g. \cite[p. 3]{CFM}), even in the Euclidean case. Here, we prove that the refined second law still holds if the domain is not too far from convex and the negative part of the Ricci curvature is not too large, in an explicit, nonperturbative sense, thus realizing some of the expectations. The proof is based on a new, sharp second order log Poincaré inequality that does not require explicit curvature conditions of the manifold. If the negative part of the Ricci curvature is too large, a counterexample to the concavity is given. Some other related estimates for the entropy density (Hamilton type estimates) are also proven.

We construct the first examples of one-dimensional long-range crystals exhibiting arbitrarily fast polynomial dispersive bounds. The Floquet functions of our Hamiltonians are highly oscillatory Weierstrass functions, whose rough autosimilar structure drives the fast dispersion. The proof develops a new Fourier decay theory for $C^\alpha$ images of the Lebesgue measure, based on a Dolgopyat-inspired transfer operator method, and yields a van der Corput lemma for Weierstrass functions. As a consequence, the local time of classical Weierstrass functions of sufficiently large lacunarity exists and is $C^k$, answering a question raised by Geman and Horowitz in 1980.

We prove a sharp partial regularity result for Hamiltonian stationary Lagrangian Lipschitz submanifolds in arbitrary smooth almost Kähler manifolds: every weak solution of the corresponding equation is smooth away from a relatively closed singular set of Hausdorff dimension at most $n-5$. We show that the estimate is optimal by constructing a nonzero two-homogeneous viscosity solution \[ U\in C^{1,1}(\mathbb{R}^5)\setminus C^2(\mathbb{R}^5) \] of the phase-zero special Lagrangian equation, whose level sets on $\mathbb{S}^4$ are the leaves of Cartan's isoparametric foliation. Its gradient graph is a non-flat calibrated cone, real analytic away from the vertex. This also gives the first $C^{1,1}$ but non-$C^2$ solution of the special Lagrangian equation, and shows that the same dimensional estimate is sharp in the case of special Lagrangian graphs.

Denote by $\mathcal{P}_r$ an almost-prime with at most $r$ prime factors, counted according to multiplicity. In this manuscript, it is established that, for any fixed $0.98353<\gamma<1$, there exist infinitely many primes of the form $p=[n^{1/\gamma}]$, where $n$ is an almost-prime $\mathcal{P}_7$. This result constitutes an improvement upon the previous result of Baker, Banks, Guo and Yeager [1], who showed that there exist infinitely many primes $p$ such that $p=[n^{1/\gamma}]$ with $n\in\mathcal{P}_8$ for $\gamma$ near to one.

This blueprint serves as a companion to a forthcoming, shorter traditional mathematical paper. The purpose of this blueprint is two-fold: first, it has served as the foundation for a formalization in Lean 4 of these results. This formalization has been completed largely automatically, making essential use of current frontier large language models. Second, it will serve as a resource to readers of the main paper who are interested in further technical details of the proofs. The main result concerns norm-variation estimates for multiple ergodic averages associated with $n\ge 2$ commuting measure preserving transformations, providing a quantitative strengthening of Tao's norm-convergence theorem and answering an open question of Avigad and Rute. At the core of the analysis lies an explicit real-variable estimate for twisted multilinear averages that is closely related to certain singular Brascamp–Lieb inequalities.

In this article, we establish a bound on $\theta_q$ that guarantees the existence of a pair of consecutive non-primitive elements in $\mathbb{F}_q$, with the exceptions $q=4$ and $q=8$. We first derive a sufficient condition for the existence of such a pair using character sums and then obtain the stated bound by considering several cases according to the least prime divisor of $q-1$.

In this paper, over the field of complex numbers, we prove that a numerical Godeaux surface contains at most six pairwise disjoint $(-2)$-curves, and that this bound is sharp. As an application, we refine the classification of involutions on smooth minimal surfaces of general type with $p_g=0$ and $K^2=7$: the divisorial fixed part $R$ satisfies $R^2=-1$, the involution acts trivially on $H^*(S,\mathbb{Q})$, and, if the minimal resolution of the quotient is of general type, it is a numerical Campedelli surface containing five pairwise disjoint $(-2)$-curves. Another application concerns commuting involutions on smooth minimal surfaces of general type with $p_g=0$ and $K^2=8$.

In this note, we construct non-decomposable Lagrangian endoconcordances of Legendrian knots. We prove that for any positive integer k, there exist a (stabilised) Legendrian knot and a Lagrangian fillable Legendrian knot in the standard contact 3-sphere, each admitting at least k pairwise Hamiltonian non-isotopic relative boundary, non-decomposable Lagrangian endoconcordances. The obstruction to decomposability that we rely on is based on Khovanov homology.

The presence of bilinear terms in mathematical modeling generally yields nonconvex quadratic programs (QPs) that remain computationally challenging to solve to global optimality. While continuous piecewise linear (CPWL) approximations can reformulate these nonlinearities into mixed-integer linear programs (MILPs), the geometric construction of the domain partition heavily dictates the resulting solver efficiency. In this paper, we present efficient MILP formulations for approximating bilinear terms and rigorously evaluate their computational merits against direct QP solvers. First, we introduce CPWL approximations with arbitrary high degrees of accuracy that explicitly account for and exploit the inherent symmetries of the unit bilinear function. Second, we analytically establish the exact maximum and average approximation errors of the CPWL approximations. Third, we construct and compare three distinct MILP formulations (a “Triangle”, a “Square”, and a difference-of-convex “DC” formulation) of the CPWL functions. Fourth, we formalize a highly relevant class of optimization models, Sequentially Coupled Bilinear Programs (SCBP), where variables represent system states and state changes. Finally, through extensive numerical experiments, we demonstrate that our compact “Square” MILP formulation achieves superior computational performance on SCBP instances with long sequence horizons, allowing mature open-source MILP solvers to effectively outperform QP solvers.

The connections between conformal prediction and permutation tests are already widely-known in the literature. Some authors motivate conformal prediction by saying that it computes a permutation p-value for the hypothesis $H_0 : Y_{n+1} = y$, and then inverts this to form a prediction set for $Y_{n+1}$ (i.e., accepts all values $y$ into the prediction set for which the p-value is large). In this paper, we examine an alternative view, which is less well-known: we again cast conformal prediction via the inversion of a permutation test, but for the null of exchangeability of the joint distribution of the $n+1$ samples. This change in perspective, while simple, adheres more closely to traditional formalization in hypothesis testing, which offers several benefits. First, we use the duality between conformal sets and testing to show that foundational universality and impossibility results in the conformal prediction literature can be reproduced directly using classical hypothesis testing theory (due to Neyman, Lehmann, Scheffé, Kraft, Le Cam, and others). Furthermore, we show that an optimality result for conformal prediction can be derived using standard Neyman-Pearson theory: for any joint distribution of the covariates and response $X,Y$, and any sample size, the optimal method for prediction sets—delivering the most efficient set among all methods with valid coverage for exchangeable distributions—is a conformal predictor whose score is the inverse conditional density of $Y|X$.