Papers

Nudged elastic band (NEB) calculations of small-polaron hopping are demanding because nuclear motion is coupled to a self-consistent localized electronic state. Near the charge-transfer region, slow self-consistent-field convergence, localization switching, and competing electronic solutions complicate surrogate-assisted optimization. We develop a Gaussian-process-assisted NEB workflow for this regime and demonstrate electron small-polaron hopping in rutile TiO2 using spin-polarized density functional theory with a Hubbard U correction (DFT+U ) with SIESTA. A joint energy-force Gaussian process (GP) with a Matérn covariance model advances the elastic band, while active-atom root-mean-square and maximum-atom displacement limits constrain surrogate motion relative to evaluated configurations. Reaching these limits triggers first-principles sampling without requiring full surrogate convergence. Three independently DFT-verified 96-atom [001] calculations using 11, 13, and 15 images and different surrogate settings and GP parameterizations yield barriers of 0.1645, 0.1637, and 0.1640 eV, respectively, spanning 0.793 meV. In the 15-image calculation, repeated central-image sampling activates a separately calibrated local GP during surrogate optimization. Final full-band DFT verification gives a maximum projected NEB force of 0.0346 eV/Å. A 324-atom calculation yields a lower barrier of 0.1302 eV, indicating supercell sensitivity and demonstrating feasibility at larger GP observation dimension. The workflow combines controlled surrogate extrapolation, targeted charge-transfer-region interpolation, and independent DFT verification to obtain polaron migration pathways, emphasizing reproducible barriers and full-band force convergence.

Face recognition systems often experience performance degradation under large pose variations due to changes in facial appearance and visibility of discriminative features. This study presents a multi-task deep learning framework that jointly addresses pose-invariant face recognition and facial landmark localization within a unified architecture. To enable simultaneous optimization of identity recognition and facial structure learning, the original Cross-Pose Labeled Faces in the Wild (CPLFW) dataset was adapted into a multi-task learning dataset by associating facial images with identity labels and five-point landmark annotations. The proposed framework consists of two task-specific branches operating on a common facial input: a ResNet50-based branch for identity recognition and an EfficientNetV2B0-based branch with spatial attention for facial landmark localization. The two objectives are jointly optimized using a weighted multi-task loss, allowing the model to learn complementary semantic and spatial facial representations. The proposed framework was evaluated on the prepared CPLFW-based multi-task dataset. Unlike the original CPLFW benchmark, which focuses primarily on identity recognition, the proposed dataset adaptation incorporates five-point facial landmark annotations to enable joint recognition and localization learning. Experimental results demonstrate that the model achieves a validation recognition accuracy of 81.58% and a facial landmark localization mean squared error of 96.38. The results indicate that combining identity classification and landmark regression within a single optimization framework provides an effective approach for learning robust facial representations under cross-pose conditions The proposed method demonstrates the potential of multi-task learning and attention-based feature refinement for unconstrained facial analysis.

We prove that, for every fixed rational $c>0$, computing a $2^{-n^c}$-approximate Nash equilibrium in an $n$-player two-strategy anonymous game is $\mathsf{PPAD}$-complete. This holds even when one strategy always pays $1/2$ and, for all but two players, the other is constant outside at most five opponent counts. The reduction uses coefficient estimates to simulate circuit constraints approximately. Exact computation requires eliminating the simulation error: with three strategies, a separate construction uses polynomial roots to implement circuit gates exactly. This yields $\mathsf{FIXP}$-completeness of exact equilibrium computation under separable-linear reductions and $\mathsf{FIXP}_a$-completeness of approximation in distance to an exact equilibrium.

We study the amplification and concentration of wave energy in smooth, periodic space-time media through the dynamics of the characteristic curves of the associated wave equation. In an impedance-matched setting, we show that stable limit cycles attract neighbouring characteristics, focusing the wave energy density. The exponential attraction rate is estimated analytically from the system's Lyapunov exponent. We further show that the attraction of characteristics and energy accumulation persist under a range of nonperiodic perturbations. Finally, we present a Floquet-Fourier numerical method for computing stable and unstable limit cycles and their attraction rates. Throughout, we exemplify the results using the example of modulation given by a superposition of travelling waves, revealing saddle–node bifurcations where limit cycles may appear depending on the relative wave speeds. Full-wave simulations confirm that the characteristic mechanism continues to describe energy accumulation beyond the impedance-matched case.

This paper studies fundamental solutions of uniformly parabolic systems with stationary random coefficients depending on space and time. Under ergodicity and local regularity, we establish quenched Gaussian bounds for the kernel and its spatial derivatives, with stationary random prefactors. A spectral gap assumption gives moments of every finite order for these prefactors and hence annealed Gaussian bounds. We also prove quantitative two-scale expansions for the kernel, its spatial gradient, and its mixed spatial derivative. The error estimates hold in every finite probability moment and retain Gaussian decay. Their dimension-dependent growth is described by a corrector weight, with a single additional logarithmic factor in the derivative estimates. The proofs combine large-scale regularity, extended correctors, weighted estimates, and a parabolic weak-norm estimate. The latter supplies the passage from deterministic pairings to local annealed estimates without interchanging expectation and a supremum over test functions. The mixed-derivative estimate builds on the gradient expansion and does not introduce a further logarithmic loss.

Standardizing a rare feature can help stochastic gradient descent learn its effect within a finite budget, but it also changes the noise cost of learning. We characterize the lowest rare-subgroup risks attainable by standardized and unscaled training when each chooses a nonnegative constant learning rate and a deterministic stopping time under the same observation cap. For two independent binary features, we derive both optimized risks as prevalence vanishes while the expected number of rare observations remains finite. Uniform control over rates and stopping times reduces each limiting oracle to a scalar minimization. With positive noise, the preferred scaling changes exactly once as we move signal to the rare feature while the limiting risk at initialization stays fixed and positive. The formulas also explain how noise and budget affect the comparison. Separate finite-prevalence bounds favor standardization on a rare-dominant signal family and unscaled training on a common-dominant family. The results concern final iterates with population preprocessing and fresh observations. Exact-moment searches assess the approximation at finite prevalence. In a controlled study, both orderings persist under observation reuse, estimated preprocessing, and intercept fitting when tuning choices are fixed before evaluation. Selecting a predictor after observing training can reverse an ordering, even when selection uses exact population risk.

We construct a meromorphic Wilf–Zeilberger pair for the hypergeometric series for \(3\ln 2\), completing the WZ proof left open by Guillera and Rogers. The same pair yields a one-parameter identity and, after a half-integer shift and differentiation, an inverse-binomial series for \(\zeta(3)\). We also study the finite truncations of the logarithmic series. For every prime \(p\ge 5\), we prove that the lower truncation is congruent to \(q_p(2)-1\) modulo \(p\), while the upper truncation is congruent to \(-\frac{7}{16}p^2B_{p-3}\) modulo \(p^3\). The proofs combine the WZ method with terminating hypergeometric identities, the Pfaff–Saalschütz identity, and classical congruences for harmonic sums.

Searches for new contributions to the invisible width of the $Z$, $\Delta\Gamma_{inv}$, can be used to provide model-dependent constraints on new physics that may arise from an unknown dark sector. In the vector portal setup considered here it is shown that loops of portal matter (PM) fields, having both Standard Model (SM) and dark sector quantum numbers, can lead to new contributions to $\Delta\Gamma_{inv}$ that satisfy current experimental constraints while being simultaneously accessible to measurements at the future Tera-Z. Such PM fields, which here take the form of vector-like fermions, are well known to be able to induce kinetic mixing (KM) between the SM and dark photons (DP) at low energies. This analysis is performed within a simplified version of a previously examined `$E_6-$like' setup that predicts a calculable value for the strength of KM in a phenomenologically interesting range. On the other hand, owing to the constraint on the invisible branching fraction of the Higgs, the tree-level mixing of the SM and a weak isosinglet dark Higgs fields yields a value of $\Delta\Gamma_{inv}$ which is far too small to be observable. A similar result holds for the shift in the $Z$'s invisible width induced by the mass and/or KM of the $Z$ with the light DP which then subsequently decays to dark matter. On the other hand, PM models wherein the light dark Higgs carries a significant admixture of a $T_3\neq 0$ weak isospin component are shown to already violate existing $\Delta\Gamma_{inv}$ constraints in the absence of parameter tuning. Future measurements of $\Delta \Gamma_{inv}$ can thus play an important role in providing indirect constraints on dark sector interactions and PM model building, in particular, on PM loops not necessarily associated with the generation of KM of the DP with the corresponding SM field.

For $T>0$, we consider the Benjamin–Ono-type Dirac equation $$ i\partial_t\psi(x,t) + D(-\Delta)^{\frac{1}{2}}\psi(x,t) = D\,G(\psi(x,t)), \qquad (x,t)\in\mathbb{R}^n\times[0,T], $$ where $(-\Delta)^{\frac{1}{2}}$ is the fractional Riesz operator, $D$ is the Dirac operator associated with the Clifford algebra of signature $(0,n)$, and $G(\cdot)$ a scalar-valued quadratic nonlinearity. Under a pointwise anti-commutation condition on $\psi$ and a uniform dyadic regularity assumption on its spatial Fourier profile, we construct a unit-modulus gauge map $u$ by $ u=\exp\bigl(-iw\bigr),$ where $w$ is a solution of $Dw=\psi$. Assuming, in addition, the stated regularity and commutation conditions on $w$, we show that the Plemelj components $u_{\pm}=P_{\pm}u$ solve a coupled Schrödinger system with gauge nonlinearities $P_{\pm}N(\psi,u)$, where $$ N(\psi(x,t),u(x,t)) = -2P_{-}\!\left[ G(\psi(x,t)) \right]u(x,t). $$ We also obtain a priori estimates for the homogeneous Sobolev seminorms $\dot H^\sigma$ of $\psi(\cdot,t)$, with at most exponential growth on $[0,T]$, together with estimates for $P_{\pm}N(\psi(\cdot,t),u(\cdot,t))$. Combining these bounds with the homogeneous and inhomogeneous Strichartz estimates for the Schrödinger system, we obtain, for $n\geq 3$ and every sharp Schrödinger admissible pair $(q,p)$, an a priori Strichartz-type estimate for $\psi$ in $L_T^q\dot W_x^{\sigma,p}$, expressed in terms of the initial datum $\psi(\cdot,0)$ and the a priori controls of $\psi$ in $L_T^\infty H_x^{\sigma+1}$ and $L_T^\infty L_x^\infty$.

Three spectacular LHCb measurements in 2026—the observations of $\Omega_{cc}^{+}$, $\Xi_{cc}^{+}$, and the beauty-strange structure $B_{s0}^{*}(5700)^0$—make possible an experimentally constrained analysis of chiral partners of strange doubly heavy baryons. The doubly charmed masses test the earlier Ma–Harada prediction and support a compact, internally S-wave $cc$ color source. We combine this comparison with the heavy-quark–diquark hyperfine relation and with chiral gaps from the theory of chiral doublers, to construct consistent $ccs$ and $bbs$ parity multiplets and a conditional extension to $bcs$. Under meson-to-baryon parity-gap transfer, the $349$ MeV input from the $D_s$ sector gives negative-parity $ccs$ targets near $4.075$ and $4.183$ GeV. The separate $332$ MeV input from the $B_s$ sector fixes the $bbs$ partner masses relative to their unknown ground-state mass, conditional on the new beauty-strange state's chiral-partner assignment. We relate the differences of hyperfine splittings in the two parity sectors of mesons and baryons, and identify equal reduced single-pion widths, an E1 reduced-width ratio $1:8:4:5$, and a kaon selection rule whose leading predictions contain no new transition parameters. Absolute widths are estimated by matching the published Bardeen–Eichten–Hill and Harada–Rho–Sasaki meson calculations to a common convention. In the reference calculation, all members of the predicted $ccs$ and $bbs$ multiplets are narrow. In the proposed $\Omega_{QQ}\pi^0$ and $\Omega_{QQ}\gamma$ searches at LHCb, these states are expected to produce peaks dominated by detector resolution, allowing their masses and decay patterns to be investigated even when their intrinsic widths remain unresolved. The upper charm state is above kaon threshold but protected in the spectator limit.

Real-world systems often serve many objectives but can deploy only a few options, whose performance must be learned from noisy feedback. Modeling each option as an arm in a multi-objective bandit, we introduce few-for-many best-arm identification, which seeks a small set of arms minimizing the worst weighted degradation from the objective-wise optima, rather than the best arm for every objective. We consider both known objective groups and latent groups specified only by a representative budget, and propose GFM-Elim and AFM-Elim with instance-dependent sample-complexity guarantees. For Gaussian rewards, we further derive a lower bound for the latent-group setting and show that it is asymptotically attainable with high probability by an oracle-based Track-and-Stop procedure. Our results show that sharing observations across objectives and ignoring irrelevant objective-wise distinctions can substantially reduce the sampling cost, as confirmed by experiments on synthetic and real-world tasks.

In-context learning (ICL) applies large language models to new tasks by prepending a few labeled demonstrations to a query. Prior work has studied how demonstration ordering affects ICL, but the interaction between label arrangement patterns and model biases has not been systematically examined. We define label collapse as the failure mode in which the model generates query-independent labels, and show that it is architecturally universal at the tested 8-9B scale: five models spanning autoregressive (Llama-3, Qwen3.5) and discrete diffusion (LLaDA, iLLaDA, LLaDA-Instruct) families all exhibit label collapse even under the class-balanced alternating arrangement, with some label word pairs collapsing in virtually every tested condition and others flipping their collapse direction with arrangement. Critically, accuracy alone cannot detect this collapse: models can achieve near-chance accuracy while generating query-independent labels, a score indistinguishable from balanced performance, so the failure can persist undetected in deployment. We measure separation, the log-odds of the two label words at the query position (the quantity that label-bias calibration operates on), as a complementary diagnostic that exposes collapse hidden beneath surface accuracy, together with a measurement protocol that guards label extraction against tokenizer and formatting artifacts. For autoregressive models, collapse persists where the majority-label-bias account does not apply, and arrangement patterns can reverse the collapse direction; for diffusion models, we present a systematic evaluation of ICL failure modes, complementing contemporaneous studies of dLLM context comprehension and demonstration-order sensitivity, and identify open questions about how iterative denoising interacts with label biases.

As language-model agents write code, move money and change records on behalf of users, deciding whether an action is authorized has become a central safety problem. Guardrails are often placed in front of these actions, yet few evaluations ask whether their decisions follow authorization: blocking unauthorized actions while still allowing the same actions under valid permission. We introduce \bench{}, a benchmark that contrasts authorization evidence for the same operation in its language track. It combines expert-annotated natural-language scenarios with executed and verified operations, plus externally sourced native-policy requests. \bench{} checks each decision for correctness, usability and resulting effect, and tracks each request across guardrail changes. Our experiments show that many blocked operations stem from failed reviews, not correct denials. Repairing the output format restores usable decisions but changes which authorized requests are honored, and adding a second reviewer removes some errors while rejecting valid authorization. Guardrails should therefore be evaluated on both the unauthorized actions they stop and the authorized actions they let through.

Syntactic language models (SLMs) that generate sentences along with their syntactic tree structures have shown promise in both syntactic generalization and downstream tasks. Prior studies, however, have mostly evaluated smaller models trained on less pretraining data and tested on narrower task suites. In addition, the design space of tree linearization and attention masks remains underexplored. We conduct a large-scale empirical study of 12 controlled configurations—one terminal baseline and 11 SLM variants spanning different linearizations and masks—at scales up to 1B parameters and 100B training tokens, evaluated across 11 downstream tasks and two syntactic benchmarks. The experimental results show that SLMs with standard causal attention match or exceed structural-mask variants across downstream and syntactic benchmarks. They require no architectural modifications and remain compatible with FlashAttention. We also observe that when scaling up to the 1B-parameter scale, the top-performing SLM variant no longer shows a clear advantage over a compute-matched non-structural baseline, suggesting that SLMs may become less competitive with scale. Our code is available at https://github.com/qlwpc/TG-Interpolation.

Neutrinos and muons produced in the forward direction of LHC collisions and detected by the FASER and SND@LHC experiments enable a comprehensive deep-inelastic scattering (DIS) programme, covering both neutral and charged currents, at TeV energies. This programme aims to provide stringent constraints on proton and nuclear structure as well as on hadron fragmentation, and to offer novel insight into neutrino interactions at the highest energies achieved in a laboratory. Realising this potential requires accurate and precise simulations of neutral- and charged-current DIS. We systematically compare the predictions of NLO general-purpose Monte Carlo (MC) event generators for neutrino- and muon-initiated DIS at the LHC (Sherpa, Herwig7, and POWHEG) against analytical structure-function calculations (YADISM) and dedicated neutrino simulation packages (GENIE, NuWro, GiBUU). Within this framework, we quantify the role of higher-order QCD corrections, (nuclear) parton distribution functions (PDFs), parton shower and hadronisation modelling, and QED radiation both for inclusive observables and for the FASER fiducial selections. We also present representative applications to the FASER neutrino programme: comparisons with its most recent data from the emulsion and electronic sub-detectors, a study of strange PDF sensitivity in the dimuon final state, and predictions for charged-hadron production in semi-inclusive DIS. Our analysis highlights the agreement of modern event generators for inclusive and differential DIS observables once common settings are adopted, while differences remain for observables sensitive to the hadronic final state.

Model providers can support image protection, but open-weight deployment introduces practical constraints. We propose four requirements: protection effectiveness, clean fidelity, compatibility, and tamper resistance. We incorporate long-horizon attack simulations to strengthen resistance to protection removal. Experiments with a single protection tool show improved resistance to re-alignment at a cost to clean fidelity, motivating further research toward practical model-side protection.

The total variation profile $\mathrm{TV}_n$ of a random walk measures how far its distribution at time $n$ is from being invariant under the generators. Three lower bounds are given: an elementary bound $\mathrm{TV}_n \geq 1/\big(44\,L(n)\big)$ in terms of the speed $L(n)$; a bound by the entropy increment $H(P^{n+1}) - H(P^n)$, which is a reverse Pinsker inequality; and a bound in terms of the entropy and the isoperimetric profile. Deciding whether the entropy-increment bound always improves on the isoperimetric one is related to conjectures on the growth of the entropy and of groups. The motivation comes from the question of whether the distributions of a random walk can be transported back onto their translates at bounded cost.

The purpose of this note is twofold. First, we revisit the classical regularity theory of harmonic maps in the setting of maps defined on a lattice $\mathbb{Z}^d$ instead of $\mathbb{R}^d$ or some other manifold. The target space of our harmonic maps will be spheres $\mathbb{S}^{N-1}, N\geq 2$. A harmonic map $\sigma : \mathbb{Z}^d \to \mathbb{S}^{N-1}$ is a map which in any finite subgraph $\Lambda$ minimizes the internal Dirichlet energy $\frac 1 2 \sum_{i \sim j \in \Lambda} \|\sigma_i -\sigma_j\|^2$ given the boundary condition $\sigma_{|\partial \Lambda}$. One key difference between our setting and the continuum is that our main Hölder regularity statement in a large domain $\Lambda\subset \mathbb{Z}^{d=2}$ does not require any regularity assumption on the map $\sigma$ restricted to the boundary of the domain. For example, when $d=2$ and $N=3$, the Hölder regularity induced in the bulk of the domain holds uniformly in all boundary configurations $\sigma \in (\mathbb{S}^{2})^{\partial \Lambda}$ (this is in contrast with the continuum which requires extensions in suitable Sobolev spaces). The fact we obtain results uniformly in the boundary conditions allows us to extend our results for positive temperatures. Namely, as $T \searrow 0$, we obtain a local regularity theory for all infinite volume Spin $O(N)$ Gibbs measures, at least when $d=2$ and $N\geq 3$. We also discuss implications of our analysis in dimensions $d\geq 3$.

We consider stochastic nonconvex optimization with sample-wise Lipschitz gradients and Hessians, and reusable opaque sample handles. Gradient and complete-Hessian evaluations are charged separately, including both endpoints of a same-sample difference. For fixed positive problem parameters and in the high-dimensional worst case, we prove that the minimax total derivative-query complexity of finding an $(\varepsilon,\gamma)$-second-order stationary point is \begin{equation*} \widetilde\Theta(\varepsilon^{-3}+\gamma^{-3}). \end{equation*} The upper bound combines curvature-first randomly signed negative-curvature steps with binary-ancestor derivative estimators; a truncated increment estimate charges predictable drift linearly and martingale motion quadratically. The curvature lower bound uses a deterministic smooth bounded-gradient chain, a radial barrier, and a random rotation. The new first-order lower bound is specifically robust to persistent sample reuse and to complete-Hessian access. It uses a slowly parameterized conservative sign chain, a third-order reverse stick-breaking localization inequality, and stateful relative-entropy accounting for reusable handles. Thus the total rate is near-optimally matched for every relation between $\varepsilon$ and $\gamma$.

The structural characteristics of carbon nanotubes are usually discussed in terms of processes occurring during nanotube nucleation and subsequent growth from an already formed catalyst nanoparticle. Here, we show that their structural trajectory can begin much earlier, within the first nanoseconds of precursor decomposition and catalyst formation. We combine experiment and reactive molecular dynamics to compare nitrogen-free CNT synthesis with two closely related aromatic nitrogen precursors, pyridine and pyrazine, containing one and two nitrogen atoms per molecule, respectively. Experimentally, both nitrogen-containing precursors promote bamboo-like CNT formation and produce substantially more structurally disordered CNTs than the nitrogen-free reference. Notably, nitrogen incorporation does not scale simply with the amount of nitrogen supplied, demonstrating a strong contribution from precursor chemistry. Reactive molecular dynamics reveals that nitrogen-containing precursors significantly modify local Fe-carbon and Fe-SiO2 interactions during catalyst evolution, while pyridine and pyrazine exhibit distinct decomposition pathways, with pyrazine showing significantly greater decomposition within the simulated timescale. These results demonstrate that precursor chemistry influences the local chemical environment and early catalyst evolution, highlighting its potential role in the initial stages of CNT growth.

We study the two-player communication complexity of the Coin problem. Here, two players, denoted Alice and Bob, are given up to $n$ individual i.i.d. random coins, and asked to output whether there are more heads or tails. An immediate approach is for Alice to send Bob her exact numbers of heads and tails, yielding a deterministic protocol with one-way communication cost $O(\log n)$ bits. Indeed, this is provably optimal in the related multi-player and streaming settings [Braverman, Garg, Woodruff, FOCS'20]. Our contribution is to show that this is not the right answer in the one-way two-player setting. Instead, we prove a $\Omega(\log\log n)$ bit lower bound for the constant error-rate regime, and a $\Omega(\log(1/\delta))$ bit lower bound for arbitrarily small error rates, $\delta \geq n^{-0.1}$. We also complement these with a tight one-way protocol using $O(\log\log n + \log(1/\delta))$ bits of communication. Unlike the previous works, our lower bounds avoid intricate direct information-theoretic arguments. Instead, our $\Omega(\log(1/\delta))$ bound is via a standard fooling-set style argument, whilst our $\Omega(\log\log n)$ bound is via a novel yet simple reduction from the Gap-Greater-Than problem. This reduction is our main technical contribution, and can be seen as loosely inspired by the Robust Communication Complexity arguments of [Chakrabarti, Cormode, McGregor, STOC'08]. Finally, we also show that our constant-error lower bound and protocol can be adapted to the multi-round setting: up to polynomial factors in $r$, the $r$-round communication complexity of the coin problem is $\Theta\left((\log\log n)^{1/r}\right)$ bits.

We study the coupled diffusion of baryon number, electric charge, and strangeness in hot and dense quark matter using a three-flavor Nambu-Jona-Lasinio model with self-consistently generated constituent quark masses. The complete diffusion matrix is evaluated within a relaxation-time kinetic approach, including its diagonal and off-diagonal components. We find that the off-diagonal transport coefficients exhibit distinct temperature and baryon-chemical-potential dependence associated with the flavor-dependent composition of the charge carriers. We further investigate generalized charge diffusivities and mobilities obtained from the corresponding susceptibilities. At finite baryon chemical potential, the baryon and strangeness mobility scales cross, reflecting the different density dependence of their transport coefficients and susceptibilities. The results provide model benchmarks for coupled conserved-charge transport in baryon-rich QCD matter.

When does a shared representation become insufficient for jointly supporting multiple tasks? This question lies at the heart of multi-task learning models. Although task interference in shared representations has been widely observed, its onset, measurable form, and capacity boundary remain poorly understood. In this paper, we revisit this problem through the lens of Bell's theorem. We show that if four binary task outcomes can be jointly represented by a common shared representation, their correlations must satisfy the classical CHSH bound $|S|\leq2$. We introduce NCnet, a controlled multi-task model for studying how representational capacity affects task compatibility. Experiments reveal a clear capacity-dependent transition. Once nontrivial task fitting emerges, constrained capacity can yield $|S|>2$. As capacity further increases, these exceedances progressively disappear and $S$ returns to the bounded region. This systematic pattern provides evidence that limited representational capacity is an important source of the observed incompatibility. We observe similar transitions across CNNs, PLMs, and LLMs, suggesting that the phenomenon extends beyond the controlled NCnet setting. We refer to this capacity-dependent transition as the Bell Boundary, providing a measurable diagnostic for representation bottlenecks, task incompatibility, and capacity allocation in multi-task models.

Bastidas et al. (2025) highlighted the importance of exploiting group symmetries to reduce computational overhead and unify simulation methods across both condensed matter physics and ab initio quantum chemistry. I discuss the discrete character transform, which is the group Fourier transform restricted to $Z(\mathbb{C}[G])$, the center of the group algebra, where class functions form a commutative convolution algebra and irreducible characters are its spectral basis. The normalized character table of a non-abelian group can be obtained through a restriction operator, which means an efficient quantum character transform for any finite group with an efficient quantum Fourier transform. I give complementary quantum algorithms for class convolution based on direct class-sum block encoding and spectral implementation in the character basis, explaining their applications to symmetry-adapted many-body dynamics, molecular and spectroscopic symmetry analysis, and quantum walks, as well as characterizing the central information used in hidden-subgroup problems, including its limitations for graph isomorphism.

Deepfake speech detectors are typically evaluated as binary classifiers on fixed benchmarks, but deployed systems face evolving attacks, codecs, speakers, and recording conditions, with labels unavailable during operation. This makes it difficult to know when a detector operates in familiar conditions and can be relied on. We propose an unsupervised drift monitor that operates on the internal embeddings of a deployed detector. The monitor compares each input against bona fide and spoof references using regularized Mahalanobis distances. These distances define trusted, ambiguous, and novel regions of the detector embedding space, allowing unfamiliar inputs to be routed for review. We evaluate the monitor on four modern detectors on three datasets. The monitor raises alerts for dataset mismatches and substantially reduces the security-critical case of accepting deepfakes at the cost of reduced automatic coverage. The results also show that drift affects both classes: bona fide recordings can move away from the training reference, increasing alert rates even in in-domain settings. Our findings suggest that deployed deepfake speech detectors should be treated not only as a binary classification, but also as a reliability problem.

A release decision about a fine-tuned model is an operational commitment about the next rerun of its training recipe. Yet standard benchmark reporting gives only the mean accuracy of past seeds. These target fundamentally different questions: release decisions depend on catastrophic tail risk, whereas standard evaluation reports only central tendency. Because fine-tuning induces bimodal mode collapse, high average accuracy frequently conceals severe failure rates. Auditing $2\,124$ fine-tuned models across eleven language models, we show that shipping models based on mean accuracy violates a $10\%$ deployment failure budget $48.0\%$ of the time. Common heuristics like $\bar b - 2\sigma$ prove equally dangerous: by underestimating variance on small samples, they become most permissive when evidence is weakest. We introduce recipe cards, an evaluation framework that directly certifies rerun tail failure rates using exact finite-sample binomial bounds. Recipe cards reduce held-out budget violations to $0.6\%$, enforce a mathematical sample floor ($N \ge 22$) that prevents certification on thin evidence, and explicitly refuse unverified claims under domain and task shifts.

Polyakov predicted in 1975 ([Pol75) that the spin $O(3)$-model (or classical Heisenberg model) should have exponential decay of correlations at all positive temperatures on $\mathbb{Z}^2$. Our main result is as follows: if we replace the Euclidean scalar product \begin{align*} \langle \sigma_i , \sigma_j\rangle = \sigma_i^x \sigma_j^x + \sigma_i^y \sigma_j^y + \sigma_i^z \sigma_j^z \end{align*} by the $\lambda$-modified scalar product \begin{align*} \langle \sigma_i , \sigma_j\rangle_\lambda := \sigma_i^x \sigma_j^x + \sigma_i^y \sigma_j^y + \lambda \sigma_i^z \sigma_j^z \end{align*} we show that for all $\lambda\in(-1,1)$, the $\mathbb{S}^2$ valued spin system with Gibbs measure $\exp(\beta \sum_{i\sim j} \langle \sigma_i,\sigma_j\rangle_\lambda)$ has power-law decay of correlations if $\beta$ is chosen large enough. This is the first evidence that arbitrary small smooth modifications of the classical Heisenberg model produce a BKT phase in $2d$. It was proved in the 1970s and 80s that there exist arbitrary small smooth modifications of the classical Heisenberg model which produce instead an Ising phase. In these work, they considered the regime $\lambda>1$ while in this paper, we consider the regime $\lambda<1$ which cannot be analyzed via reflection positivity and chessboard estimate. The two main ingredients used in our proof are the stability of the BKT phase under percolation type disorder ([DG25]) and weak regularity type properties of the groundstates or $\mathbb{S}^2$-valued harmonic maps associated to our $\lambda$-energy functional. This analytic step is interesting on its own. Finally, the exponential and power-law decay regime should be separated by a curve $\lambda \mapsto \beta_c(\lambda)$. We prove a rigorous lower bound in the case of the $\lambda$-anisotropic hemisphere model $\mathbb{S}^2_+$ which matches with RG flow predictions.

This paper introduces and systematically investigates several classes of numerical ranges associated with real-linear operators on Hilbert spaces, including the essential numerical range, maximal numerical range, essential maximal numerical range, $q$-numerical range, and trace numerical range. We establish fundamental structural properties of these sets and characterize their geometric shape. In particular, we analyze how the real-linear nature of the operators influences the geometry and spectral behaviour of the corresponding numerical ranges, thereby extending and enriching the theory of numerical ranges from complex-linear to real-linear operator settings.

Architecture attribute predictors reduce the cost of neural architecture search by estimating accuracy or latency from a small set of evaluated candidates. The key difficulty is that neural architectures are operation-labeled DAGs: edge direction defines computational flow, but existing predictors often treat direction as a local structural cue. We propose DTA-GT, a Direction- and Topology-Aware Graph Transformer based on encoder-wide directional consistency, which preserves directed computational semantics across node initialization, global spectral encoding, pairwise interaction, and feature update. DTA-GT realizes this principle with direction- and topology-aware node initialization, Magnetic Laplacian spectral encoding, direction-dual structural attention, and a Direction-Sensitive MoE-FFN. Across accuracy and latency prediction on NAS-Bench-101 and NAS-Bench-201, DTA-GT consistently outperforms representative sequence-, GNN-, Transformer-, and hybrid predictors under limited-budget settings. Ablations and Magnetic spectral controls show that the improvements come from coordinated direction-aware stages and edge-orientation-sensitive spectral information, while mechanistic diagnostics confirm that the modules exhibit the intended directional behavior. These results support encoder-wide directional consistency as an effective design principle for architecture DAG representation.

Let $G$ be a finite group and let $\pi(G)$ be the set of prime divisors of $|G|$. For each $p\in \pi(G)$, choose a Sylow $p$-subgroup $G_p$ of $G$. The subgroup $\Psi(G)= \prod_{p\in \pi(G)} \Phi(G_p)_G$ is called the local Frattini subgroup of $G$. Here, for $J \leqslant G$, the symbol $J_G$ denotes the core of $J$ in $G$. The subgroup $\Psi(G)$ is a characteristic subgroup of $G$ that is contained in $\Phi(G)$, the Frattini subgroup of $G$. We prove that $\Psi(G)$ satisfies many properties similar to $\Phi(G)$. Let $J$ be a normal $p$-subgroup of $G$ with $J \leqslant \Phi(G)$, and let $S$ be a Sylow $p$-subgroup of $G$. John S. Rose proved that if $J \leqslant \mathbf{Z}(S)$, then $J \leqslant \Phi(S)$. So, in fact, $J \leqslant \Phi(S)_G \leqslant \Psi(G)$. Kaplan, Lev, Mann, and Solomon found various other conditions that would ensure $J$ is contained in $\Phi(S)$. In this article, we add a few more conditions to this list.