Papers

In this article, we investigate the existence and nonexistence of weak solutions to higher-order doubly critical elliptic problems with weights, driven by a polyharmonic double phase operator. More precisely, we deal with the following problem \begin{equation} \begin{cases} \mathcal{L}^m_{p,q}(u) = f(x,u) ~&\text{in } \Omega,\\[6pt] u=\nabla u=\cdots\nabla^{m-1} u=0 &\text{on }{\partial\Omega}, \end{cases} \end{equation} where $\Omega \subset \mathbb{R}^N$ with $N \geq 2$ is a smooth bounded domain with Lipschitz boundary $\partial\Omega$, $m \in \mathbb{N}$, $1 < p < q < \frac{N}{m}$ with $(N-1)q\leq Np$, the nonlinear term $f\colon\Omega\times\mathbb{R}\to \mathbb{R}$ is a Carathéodory function, which has doubly critical growth, and $\mathcal{L}^m_{p,q}$ represents a polyharmonic double phase operator. By establishing new compactness results within a suitable Musielak–Orlicz–Sobolev framework and applying variational methods, we prove the existence of nontrivial weak solutions. In addition, we derive nonexistence results under appropriate assumptions by establishing a Pohozaev-type identity for higher–order derivatives. Our approach extends classical techniques to capture the intricate features of the double-phase operator for higher–order derivatives, and addresses the difficulties arising from critical nonlinearities, in particular extending the results of [F. Colasuonno, K. Perera, J. Differ. Equ., 422 (2025), 426–488] in a polyharmonic double phase setup overcoming the non-closedness of truncations in higher-order Sobolev spaces.

We decompose a symplectic column by detaching its maximal nonempty intervals that are fixed by the parity involution, this is the part of the symplectic column that is fixed by that involution, and the remaining part consisting of the subword whose image under the parity involution is disjoint of the symplectic column. The parity involution swaps an even number with the previous odd number and an odd number with the next even number. That decomposition is uniquely determined by the symplectic column and the symplectic conditions satisfied by those pieces pack the needed information to explicitly write the inverse of the reduction map in the quantum Littlewood-Richardson bijection. The tools provided here can be combined with the composition of the inverses of the several maps in which the reduction map decomposes, given by Watanabe, namely, among them, combinatorial $R$-matrices and reduction maps on shorter symplectic columns.

We classify possible Amitsur groups of smooth primitive Fano threefolds defined over complex numbers that admit a faithful action of a finite group. We also classify the Amitsur groups for toric Fano threefolds.

The notion of $L_\infty$ spaces over dg manifolds is developed. An equivalence between the category of transitive $L_\infty$ algebroids and that of $L_\infty$ spaces is established, and this equivalence detects weak equivalences. Moreover, a faithful functor from $L_\infty$ algebroids to $L_\infty$ spaces is constructed, which also detects weak equivalences.

A classical question in the theory of hyperbolic surfaces is the study of lengths of closed geodesics under various constraints. A celebrated result in this area is M. Mirzakhani's asymptotic formula for the number of simple closed geodesics of length $\le L$ on a hyperbolic surface of genus $g$ with $n$ punctures. We investigate the number of simple closed geodesics of length $\le L$ representing a fixed primitive nonzero homology class $x$ on a hyperbolic surface $S$. We denote this number by $h_{S}(L, x)$. It follows from Mirzakhani's result that $h_{S}(L, x) \le C L^{6(g-1) + 2n}$. However, numerical evidence suggests that this bound is apparently not asymptotically sharp. We prove that for a surface $S$ of genus $g$ with $n$ punctures and $b$ geodesic boundary components, under the condition that $g \ge 1$ and $g+n+b \ge 3$, there exists a constant $C_1 > 0$ such that for sufficiently large $L$ the inequality \[ h_{S}(L, x) \ge C_1 L^{6(g-1) + 2(n + b-1)} \] holds. In the special case of a torus with two punctures $S_{1, 2}$, we obtain the following stronger result: there exists a constant $C_2 > 0$ such that for sufficiently large $L$ the inequality \[ h_{S_{1, 2}}(L, x) \ge C_2 L^{3.011057 \ldots } \] holds.

We prove that for all degree $d\geq 2$ and all bounded type irrational $\theta$, in the space of monic polynomials having a period $1$ Siegel disk $\Delta$ of rotation number $\theta$, the maximum locus of the conformal radius of $\Delta$ with respect to its fixed point contains polynomials having all critical points on the boundary of $\Delta$. We apply this to reduce a conjecture of Douady (optimality of the Bruno condition) to a weaker statement.

Following the work of Deift and Zhou (DOI:10.1007/978-1-4615-2474-8_15), we analyze the long-time asymptotics of the autocorrelation function of the transverse Ising chain at the critical magnetic field (a special case of the spin-$\frac12$ XY model in a magnetic field) via the associated Riemann-Hilbert problem. We refine the original Deift-Zhou's result by determining the subleading growing term in the asymptotics.

In this paper, we study a class of polyharmonic Kirchhoff problems driven by a double phase operator. The reaction term has subcritical growth but does not satisfy the Ambrosetti–Rabinowitz condition. Motivated by the work of Harrabi-Hamdani-Fiscella \cite{Harrabi-Hamdani-Fiscella-2024} on m-polyharmonic Kirchhoff problems without Ambrosetti–Rabinowitz conditions, we extend their analysis to a nonhomogeneous double phase setting. We study the problem in the natural Musielak–Orlicz–Sobolev framework associated with the double phase structure. The main novelty of the paper lies in combining the nonlocal Kirchhoff term with a higher-order double phase operator under assumptions weaker than the classical Ambrosetti–Rabinowitz condition. By developing suitable modular estimates and compactness arguments, we establish the variational setting and obtain existence and multiplicity results by minimax methods.

We give a modular proof of the measurable Turán-Nazarov inequality for exponential polynomials. The proof first establishes a Remez principle for holomorphic functions satisfying two disk-growth assumptions. The global growth assumption controls the number of relevant zeros, while the local growth assumption gives an effective degree. This yields Cartan coverings, sublevel estimates, and a geometric-mean Remez inequality. For exponential polynomials with bounded spectral diameter, the required disk growth follows from the classical interval Turán inequality. For large spectral diameter, we use a first-order pruning step. If $\rho = \diam(\spec p)$ and $a\in\spec p$, then $$ Q_a = \rho^{-1}(D-a)p $$ has one fewer exponential term, and the quotient $Q_a/p$ satisfies an absolute weak distribution estimate away from the zero set of $p$. Writing $$ Q_a = \rho^{-1}(D-a)p, \quad Q_b = \rho^{-1}(D-b)p $$ for two farthest spectral points $a,b$ gives $$ Q_a-Q_b = \frac{b-a}{\rho}p, \quad |b-a| = \rho, $$ and hence $|p|\le |Q_a|+|Q_b|$. The induction is carried out in geometric-mean form on the original measurable set. This avoids losing a fixed proportion of the set at each step and gives the classical measurable Turán-Nazarov inequality with the sharp algebraic exponent $m-1$. The final measurable $L^\infty$ estimate is classical; the point here is the modular proof and the geometric-mean induction. The only Turán-type input is the classical interval Turán inequality.

We determine the full graded Betti tables of graph families that subsume several classes studied recently in the literature, namely the generalized multiple complete split-like graphs and the generalized clique-star graphs with arbitrary clique block sizes. The method combines Hochster's formula with a precise decomposition of the associated independence complexes into disjoint unions of simplices and iterated joins of discrete complexes. This reduces every graded Betti number to an explicit coefficient extraction formula and yields closed expressions for the linear strand, higher strands, Hilbert series, regularity, projective dimension, and extremal Betti numbers. In particular, we prove a sharp criterion for $2$-linear resolution and identify the regularity corner in terms of the number of nontrivial clique blocks. As applications, we recover and extend earlier results on equal-block split-like graphs, obtain complete formulas for pineapple graphs, and derive consequences for power graphs of cyclic groups, elementary abelian groups, and prime-power dihedral groups.

We study the strong Feller property and irreducibility for continuous-state nonlinear branching processes defined as solutions to stochastic differential equations with jumps. Due to boundary degeneracy and discontinuous jump coefficients, classical methods do not apply. We develop a pathwise approach combining state-dependent time change, truncated auxiliary processes, and localized coupling to establish these two properties. As applications, we obtain exponential convergence to a unique quasi-stationary distribution in the absorbing case, and uniform exponential ergodicity in the non-absorbing case. This pathwise approach is flexible and can be adapted to a broader class of jump-diffusions without relying on specific coefficient structures.

Modern finite element libraries allow users to express partial differential equations directly in variational form, with the added convenience of automatic quadrature selection. In the context of Hamilton-Jacobi-Bellman (HJB) equations, automatic quadrature selection can result in nonmatching quadratures between different terms that may lead to loss of convergence of the policy iteration, which is otherwise expected from theory to converge superlinearly. The simple remedy of enforcing matching quadrature recovers the expected superlinear convergence.

We develop the $L^p$ space and approximation theory for shallow neural networks with $\mathrm{ReLU}^k$ activations. The central object is the Radon-domain $L^p$ space $\mathcal{R}L^p_k(\Omega)$ containing all functions on a bounded domain $\Omega$ that admit a ridge integral representation whose coefficient density belongs to $L^p$ in the Radon domain. In the Hilbert case $p=2$, we prove by elementary Fourier analysis that this space recovers the critical Sobolev space $H^{k+(d+1)/2}(\Omega)$. For general $1<p<\infty$, the identity becomes a sandwich for Bessel-potential Sobolev spaces. The sharp gap of each side is exactly the Seeger–Sogge–Stein loss for the Radon transform as a Fourier integral operator. This also clarifies how the activation regularity and Radon back-projection jointly produce the regularity. As an application, we discretize the integral representation using a deterministic interpolation skeleton plus uniform sampling. This yields high-probability $L^p$ approximation rates and the optimal Hilbert rate $O\!\big(n^{-\frac12-\frac{2k+1}{2d}}\big)$ at $p=2$ for linearized neural networks.

We study a nearest-neighbor coloring process in which vertices are revealed in random order and inherit the color of the closest vertex revealed before them. This model is a discrete analogue of coloring processes previously studied by Preater (2009) and Aldous (2018) in Euclidean spaces. We focus here on regular trees and analyze the associated genealogy of color inheritance. In contrast with the Euclidean case, the genealogical graph on an infinite regular tree is not connected: it has infinitely many infinite one-ended components, each with a distinct asymptotic direction, while every vertex has only finitely many descendants. We also describe how this structure is modified in the presence of finitely many initial seeds. Finally, we study local limits of the coloring on finite regular trees as their height tends to infinity, for two natural seed configurations: two fixed seeds, and one blue seed at the root with red seeds at the leaves.

Graph-Laplacian systems $L\phi=b$ underlie spectral clustering, semi-supervised learning, finite-element analysis, and network-flow solvers. We present LAMG+, a lean, parameter-free, empirically linear-time algebraic multigrid solver: a Julia re-derivation of Lean Algebraic Multigrid (LAMG) with two targeted refinements. We establish three facts. (1) Benchmarking against approximate-Cholesky (AC) and four other solvers (BoomerAMG, PETSc GAMG, pyAMG, CMG): LAMG+ and AC are complementary peers – AC is faster on social/citation graphs; LAMG+ is faster on finite-element/structural matrices (fastest robust solver, most memory-frugal, $2.2\times$ faster than the robust AC variant on large graphs). Only LAMG+ and AC converge across all 13 test classes; the others fail or slow by an order of magnitude off their home turf. (2) Linear scaling: LAMG+ is empirically $O(m)$ with $m$ nonzeros over the full 1,711-graph SuiteSparse set (100% converged, median 4 cycles, log-log slope 1.01), verified up to $2.4\times 10^8$ nonzeros. (3) Robustness: prior benchmarking reported LAMG non-convergent on certain families; running the unmodified LAMG 2.2.1 under identical conditions establishes full convergence, indicating an evaluation artifact. A Local Fourier Analysis proves a strict interpolation-order deficit on grid-aligned anisotropy. Two lean local refinements – a strength-of-connection aggregation veto and selective caliber-2 interpolation – resolve LAMG's anisotropy failure (convergence factor $\approx 0.99 \to 0.11$) with negligible overhead.

We study finite-band optimal control of acoustic waves actuated by local clusters of subwavelength resonators. The acoustic problem reduces to a time-domain Foldy-Lax approximation capturing wave-structure interaction. Spectral analysis of the delayed transfer matrix isolates collective scattering resonances corresponding to weakly damped poles $s_\alpha^\epsilon=-\gamma_\alpha^\epsilon+i\omega_\alpha^\epsilon$ with radiation damping $\gamma_\alpha^\epsilon>0$. Projecting onto a finite band yields the coupled system $\ddot{a}+\Lambda a=C_\epsilon\eta$, $\ddot{\eta}+2\Gamma_\epsilon\dot{\eta}+K_\epsilon\eta=u$, where $a$, $\eta$, and $u$ are modal coefficients, microstructural states, and control. For a tracking functional $\mathcal{J}_{\mu}$ with regularization $\mu>0$, we prove existence and uniqueness of the optimal control and derive the adjoint system. Our main quantitative result is a resonant source-lifting estimate: if a source profile $\eta_r$ is spectrally concentrated in bands $I_\alpha$, the input $u_r=(\partial_t^2+2\Gamma_\epsilon\partial_t+K_\epsilon)\eta_r$ satisfies $\|u_r\|_{L^2(0,T)}^2 \le \sum_\alpha \left(\sup_{\nu\in I_\alpha} |(\omega_\alpha^\epsilon)^2+(\gamma_\alpha^\epsilon)^2-\nu^2 +2i\gamma_\alpha^\epsilon\nu|\right)^2 \|(\eta_r)_\alpha\|_{\mathcal{B}_T(I_\alpha)}^2$. This provides an upper bound for the optimal value function. At exact matching $\nu=\omega_\alpha^\epsilon$, the multiplier equals $2\gamma_\alpha^\epsilon\omega_\alpha^\epsilon+O((\gamma_\alpha^\epsilon)^3)$, showing clustering yields a finite resonant gain governed by the pole's real part. Finally, this attenuation enables finite-band stabilization under an explicit modal coupling condition, with a decay rate proportional to the cluster damping scale.

One of the classical open problems in the theory of skew left braces is the study of their representation theory. We propose in this paper a definition of representation of a skew left brace and study its properties. Representations of the trifactorised groups associated with skew left braces play a fundamental role.

A variational minimum principle for linear elastodynamics of a possibly heterogeneous material without a stored energy function is developed. It involves a change of variables to dual fields, and results in a degenerate elliptic Euler-Lagrange system, even when the primal elastodynamics is hyperbolic. Uniqueness assertions for the dual dynamic and static problems and implications of the degenerate ellipticity are sketched. Some implications pertaining to heterogeneous materials and ones with indefinite elastic moduli are discussed.

The study of $(d+1)$-uniform set systems with VC-dimension at most $d$ links the Erdős–Ko–Rado theorem with VC-dimension. But already in 1997, Ahlswede and Khachatrian showed that this is not the right extension of the Erdős–Ko–Rado theorem. In 2025, Chao, Xu, Yip and Zhang proposed the uniform witness conjecture as a possible right extension: for $0\le s\le d$, if every set of a $(d+1)$-uniform family has a missing trace of the same fixed size $s$, then the family should have size at most $\binom{n-1}{d}$. They proved the conjecture when $s=d$, and when $s=1$ and $n$ is large. Very recently, Chao, Xu and Zakharov proved the conjecture when $s\le \frac{d}{2}$ and $n$ is large. We fill in the missing half of the picture, although the picture is not the one suggested by the conjecture. More precisely, for $d\ge 4$ and $\left\lceil \frac{d+2}{2}\right\rceil\le s\le d-1$, we construct such a family $\mathcal{F}\subseteq\binom{[n]}{d+1}$ with $|\mathcal{F}|=\binom{n-1}{d}+\binom{n-2(d+1-s)-2}{2s-d-2}$ for every $n\ge2(d+1)$, thereby disproving the uniform witness conjecture.

This paper introduces a Riemannian autoregressive (R-AR) model of order one for manifold-valued time series. The model is specified through an autoregressive process in the tangent space at a reference point, which is mapped to the manifold via the exponential map. It is characterised by two parameters: a reference point $\mu$, representing the intrinsic central tendency through the Fréchet mean, and an autoregressive parameter $\phi$, governing the dependence structure and stationarity properties. When these parameters are unknown, their estimation introduces geometric and probabilistic challenges arising from the intrinsic estimation of $\mu$ and from estimating $\phi$ as the base point varies over the manifold and the associated tangent spaces change. The estimation theory is developed using a strong law of large numbers for the sample Fréchet mean set of ergodic Markov chains, together with ergodic arguments for the strong consistency of the autoregressive parameter estimator. The framework is validated through numerical simulations in the hyperbolic plane and an application to aerosol size distributions on the Fisher-Rao manifold, demonstrating how the proposed model can characterise mean-reverting dynamics in nonlinear geometries.

Even when Dirac-Frenkel dynamics determine a well-defined evolution in function space, the corresponding parameter dynamics can be non-unique or ill-conditioned for redundant nonlinear parametrizations, such as typical neural networks or mixture models. We propose to add inertia to the Dirac-Frenkel dynamics and show that this allows useful parameter velocity information to persist from the past trajectory in directions that are weakly informed, while well-informed parameter velocity directions continue to follow the Dirac-Frenkel dynamics. We prove that the inertial formulation yields well-posed parameter dynamics and provide a posteriori error bounds. After time discretization, the method requires the solution of the same type of regularized linear least-squares problem as standard Dirac-Frenkel dynamics, but with the previous velocity appearing as an anchor. Numerical experiments demonstrate the increased robustness obtained with inertia.

This paper presents general strong duality results when testing hypotheses by betting against them. A bet is an e-variable for a composite null hypothesis $\Pcal$: a nonnegative random variable $X$ whose expected value is at most one under every $P \in \mathcal P$. Following Kelly, Breiman, Cover, Shafer, and Grunwald et al. (2024), we study a natural minimax log-optimality criterion: given a composite alternative $\Qcal$, we characterize the “GROW value” $\sup_{X} \inf_{Q} \E_{Q}[\log X]$. This paper generalizes the results of Larsson et al. (2025) from (arbitrary $\mathcal P$ and) simple $\mathcal Q$ to arbitrary $\mathcal Q$. We prove that there always exists a minimizing information-projection pair between the weak-$*$ closures of the convex hulls of arbitrary $\mathcal P$ and $\mathcal Q$, and show that the GROW value for bounded e-variables always equals their relative entropy. We also prove a similarly general strong duality for the REGROW criterion with bounded e-variables and arbitrary bounded offsets. Under various assumptions our results extend to unbounded e-variables, and examples show that without any assumptions such extensions fail. Our results are analogous to those in Larsson et al. (2026), swapping tests for bounded e-variables, minimax risk for the GROW criterion, and total variation for relative entropy.

Concentration inequalities for sample covariance matrices are fundamental tools in high-dimensional probability. Classical results typically assume that the selected random vectors are independent of the selection rule. In this paper, we study spectral concentration for sample covariance matrices formed from arbitrary, possibly data-dependent subsets of i.i.d. random vectors. Such data-dependent selection destroys the usual independence structure and makes standard covariance concentration bounds inapplicable. For i.i.d. Gaussian random vectors, we prove high-probability lower and upper bounds for the minimal and maximal eigenvalues of such selected covariance matrices. Compared with a direct union-bound argument, our results provide substantially sharper guarantees and allow much smaller subset proportions. We further discuss extensions from Gaussian to sub-Gaussian random vectors, and beyond independence to weakly dependent observations, with geometrically strong-mixing Gaussian sequences serving as a representative example of the latter. Finally, we apply the developed concentration inequalities to the K-subspace clustering problem under a low-rank Gaussian mixture model, where the optimal clusters are inherently data-dependent. Our results yield recovery guarantees showing that the clustering error of global minimizers decays polynomially with the signal-to-noise ratio.

For which graphs $H$ does there exist a graph $G$ with at least one edge and no induced subgraph isomorphic to $H$, such that deleting any edge of $G$ creates an induced copy of $H$? We call such a graph "$H$-deletion-saturated". This version of the well-studied notion of "$H$-induced-saturated" graphs – where both adding and deleting any edge creates an induced copy of $H$ – appears more tractable. For example, while it remains wide open whether $H$-induced-saturated graphs exist for every even cycle $H$, we proved recently that deletion-saturated graphs exist for all even cycles. In fact, apart from complete graphs, no graph $H$ is known for which $H$-deletion-saturated graphs do not exist. We conjecture that $H$-deletion-saturated graphs exist for every non-complete graph $H$, and prove this conjecture for several types of graphs, including: complete bipartite graphs with parts of unequal size, triangle-free graphs with one cycle, graphs with two leaves at distance at most three, and line graphs of trees. In fact, in all cases, we prove the conjecture for substantially more general families. We also verify our conjecture for every graph $H$ on at most six vertices.

The Chen-Teo geometry is a Riemannian, Ricci-flat ALF 4-manifold, containing an AF gravitational instanton that gives the first counterexample to the Euclidean black hole uniqueness conjecture. We investigate the problem of constructing an Einstein analogue with a non-zero cosmological constant $\lambda$. We show that the solution is either the Plebański-Demiański metric with $\lambda$, or it has an anti-self-dual Weyl tensor. We study the latter case in detail: we prove that for $\lambda<0$, there is a conformal infinity separating two asymptotically hyperbolic metrics; we show that one of them is globally conformal to an ALE scalar-flat Kähler metric; we construct gravitational instantons with different topologies; and we show that the geometry is a 4-pole solution in the Calderbank-Pedersen classification.

We prove that every closed Riemannian $4$ or $5$-manifold $M$ contains a branched immersed closed minimal surface. That is, there exists a non-constant weakly conformal harmonic map from some closed Riemann surface into $M$. We rely on the existence of multisections in dimensions $4$ and $5$ to generate a non-trivial class of sweepouts of $M$ by mappings from a closed surface $S$ of genus at least $2$. To each sweepout in a minimizing sequence within the class, through the intermediary of quasiconformal maps of the upper half-plane, we associate a family of hyperbolic metrics on $S$ with respect to which the mappings in the sweepout have nearly equal energy and area. The harmonic replacement method of Colding and Minicozzi is then applied to obtain a min-max sequence that converges to a bubble tree of branched minimal immersions.

Consider the Dunkl Laplacian $\Delta_k$ associated with a root system $\Phi$ in $\R^d$ and a nonnegative multiplicity function $k$ on $\Phi$. In this paper, we establish a Caffarelli-Silvestre characterization for the fractional Dunkl Laplacian through an extension problem. We also express the corresponding fundamental solution in terms of the $\Delta_k$-Riesz kernel and prove a fractional Nash-type inequality.

For many information processing tasks, de Finetti-style theorems can often simplify the analysis in worst-case input scenarios for which the task exhibits some permutation-invariance symmetry, as they can allow for a reduction from an analysis on worst-case inputs to that of i.i.d. inputs. If further information is available on the inputs, it might be advantageous to reflect this information in the de Finetti reduction. In our work, we focus on a form of such constraint, based on the type of the input. This allows us to obtain a conceptually simple proof of a new de Finetti reduction for classical probability distributions, derived from elementary properties from the method of types. We apply our constrained de Finetti reduction to the compression of quantum interactive communication protocols with classical inputs, and prove that the prior-free quantum information cost equals the worst-case input amortized quantum communication cost.

We show that if a group admits an action with no algebraicity then all of its mixed identities are singular. Previously, such groups were only known to be lawless by a theorem of Abért. Our result confirms, in particular, a conjecture of Bodirsky, Schneider, and Thom for a large class of oligomorphic permutation groups. It thereby not only subsumes numerous results from the literature in a simple uniform theorem, but also settles the question for prominent groups for which the conjecture was an open problem, such as the automorphism group of $(\mathbb{Q}; <)$. It also applies outside the oligomorphic context, e.g. to much-investigated groups such as Thompson's groups $F, T$, and $V$, to Grigorchuk's group, and to the homeomorphism groups of any manifold of dimension $\geq 1$. More generally, we prove that all mixed identities of a group are singular as long as it has an action satisfying certain geometric conditions. This additionally covers, for example, the infinite-dimensional general and projective linear groups.

Let $D$ be an oriented graph, and let $f_T(D)$ denote the number of copies of $D$ in a tournament $T$. We say that $D$ is converse invariant if $f_T(D)=f_T(\overline D)$ for every tournament $T$, where $\overline D$ is obtained from $D$ by reversing all arcs. Ai, Gutin, Lei, Yeo, and Zhou introduced a digraph polynomial for studying this property and conjectured that an orientation of a tree of maximum degree at least $3$ is converse invariant if and only if it is self-converse or can be obtained recursively by bridge-mirroring from an orientation of a path. We disprove this conjecture. More precisely, we characterize converse-invariant orientations of trees of diameter four and exhibit non-self-converse examples that do not arise from the recursive bridge-mirroring construction. To prove the classification, we introduce a multilinear polynomial $P_D$ encoding the difference $f_T(D)-f_T(\overline D)$ over all tournaments $T$, and we give a coefficient formula for $P_D$ as a signed sum over copies of subgraphs of the underlying graph of $D$. This polynomial method yields parity obstructions, gives new proofs that oriented paths and cycles are converse invariant, and provides the main tool for the diameter-four classification.