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arXiv2026

Conjugation Differential Invariants of $\mathrm{SL}_2(\mathbb{F}_q)$ on trace-free matrices

Let $q=p^k$ be prime power, let $F= \mathbb{F}_q$ and let $V$ be the vector space of 2 by 2 matrices over $F$ with trace zero. Let $G = \mathrm{SL}_2(F)$. Then $G$ acts on $V$ via conjugation. Let $Ω= S(V^*) \otimes Λ(V^*)$ be the algebra of differential forms on $V$. A minimal generating set for $Ω^G$ when $q=3$ was computed by the author and Meyer. In this article we compute a minimal generating set for $Ω^G$ for all $q$.

math.AC↗
arXiv2026

Improved convergence radius of the Fer expansion for Hermitian generators

The Dyson series expands the propagator of a time-dependent Hamiltonian in powers of the Hamiltonian, but its truncations are in general not unitary. The Fer expansion writes the same propagator as an infinite product of matrix exponentials, each of them unitary, and its remainder decays doubly exponentially with the number of factors. Convergence, however, is guaranteed only within a finite radius: the time integral of the norm of the Hamiltonian must be smaller than $2$. This is the best value known to date. Here we improve it by about $30\%$, raising it to about $2.6058$. The result also holds for non-Hermitian Hamiltonians that are Hermitian with respect to a fixed metric.

quant-ph↗
arXiv2026

Hutch#: Optimal non-adaptive Frobenius norm estimation

The Girard--Hutchinson estimator provides an extremely simple randomized estimate of the Frobenius norm of a matrix $A$ that can only be accessed implicitly via matrix-vector products. In particular, if $Ω$ is a random Gaussian matrix with $r = O(1/\varepsilon^2)$ columns, than $\frac{1}{r}\|AΩ\|_F^2$ provides a $(1\pm \varepsilon)$ multiplicative approximation to $\|A\|_F^2$ with high probability. In this work, we introduce a closely related estimator, given by \begin{align*} {\frac{1}{r}\|AΩ\|_F^2 + \frac{1}{r}\|Ψ^T A\|_F^2 - \frac{1}{r^2}\|Ψ^T AΩ\|_F^2}, \end{align*} where $Ψ$ is a second, independent random Gaussian matrix with $r$ columns. We prove that this estimator yields a $(1\pm\varepsilon)$ multiplicative approximation to $\|A\|_F^2$ when $r = O(1/\varepsilon)$, a quadratic improvement over Girard--Hutchinson. This dependence on $\varepsilon$ is optimal. Our method, which we call Hutch# (pronounced ``Hutch sharp''), matches the complexity of the Hutch++ algorithm [Meyer, Musco, Musco, Woodruff, 2021]. However, unlike Hutch++, Hutch# uses only \textit{non-adaptive} matrix-vector products with $A$ and $A^T$ and requires no orthogonalization or other advanced linear algebra steps. Thus, Hutch# combines the simplicity of the Girard--Hutchinson estimator and the optimal query complexity of Hutch++.

math.NA↗
arXiv2026

Rotating black holes surrounded by PFDM in STVG: Shadows and eikonal quasinormal modes

We investigate the optical and dynamical properties of a rotating black hole in scalar--tensor--vector gravity (STVG) surrounded by perfect fluid dark matter (PFDM). Starting from the static STVG--PFDM geometry, we consider a Kerr-like rotating extension characterized by the spin parameter $a$, the STVG coupling $α$, and the PFDM parameter $λ$. We first examine the horizon structure and a radial effective mass function, then analyze null geodesics and unstable spherical photon orbits. We use these results to construct the black-hole shadow and study its size, distortion, and dependence on model parameters and observer inclination. We then compare the predicted angular shadow diameter with the Event Horizon Telescope observations of M87$^*$ and Sgr~A$^*$ in the $(a,α)$ parameter space for representative values of the PFDM parameter. The comparison shows that the PFDM contribution can shift the parameter-space region compatible with the observed shadow diameter, while the shadow size is generally less sensitive to spin than to deformation. We further estimate the Hawking energy-emission spectrum in the high-frequency geometric-optics approximation, using the shadow radius as an effective absorption scale. Finally, we study the quasinormal-mode spectrum in the eikonal limit through the correspondence between unstable photon orbits and ringdown frequencies. The numerical results recover the expected Schwarzschild limit and show a clear separation between the prograde and retrograde modes as the spin increases. The combined analysis provides a consistent optical, energy emission, and ringdown characterization of the rotating STVG--PFDM geometry.

gr-qc↗
arXiv2026

The Siren Call of Silicon Leviathan: Reflections on blowup and Aufklärungsdämmerung

Over three centuries ago our Enlightenment forefathers brought forth in Europe--and later on this continent--a new approach to knowledge production and dissemination, conceived in distrust of authority, and dedicated to the proposition that truth is accessible to human reason. Mathematics, the Enlightenment's firstborn, was its existence proof--knowledge that compels assent without appeal to authority--because the only way to make a theorem certain was to show it to a human mind. Now we are engaged in an existential struggle testing whether mathematics, or any discipline so conceived and so dedicated, can long endure. We are met in the wake of the OpenAI press release: 166 pages, 616,000 lines of Lean, produced in 88 hours by some 10,000 software agents, and read in full--at the time of this Siren Call--by no human being. A proof has historically been two things: a certificate that a theorem is true, and a demonstration that lets another person see why. OpenAI delivered the first without the second. But, in a larger sense, what matters is not what the Silicon Leviathan (OpenAI and its peer competitors) produces but rather what we accept. By accepting the certificate without the demonstration, we adopt the Hobbesian bargain: authority, not truth, makes the law. The age that this augurs is--in Panofsky's phrase--a Middle Ages in reverse--its authority a subhuman superintelligence. The great task before us is to frame a covenant under which what has become optional--understanding--remains required: that nothing counts as mathematics until a human being has understood it--and can show the next person why. Whether we will safeguard and keep our sovereign inheritance--decided at the moment of acceptance--which is not yet past--will determine if this Aufklärungsdämmerung spells the Enlightenment's end.

math.HO↗
arXiv2026

Characterisations of Planar Galled Networks

Rooted phylogenetic networks are widely used to represent the evolution of species that have undergone reticulate processes. However, these networks can be highly non-planar, making them more difficult to visualise and interpret than evolutionary trees. In this paper, we investigate planarity properties of galled networks, an important subclass of phylogenetic networks. We show that all planar galled networks are necessarily upward planar. Furthermore, by leveraging recent results on planar phylogenetic networks, we provide three characterisations for each of the outerplanar and terminal planar galled network classes in terms of forbidden vertex configurations, forbidden directed subgraphs, and forbidden structures in their associated underlying undirected graphs. These results contribute to a deeper understanding of the structural properties of galled networks and may inform future methods for their construction and visualisation.

math.CO↗
arXiv2026

fable.intermittent: benchmarking probabilistic forecasting methods for intermittent time series

Intermittent time series are common in spare-parts demand and retail sales. Since the cost of forecast errors is typically asymmetric, decisions such as inventory control require the full predictive distribution rather than a point forecast. Many probabilistic forecasting methods have been proposed; their implementations, however, are scattered across different software frameworks, making it difficult to compare them systematically. We introduce fable$.$intermittent, an R package that implements several probabilistic forecasting methods for intermittent series within the fable framework. The package allows several models to be fitted and evaluated on a collection of time series through a single, simple forecasting pipeline. We also introduce TWEES, a new exponential smoothing model with a Tweedie predictive distribution. Fitting TWEES requires repeated evaluation of the computationally demanding Tweedie density. We also release the R package tweedieDistr, whose implementation of the Tweedie distribution is substantially faster than the existing one while preserving the same numerical accuracy. We evaluate the methods implemented in fable$.$intermittent on four datasets, also released in the package.

cs.LG↗
arXiv2026

Global Convergence of Third-Order Langevin Dynamics for Non-Convex Optimization via Simulated Annealing

We study global convergence guarantees of third-order Langevin dynamics for non-convex optimization via simulated annealing with fixed friction and decreasing noise. An explicit three-block distorted entropy transfers dissipation from the noisy auxiliary variable to the full state. Under dissipativity, regularity, and low-temperature functional-inequality assumptions, logarithmic cooling drives the objective values to the global minimum in probability at the barrier-controlled kinetic rate. For the exact-force-integral and midpoint three-stage discretizations, polynomially decreasing steps preserve this rate on the physical time scale. The cubic local endpoint estimate gives a less restrictive sufficient step-size condition than the available frozen-force kinetic result. A comparison with the one-gradient UBU integrator shows how its centered stochastic local error leads, under the same strong-coupling analysis, to a smaller sufficient iteration exponent. Numerical experiments are conducted to illustrate our theory. For a double well objective, third-order Langevin terminal-success point estimates are higher than UBU at both a common horizon and an equal gradient budget. For a high-dimensional nonconvex neural-network objective using synthetic data, independently tuned UBU and third-order Langevin schemes both outperform overdamped Langevin dynamics; the third-order Langevin point estimate is higher. For the same neural-network objective on real data, we show the same point-estimate ordering for best-basin probability and post-quench test accuracy. Numerical code and associated experiment results are publicly available at https://github.com/gagawjbytw/simulated-annealing-third-order-langevin.

math.NA↗
arXiv2026

Continuity of Regularized Channel Rényi Divergences

We prove that the regularized, stabilized sandwiched Rényi divergence of finite-dimensional quantum channels converges to their regularized relative entropy as the Rényi order tends to one. The key tool is the channel hockey-stick divergence: Gour's Stinespring approximation bound and a Schatten norm estimate amplify an asymptotic bound below one into exponential decay at higher threshold rates. For channel pairs with finite max-relative entropy, known operational connections then give exponential strong converses for parallel and adaptive discrimination, a sharp zero--one testing law, and the subchannel asymptotic equipartition property.

quant-ph↗
arXiv2026

When Fancy Eviction Fails: Rethinking Cache Replacement For LLM Prefix Reuse

Long-running LLM applications repeatedly send growing context, making prefix caching critical for reducing prefill cost. Yet prefix-cache behavior under agentic workloads remains poorly understood. We study production traces from two companies and evaluate 14 eviction algorithms across HBM-constrained and large memory-pool settings. Despite a large gap to Belady, sophisticated policies designed for traditional caches provide little benefit over LRU. The reason is structural: prefix reuse is dominated by the regular pacing of active sessions, making recency unusually predictive. Prefix caching nevertheless introduces new challenges, including heavy-tailed session footprints and highly variable miss costs as attention computation grows with sequence length. We introduce the compute-savings ratio and two offline oracles to quantify these effects. Our results show that effective prefix-cache management should retain recency as its foundation while selectively adding quick demotion for one-hit prefixes, compute-aware partial eviction for expensive misses, and capacity-dependent eviction granularity. We will release the traces and simulator to support future research.

cs.DC↗
arXiv2026

Nearly optimal packings of equally sized rainbow forests

A forest in an edge-colored graph is rainbow if its edges have pairwise distinct colors. We prove that for every $\varepsilon>0$ and all sufficiently large integers $m$, every properly edge-colored simple graph with $km$ edges, where $1\leq k\leq 2m$ and every color class has size at most $m$, contains at least $(1-\varepsilon)m$ pairwise edge-disjoint rainbow forests, each with exactly $k$ edges. The range $k\leq 2m$ is best possible: for every $k>2m$ there are such graphs containing no $k$-edge forest. Thus the conjecture of Montgomery, Pokrovskiy, and Sudakov fails beyond this range, while our theorem establishes its predicted conclusion throughout the largest possible range of $k$. The number of forests is asymptotically optimal. The proof uses an orientation dichotomy, hypergraph matching, matroid intersection, and martingale concentration.

math.CO↗
arXiv2026

Limits on Primordial Black Hole Evaporation from LUX-ZEPLIN

The LUX-ZEPLIN (LZ) collaboration recently reported a singular anomalous nuclear recoil event at $248 \pm 23 \, \rm keV$, accompanied by a strict null result for unexplained excesses in their primary $5.4-50 \, \rm keV$ search window. The macroscopic momentum transfer required to generate this event is kinematically inaccessible to standard halo cold dark matter (CDM), naturally motivating models featuring light, relativistic relics. In this work, we investigate whether a contemporary flux of Hawking-boosted dark matter (DM) emitted by evaporating Primordial Black Holes (PBHs) in the $10^{11}-10^{13} \, \rm g $ mass range can source the anomaly. By parameterizing the nuclear scattering with heavy non-relativistic effective field theory (NREFT) operators and endothermic inelastic mass transitions, we formulate a strict kinematic exclusion. We demonstrate that the inherent thermal nature of the Hawking emission, coupled with the relativistic kinematics of the incident flux, inevitably overproduces low-energy recoils, aggressively violating the LZ background bounds. Utilizing this failure, we map the LZ low-energy null results into novel, highly stringent upper limits on the PBH abundance fraction, excluding $f_{\rm PBH} \gtrsim 10^{-6}$ for $M_{\rm PBH} \sim 2 \times 10^{11} \, \rm g $ at $ 90 \% $ confidence level.

astro-ph.CO↗
arXiv2026

Artificial Societies Benchmark: A Validation Framework for Synthetic Research

A synthetic survey can reproduce the average answer while misrepresenting how people differ, how their answers relate to one another, or how they respond to changes in conditions. We introduce the Artificial Societies Benchmark to help researchers assess whether synthetic populations support their intended analyses. The framework combines eleven tests across internal, construct, and external validity, drawing on twenty human sources and comparing nine language models. It connects each research use to the evidence it requires and tests how results change with the information we supply about respondents. Importantly, strong performance in one domain does not establish fidelity in the others. Models often answer too consistently, compress response scales, and alter relationships between traits whilst richer profiles improve prediction for some models and worsen it for others. The resulting scorecard helps researchers identify which aspects of a synthetic population can support their analysis and where researchers need further human evidence.

cs.CL↗
arXiv2026

Superintegrability of discrete-time rational Ruijsenaars-Schneider model and deformed polynomial symmetry algebras

We explicitly construct the additional integrals of motion, ensuring maximal superintegrability of the discrete-time rational Ruijsenaars-Schneider model. Using them, we investigate the algebraic aspects of superintegrability in both continuous- and discrete-time settings. In particular, we determine the complete structures of the polynomial symmetry algebras associated with both the rational Ruijsenaars-Schneider model and its discretization. We demonstrate that discretization leads to a nontrivial deformation of the continuous symmetry algebra with respect to the discretization parameter, thereby extending recent analogous results from the rational Calogero-Moser system to its relativistic generalization.

nlin.SI↗
arXiv2026

Binary black hole scattering in the extreme-mass-ratio limit: time-domain waveform

The time-domain gravitational waveform emitted during the scattering of two nonspinning compact bodies is computed in the extreme-mass-ratio limit. It provides a time-domain description of the gravitational radiation emitted during the encounter and complements previous first-order self-force calculations, which have primarily been formulated in the frequency domain. In this way, the present results provide a complementary representation of the radiative dynamics and facilitate a direct comparison between time-domain and frequency-domain approaches to the gravitational-wave signal. The waveform is accurate to the fifth post-Minkowskian (three-loop) level and seventh post-Newtonian order, and will serve as a benchmark for future calculations by other methods, to first order in the mass ratio. The results have been already tested in a previous work by constructing the energy and angular momentum fluxes, thereby computing the corresponding radiative losses at the 5PM and 4PM level, respectively, with the same PN accuracy, which are in agreement with recent amplitude-based calculations.

gr-qc↗
arXiv2026

A Living Benchmark for Information Retrieval from Electronic Health Records

Large language model (LLM)-based clinical assistants are increasingly being integrated into electronic health record (EHR) systems, transforming how clinicians retrieve and synthesize information from patient records. Their safety and utility depend on rigorous evaluation, yet existing benchmarks are manually curated, costly to update, and rapidly become obsolete with evolving technological advancements. We present a scalable framework that automatically generates question--answer pairs from longitudinal EHR notes. Nineteen clinicians validate the benchmark generator, producing the Benchmark for Retrieving Information in EHRs (BRIE), a continuously maintainable evaluation dataset. Across nine LLMs and five inference strategies, state-of-the-art systems frequently omit clinically important information, particularly for questions requiring synthesis across multiple documents and encounters. Because the generator itself is validated, BRIE supports evaluations that static benchmarks cannot, including the generation of multiple answers that reflect variation in clinician reasoning for robust performance assessment and continuously refreshing benchmark content to guard against leakage. Our results demonstrate that scalable benchmark generation enables rigorous, up-to-date evaluation of clinical LLMs as they are deployed in rapidly evolving healthcare settings.

cs.AI↗
arXiv2026

Planar Contact Structures with Calabi-Yau Fillings and Topological Quantum Computation

We study planar openbooks obtained by lifting braids through branched covers of D^2, together with the quantum operations in the Ising representation. We give a criterion for the Stein fillings to be Calabi-Yau (CY). Among positive factorizations of a fixed monodromy, a CY one has minimal length and its filling minimizes χand b_2. Applied to Baykur's recent examples, this gives a planar contact 3-manifold with infinitely many non-homeomorphic CY fillings, the cover there having degree \ge 6. At degree 4 a single CY filling forces every filling to be CY, and in one case the filling is unique; at degree \le 3 it is unique, and CY under a mild condition. Admissible cuts of D^2 decompose the openbook into subopenbooks. Every positive factorization then localizes, and the CY condition holds exactly when it holds locally. The state space is then a direct sum of tensor products indexed by the compatible parity choices, with at most two qubits per factor when the pieces have degree $\le 4$, the same range in which the CY condition depends only on the monodromy. For one degree 4 cover, the points, lines and flags of the two-qubit doily are realized by its subopenbooks. Fifteen liftable braids there share the same quantum operation and Stein filling, and are separated only by the subopenbook systems they admit. A choice of tensor-product structure is thus carried by the lift, not by the braid group representation, suggesting a link between contact topology and quantum entanglement.

math.GT↗
arXiv2026

Rational homotopy theory of flag manifolds

Let $G/P$ be a flag manifold with $G$ simple. We determine the rational homotopy groups $π_{\ast }(G/P)\otimes \mathbb{Q}$ and describe the rational cohomology ring $H^{\ast }(G/P;\mathbb{Q})$ solely in terms of the rational homotopy types of $G$ and $P$, respectively. The proof is based on the restrictions of basic Weyl invariants of $G$ to suitable simple factors of $P$, which also address a key nonvanishing issue not settled by earlier approaches of Meier, Shiga-Tezuka, Kotschick, and Terzic.

math.AT↗