Papers

This paper gives a readable and accessible version of the profile-construction part of OpenAI's manuscript "Finite Time Blowup for Navier-Stokes". We regard OpenAI's work as a major advance on the Navier-Stokes Millennium Prize Problem. The profiles are smooth and axi-symmetric. Inserting them into the Navier-Stokes equations gives a residual consisting of a divergence-form term and a remainder vanishing to infinite order in $1-t$ on fixed similarity sectors. The associated stress and radial shear satisfy the admissible cone condition. The stress need not be small. We introduce a new linear model that provides a clearer explanation of the inner core construction. The cancellation by oscillatory pulses will be treated in the companion paper "Finite-Time Blowup for Navier-Stokes with Smooth Forcing, Part II: Residual Correction via Oscillatory Pulses".

Pretrained 2D foundation models offer a practical alternative to dedicated 3D pretraining for brain structural magnetic resonance imaging (sMRI), but their use on volumetric data requires bridging the mismatch between a 2D encoder and a 3D volume input. Existing methods typically encode slices independently and integrate their features afterwards. We introduce Multiscale Volumetric Reduction (MVR), a reduce-then-encode approach that compresses each anatomical view from (D) slices into (M << D) complementary 2D components before foundation-model encoding. MVR combines an uncentered-PCA base component derived from the original through-plane intensities with residual detail components constructed from multiscale spatial descriptors. The reduction is estimated from the training volumes without diagnostic labels or gradient-based optimization and remains fixed thereafter. The resulting components are independently processed by a shared frozen 2D foundation model and concatenated for linear probing. Under this frozen-encoder setting, MVR achieves strong overall performance across ADNI, OASIS, and ABIDE relative to the evaluated 2D-to-3D adaptation methods and simple input-reduction baselines, while also generalizing strongly from ADNI to AIBL.

Let $A$ be an abelian variety of dimension $g$ over an algebraically closed field and let $\ell$ be a prime invertible in the field. For every constructible $\mathbb{F}_{\ell}$-sheaf $F$ on $A$ we prove that there is an integer $e$, depending on $F$, such that the pullback map $[\ell^{e}]^{*}\colon\mathrm{H}^{i}(A,[\ell^{n}]^{*}F)\to\mathrm{H}^{i}(A,[\ell^{n+e}]^{*}F)$ is zero for all $n\geq0$ and all $i>\dim\operatorname{Supp}F$. In particular $\varinjlim\limits_{n}\mathrm{H}^{i}(A,[\ell^{n}]^{*}F)=0$ for $i>\dim\operatorname{Supp}F$, which answers a question of Bhatt–Schnell–Scholze. Our methods also give sharp codimension estimates for the supports of the cohomology sheaves of the Fourier–Mellin transform of a perverse sheaf. With $\ell$-adic coefficients, we remove the arithmeticity hypothesis from the estimates of Esnault–Kerz. The corresponding estimates for the completed transform with $\mathbb{F}_{\ell}$-coefficients are new and hold even when Hard Lefschetz fails.

Connecting mathematical formalism in open quantum systems to its underlying physics necessitates the notion of a dilation, a way to bridge stochastic dynamics with deterministic Schrödinger evolution on a larger space. Although dilations are well known in the literature for states, channels, and quantum combs, there is a substantial gap when it comes to the fully general setting of non-Markovian dynamics on a continuous interval. In an accompanying work, we introduce a continuous process tensor (cPT) framework wherein these processes are represented by vectors in multi-species bosonic Fock spaces of $L^2$ functions. This accommodates all multi-time statistics of a quantum system on an interval. Here, we present a corresponding dilation theorem and show that all cPTs can be approximated arbitrarily well by an explicit closed description with system and environment coupled by a bounded Hamiltonian. As well as its physical appeal and demonstrating completeness of the framework, we show that such regular representations actually underpin the formalism, defining a dense subspace of all cPTs that is easy to handle. This leads to a continuum-suitable version of Choi duality, and permits the rigorous treatment of continuum controls.

When pricing agents meet repeatedly on a platform, the platform decides who faces whom. We ask whether that choice moves the prices the agents learn, and whether a rise comes with learned punishment. In a pre-registered randomised experiment in the Bertrand duopoly of Calvano et al., each agent's price is set by a tabular Q-learning module, not by the small language model attached to it, and we randomise whether each agent keeps its partner, sees its rival's prices and can send messages. Keeping the same partner raises the level of profits, averaged over training, by 0.27 of the gap between competitive and monopoly profit (95% CI 0.20 to 0.35, all twenty paired runs positive), our registered primary result, and the resting price by 0.17 of the Nash-to-monopoly range (post hoc). A plain tabular learner reproduces the effect in all 25 further blocks, and there one permanent partner raises the level more than about three do (+0.23 against +0.05, exploratory). Where rival prices are hidden, the price-setting module cannot see a cut, so cannot punish it, yet the resting price rises as much and the rise lasts to the end of training, while with visible rivals it shrinks with longer training (post hoc). Where the rival is visible, a static best responder accounts for a third to a half of what a forced-deviation probe reads as punishment, on the starts where the rival can see the cut, and net of it the registered test of learned punishment is inconclusive. A test that looks only for punishment would thus miss the rise where the rival is hidden, while a check for profitable deviations flags most of those prices (post hoc). In an exploratory extension, untrained Qwen2.5 7B and 14B models under one prompt show the effect when the rival's price is left out of the prompt and inconsistently when it is shown, the 7B result replicating on fresh blocks, while two other model families show none.

The Qprop 3.0 package for the solution of the time-dependent Schrödinger equation for an electron in a laser field and atomic potential is extended to incorporate nondipole effects related to the laser wave propagation. The developed code can be used for the simulation of strong field atomic phenomena in the nondipole regime, such as above-threshold ionization, ionization stabilization in high-frequency laser fields, high-order harmonic generation, when the effects connected with the magnetic field are non-negligible or when the laser wavelength is comparable with the atomic size. Similar to the dipole Qprop the wave function is expanded in spherical harmonics, the time-propagation of the wave functions is performed using the Crank- Nicolson approach, and the photoelectron spectra are calculated employing the t-SURFF or i- SURFV methods. The validity and accuracy of the implementation are demonstrated through several illustrative examples.

Artificial intelligence is increasingly deployed in critical industrial domains, including healthcare, energy grids, subsurface exploration, where failures can have severe consequences for human safety, system stability, and economic outcomes. Yet AI is still evaluated primarily through benchmark accuracy, a model-centric metric that fails to capture the structural complexity and risks of real-world deployment. We propose a framework that views industrial AI reliability as a problem of structural alignment across four interacting worlds: physical, representational, machine, and human cognitive. These worlds are connected through two interfaces: digitalization, linking physical reality to computational representations, and goal encoding, translating human cognition to the machine objectives. Together, they define the space of admissible solutions. We characterize the solution space through four attributes: existence, non-uniqueness, robustness, and interpretability and show how mismatches arise at interfaces and propagate across worlds to produce reliability failures. Applications to healthcare, energy grids, and subsurface exploration illustrate that although dominant failure modes differ across domains, for example, interpretability in healthcare, robustness in energy grids, and non-uniqueness in subsurface exploration, all originate from a shared structural mechanism. By shifting the focus from model-centric evaluation to system-level alignment, this framework offers a principled foundation for assessing and governing reliability in industrial AI systems.

We propose candidate lower bounds on the real Grothendieck constants $K_{\mathrm G}(d)$ and their infinite-dimensional limit $K_{\mathrm G}$ using rotationally invariant cubic kernels. For the continuous construction, we identify an explicit threshold above which hemispherical binary strategies are unstable under cubic boundary deformations. Supported by numerical evidence, we conjecture that hemispherical strategies are globally optimal at this threshold. This conjecture would imply $K_{\mathrm G}\geq9\pi/16\simeq1.767146$, improving the rigorous lower bound $6\pi/11\simeq1.713596$. We also construct finite symmetric coefficient matrices from unit vectors in dimensions up to $24$. Large-scale see-saw optimization supports the continuous predictions and suggests improved lower bounds on $K_{\mathrm G}(d)$ in the studied dimensions from $4$ to $24$, conditional on the conjectured binary optima being exact. Selected matrices also yield finite-setting candidate dimension witnesses when interpreted as bipartite correlation Bell functionals. If the corresponding binary upper bounds hold, these witnesses would certify a local Hilbert-space dimension of at least five for each party.

We consider the problem of minimizing the Ginzburg-Landau functional among $\mathbb{R}^n$-valued maps from the unit ball $B \subset \mathbb{R}^n$ with vortex boundary condition $u(x) = x$ on $\partial B$. We show that, for every Ginzburg-Landau parameter $\varepsilon > 0$, there exists a unique minimizer given exactly by the rotationally symmetric vortex solution in dimension $n\geq 2$.

Real-time music source separation must satisfy two constraints: a bound on algorithmic latency and a bound on computational cost. Offline separators are usually omitted from real-time comparisons or credited with a latency equal to their full input length. We show that this latency is set by where the output is read, not by the length of the separator's input. An unmodified offline model can therefore run in a streaming setting, without retraining. At each step, the input slides by one STFT hop, and one output hop is read out. The resulting latency can be as low as one STFT hop (23 ms), and the computational cost does not increase as latency shrinks. We identify a theoretical model-dependent latency boundary below which separation quality should drop steeply, and confirm this experimentally across three architectures. At equal algorithmic latency, streamed off-the-shelf checkpoints for HT-Demucs and SCNet match the published results of dedicated real-time models in terms of separation quality. Streamed models remain far less computationally efficient: only HT-Demucs runs faster than real time on a GPU.

We extend recent work on analogues of Grün's Lemma from skew-braces to all semiassociative Mal'cev algebras which is a broad class of algebras encompassing all expansions of groups. The direct analogue of Grün's Lemma is known to fail for skew-braces; thus, the more general result is a characterization for perfect algebras of when the upper-central series stabilizes at the second-center. The proof is an application of the description of generating sets for commutators in Mal'cev algebras.

Power laws are often claimed for Bitcoin quantities from the slope of a log-log plot. We test one such claim with a formal protocol: whether the upper tail of the values of live unspent transaction outputs (UTXOs) follows a power law. We reconstruct the exact UTXO set at eight block heights from 2012 to 2025, validate it against an independent Bitcoin Core snapshot, and apply the Clauset-Shalizi-Newman protocol with three adaptations the data require: exact discrete support, several near-tied cutoffs retained and reselected in every bootstrap replicate, and resampling by transaction, because the outputs of one transaction are not independent. The protocol was fixed before inference and every later change is recorded. The pure power law is rejected at every height, with observed Kolmogorov-Smirnov distances five to forty times those the fitted law produces, and where the exponent is below 2 its extrapolation predicts, in expectation, several outputs each larger than the total coin supply. A lognormal, a stretched exponential and a power law with an exponential cutoff near a thousand bitcoin fit better in sample at every height, as families that contain the power law must, by margins that exceed those seen on simulated power-law tails at six of the eight heights, and they predict held-out transactions better at nearly every height. They rank differently from height to height, and the lognormal and the cutoff power law differ clearly at only two. The tail's shape and the share of supply it holds drifted slowly while the number of outputs grew seventy-fold. The rejection survives a null that reproduces the dependence measured within transactions, which halves the one exceptional effect size in the panel and changes no verdict. Bitcoin UTXO values are heavy-tailed but not power-law distributed.

Video Large Language Models (Video LLMs) have achieved remarkable progress in video understanding, but their inference efficiency is constrained by the large number of visual tokens produced by long videos. Recent video token compression methods increasingly introduce sophisticated strategies for token selection, pruning, and merging. This raises a fundamental question: how much of compression performance can be obtained by simply preserving the structure encoded in the visual representations? We investigate this question with SimpleCluster, a simple and training-free baseline that performs position-aware cross-frame clustering in the visual feature space and represents each cluster using the mean of its original visual features. Extensive experiments across four video understanding benchmarks and three representative Video LLMs show that SimpleCluster achieves competitive or superior performance over recent compression methods across a wide range of token retention ratios, with particularly strong robustness under extremely low retention rates (e.g., 1%). To understand this behavior, we analyze the feature space preserved by different compression methods in terms of local approximation fidelity and global coverage. The results show that stronger downstream performance is consistently associated with better preservation of the original visual feature distribution, especially its global coverage. These findings highlight feature-space preservation as an important consideration for video token compression under highly constrained token budgets. Our code is available at https://github.com/xiaozhang79/SimpleCluster.

Let $\mathcal{O}$ be a complete discrete valuation ring with a finite residue field. We introduce a simple framework of constructing smooth representations of $\mathrm{GL}_n(\mathcal{O})$ modulo the knowledge of nilpotent orbits, called Jordan induction, which enjoys several remarkable properties: It yields only irreducible representations, it yields all the even level irreducible representations, and up to conjugation it yields non-isomorphic representations. In the even level case, this gives an explicit realisation of Hill's analogue of Lusztig's Jordan decomposition, as well as a vast generalisation of Gérardin's construction for $\mathrm{GL}_n(\mathcal{O})$. As a simple application, we construct an explicit section to the orbit map. We also propose a conjecture linking Jordan induction and Lusztig induction, aiming at generalising the algebraisation theorem obtained in recent joint works with Stasinski.

Adam is the default optimizer for training modern deep neural networks, yet its adaptive behavior remains poorly understood due to the complex interaction between its first- and second-moment exponential moving averages (EMAs). We study Adam in the tied-$\beta$ regime, where the two EMA decay rates are equal, and show that its adaptive dynamics can be expressed through a transformed ratio with approximately scale-stable behavior. Empirically, this transformed ratio exhibits a stable, heavy-tailed distribution across tasks, model scales, and training stages, in contrast to the variability of raw moment magnitudes. This empirical stability has both practical and conceptual consequences. First, we derive a recurrence for the transformed ratio, yielding a reparameterization of Adam that replaces the second moment with a compressible state. Leveraging its stable distribution, we show that a fixed 4-bit codebook is sufficient in our experiments to store this state without auxiliary scaling, achieving performance competitive with full-precision Adam. Second, the transformed ratio view clarifies Adam's connection to sign-based methods: Adam reduces to sign-based momentum modulated by the transformed ratio, and replacing it with a constant recovers Signum as a limiting case. This perspective further provides a simple rule for transferring learning rates between the two methods. Together, these results suggest that tied-$\beta$ Adam admits a simple and approximately stable ratio structure underlying its adaptive behavior and demonstrate its utility for both analysis and efficient implementation.

We describe and use a macroscopic linear quadrupolar trap to explore the trapping and equilibrium properties of charged particles. These devices, when operated to trap atomic ions, are at the core of ion based quantum computer and optical atomic clock. To illustrate the characteristics of few particle trapping, and observe different force balance on a charged particle, a macroscopic linear trap, operating at room condition, is trapping micrometer-sized particles. Compared to atomic or molecular ions, their mass can not be neglected and requires an extra voltage for gravity force compensation. We show how this compensation can be demonstrated experimentally and that the choice of hollow-core glass spheres has the advantage of a good reproducibility of the experiments, that allows one to deduce with good precision the charge-to-mass ratio of the trapped particle.

Verbal feedback can identify errors and prescribe corrections, providing rich supervision for language-model post-training even when reliable programmatic verifiers are unavailable. Such feedback, often generated by a capable model, can be used to condition the teacher in on-policy distillation, which trains the student to match the teacher's predictions on student-generated rollouts. However, this approach can transfer teacher preferences that the feedback did not motivate, while leaving much of the feedback's guidance unused. We find that both problems come from the standard on-policy distillation objective, specifically the divergence it minimizes and the distribution it uses as its target. Our proposed method addresses both limitations. First, to isolate the information conveyed by the feedback from the teacher's inherent preferences, we treat verbal feedback as evidence for or against the hypothesis that a particular token comes next at a given prefix. We then adopt a probabilistic confirmation framework which uniquely determines an ordering over the vocabulary based on the teacher's predictions before and after it receives feedback. Using a confirmation score consistent with this ordering, we construct a target distribution within a trust region of the student. Second, to learn from guidance that student rollouts can leave unused, we derive a simple shared-rollout estimator of a symmetric divergence between the student and target distributions over rollouts, reusing student and feedback-conditioned teacher rollouts in both directions through importance weighting. Empirical evaluations show that our method outperforms the common on-policy distillation recipe and a recent contrastive variant on knowledge-based and agentic benchmarks.

We study the problem of extracting reliable and interpretable linear models from nonlinear dynamical systems with periodic behavior. Dynamic Mode Decomposition (DMD) is a widely used data-driven method for approximating dynamics by a linear system and is commonly interpreted through Koopman operator theory. However, because standard DMD assumes linear evolution in the chosen observables, it can produce misleading results on nonlinear systems. Recent extensions address this limitation by modeling the dynamics as a linear system driven by a nonlinear forcing term, but their relationship to Koopman theory and their reliability remain unclear. We observe that the Koopman theoretical framweork allows for explicit, finite-dimensional, forced linear representations of dynamical systems, independently of the chosen observable space. The main question is therefore how to choose a useful, computable, and interpretable model for the forcing term, since forced linear representations are generally nonunique. In the framework considered here, this modeling question is related to the choice of a defect space: a complementary space that closes the action of the Koopman operator on the chosen observables. Existing forced-DMD approaches can be related to different modeling choices for this defect space, where the structure of the learned linear operator depends on the choice of the defect space. As an example of defect-space modeling for algorithm design, we introduce Correlation-Basis enhanced DMD (CB-DMD), a data-driven method whose goal is to learn an effective autonomous linear operator for periodic nonlinear dynamics. The forcing term is used to absorb nonlinear components that cannot be represented by the learned linear operator. Experiments on three nonlinear periodic systems demonstrate that CB-DMD yields more reliable linear models than standard DMD and related variants in the tested examples.

We show that an almost-everywhere upper bound $b\geq 1$ for the Hausdorff dimension of visible parts of all compact planar sets implies the upper bound $d-2+b$ for visible parts of all compact sets $K$ in $\mathbb{R}^d$, $d\geq3$. Inserting the sharp planar bound $b=3/2$ of Orponen and Rutar \cite{OrponenRutar2026} gives $\dim_H \mathrm{Vis}_e(K)\leq d-1/2$ for almost every $e\in S^{d-1}$. This would answer their Question 1.9 sharply in all dimensions.

Reliable confidence estimation is essential for large language model deployment. However, answer-level calibration remains challenging because generation errors are often localized: a response may be fluent and high-probability overall while still failing at a critical number, entity, or factual claim. Existing estimators compress token probabilities, sequence likelihoods, entropy, or beam statistics into a global score, which can dilute such local risk signals. We propose TRACE, a single-pass, decoded-answer-preserving confidence estimator that treats decoding-time uncertainty as a trajectory through three steps: (i) recording token-level surprisal and predictive entropy during decoding, (ii) applying local risk operators to preserve uncertainty spikes, and (iii) converting localized trace risk into answer-level confidence. TRACE produces a label-free risk score, while TRACE+ calibrates trace-only features into probabilities using a held-out split, without extra generations or external verifiers. We evaluate four tasks against 19 calibration baselines, and TRACE+ reduces Brier from 0.149 to 0.137 and improves AUROC from 0.758 to 0.792 over the strongest likelihood baseline. Across seven LLMs, TRACE+ improves over the best non-TRACE baseline pool from 0.136 to 0.120 Brier and from 0.764 to 0.817 AUROC. Results show that localizing decoding-time risk provides a general approach to calibration.

We study the problem of covering binary strings of the same length, possibly with missing entries, by a fixed number of center strings, such that each input string is within a given relative distance of the closest center. The problem has applications in binary $k$-center clustering and bioinformatics, but our main motivation comes from computational social choice. In the setting of multi-issue approval voting, we consider an algorithmic question that precedes any election: how should representatives be designed so that as many voters as possible are willing to delegate? We study the problem in two dimensions, namely the number of centers and the agreement threshold, parameters that are application-specific, and provide a complete picture of its computational complexity when entries may be missing. For the special case without missing entries, we extend our hardness results to any number of centers and large values of threshold, leaving a narrow range of thresholds unresolved.

Collective attention is a finite shared resource, and social media posts compete for limited opportunities to be seen. On X, users can undo a repost and repost it again. By repeating this cycle, a user can put the same post back into followers' timelines any number of times without making new content. We call this procedure repeat reposting, and we read it as placing repeated demand on this shared resource (Attentional DoS). We formalize this idea and explore repeat reposting in a large-scale dataset of cascades with at least 1,000 reactions, originating from posts classified as Japanese on X, covering April 2025 to March 2026. Repeat reposts are rare, appearing in a small share of all (user, post) pairs. Even so, close to a million posts have at least one repeater, and most repeats come from a small group of habitual accounts. The central result is that subsequent audience growth is associated less with the number of repeats than with the estimated reach of the repeating accounts. Through the lens of seriality, how habitually the same groups of accounts repeat reposts across many posts, a distinct distributed form emerges: several serial amplifiers converge on the same post (Attentional DDoS). Its synchrony, how closely their actions are timed together, forms a continuum from bursts on the scale of minutes to a daily clock. A check of the content of 99.7% of amplified posts shows that most ADDoS repeat events are directed at Chinese-script content, which our vocabulary matching and sample inspection indicate is predominantly commercial spam. Repeat reposting thus lets a post re-enter the competition for visibility. The observed pattern is better characterized as repeated temporal coverage of an existing audience and the self-reinforcement of posts that are already growing, rather than as evidence that repeat reposting takes reach from other content.

Online Resource Allocation (ORA) is a fundamental framework for sequential decision-making problems under budget constraints. Classical ORA models typically assume that resources are monotonic, meaning that selecting actions can only decrease the available budget. In this work, we study a more general setting with replenishable budgets, in which actions may either consume or replenish resources over time. This extension is necessary to capture scenarios such as inventory systems or energy markets in which capacity can be actively recovered. We develop a dual-based algorithm that recovers best-of-both-worlds guarantees for standard ORA when the replenishment factor $\beta = 0$, and improves them when $\beta > 0$. In particular, our algorithm attains $\widetilde{\mathcal O}(\sqrt{T})$ regret in the stochastic setting and $\widetilde{\mathcal O}(\sqrt{T})$ $\alpha$-regret in the adversarial setting, where $\alpha$ depends on the per-round budget and on the replenishment factor of the void action. Moreover, the algorithm ensures strict satisfaction of the budget constraints.

We study the edge scaling limit of the logarithms of the squared singular values of multiplicative Brownian motion on the general linear group $\mathrm{GL}_N(\mathbb{C})$ with general initial conditions. Building on earlier work [arXiv:2605.06429], we prove that, for every initial condition in the enhanced state space, the limiting dynamics are determinantal and derive an explicit double-contour formula for their extended multitime correlation kernel. The dependence on the initial state, including information not determined by the particle coordinates, is encoded by the associated Laguerre–Pólya entire function. We further extend the construction of solutions to the corresponding infinite-dimensional system of singular stochastic differential equations to arbitrary initial configurations, including those with coinciding coordinates.

Let $\Gz$ be a cofinite Fuchsian group whose quotient $\Gz\backslash\HH$ has genus one and a single cusp, normalized to be at $\infty$ with width one. Let $\Xg$ be its smooth compactification, which is an elliptic curve over $\CC$. Let $x,y$ be canonical generators of the function field on $\Xg$, which have poles of order $2,3$ respectively at the cusp and which {satisfy} a generalized Weierstrass relation $y^2+(a_1x+a_3)y=x^3+a_2x^2+a_4x+a_6$. Let $f$ be the unique (up to scale) weight two cusp form for $\Gz$, normalized so that the associated holomorphic differential is $\omega=dx/(2y+a_1x+a_3)$. With all this, we prove that the generating function identity \[ \frac{\bigl(y(\tau)+y(z)+a_1x(z)+a_3\bigr)f(z)}{x(z)-x(\tau)}=\sum_{m\ge0}\Phi_m(\tau)q_z^m \] defines a family $\{\Phi_m\}_{m\ge0}$ of weakly holomorphic modular functions on $\Xg$ such that the set $\{\Phi_0\}\cup\{\Phi_m\}_{m\ge2}$ is a canonical basis of the space $M_{0,\Gz}^{!,\infty}$ of weakly holomorphic modular functions on $\Xg$ with poles supported only at the cusp. As an application, by comparing the generating function of the family $\{\Phi_m\}_{m\ge0}$ with the generating function of the Niebur–Poincaré series, we show that the generating function of the special values of the Kloosterman zeta function at $1$ is the holomorphic part of a certain weight two harmonic Maass form up to an additive constant.

Many Model Context Protocol (MCP) servers wrap web APIs built for human developers, and their error messages tell the reader to run a command, edit a configuration, open a web page or wait. Many agents that read them can only call the server's tools. In 150 widely used MCP servers, 949 of 3,001 error messages tell the caller what to do next, and half of these steps depend on something the server cannot see about the caller. On credential errors, 62 of 67 steps ask for a terminal command, a configuration change or a web page; on rate limits, 20 of 30 say to wait and retry without naming the call to repeat. We tested five OpenAI models that act only through the tools of Berkeley Function Calling Leaderboard tasks, and the agents did what the step said. On expired credentials, a terminal command in the step left 45% of tasks recovered, and the loss it caused grew from 18 points for GPT-5.5 to 69 for GPT-6 Astra. On a rate limit, GitHub's "Wait before retrying." left 6%. We tested two remedies. For MCP developers, naming a server tool in the step raised recovery on expired credentials to 84%, with the login tool in place of the command, and on a rate limit to 88%, with the call to repeat in place of the bare wait. For agent developers, deleting the step with a one-sentence prompt before the model reads it raised recovery on expired credentials to 82%.

Extreme compression can reorganize atomic valence states, in some regimes driving charge from spatially extended orbitals into more compact ones. We investigate this effect in amorphous Zr67Cu33 metallic glass by combining high-pressure X-ray absorption fine structure measurements up to 69 GPa with molecular dynamics simulations and density functional theory calculations. The Zr K-edge XANES spectra show a pronounced increase in the pre-edge intensity and a change in its pressure dependence, whereas the Cu response is markedly weaker. Molecular-dynamics simulations reproduce the measured EXAFS spectra and provide reliable input for the electronic-structure calculations. The calculated Mulliken populations reveal pressure-induced depletion of Zr s and p-states, accompanied by an increase of the d-state population, while the density of states indicates enhanced p-d hybridization under compression. These results provide experimental and computational evidence for pressure-induced s,p-d electronic redistribution in zirconium and link this effect to the previously reported anomalous structural evolution of the compressed amorphous alloy.

Electroencephalography (EEG) records mixtures of brain-source activity. Even with a known anatomical forward model, experiments that excite only part of the source-state space leave the dynamics unidentified, and repetition cannot resolve the ambiguity. We show that unknown local mechanism changes can supply the missing information. We consider linear dynamics among fixed anatomical sources with known source-state initialization patterns. Changing one source's update rule for one transition leaves a rank-one, source-specific signature in subsequent EEG: subtracting matched baseline responses isolates it, and the forward model identifies the source and calibrates its response history. Combining these histories with initialization responses recovers source interactions without baseline reachability and without first identifying the intervention coefficients. We establish sufficient recovery conditions, a direct estimator, and a noise-sensitivity bound conditional on correct source labels. Simulated EEG on anatomy derived from magnetic resonance imaging confirms the information gain: with baseline excitation confined to four of twelve source coordinates, eight unknown changes recover all dynamics in 32/32 systems, whereas baseline realization, baseline regression through an invertible forward model, and changes that leave the tested states unexposed all fail, and explicitly constructed alternative dynamics reproduce every baseline mean. Where baseline information suffices, direct reconstruction is also more reliable than a matched-information spectral estimator. Nonlocal changes and forward-model error limit accuracy even when source labels are correct.

In response to the growing demand for long sequences in agentic and reasoning use cases, many state-of-the-art LLMs combine multiple variants of attention to mitigate the quadratic complexity of traditional softmax attention. These hybrid attention LLMs aim to balance the strengths and limitations of full attention and alternatives based on recurrence. This work presents a first study of how hybrid attention impacts the multilinguality of LLMs. Beyond the impact on long sequences in poorly tokenized languages, our study is motivated by the possibility that the inductive biases of the recurrent state alter linguistic processing. Our interpretability analysis confirms this, showing that cross-lingual representations in hybrid models develop in patterns tied to the ordering of recurrent and full-attention layers. Across diverse models, we notably observe a pronounced spike in cross-lingual alignment around the first full-attention layer. These findings lead us to question the conventional ordering of attention layers. In distillation experiments on multilingual data, all alternative layer orderings outperform the standard throughout training, learning up to 2.5X faster. These stark, replicable results prompt our theory that multilingual models would benefit from starting with a full-attention layer rather than recurrent layers.

Diffusion models trained on a finite dataset first learn to generate novel, high-quality samples and only much later collapse onto their training set. We identify the mechanism behind this separation of timescales and the object that probes it. The training dynamics of the score function are governed—exactly, and at any width—by the Gram matrix of the Neural Tangent Kernel (NTK) evaluated on the noisy training data, so the timescales of generalization and of memorization must be encoded in its spectrum. We show that they are, and that the structure responsible has no analogue in standard kernel settings. The use of multiple noise realizations per sample ($m$ noised copies at a fixed noise level) in the score-matching loss is what restructures the Gram matrix spectrum into two distinct parts. The first, of large eigenvalues, carries the global features of the target distribution and is present already for $m=1$. The second, which the repeated noising creates, consists of the smallest eigenvalues and is supported on eigenvectors aligned with the sample-specific noise directions; it sets a memorization timescale parametrically larger in the training set size $n$. We establish this picture on two fronts. Analytically, we solve the spectrum in the lazy high-dimensional limit for both linear ($n \asymp d$) and polynomial ($n \asymp d^k$) sample complexities, and prove through a bias–variance decomposition that the first bulk minimizes the approximation error while the second drives the error associated with memorization. Empirically, we show the same two-bulk structure in Convolutional NTKs on CelebA and in finite-width U-Nets trained well beyond the lazy regime, and we make the link causal: truncating the Gram matrix at rank $r$ tunes the generalization–memorization transition, and an $L_2$ penalty targeting the second bulk suppresses memorization in feature-learning U-Nets.