Papers

We study weighted Carlson-Levin inequalities on cones, computing the optimal constant and classifying optimizers via an Entropy method. The limiting case of a logarithmic Carlson-Levin inequality is also treated. As a Corollary, we obtain a rigidity result for a weighted Gagliardo-Nirenberg inequality. We conclude with some examples involving monomial weights.

The Hodge decomposition at infinity gives rise to distinct tangential scattering channels for differential forms on asymptotically conic manifolds. We construct co-closed generalized eigenforms for the Hodge Laplacian on $p$-forms in dimension $n\geq 3$ and analyze the resulting Bessel and Hankel systems. From these we determine the scattering matrix modulo smoothing operators, identifying it as a Fourier integral operator associated to the time $\pi$ geodesic flow and whose principal symbol encodes fibre-wise parallel transport with respect to the Levi-Civita connection on differential forms. In this setting this lifts the main result of Melrose and Zworski in [24] to differential forms and identifies additional scattering phase. The construction also yields a spectral representation in the co-closed sector, including a Stone-type formula and a functional calculus. In dimension $3$ and degree $p=1$, this framework describes divergence-free fields and thus the electric-field component of Maxwell's equations.

We classify, up to PI-equivalence, associative algebras over a field of characteristic zero whose codimension sequences have quadratic growth. More precisely, making use of fundamental algebras, we prove that every such algebra is PI-equivalent to a finite direct sum of algebras generating minimal varieties of at most quadratic codimension growth, together with a possible nilpotent summand.

We study estimation of the trace polynomial $\operatorname{tr} p(P\rho P)$ from global classical shadows, where $\rho$ is an unknown quantum state and $P$ is a fixed projector. Disjoint batching and complete U-statistics yield unbiased estimators of the same trace moments, but assign different sample-size factors to the degenerate terms in their Hoeffding decompositions. Under the global Clifford protocol, exact degree-two variance formulas show that, on a null projected block of rank $s$, the quadratic degenerate term has order $s^2/N$ under batching and $s^2/N^2$ under complete symmetrization. For a logarithmic-degree polynomial used in entropy approximation, the quadratic coefficient raises the batched variance to at least order $s^2N\log^2N$ at the classical entropy cutoff. For complete U-statistics, we derive a cross-degree covariance identity and an exact variance decomposition for polynomial estimators. We also bound every Hoeffding order at a fixed degree and obtain a growing-dimensional risk bound for a small-spectrum entropy functional. The higher-order bounds retain a polynomial dependence on the ambient dimension and therefore do not cover logarithmically increasing degrees. Monte Carlo experiments confirm the degree-two formulas, and exact calculations illustrate the entropy risks.

Let $q$ be an odd prime power, and $\mathcal{C}, \mathcal{D}$ be two smooth plane conics defined over $\mathbb{F}_q$ with transversal intersection. We present an exact formula for the number of points in $\mathbb{P}^2(\mathbb{F}_q)$ that are internal/external to $\mathcal{C}$ and internal/external to $\mathcal{D}$. This refines the $\frac{q^2}{4}+O(q^{3/2})$ asymptotic estimate for this quantity due to Asgarli and Yip \cite[Theorem 1.2]{Asgarli}. In particular, we show that the error term is of size at most $q+\sqrt{q} + 1$. By studying the geometry of the incidence variety related to this problem, we link the exact point counts directly to the Frobenius traces of two associated elliptic curves, and the number of $\mathbb{F}_q$-rational intersection points of $\mathcal{C}$ and $\mathcal{D}$ and of the corresponding dual conics $\mathcal{C}^*$ and $\mathcal{D}^*$. Lastly, We provide a remark explaining the challenges in generalizing this method to the study of higher-dimensional quadrics.

Let $p(m,n)$ denote the number of partitions of a rectangle $m\times n$ into integer-sided rectangular blocks, where two partitions are indistinguishable if they consist of the same multiset of blocks, regardless of their geometric arrangement. We present an elementary approach to show that, for every fixed positive integer $m$, $$ \log p(m,n)=\pi\sqrt{\tfrac{2mH_m}{3}}\sqrt{n}+O(\log n), \qquad \text{as }n\to\infty, $$ where $H_m$ denotes the $m$-th harmonic number. This confirms a conjecture recently posed by the authors and generalizes the Hardy–Ramanujan formula for integer partitions.

We analyze a variant of stochastic gradient descent with initial regularization (SGDIR) and derive dimension-free upper bounds on its expected excess risk for the squared loss. In the noiseless case, we obtain new bounds for both averaged and non-averaged SGDIR under moment, source, and capacity assumptions. For a particular value of the source parameter, these bounds are of order $m^{-2}\log^{2}m$, where the number of training samples is of order $m$. For another value of the source parameter, we obtain, for any $\epsilon>0$, bounds of order $m^{-3+\epsilon}$, provided that the capacity parameter exceeds $\epsilon^{-1}$. We also establish a lower bound that matches our upper bounds in certain regimes up to a polylogarithmic factor. In the noisy case, we provide an instance-based comparison between SGDIR and ridge regression. Under general assumptions and a mild lower bound on the regularization parameter, we show that the expected excess risk of SGDIR is no larger than that of ridge regression, up to a polylogarithmic factor. Numerical experiments on synthetic and real data are consistent with our theoretical findings.

Accurate simulation of three-dimensional time-harmonic wave propagation over many wavelengths requires control of phase error and efficient solution of large indefinite systems. We develop a three-grid solver based on the compact 27-point interpolated optimized finite-difference (IOFD) discretization. Its wavenumber-dependent stencil supports a fine-grid resolution of six points per shortest wavelength and an unshifted physical correction on the \(2h\) grid at only three points per shortest wavelength. The method retains unshifted IOFD operators on the \(h\) and \(2h\) grids, while a complex-shifted \(2h\)–\(4h\) auxiliary cycle preconditions a factorization-free iterative approximation of the coarse inverse. Restricting the shift to this auxiliary cycle preserves the propagative character of the coarse correction. Comparison with the outgoing Green function confirms phase and relative-amplitude accuracy on a sequence of meshes up to \revision{\(10240^3\)}, \revision{exceeding one trillion unknowns,} with the largest problem spanning approximately \revision{1704} wavelengths per coordinate. The same fixed solver configuration retains robust convergence across smooth, discontinuous, high-contrast, and geophysical velocity models and exhibits scalable parallel performance. In particular, a \revision{\(2560^3\)} problem spanning approximately \revision{425} wavelengths in each coordinate direction is solved in \revision{43.8 seconds} on just 64 NVIDIA A100 GPUs.

Federated learning has traditionally been formulated as a single-objective optimization problem, primarily focused on maximizing model utility. In real-world applications, however, machine learning models often need to optimize multiple and potentially conflicting objectives simultaneously. This motivates federated multiobjective optimization (FMOO), which provides a natural framework for jointly handling multiple task-specific objectives in federated learning. In this paper, we propose a momentum-based variance-reduced algorithm for federated multiobjective optimization. The method incorporates a momentum-driven gradient estimator into the local updates to reduce the variance of stochastic updates, leading to an improved convergence rate. We establish theoretical guarantees showing that the expected Pareto stationarity measure of a randomly selected output iterate decays at a rate of $\mathcal{O}(T^{-2/3})$, improving upon the $\mathcal{O}(T^{-1/2})$ rates established for existing methods such as FSMGDA and FedCMOO. Numerical experiments on federated multiobjective optimization benchmarks demonstrate the effectiveness and competitive performance of the proposed algorithm.

Let $D_n$ denote the supremum of the orthogonality defects over all $n$-dimensional Hermite–Korkine–Zolotarev (HKZ)-reduced bases. This paper shows that, as $n\to\infty$, \[ \log D_n=2n\log n-n\log\log n+O(n). \] We also determine the exact value $D_4=4375/1024$.

We prove that if $M$ is a connected complete parabolic Riemannian manifold and $N$ is a connected complete stochastically complete Riemannian manifold, then every bounded harmonic function on $M\times N$ is independent of the $M$-variable. Equivalently, pullback by the second projection induces an isometric isomorphism from the space of bounded harmonic functions on $N$ onto that on $M\times N$. In particular, the product of two parabolic manifolds has the bounded Liouville property, thereby answering Problem~16 of Grigor'yan's survey in the affirmative. The key analytic input is a total-variation memory-loss property of the heat kernel on a parabolic manifold. We establish this property by showing that the time-one heat kernel defines an aperiodic Harris recurrent transition kernel and then applying the row-merging theorem of Jamison and Orey.

Signed difference sets (SDSs) extend ordinary difference sets by allowing negative coefficients. We establish new nonexistence criteria and existence constructions for SDSs in finite abelian groups. Using the classical self-conjugate-prime method and a support-refined quotient method, we derive four obstructions, including the $C_2$-quotient, near-full-support, and small-defect obstructions. Applied cumulatively to the $67{,}823$ open group-specific cases in the database with $9\leq v\leq499$, these criteria rule out $45{,}361$ cases. For existence, we construct an infinite family from PCP-type regular partial difference sets arising from Desarguesian spreads and an explicit $(125,28,3)$-SDS in $C_5^3$ using quartic multiplicative characters. These constructions settle three further open cases.

We give examples of two derived equivalent gentle algebras which are not tilting-cotilting equivalent.

Dense IoT networks require reliable communication despite limited spectrum and substantial multi-user interference while maintaining manageable receiver complexity. This work introduces a deep-learning-based end-to-end multi-user communication design for interference-limited finite-blocklength IoT scenarios, focusing on short and medium blocklengths. We extend a prior 2-user SiameseNet transceiver framework to accommodate 2, 4, and 8 users, leveraging learned redundancy for interference suppression and noise robustness. Compared to conventional non-orthogonal access baselines, our method demonstrates strong Block Error Rate (BLER) performance across various scenarios without resorting to joint detection; the per-user decoder scales roughly linearly with the number of users. Further, we examine the robustness under interference mismatch and unequal interference strengths, critical for practical deployments with heterogeneous devices. The Latent-space analysis reveals that the learned codeword distance increases as the effective per-user rate decreases, corroborating with the observed BLER improvements. In addition, we also present preliminary results for a 2X2 MIMO setup under fixed-channel CSIT and CSIR, indicating potential for extending the framework to IoT gateways with multiple antennas.

The Berry phase arises naturally from the adiabatic dynamics of a pure quantum state, whereas the Uhlmann phase is usually formulated kinematically for a prescribed path of mixed states. For a qubit subject to a uniformly gapped conical drive, we obtain the Uhlmann connection and holonomy in closed form for the equilibrium Gibbs cycle. We then solve the corresponding Lindblad equation exactly in the rotating frame and show that its steady state forms a nonequilibrium limit cycle that lags the instantaneous Gibbs cycle. As the driving becomes slow, the Uhlmann phase approaches the equilibrium Uhlmann phase, which cooling then reduces to the Berry phase: in the joint adiabatic and low-temperature limit, the dynamical Uhlmann phase therefore converges to the Berry phase. This grounds the Uhlmann–Berry correspondence in the physical dynamics of an open system. We also characterize the finite-driving corrections, together with the discontinuous $\pi$ jump that the Uhlmann phase undergoes on the equatorial cycle as the temperature is varied. Finally, we examine an isolated transversal gap closing, where this zero-temperature correspondence can break down: the pure-state path becomes open, its closure is ambiguous, and gap-opening regularizations close it along a geodesic set by the direction of the bias field, splitting the Berry phase into a one-parameter family of values. The Gibbs path instead closes smoothly through the maximally mixed state, so the Uhlmann holonomy requires no regularization and remains unique and continuous at every finite temperature. In the low-temperature limit the Uhlmann phase selects a single member of the Berry family—the geodesic closure in the osculating plane, fixed by the velocity and acceleration of the driving field. The Uhlmann–Berry correspondence of the gapped regime thus survives the gap closing as a geodesic selection rule.

For next-generation air interfaces, two-sided artificial intelligence (AI) models have received growing attention, with AI models deployed at both the transmitter and receiver for efficient channel feedback and data communication. However, their practical deployment is complicated by assumptions commonly made in existing studies, including isolation from legacy users, training under predefined channel conditions, and gradient-based fine-tuning requiring substantial cross-vendor communication. This article revisits these assumptions and presents practical alternatives. First, for legacy coexistence, we integrate two-sided model processing into the 5G New Radio (NR) protocol stack and validate its operation alongside conventional NR on a real-world testbed. Second, instead of training under a massive number of predefined channel conditions, we construct a compact model table by jointly optimizing two-sided models with trainable surrogate channels, and select the best model according to the current channel condition to enable channel adaptation with high task performance and low training/storage overhead. Finally, unlike existing fine-tuning that exchanges large gradient vectors containing potentially private model information, we present gradient-free zeroth-order fine-tuning that requires only scalar feedback, facilitating multi-vendor interoperability. Together, these approaches advance the practical deployment of two-sided AI models while highlighting key open challenges.

We prove that, in each fixed degree, the exponent of the integral homology of a finite group is bounded solely in terms of the degree and the exponent of the group. The proof combines the solution of the restricted Burnside problem with a representability property of the bar construction and may be viewed as a torsion analogue of the method of acyclic models. We also use the Lyndon–Hochschild–Serre spectral sequence to obtain explicit bounds for finite solvable and nilpotent groups in terms of their derived length and nilpotency class.

We show a priori bounds for the generalized Parabolic Anderson Model $(\partial_t - \Delta)u = \sigma(u) \diamond \xi$ in finite volume in the full subcritical regime. The approach is based on the observation that, for constant initial data, the time interval over which the solution remains close to its initial condition can be chosen uniformly in the value of the initial condition. A comparison principle is then used to extend this estimate to general bounded initial data.

We study a resolution problem in local asymptotic decision theory: individual risks may admit Gaussian approximations that do not determine their vanishing difference. A finite menu of experiments is installed before context disclosure, although observations may be routed adaptively afterward. A greatest-element Blackwell order collapses adaptive routing to the best installed experiment and reduces the fixed-menu excess to an inverse-information distortion with frontier $A_k$. We develop differentiated, all-prior posterior transfer along a one-dimensional degradation chain and establish $F_{n,k_n}(H_n)=A_{k_n}\{1+o(1)\}$ for every diverging menu sequence with positive frontier and every admissible localization radius, without an additional direct sample-menu restriction. The transfer is exact under Gaussian degradation. Prior-free likelihood-generator conditions imply it for jump generators and are verified for binary attenuation, Poisson thinning, and negative-binomial thinning. If the distortion is uniformly quadratic on an Ahlfors-regular oracle image of dimension $r$, then $A_k\asymp k^{-2/r}$, and the original-scale excess mean squared error is of order $n^{-1}k^{-2/r}$. Calibrated Poisson sensor and radial-qubit measurement menus illustrate the result. A triangular Gaussian counterexample shows why pointwise Gaussian convergence is insufficient.

Three-dimensional time-harmonic Maxwell simulations generate massive complex indefinite systems whose mesh coarsening is strictly limited by phase accuracy. Although matrix-free finite element kernels utilize GPU throughput efficiently, standard multilevel solvers are ultimately bottlenecked by the memory and communication costs of exact coarse-grid factorizations. We present a fully matrix-free, factorization-free three-grid preconditioner for curl-conforming Nédelec discretizations with perfectly matched layers (PML) and optimally blended quadrature. The method employs an outer FGMRES to solve the unshifted fine-grid equation, while an intermediate-grid correction is computed by a fixed-work FGMRES preconditioned with a complex-shifted $2h$–$4h$ cycle. This strategically confines the complex shift to an auxiliary preconditioner, preserving the physical Maxwell operator. A local Fourier analysis derives the blended Maxwell branches and compatible edge transfers, identifying robust shift and Jacobi damping parameters. Validated against the analytical Maxwell Green tensor, our approach demonstrates extreme scalability: using a single solver configuration, both homogeneous and highly heterogeneous systems with approximately 10.89 billion complex edge unknowns are solved in 42.0–72.0 seconds on just 64 NVIDIA A100 GPUs.

A wireless sensor must alert a remote monitor before a monitored process crosses a safety threshold; an alarm arriving afterward may be too late. The sensor can react to its current estimate or predict ahead and trigger earlier, but the value of such lookahead is not obvious. In some systems it creates an early-alarm opportunity unavailable to the current test, while in others it cannot cross the alarm boundary. This letter gives a practical three-stage rule for deciding when to predict. First, an algebraic spectral test decides at design time whether lookahead is structurally useful: it is redundant exactly when the threshold direction is a left-eigenvector of the dynamics with a non-negative eigenvalue. Second, a closed-form channel decomposition shows that deeper prediction becomes more valuable as the channel degrades, because longer lead windows permit more pre-crossing transmission attempts. Third, simulations show that large gains also require retained prediction magnitude; oscillatory dynamics amplify the benefit through rotation, and a two-sensor setting reveals a sensing-channel tradeoff.

We prove extremality results concerning specific quasiconformal mappings in the hyperbolic plane, namely, hyperbolic spiral and stretch maps.

The List Edge-Coloring Conjecture predicts that any graph whose edges can be colored with $k$ colors can also be colored from arbitrary lists of $k$ colors. We prove its stronger online form for two new infinite families, $K_{p-1}$ and $K_{2p}$, where $p$ is an odd prime. For even $n$, order the vertices of $K_n$ and draw each perfect matching as arcs above them. Count crossings separately within each matching, and let $S_n$ be the number of decompositions into perfect matchings having an even total crossing count minus the number having an odd total. Then \[ S_{p-1}\equiv\left(\frac{-2}{p}\right)\pmod p, \qquad S_{2p}\equiv-p\pmod {p^2}. \] The two congruences are governed by the same elementary matching sum over $\F_p$, although their proofs use the prime $p$ differently. Their nonzero residues give the conjectured values even in the online game. They also treat the corresponding complete graphs with one perfect matching removed, as well as $K_{2p}$ after deleting some, but not all, of a natural cyclic family of $p$ disjoint perfect matchings.

For a graph $H$ and a family of graphs $\mathcal F$, let $\text{ex}(n,H,\mathcal F)$ denote the maximum number of copies of $H$ in an $\mathcal F$-free graph on $n$ vertices. For every integer $i\ge 3$, let $C_i$ denote the cycle of length $i$. For $r\ge 3$, set $\mathscr {C}_r=\{C_3,C_4,\ldots,C_r\},$ and set $\mathscr {C}_2=\varnothing$. In this paper, we prove that, for all integers $l>k\ge 2$, $$ \text{ex}(n,C_{2k+1},\mathscr {C}_{2k}\cup\{C_{2l+1}\}) =O_{k,l} \left(n^{2-\frac{1}{k(k+1)(l-k)}}\ \ \right). $$ Together with the known upper bounds for the number of triangles in $C_{2l+1}$-free graphs, this confirms a conjecture of Gerbner, Győri, Methuku, and Vizer.

Building on the generalized do Carmo-Wallach theory, we construct a functorial framework for holomorphic maps from flag manifolds into Grassmannians and quadrics. Through the direct image sheaf and its inverse, we construct Penrose-type transforms that allow both the domain and target to vary. These transforms provide functors between categories of full holomorphic maps satisfying the gauge condition for semi-positive homogeneous bundles. Within this framework, Einstein-Hermitian holomorphic maps form a natural subcategory, stable under all transforms, and the minimality of L^2-norm of the mean curvature operator is preserved. For Grassmannian targets, the moduli of Einstein-Hermitian maps from a flag manifold are identified with r-tuples of non- positive integers modulo symmetry, where r is the rank of the group of holomorphic isometries. For quadric targets, we obtain a complete geometric description of the moduli space. The center of this moduli space corresponds to the Einstein-Hermitian map into the projective space, and the tower of transforms identifies all intermediate moduli spaces with the moduli of holomorphic isometric embeddings at the bottom level. The results yield a unified categorical and geometric description of holomorphic isometric embeddings of flag manifolds.

For $\alpha>1$, we study the weighted multiplicative Hilbert operator \[ (T_\alpha f)(s) = \int_{1/2}^{\infty} f(t)\bigl(\zeta(s+t)^\alpha-1\bigr) \left(t-\frac12\right)^{\alpha-1}\,dt \] on the Hilbertian Bergman space $A_{\alpha,0}^2$ of Dirichlet series with zero constant term. In the natural orthonormal basis, $T_\alpha$ is represented by the weighted multiplicative Hilbert matrix \[ \left( \frac{\Gamma(\alpha)\sqrt{d_\alpha(m)d_\alpha(n)}} {\sqrt{mn}\,(\log(mn))^\alpha} \right)_{m,n\ge2}, \] where $d_\alpha$ is the generalized divisor function defined by $\zeta(s)^\alpha=\sum_{n\ge1}d_\alpha(n)n^{-s}$ for $\Re s>1$. Set $\Lambda_\alpha=\Gamma(\alpha/2)^2/\Gamma(\alpha)$. We prove that the essential spectrum of \(T_\alpha\) is \([0,\Lambda_\alpha]\) and that the spectrum in this interval is purely absolutely continuous with multiplicity one. Above \(\Lambda_\alpha\), there are at most finitely many eigenvalues, all of which are simple. Moreover, there is a unique \(\alpha_*\in(1,2)\) such that \(T_\alpha\) has eigenvalues if and only if \(\alpha>\alpha_*\). Consequently, the associated sharp embedding inequality admits a non-zero extremizer precisely when $\alpha>\alpha_*$, and this extremizer is unique up to scalar multiplication. We also establish the asymptotic formula $\|T_\alpha\|\sim\Gamma(\alpha+1)/(2(2\log 2)^\alpha)$ as $\alpha\to\infty$. A key step in the proof combines Mellin-transform asymptotics, strict total positivity, and eigenvalue counting to exclude embedded eigenvalues. The proof includes a general result on the essential and singular continuous spectra of weighted integral Hankel operators.

We provide a geometric foundation for the kinetic theory of photons. Although the induced metric $\hat h$ on the light cone bundle $\Gamma_0^+$ is degenerate, which causes the corresponding volume element to vanish, we can still use a method similar to the Hodge dual to construct a volume element $\eta_{\Gamma_0^+}$ for the light cone bundle. Based on this geometric structure $(\Gamma^+_0,\eta_{\Gamma_0^+},\hat h)$, we establish the fully covariant Boltzmann equation for photons. More importantly, the volume element and the induced metric are linked in a nontrivial way, which allows the physical distributions to be defined consistently. This yields the corresponding hydrodynamic quantities and their divergences, which take the same form as in the case of massive particles.

The collision operator for the Boltzmann equation is a nonlinear nonlocal operator. When lifted in the extended 2-particle space, it is viewed as the projection of a collisional linear operator. In this paper, we propose a particle method that samples the post-collision relative velocity directly from the Green's function of this operator (the transition probability of the generated time-continuous Markov chain). The normalizing flow amortized sampling is then proposed to reduce the sampling complexity. The resulted method takes $O(N)$ each time where $N$ is the particle number, and conserves momentum and energy exactly. This method does not require the boundedness of the kernel and, more importantly, it allows learning the Green's function directly from the scattering data without selecting the kernel in a specified family.

Space situational awareness (SSA) requires purpose-matched models across spatial, temporal, and fidelity scales. Building on our previously reported three-dimensional (3D) cell formulation and implementation, this study establishes a reproducible, resolution-aware, catalog-conditioned framework for macroscopic assessment of the low Earth orbit (LEO) collision environment. The framework maps supplied catalog or scenario populations to time-averaged spatial density and target-specific impact metrics while retaining individual-object information. Using a 2025 Space-Track snapshot, we evaluate radial, declination, and right-ascension resolution sensitivity and computational performance for six targets, including two Korean space assets. Normalized expected impact counts range from 0.615 to 1.599 and vary nonmonotonically; for a synthetic 500-km circular target, the result at a 0.25-km radial width is 38.5% below the 10-km reference. Runtime and memory show direction-dependent trade-offs. Ten annual snapshots show catalog growth from 15,723 objects in 2016 to 28,540 in 2025 and a 7.86-fold increase in the 500-km target metric, driven primarily by Starlink, other payloads, and unknown/TBA records. In a conditional stress test of the proposed 998,240-satellite SpaceX Orbital Data Center population, exact annual probabilities of at least one impact reach $3.75\times10^{-3}$ and $1.48\times10^{-3}$ for the 700-km and 1,000-km targets. The framework provides a reproducible, resolution-aware basis for catalog-conditioned environment monitoring, comparative scenario assessment, and prioritization of cases for higher-fidelity follow-up analysis.

A graph $G$ is apex if $G$ has a vertex $v$ such that $G-v$ is planar. We prove that every $2$-connected apex cubic graph is three-edge-colorable. This result gives the final piece of the proof for the well-known Tutte's three-edge-coloring conjecture from 1966 \cite{tutte}. The proof, as well as the result, generalizes that of the Four Color Theorem, which requires computer checks. As in the previous proof of the Four Color Theorem, the proof is constructive. More precisely, given a $2$-connected apex cubic graph $G$ on $n$ vertices, our reducibility and discharging procedure yields a three-edge-coloring of $G$ in $O(n^2)$ time. As an additional reproducibility check for our computer checks, independent implementations reconstructed from the detailed pseudocode (given in the appendix) using generative AI systems reproduced the required computational results. These reconstructions are not part of the mathematical justification of the theorem, but provide additional evidence for the reproducibility of the computations.