Papers

Chain-of-Thought (CoT) faithfulness detectors are widely used to audit reasoning models, yet a detector is itself a predictor whose verdicts are treated as stable properties. We ask whether a detector is faithful to itself under distribution shift. We formalize meta-faithfulness as an invariance principle: a valid detector must return identical verdicts on traces that differ only by transformations preserving ground-truth faithfulness. We prove three results: (i) no detector using only intervention-response profiles can separate faithful from epiphenomenal mechanisms with identical signatures; (ii) any detector relying on shift-sensitive features violates invariance at a rate independent of its in-distribution accuracy; (iii) an asymptotic certified selective-risk guarantee enables confident abstention. We operationalize the principle in FaithShift, a stress-test protocol spanning ten shift axes, and propose SIFT, a hidden-state trajectory detector trained with cross-environment invariance objectives and certified abstention. Across 14,996 traces, four domains, and eight models, three findings emerge. First, transfer collapse is real: all existing detectors show gaps $\geq 0.15$ AUROC. Second, the dominant bottleneck is sampling stochasticity, not shift: over 80% of detector instability stems from random seed variation, falsifying our preregistered prediction that shift-attributable violations exceed 0.25. Third, SIFT cuts invariance violations by 64% over the best single-seed baseline, but a four-seed ensemble of any detector narrows the margin to 0.01 (indistinguishable at matched coverage, $p=0.21$), and SIFT needs a 51% abstention rate. Cross-model transfer degrades from within-family to cross-family to open-weight-to-API, partly closed by multi-model training. We offer a framework for auditing auditors: the real barrier is detector variance, not distribution shift.

Based on our previously proposed direct approach to central configurations in the full configuration space, we study the degeneracy of planar four-body kite central configurations with arbitrary positive masses both in kite symmetry restricted subspace and in full configuration space. We derive a factorization of the determinant of the Jacobian that substantially simplifies the subsequent analysis. The methods used to study the singularity of the Jacobian include rigorous interval arithmetic, the Krawczyk operator, a quantitative implicit function theorem, and computer-assisted proofs. Both convex and concave kite central configurations are considered. In the convex case, we prove the nondegeneracy in both the restricted subspace and the full configuration space. In the concave case, there is a unique smooth degeneracy curve in the restricted subspace; an additional degeneracy curve emerges in the full space setting, which is hidden in the subspace of kite configurations and makes it unavoidable to study the degeneracy in the full configuration space. We also establish several properties of these degeneracy curves. As an illustration to this complete characterization, we exam the well-known central configuration of equilateral triangle with equal masses at the vertices and a fourth mass at the center, which reveals additional features of degeneracy captured by our full-space framework.

KRATOS is a group-structured bibliometric diagnostic that compares the distribution of documents (participation) with the distribution of citation weight (recognition) over a fixed, finite universe of analytical groups. This note gives its complete mathematical specification and derives the properties that govern its interpretation. Beyond bounds and equality conditions for each component, we show that the recognition-alignment score depends on citation data only through ratios of group mean citation rates to the corpus mean; that the composition ceiling of the participation–recognition factor is an affine function of the total variation distance to the uniform reference; and that the factor itself satisfies Fréchet-type bounds in terms of this ceiling and a participation-weighted recognition score. We establish an exact logarithmic decomposition of the composite index, an ordering between the primary and a reciprocal-symmetric recognition score, invariance properties, and closed-form first and second moments of the recognition ratios under global and stratified permutation nulls. These results explain why composite orderings can be sensitive to small groups, heavy-tailed citation distributions and metadata reassignment. All results are verified numerically with an accompanying script. The specification concerns measurement structure only; it does not define a measure of epistemic change or justice.

Since its original publication in 2020, the SWAN-SF dataset has been cited in over 300 research publications. Here we describe a comprehensive update to the original multivariate time series (MVTS) dataset extracted from solar photospheric vector magnetograms in the Spaceweather HMI Active Region Patch (SHARP) series. As was performed in the original release, our dataset includes a cross-checked NOAA solar flare catalog that facilitates the comparative evaluation of methods designed for solar flare prediction. We discuss the methods used for data collection, cleaning, and pre-processing of the solar active region and flare data, which includes improvements to the data integration and sampling methodology. Additional parameters have been derived from the Solar Dynamics Observatory's Atmospheric Imaging Assembly to augment the original features derived from magnetogram data. Our dataset expands the period of observations to cover 7,730 MVTS data collections from active regions occurring between May 2010 and December 2025, each of which is composed of 182 flare-predictive parameters. We continue to integrate flare reports, with over 39,000 multi-source verified flare reports providing flaring history information about the source active regions. The primary tasks enabled by the disseminated dataset continue to include optimization of solar flare prediction and detailed investigation for elusive flare predictors or precursors, with both operational (research-to-operations), and basic research (operations-to-research) benefits.

Let $\Omega\subset\mathbb C^n$, $n\ge2$, be a bounded connected pseudoconvex domain whose boundary contains a smooth strongly pseudoconvex point. We prove that if the Bergman metric of $\Omega$ is extremal, then its scalar curvature is identically $-n$, and that constant scalar curvature forces the Bergman metric to be Kähler–Einstein. Consequently, extremality, constant scalar curvature, and the Kähler–Einstein condition are equivalent in this setting. A key analytic ingredient is a local unique-continuation theorem for the Bergman Laplacian at an ACH boundary, proved using Hörmander's Carleman estimate. We also obtain an extension for possibly unbounded pseudoconvex domains: if the Bergman metric is cscK or extremal near such a boundary point, then it is well-defined and Kähler–Einstein throughout the locus where $K_\Omega(z,z)>0$.

Is agentic memory robust against stealthy steganographic attacks? We carry out a large-scale red-teaming exercise to test whether agents can encode attacker-controlled strings in one session and recover them in another without triggering safety oversight. Following SHADE-Arena-style tasks, we embed malicious side tasks to encode secret strings using steganography within otherwise benign tasks and evaluate them using independent task-completion and safety oversight. We test 14,000 attack trials spanning 91 model-attack configurations, 13 models, and 7 steganographic schemes: Acrostic, Mesostic, Telestich, Word-choice, Capitalization, Position-based, and Homoglyph. 41.2% of trials successfully encode the target payload in the generated response, while 29.8% survive the memory-write step with the encoding intact. Overall, 25.4% of trials yield a recoverable payload at retrieval, and 20.1% achieve exact-match recovery, including synthetic API keys and cryptographic material never exposed to the receiver. Among the exact-match transmissions, 71.4% also pass primary task-completion scoring and are independently judged safe by the oversight model, corresponding to 14.4% of all trials in which a successful covert transmission would appear to be an ordinary, benign interaction under task-level evaluation. Our results demonstrate that agentic memory can function as a persistent cross-session covert channel. The results further show that the principal bottleneck occurs at memory persistence rather than retrieval: once a steganographic payload survives the memory-write stage, a substantial fraction remains recoverable. We therefore argue that memory integrity, information-flow control, and covert-channel detection should be explicit security requirements for agentic systems.

Molecular property prediction is central to computational drug discovery, where graph neural networks learn to weight neighboring atomic environments during message passing. Yet it remains unclear how variational quantum circuits alter learned attention behavior in molecular graphs. We introduce an edge-aware variational quantum attention mechanism for molecular graph learning, in which the receiving atom, neighboring atom, and connecting bond jointly determine the quantum attention state. Across five molecular property and bioactivity prediction tasks, QGAT achieves competitive performance relative to GATv2, with a consistent improvement on BBBP across all six evaluated circuit ansatzes. We further compare how the quantum and classical attention scores weight molecular structure beyond accuracy. In the Verubecestat BACE1 inhibitor series, QGAT achieves a higher Spearman correlation than GATv2 and assigns positive attributions to several structural changes consistent with reported structure-activity relationships (SARs). This case study shows that the two attention mechanisms can exhibit different prediction and attribution behavior across structurally related BACE1 analogues, while broader validation is required to determine how consistently these differences generalize across chemical series and targets. Circuit ablations further show that performance depends on the circuit design. Together, these results show that variational quantum attention can serve as a viable alternative molecular attention parameterization while inducing circuit- and chemistry-dependent behavior distinct from a matched classical scorer.

Programmable Data Plane (PDP) switches enable flexible packet processing but remain limited in scheduling capabilities, particularly for priority-based policies that require strict ordering of packets according to user-defined ranks. An earlier work, called Push-in First-out (PIFO), proposed a priority queue that enables ordering enqueued packets based on their priority. It provided an ideal abstraction for programmable packet scheduling; however, its requirement for line-rate packet sorting makes it impractical at high network speeds (100 Gbps and beyond). Approximate schedulers such as SPPIFO, AIFO, and RIFO reduce complexity but introduce priority inversions, thus degrading latency, fairness, and Flow completion times (FCTs) since they rely on First-in First-out (FIFO) queues or use Active Queue Management (AQM) as a substitute for scheduling. To address the scalability limitations of accurate schedulers while avoiding the correctness issues of approximate schedulers, this work proposes two packet queuing architectures, Parallel Processing for Priority Ordering (P3PO) and Parallel PIFO Queuing Architecture (PPQA). Both architectures decompose a large global priority queue into multiple smaller bounded-capacity priority queues that operate in parallel. P3PO uses a cascading set of priority queues with priority-based bounds to ensure that packets are inserted into the earliest queue capable of maintaining ordering. PPQA distributes packets across multiple independent priority queues based on occupancy, using a demultiplexer for balanced queuing and a multiplexer for globally selecting the highest-priority packet at dequeue. The architectures are implemented in the NetBench packet-level simulator and evaluated using empirical datacenter workloads (Web-search and Data-mining).

We investigate the damping rate of Alfvén waves by means of the linear analysis of the two-fluid ten-moments system. To treat effects of viscosity in a wide frequency range, we take account of the time evolution of the traceless part of the pressure tensor. In addition, to treat effects of viscosity in a wide range of the ionization degree, we consider interactions between hydrogen atoms and between protons and hydrogen atoms in the equation of the traceless part of the pressure tensor. We find approximate analytical solutions to the dispersion relation for some limits. Our analysis in the low-frequency limit shows that the viscous damping dominates over the frictional damping in the high-beta plasma, which is independent of the ionization degree. Therefore, the viscosity of hydrogen gas significantly affects cosmic-ray propagation in a partially ionized high-beta plasma.

Joint-embedding predictive architectures (JEPAs) for world modeling train an encoder so a predictor maps a current embedding and action to the next frame's embedding, always from a single rendered frame. This has a structural blind spot: a renderer without motion blur draws a scene from configuration alone, so a single-frame embedding carries no velocity information, for any encoder, including the official released LeWM weights. We confirm this on official checkpoints across four real benchmarks (PushT, Reacher, Cube, TwoRoom): every linear velocity probe sits at or below chance while position probes reach R^2 about 0.95. We introduce RateIdent, a three-stage diagnostic protocol, and TI-JEPA, a lightweight fix splitting the latent into a pose code and an explicit finite-difference motion code, predicted jointly. Across three physically grounded environments, TI-JEPA gives a significant, seed-robust gain on a stop-at-goal planning task over a matched-memory baseline, e.g. 55% lower final distance on Pendulum (p=3.2x10^-10) and 64% on CartPole (p=5.1x10^-15). We reproduce this at official ViT-Tiny plus AdaLN-transformer scale, then push the same recipe onto real dm_control Reacher photographs trained from scratch, where TI-JEPA's branch separation exceeds the memory-having baseline's by roughly 38x, the paper's largest margin. Against a same-footprint recurrent RSSM-style predictor, TI-JEPA matches or beats its rollout accuracy on two of three environments, stays separately probeable for pose and motion, and wins outright on the most coupled one. A checkable formal argument and six evaluated environments show single-frame targets are the wrong object to predict when velocity matters, and a small, interpretable structural change fixes it with no privileged supervision. Code, checkpoints, and the project page are linked below the title.

Phosphorus donors in silicon are one of the promising platforms for fault-tolerant quantum computing, owing to their long coherence times and compatibility with complementary metal-oxide-semiconductor (CMOS) technology. However, a key bottleneck for scaling this platform lies in the independent addressability of individual spin qubits. Here, we present a scalable quantum processor architecture that integrates scanning tunneling microscope (STM)-defined phosphorus donor-cluster arrays with on-chip micromagnets. By generating magnetic field gradients, the micromagnets enable a broader addressable frequency range compared to the addressing scheme based solely on variations in hyperfine (HF) interactions, which arise from the randomness of intra-cluster configurations. Through micromagnetic simulations to optimize micromagnet geometry, we demonstrate that these structures are compatible with customized array designs, facilitating the integration of over 120 atomic spin qubits and enabling frequency multiplexing with a single microwave antenna, which greatly reduces the overhead of fan-out addressing lines. This design is further validated via magnetic measurements, which generally align with our micromagnetic simulations. These results establish a critical foundation for the development of scalable silicon quantum computing with atomic qubits.

We prove a simultaneous Williamson's normal form for two symplectically commuting positive operators $A, B$ on a real symplectic Hilbert space $(\mathbf{H}, J)$. Under a natural commutativity condition, there exists a (generally unbounded) operator $\Phi$ that simultaneously diagonalizes $A$ and $B$ in the symplectic setting as $A = \Phi^t M_A \Phi$, $B = \Phi^t M_B \Phi$, and maps $J$ to a canonical symplectic operator. The result generalizes Williamson's normal form to infinite dimensions and yields a complete spectral invariant for the pair $(A,B)$.

Storage functions are widely used to relate a control system to a simpler model for analysis and controller design, and to quantify model mismatch when a controller is designed on one model but implemented on another. In applications, several models of the same physical process are available at different levels of refinement, while refinement relations, consisting of a storage function and an associated interface, are constructed only between neighboring levels. Thus, a direct refinement relation may be unavailable between the model used for controller design and the model used for implementation. To address this gap, we consider open control systems with internal ports for interconnection and external ports for input-output analysis. We define a category, $\mathsf{OCS}_{\mathrm{chain}}$, whose objects are open control systems and whose morphisms are finite chains of refinement relations. Using this category, we derive an output mismatch bound and a closed-loop performance result for controllers refined through a sequence of interfaces. We then define a functor that transfers refinement relations from open control systems to systems without internal ports. We also develop a network lifting result showing that, under standard interconnection compatibility conditions, subsystem storage functions can be combined to obtain refinement relations and performance bounds for composite networks. Two thermal network examples illustrate recursive controller refinement and the propagation of model mismatch along refinement chains. Simulations show that an HVAC system tracks a time-varying reference temperature with error below $0.2^\circ\mathrm{C}$ even when the controller is designed on the coarsest model and deployed on the finest model.

Photonic qubits play an important role in quantum information processing. However, to date, efficient high-speed single photon sources and photon number resolving (PNR) detectors are still difficult to make. Here, we use phase-randomized coherent states and threshold detectors, combined with decoy-state post-processing, to simulate single photon sources and PNR detectors and perform various quantum information processing tasks. We propose a simple linear regression method to predict Fock state statistics based on observable statistics from coherent-state probes, combined with random/structured sampling of probe settings. Compared with traditional decoy-state post-processing based on linear programming and Cartesian product grid sampling, our method enables drastically improved accuracy and much reduced probe resources. It works particularly well for unknown unitary circuits of interest with fixed-size Hilbert space dimensions. For instance, it can accurately simulate 4-photon interference in linear optics with just 32 input/output probe settings and 6-photon interference with 256 probe settings. Moreover, it also works for non-unitary (such as lossy, nonlinear, or arbitrary) quantum channels. This enables a wide variety of applications that can be implemented with easily attainable coherent light sources and threshold detectors, such as characterization of small-scale linear optical quantum computing circuits or quantum sensing/metrology devices.

Multi-modal large language models (MLLMs) achieve strong modality understanding by pairing a large language model (LLM) with an encoder for a target modality such as vision, video, or audio. However, improving an MLLM's capability for a given modality typically requires additional training on large modality-specific datasets, incurring substantial data collection and compute costs. Model merging offers an alternative, but it is often infeasible for data-scarce, large per-sample size, or domain-specific modalities (e.g., audio and video), where same-modality model variants are rarely available. In this work, we characterize an intriguing asymmetric phenomenon: merging a well-aligned, data-rich source-modality MLLM into a data-scarce target-modality MLLM substantially improves the target on its own benchmarks. Our theoretical and empirical analyses show that this gain stems from enhanced alignment between modality-specific and textual tokens, induced by the stronger donor modality. Specifically, we derive a mutual-information lower bound that is monotonic in alignment-related quantities and strongly correlated with downstream MLLM performance. Building on this principle, we propose Directional Cross-modal Alignment Transfer (DCAT), a novel framework that transfers textual alignment from a strong, well-aligned source (donor) modality to a weak target (recipient) modality, boosting target-modality performance without further fine-tuning. We further show that the alignment-enhancing objective admits a closed-form weight-space solution computed from only a small calibration set. DCAT outperforms existing model-merging methods, offering an efficient path toward cross-modal alignment transfer. Project page with code is available at \url{https://seohoiki3215.github.io/DCAT_project_page}

Let $d(n)$ be the divisor function and denote by $[t]$ the integral part of the real number $t$. We prove that \begin{align*} \sum_{n\leq x^{1/c}}d\left(\left[\frac{x}{n^c}\right]\right)=d_cx^{1/c}+O_{\varepsilon,c}(x^{\theta_c+\varepsilon}), \end{align*} where $d_c$ is a suitable constant, $\theta_c<2/(3c+2)$ for $0<c<2/11$. This result constitutes an improvement upon that of Y. Feng (2024).

Recent advances in artificial intelligence-radio access network (AI-RAN) are placing large language model (LLM)-driven agents within the open RAN (O-RAN) control hierarchy. A promising deployment for this agentic AI-RAN co-locates LLM inference with physical-layer (PHY) communication processing over the same accelerated compute pool for infrastructure sharing, data locality, and low control latency. However, this co-location induces bidirectional control-compute coupling, as the agent competes with PHY processing for compute, while its reasoned gNB control actions may alter future PHY workload and hence the compute available to its next inference. To this end, this article proposes the control-compute contract, a coordination protocol linking AI-RAN workload governance with O-RAN control execution. Specifically, we first expose direct compute contention on an over-the-air (OTA) O-RAN testbed, where continuous LLM inference increases the PHY decoding time approximately eightfold. We then formulate the contract as three protocol rules with operator-requirement protection, and map it onto O-RAN and AI-RAN functions as a five-stage workflow. Afterward, an OTA-calibrated case study of simulated uplink gNB control indicates that the contract keeps the proportion of slots violating the PHY time budget below $0.9\%$, against $20.7\%$ under ungoverned co-location, while achieving faster gNB control than a fixed compute reservation benchmark. Finally, we identify several open issues and outlooks for its practical deployment.

We construct a minisuperspace quantization of a dynamic, spherically symmetric thin-shell wormhole in the spirit of pilot-wave quantum cosmology, treating the throat radius $a(\tau)$ as the single degree of freedom of the Israel junction-condition constraint. For a pressureless shell of mass $M$ the Wheeler–DeWitt equation reduces exactly to Whittaker's equation, and to the modified Bessel equation as $M\to0$. When the exterior is asymptotically flat, boundedness of the wave function as $a\to\infty$ selects a unique state that can always be chosen real. Since any real solution has a vanishing Bohmian current, the throat is frozen for any linear equation of state and any operator ordering, independently of the choice between Euclidean and Lorentzian boundary conditions. This state is the exact $\epsilon\to0^+$ limit of the regularized Lorentzian path integral, as shown by resolvent analyticity and by an exact Picard–Lefschetz thimble calculation that agrees with the closed-form solution to $48$ digits. The near-throat self-adjointness ambiguity is resolved in closed form through the Whittaker connection formula. Replacing the exterior by Schwarzschild–de~Sitter removes the obstruction: the potential tends to $-\infty$ at large $a$, and the equation admits a genuinely dynamic, purely outgoing state. Propagated exactly through the classically forbidden barrier, it carries a conserved, nonzero Bohmian current everywhere, including the inner pocket. It describes a long-lived metastable wormhole that tunnels into unbounded expansion at a rate consistent with the WKB estimate. Airy connection formulas leave a residual prefactor constant $\kappa_\infty=0.814\pm0.004$ for sub-critical coupling, which we do not trace to its analytic origin, and the physical results do not depend on it.

We consider multiuser downlink beamforming with prescribed signal-to-interference-plus-noise ratio (SINR) requirements under imperfect channel state information and a tight transmit power budget. Robust worst-case designs often require tractable channel-error models and may not yield feasible beamformers that meet the power budget. To alleviate these issues, we propose conformal risk control for beamforming (CRCBeam), which combines the classical fixed-point minimum-power beamforming with conformal risk control to calibrate an SINR margin from calibration sample pairs of estimated and true channels. Under exchangeability, CRCBeam provides finite-sample control over served-outage risk, which is the probability of transmitting when at least one user fails to meet its SINR target. CRCBeam requires no parametric channel-error model and transmits only when the power budget is met. Numerical experiments using a pilot contamination channel error model corroborate that CRCBeam achieves the served-outage risk while meeting the power budget. In contrast, worst-case robust baselines, Gaussian-ball and conformal-ball beamforming, either do not meet the target risk or exceed the power budget.

Activation probes can predict safety-relevant properties of language models with high area under the receiver-operating-characteristic curve (AUROC), but deployed agent monitors make thresholded alarm decisions under tight false-alarm budgets. These are different estimands. We introduce an Operational Validity Contract that fixes a monitor's target, observability, identity, timing, intervention unit, comparator, calibration, and cost. We formalize risk at the semantic request or trajectory level: when one task contains repeated alarm opportunities, row-level AUROC and false positive rate do not identify semantic-unit any-alarm risk. A confidence-certified threshold also requires enough independent negative units, a tie-safe rule, and transport to deployment. Across Models Under Pressure, LASR refusal prediction, immutable AgentDojo, and a prospectively protocol-frozen ST-WebAgentBench replication, joint activation-observable monitors reach AUROC 0.957 and 0.935 on the first two benchmarks, yet their locked 5%/10% detection rates are only .642/.742 and .719/.782, respectively. On AgentDojo, the secondary mean-activation rollout monitor reaches AUROC 0.922 but detects none of 38 positive semantic cases at the locked 5% operating point; thresholds intended for 10% false alarms realize 18.7-20.0% on test. Because the test misses its prospectively frozen 40-positive support gate, we label it support-insufficient. On ST-WebAgentBench, activation reaches AUROC .874, but 23 independent calibration negatives cannot identify even a 10% controller; the locked policy abstains rather than reporting its mechanical zero FPR as a success. An exploratory counterexample also lowers full AUROC while improving realized 10% utility. The fail-closed compiler caps MUP and LASR at restricted predictive value and AgentDojo and ST-Web at representation accessibility; no setting reaches alarm-policy validity.

An $\mathcal H^2$ matrix represents well-separated interactions using nested low-rank bases while retaining nearby interactions explicitly. We consider the problem of constructing an explicit, strongly admissible $\mathcal H^2$ approximation using only products with the matrix and its adjoint, and develop two randomized recovery algorithms on an adaptive geometric hierarchy. Carried recovery begins with a single pair of dense Gaussian test matrices, without hierarchy-specific probing, and updates the resulting forward–adjoint sketches across levels. All operator samples can therefore be acquired in one batch. Resketched recovery applies the same recovery procedure using fresh Gaussian test matrices at each level. For this variant, we prove a global quasi-optimal bound on the expected Frobenius error relative to the best approximation with the prescribed hierarchy, rank bounds, and retained near-neighbor pattern. Carried recovery requires only $O(k+p)$ applications of the matrix and its adjoint, independent of tree depth, whereas resketched recovery requires $O((k+p)\log(N/k))$ on trees of logarithmic depth. Here $N$ is the matrix dimension, $k$ the maximum prescribed box rank, and $p$ the oversampling parameter. These bounds assume bounded near-neighbor counts and $O(k)$ local dimensions. Experiments on Laplace, Dirichlet-to-Neumann, and Helmholtz operators in two and three dimensions with up to $1.28$ million unknowns show that, with sufficient oversampling, carried recovery achieves nearly refinement-independent error and accuracy comparable to resketched recovery while using many fewer operator applications. Its compressed application cost scales linearly for a fixed rank.

We study the fine-grained complexity of evaluating Boolean bounded-variable first-order queries over sparse relational structures. For every fixed $k \ge 2$, every relational signature, and every fragment between $k$-variable primitive positive ($\mathrm{PP}^{k}$) and first-order ($\mathrm{FO}^{k}$) logic, we prove, assuming the Sparse MAX-$3$-SAT hypothesis, a dichotomy theorem for evaluation in $O(m^{k-\varepsilon})$ time, where $m$ is the number of tuples in the input structure. The only tractable cases fall into three families: (1) three-variable fragments, (2) two-variable fragments, and (3) fragments over unary signatures. On the hard side, for every fixed $k \ge 4$ and every $\varepsilon > 0$, there is a fixed sentence $\varphi_\varepsilon$ in $\mathrm{PP}^{k}$ over a single binary relation symbol, depending on $\varepsilon$ but not on the input structure, whose evaluation cannot be performed in $O(m^{k-\varepsilon})$ time. For the tractable cases, we show that the evaluation problem can be solved in $2^{O(|\varphi|)} \cdot m^{k-\varepsilon}$ time for some $\varepsilon > 0$. Moreover, every tractable fragment reduces, with a $2^{O(|\varphi|)}$ blowup in formula size under DAG representations, to one of the following query languages: (1) Tarski's calculus of relations, (2) a new Boolean modal logic for sparse model checking, and (3) one-variable counting logic.

Precise constraint satisfaction is a prerequisite to deploying learned models in many areas, motivating methods that repair raw neural predictions with a repair procedure. Current methods unroll multiple repair steps in training and softly penalize constraint violations that remain after the unroll. This is compute- and memory-intensive, lacks robustness when the repair fails to converge, and surrenders most of the constraint satisfaction work to the repair. Our central finding is that, contrary to common practice, a single detached projection step suffices in training. We accomplish this with a Projection-Adaptive Loss (PAL), which uses the constraint residual after this single step to adaptively weigh constraint penalties on the raw prediction. In experiments, PAL is the only method that retains virtually perfect feasibility on extremely nonlinear constraints, and matches or outperforms current methods on synthetic and engineering benchmarks. Because it only requires a single detached projection step, PAL trains 2.5x faster than the canonical repair-based method (DC3) on its own ACOPF benchmark. PAL can also be trained when constraints are expensive to evaluate (e.g., via neural surrogates), a setting where current unrolled methods are memory-intractable.

This work studies the Riemann $\zeta$ numerically with emphasis on its physical manifestations. Three areas are examined: (i) the spectral statistics of the non-trivial zeros, which we compare with the Gaussian Unitary Ensemble predictions, obtaining $\chi^2/\text{dof} = 1.2$; (ii) the role of $\zeta(\frac{3}{2})$ in determining the critical temperature for Bose-Einstein condensation, where we find $T_c = 0.13$ K for an ideal gas; and (iii) the application of $\zeta$-function regularisation to the Casimir effect, yielding the standard Casimir energy density $-\frac{\pi^2 \hbar c}{720 a^3}$, which for $a=1\,\mathrm{nm}$ gives $\sim -4.33\times10^{-4}\,\mathrm{J\,m^{-2}}$. The numerical results are consistent with the Montgomery-Odlyzko conjecture and with spectral approaches motivated by the Hilbert-Pólya conjecture, within the statistical limitations of the analysis.

In this article, we present a method of spectral serialization envisioned by G. Mahler and A. Schoenberg between the late 19th and early 20th centuries that enables the selection of timbres rooted in musical structure. Based on information geometry, we evaluate the Wasserstein distance between polyphonic timbre spectra, as demonstrated in our previous paper, in terms of the second moment of spectral chords comprising three or more notes, expressed in squared hertz. By embedding these quantities in a Banach space, we analyze specific passages from works by R. Strauss, Mahler, A. Berg, and A. Webern from the perspective of timbre, showing how the same chord exhibits different affinities and divergences depending on orchestration. Taking the wah wah mute as an example, we propose a model that treats instrumental timbre and speech within the same Banach space, and we attempt to quantify the "vowel triangle" in Hz. These efforts constitute a partial but integrated solution to the historical three problems of Schoenberg. Serialism, Klangfarbenmelodie, and Sprechgesang which neither Mahler nor Schoenberg in their lifetimes, nor the postwar generation of composers including Nono, Boulez, and Stockhausen, was able to resolve fully. We aim to reconsider musical thought as a whole from the perspective of timbre and to open new possibilities for interpretation, performance, and composition. All tools are publicly available on GitHub, so that anyone can reproduce and extend these analyses.

Large language models (LLMs) are increasingly deployed in safety-critical settings, yet their black-box nature makes it difficult to provide formal guaranties about their behavior. Existing verification approaches rely primarily on empirical probing and testing, leaving open the question of how to reason rigorously about general-purpose trans- former architectures. In this work, we establish a principled bridge between transformers and weighted automata, a classical model from formal language theory. This connection enables us to transfer verification tools from automata the- ory to the analysis of LLMs. Our contributions are twofold: First, we develop a formal correspondence between transformer architectures and weighted automata over reals, showing how distributional properties of LLMs can be captured within this framework. Second, we introduce an identity testing algorithm for weighted automata that provides a statis- tical method for distinguishing whether two stochastic models define the same distribution up to a tolerance threshold. This work provides the first formal bridge between modern neural se- quence models and classical automata theory, clarifying both the poten- tial and the computational challenges for rigorous LLM verification.

We introduce a sampling approach for energy- and score-based generative models that requires no gradient evaluations of the model. Replacing the drift term that would normally contain the score $\nabla_\mathbf{x} \log p_\theta(\bf{x})$ with a high-frequency dithered cosine of the model's value, $\sqrt{\alpha\omega}\,\cos(\omega t + k \log p_\theta(\bf{x}))$, produces, in the high-frequency averaging limit, Langevin Markov chain Monte Carlo for energy-based models and the reverse-time SDE of score-based diffusion. We prove that trajectories of the dithered Itô SDE converge to those of the target SDE, driven by the same Brownian motion, uniformly on compact time intervals in probability, by an averaging argument that extends bounded extremum seeking (ES) to Itô processes, with an explicit $O(\omega^{-1/2})$ mean-square rate under global bounds. The approach is not confined to smooth targets: it extends to $C^{1,1}$ energies with discontinuous curvature (without ellipticity requirement) and to Sobolev energies whose Hessians exist only off measure zero sets; for Lipschitz energies with gradient kinks the averaged limit remains well posed; the Krylov-Röckner integrability class is the boundary of provability. The approach provides a hard a priori bound on the per-step update rate and applies to explicitly time-varying targets on finite horizons. Gradient-free pixel-space sampling is not competitive with well-tuned backpropagation-based samplers at practical evaluation budgets; the regime where the approach offers an advantage is latent-space sampling when the model is a black box and the target drifts in time. We demonstrate latent-space tracking for time-varying images on CelebA-HQ ($256{\times}256$) from limited 1D projection measurements and latent-space EBM sampling on CIFAR-10.

For positive matrices $A,B$ and $0<t<1$, Bourin asked whether the crossed-Heinz expression \[ C_t(A,B)=A^tB^{1-t}+B^tA^{1-t} \] satisfies \[ \UIN{C_t(A,B)}\le \UIN{A+B} \] for every unitarily invariant norm. The most direct route to this inequality is to compare $C_t(A,B)$ with the ordinary Heinz expression \[ H_t(A,B)=A^tB^{1-t}+A^{1-t}B^t, \] because the classical Heinz inequality gives $\UIN{H_t(A,B)}\le\UIN{A+B}$. This route, however, encounters a genuine dimensional obstruction: the stronger comparison $\UIN{C_t(A,B)}\le\UIN{H_t(A,B)}$ is known to fail for positive definite $3\times3$ matrices, and hence, by block-diagonal embedding, in every dimension $n\ge3$. We first determine the exact threshold for this obstruction. For every pair of positive $2\times2$ matrices and every $0<t<1$, we prove the weak-majorization relation \[ s(C_t(A,B))\prec_w s(H_t(A,B)). \]

Agile flight in unknown, cluttered, and dynamic environments requires a planner that knows where and when danger will appear, not only that a trajectory is dangerous. One-stage learning-based planners trained with differentiable privileged costs are fast and expert-free, but the only signal reaching their encoder is a scalar trajectory cost with no spatial or temporal structure, so avoidance degrades into late reactive maneuvers. We present RiskFly, a one-stage planner that predicts risk in the same space in which it acts. A dual-stream observation pairs a short depth sequence with a frustum-aligned inverted spherical range-map sequence, whose angular cells match the end-state proposals one to one. An auxiliary head regresses a frustum-aligned spatio-temporal risk field, supervised by a privileged closest-point-of-approach (CPA) target. This self-predicted field is queried differentiably along the instantiated quintic trajectory at its own arrival times, and also enters the training objective, so representation supervision and planning gradients meet in a single space. Privileged signals are discarded at deployment, and the planner runs map-free from onboard depth and proprioception. Extensive simulation and zero-shot real-world flights on resource-constrained platforms show higher success rates at comparable end-to-end latency.

Engaging K-12 students in authentic scientific research remains a significant challenge, particularly at the intersection of environmental science and robotics. We introduce the Jar Jar ROV, a low-cost, open-source Remotely Operated Vehicle (ROV) platform designed for citizen science-based water quality monitoring by middle school students. This paper presents the design of the platform and the results of a large-scale deployment with over 100 students across a US state who built, programmed, and deployed the ROVs in local lakes. The educational framework yielded high student engagement in hands-on activities, with ROV construction earning a perfect average score from mentors. Scientifically, the program established a grassroots monitoring network, generating nearly eleven thousand validated measurements of temperature, pH, dissolved oxygen, and turbidity. However, our evaluation identified a critical "engagement gap," with student interest declining sharply during more complex tasks such as electronics assembly and data uploading. This paper contributes both a validated, scalable model for integrating robotics into environmental education and a data-driven roadmap for future improvements. These enhancements focus on lowering technical barriers and creating a more intuitive link between data collection and scientific discovery, addressing a key challenge in empowering the next generation of citizen scientists.