Papers

Not every moment in a video is equally useful for recognizing an action. A few decisive moments often carry the evidence: the instant an object moves, the peak of a dive, or one object placed on another. Existing video datasets rarely provide large-scale, action-conditioned labels for which short segments carry that evidence. This work asks whether vision-language models can identify meaningful moments at scale and whether those labels improve action recognition. We introduce Meaningful Moments (MM), a corpus of about 4.58 million VLM-generated per-segment importance scores for nearly 500,000 class-labeled videos spanning Something-Something v2 (SSv2), Kinetics-400 (K400), and Diving-48 across 622 action classes. We freeze the action classifier and vary only temporal sampling. An alpha-parameterized comparison of importance-led selection against an importance-inverted control separates gains from the VLM signal from gains caused by variable-density sampling alone. Across all three datasets, VLM-derived importance is informative: at the hard-cut endpoint, keeping VLM-important segments outperforms keeping VLM-designated filler. Usefulness depends on class structure. On Diving-48, importance-led fast-forwarding preserves full-video recognition while the inverted control collapses by roughly 30 percentage points. On K400, hard-cut selection beats uniform sampling by 1.45 percentage points, rising to 2.05 under threshold selection. On SSv2, the labels carry signal but do not recover full-video performance. Direct scoring matches greedy removal within its noise floor at roughly 25 times fewer oracle calls, and a four-VLM cross-oracle study shows consistent downstream trends. We release the importance labels and evaluation protocol under CC-BY-4.0; SSv2 class strings remain under CC-BY-NC-4.0.

Retrieval-Augmented Generation (RAG) systems are increasingly used for biomedical Question Answering (QA). However, evaluating their effectiveness remains challenging due to the lack of reliable reference resources and the frequent misalignment between automatic evaluation metrics and human judgments. One practical question is how evaluation results should be used to guide development decisions, especially when they correlate weakly with human judgments and may even lead to different conclusions. We explore how to use a small amount of human preference signals to calibrate direct assessment evaluation metrics, and show that calibrated metrics consistently outperform baselines that either treat all metrics equally or rely on a single best metric across multiple datasets.

We present TractorJAX, an open-source GPU reimplementation of the catalog-forced mode of The Tractor. We use it to test whether regularization or external flux priors enable reliable fitting of reference catalogs that extend below the depth directly constrained by undersampled SPHEREx images. For full-depth regularized fits, TractorJAX fits $100\times100$-pixel cutouts at 11.9 ms each on a single NVIDIA H100 GPU. In simulated images of the Abell 2537 galaxy-cluster field, extending the fitted catalog from $m_z<21$ to the full DESI Legacy Imaging Surveys DR10 catalog increases the scatter in fractional flux residuals by 34%, even with regularization. We also test the methods using SPHEREx observations in nine fields, assessing the number of reliably measured sources and photometric-redshift accuracy. Requiring S/N $>5$ in more than half of the 102 channels, we find that the regularized full-depth fit yields 40% fewer sources than the truncated fit. For sources meeting this requirement in both fits, the normalized median absolute deviation of the photometric redshifts, $\sigma_{\rm NMAD}$, is 17% larger in the full-depth fit. External flux priors recover nearly as many sources as the $m_z<21$ fit, but increase $\sigma_{\rm NMAD}$ by 0.0015 for the common calibrated sample. Neither regularization nor the external flux priors tested here simultaneously preserve the sample size and photometric-redshift accuracy achieved by the truncated fit. Our results suggest that the optimal fitted-catalog depth cannot be determined from image-level performance alone, but should be selected through end-to-end tests tailored to the intended science measurement.

We present an empirical model that analytically relates the parameters of the extended Kubo-Toyabe (KT) function $G_z^{\rm EA}(t;\Delta,\nu_{\rm i},Q)$, which describes muon spin relaxation ($\mu$SR) induced by jump diffusion of ions using the Edwards-Anderson parameter $Q$, to those of the conventional KT function $G_z^{\rm KT}(t;\Delta_{\rm KT},\nu_{\rm KT})$ that serves as an approximation of $G_z^{\rm EA}(t)$. The proposed formulae successfully reproduce the anomalous temperature dependences of the linewidth $\Delta_{\rm KT}(T)$ and the fluctuation rate $\nu_{\rm KT}(T)$ which are artifacts observed upon analyzing $\mu$SR data on ion diffusion using $G_z^{\rm KT}(t)$. Applying this model to $\Delta_{\rm KT}$ and $\nu_{\rm KT}$ published in the literature of battery materials yields activation energy $E_{\rm i}$ and pre-exponential factor $\nu_{\rm i0}$ in the Arrhenius equation for ion diffusion, $\nu_{\rm i}(T)=\nu_{\rm i0} e^{-E_{\rm i}/kT}$, that are in excellent agreement with those deduced using other techniques, transforming peculiar artifacts into a powerful diagnostic tool.

(Context) LLM agents increasingly delegate subtasks while holding authority over sensitive data, creating confused-deputy risks: an injected delegate can exfiltrate data, amplify its authority, or act after revocation. Existing leakage benchmarks expose these failures but do not constrain the authority a delegated agent may exercise. (Methods) We present \system, a reference monitor that converts the initiating user's authorization into attenuable capabilities over purpose, labels, operations, sinks, and delegation depth; every data-moving action is mediated outside the LLM, and revocation cascades through the delegation graph. We formalize a trace-level safety invariant and evaluate it with an adaptive within-authority adversary and four baselines, model-action and multi-turn ecological studies, and conformance and scale checks over deterministic traces and 517{,}401 Enron emails. (Results) \system blocks ungranted-sink exfiltration, authority amplification, revocation replay, purpose gaming, and intra-tool label violations, and dominates static allowlists on the revocation and intra-tool cases (10/10). Two open models proposed attacker-goal actions on 42/45 and 44/45 injected traces, all blocked at execution; a 90-run multi-turn study cut leakage from 0.844 to 0.20 with full task completion, at microsecond per-decision overhead. Proposition~1 characterizes exactly where attenuation-based authorization suffices, a boundary we also reproduce empirically across four observable-sink workflows. (Conclusion & Significance) Purpose-bound runtime authorization gives delegated LLM workflows a model-independent execution boundary and a precise recipe for composing it with declassification and content monitoring on externally observable sinks.

Accurate prediction of the magnetic structure of interplanetary coronal mass ejections (ICMEs) remains a key challenge in space weather forecasting. Despite several efforts using global MHD modeling, it remains difficult to deduce a flux-rope solution that can accurately and reliably reproduce the magnetic structure of a CME. We present FROST (Flux Rope embedded in the Solar Magnetic Field), a new flux-rope model implementation within the European Heliospheric Forecast Information Asset (EUHFORIA), featuring an initially force-free toroidal flux rope embedded in the low-coronal magnetic field. FROST employs a novel embedding technique that self-consistently generates a draping magnetic field around the flux-rope, ensuring continuity of the normal magnetic field component at the CME boundary. The dynamics of the flux-rope in the low and middle corona is solved by a non-uniform advection constrained by the observed kinematics of the event. This results in a global, non-toroidal, stretched loop-like magnetic structure. At heliospheric distances, the evolution is modeled as an MHD process using EUHFORIA. We validate the FROST model implementation by simulating the CME event of 2013 April 11 and comparing the results with in situ magnetic field measurements at 1 au. The modeled magnetic field profiles of the flux-rope exhibit strong agreement with the observations, with correlation coefficients ranging from 0.64-0.93 for Bx, 0.92-0.95 for By, and 0.82-0.87 for Bz component of the magnetic field. These results demonstrate the capability of the proposed flux-rope model to significantly improve ICME magnetic field prediction, highlighting its strong potential for advancing space-weather forecasting capabilities.

For context-based question answering, using one large model for both evidence extraction and answer reasoning requires substantial memory and computation. Existing approaches trade accuracy for efficiency within a single model only. We propose a multi-model approach for efficient inference: our method, PAIR, employs two differently sized models for evidence extraction and answer reasoning, respectively. A smaller Reader model processes and retains the full context, while a larger Reasoner model sees only the question and generated sequence without directly attending to the context. These capabilities are fused at the logit level: at each token, both Reader and Reasoner produce a prediction, and their logits are combined into one, behaving as a single next-token-prediction model. Fine-tuning further boosts their performance. On HotpotQA, PAIR with a 32B Reasoner retains 96% of that same 32B model's full-context accuracy while using roughly one quarter of its KV memory. At 45.7K-token contexts, PAIR achieves up to 3.2$\times$ higher end-to-end throughput than the full-context 32B model on the same GPU. PAIR also improves the accuracy-memory and accuracy-throughput trade-offs over evaluated token-reduction baselines. Its accuracy-preserving behavior generalizes across Qwen2.5, OLMo-3, and Gemma-3 model families.

A result by F.-T. Tu dating from 2011 states that the orders in a local matrix algebra of dimension four that can be written as a finite intersection of maximal orders are those having an Eichler Gorenstein closure. In a more recent paper, we gave a similar characterization for orders with a Bass Gorenstein closure. More precisely, we prove that these orders are the ones that can be written as a ghost intersection, i.e., the set of points defined over the base field of an intersection of maximal orders that is defined over a field extension. Although we proved this result only for the local four-dimensional matrix algebra, the statement makes sense also for a local quaternion division algebra, since the latter turns into a matrix algebra when we extend the base field to an even degree extension. In this work, we prove an analog of our previous result in the latter setting.

Splines on Weyl groups model the equivariant cohomology of regular semisimple Hessenberg varieties, and the natural group action on them encodes the monodromy action. The resulting representations are of great interest in algebraic combinatorics for their connections to Hecke algebra characters, LLT polynomials, and chromatic symmetric functions. We study the analogous construction for dihedral groups, which in general have no underlying geometry. Using purely algebraic methods, we show that these spline modules are free, construct explicit bases, and compute their Hilbert series, recovering the familiar properties of the geometric setting. We also determine the associated dihedral group representations completely. Unlike what is conjectured for the symmetric group case, these need not be permutation representations.

For discrete outcomes, a valid conditional quantile can have a hit probability different from its nominal level. Errors in different covariate groups can also cancel when residual signs are pooled. We study sequential audits that address both problems by testing the two inequalities that define quantile-set membership. The audit records whether each response falls below, equals, or exceeds its forecast. We derive the information projection for this three-outcome experiment and use continuous betting mixtures within the cells of a covariate tree. Context-tree weighting combines the resulting audits over all prunings, with an explicit finite-path oracle bound. Under iid alternatives with uniformly positive sign probabilities, a countable mixture of refining trees attains the full conditional sign-information rate. A matching upper bound shows that no valid evidence sequence based on the same observations can attain a larger almost-sure exponential rate. We also bound the information lost through covariate coarsening, derive a recurrence for the local mixtures, and discuss quantile selection and confidence-sequence inversion. Simulations show how partition resolution affects detection, and a held-out count-data example illustrates the difference between quantile validity and exact-hit calibration.

Cluster lens models are constrained almost entirely by observed lensed image positions, leaving a large space of mass perturbations with arbitrary amplitudes such that they reproduce every image exactly. Such degeneracies are known as shape degeneracies (ShaDes). We show that these ShaDes shift the observed time delays, by the difference of the perturbing potential between the two images, without depending on the number of source/lens planes. Bounding their amplitude by preserving the positivity of the convergence, image multiplicity, and parity, in MACS J0138.0$-$2155, they can contribute at least $\simeq7$ days to the delay between the detected images of SN-Encore (i.e., $17.5\%$ of the measured value of $39.8$ days) and $\simeq112$ days (i.e., $2.8$–$3.7\%$ of predicted value of $3,000-4000$ days) to the reappearance intervals predicted for SN-Encore and SN-Requiem. These error bars transfer one-to-one to $H_0$, and remain invisible in comparisons of independent models.

DNA methylation is a critical epigenetic modification that regulates gene expression and genome stability, primarily catalyzed by the DNA methyltransferase 3 (DNMT3) family of enzymes. Distinguishing precise sequence-specific flanking preferences between DNMT3A and DNMT3B remains computationally challenging due to the complex and high-dimensional nature of biochemical sequence data. In this study, we propose a novel Quantum Machine Learning (QML) approach to classify DNA methylation preferences, leveraging a Variational Quantum Classifier (VQC). Using experimentally derived deep enzymology data, we constructed a dataset containing 4,096 unique NNNCGNNN motifs. We evaluated a four-qubit VQC using a custom NumPy exact-statevector simulator. The described two-layer circuit contains 24 trainable rotations, eight CNOT gates, and eight input-encoding rotations, totaling 40 gates before measurement. Across five initialization seeds, the revised VQC evaluation achieved a ROC-AUC of 0.661 pm 0.009, where variability denotes sample standard deviation. The held-out test partition is approximately balanced, containing 200 DNMT3A-labeled motifs and 210 DNMT3B-labeled motifs. An always-DNMT3B predictor achieves 51.2% accuracy and a balanced accuracy of 0.500 on this test partition. This proof-of-concept establishes a foundation for utilizing near-term quantum architectures in complex epigenetic classification tasks, paving the way for hybrid quantum-classical pipelines in bioinformatics.

In the canonical Casimir effect, the quantum vacuum is isotropic, and the minimization of vacuum energy drives only a linear attractive force between the plates. However, the introduction of an axion gradient breaks this rotational symmetry. It can be demonstrated that these asymmetric zero-point fluctuations exert a macroscopic mechanical twist on the plates proportional to $\vec{b}^2 \sin (2\vartheta)$ where $\vec{b}$ is the spatial axion gradient and $\vartheta$ is the angle between the axion gradient and the normal to the plates.

We reprove Borel's computation of the rank of the $K$-groups of rings of algebraic integers $\mathcal{O}_F$ for certain number fields, e.g., $F = \mathbb{Q}(\mu_{l^k})$ for $l$ an odd regular prime number. The idea is to recover a simple formula for $l$-adic étale cohomology groups of $\mathcal{O}_F$ in terms of Iwasawa theoretic objects without using Borel's result as an input.

Geo-vision classifies vehicle-part damage using a fixed representation combining appearance, multiscale geometry, orientation, and sampled digital topology. Connected components, holes, Euler characteristics, and spatial arrangement complement amplitude-sensitive pooling and a supervised tree classifier. We establish amplitude preservation, axial sign invariance, and conditional operator stability. Retrospective checkpoint evaluation on 55,303 crops from 4,916 source groups yields average precision 0.7577, exceeding 0.7458 for an ImageNet-pretrained frozen YOLO11n comparator on identical crops. The complete added structural block, comprising topology, occupancy, and component location, improves average precision by 0.00493 beyond appearance and geometry. Geo-vision uses no external encoder pretraining and has higher AP point estimates at every matched label budget. With approximately 25% fewer task labels, its AP point estimate exceeds full-data YOLO, although the adjusted interval includes zero. Complete serial CPU inference requires 26.98 ms per crop versus YOLO's 11.20 ms, with peak memory 185 versus 437 MiB. Independent external validation remains necessary.

Reliable fetal brain biometry on magnetic resonance imaging (MRI) is essential for diagnosing congenital brain anomalies, but artificial intelligence models often lose accuracy under domain shifts. We developed BioTTA, a backbone-agnostic, source-free unsupervised test-time adaptation framework that adapts pretrained models online to individual unlabeled target volumes. This retrospective multicohort study included 394 reconstructed 3D fetal brain MRI volumes at 20-39 gestational weeks, with 11 biometric measurements annotated by radiologists. BioTTA was developed on single-center turbo spin-echo MRI and evaluated on three datasets representing cross-sequence, cross-center, and cross-nation shifts. Biometry accuracy was assessed using mean absolute error (MAE) and the percentage of correct measurements within 4 mm. A web-based reporting platform mapped measurements onto normative growth trajectories and was evaluated against radiologists in four multicenter disease cohorts covering germinal matrix-intraventricular hemorrhage, ventriculomegaly, and cerebellar abnormality. Across all three target datasets, BioTTA outperformed auto-SVRTK, source-only models, and FetalSynthGen, while matching models finetuned with target labels. MAEs ranged from 1.10 to 1.46 mm with a convolutional backbone and from 1.46 to 2.25 mm with a transformer backbone. Diagnostic accuracy ranged from 82.1% to 95.0%, comparable to a senior radiologist. BioTTA supports accurate, generalizable fetal brain biometry and standardized prenatal brain growth assessment across heterogeneous MRI sources.

Chain graphs are exactly bipartite graphs without induced $2K_2$ (a graph with four vertices and two disjoint edges). A graph $G=(V,E)$ with a given independent set $S\subseteq V$ (a set of pairwise non-adjacent vertices) is said to be a chain partitioned probe graph if $G$ can be extended to a chain graph by adding edges between certain vertices in $S$. %there exists a set $E'\subseteq\binom{S}{2}$ such that the graph $G'=(V,E\cup E')$ is a chain graph. In this note we give two characterizations for chain partitioned probe graphs. The first one describes chain partitioned probe graphs by six forbidden subgraphs. The second one characterizes these graphs via a certain “enhanced graph”: $G$ is a chain partitioned probe graph if and only if the enhanced graph $G^*$ is a chain graph. This is analogous to a result on interval (respectively, chordal, threshold, trivially perfect) partitioned probe graphs, and gives an $O(m^2)$-time recognition algorithm for chain partitioned probe graphs.

Software-defined radios (SDRs) have become an inexpensive, portable platform for replay attacks, location spoofing and jamming. The HackRF One can operate stand-alone when fitted with a PortaPack interface board running the open-source Mayhem firmware. No peer-reviewed study has addressed the forensic analysis of this device family. This work establishes, at source-code level, where forensically relevant data reside on the microSD card of a device running Mayhem v2.0.1, in which format, when they are written and how they should be interpreted. The methodology is applied to an SD card image that the author acquired from their own device after four months of real use, and every inference is drawn from the image alone. The workflow covers timestamp rules that depend on the real-time clock (RTC) battery state, traces of on-device trimming and text editing, cluster-level recovery of deleted in-phase/quadrature (IQ) captures, and decoding of recorded signals cross-checked with an independent decoder. In the example image, the final part of a deleted IQ capture and a trimmed-away capture segment are recovered, and six captures are attributed to a single rolling-code on-off keying (OOK) transmitter sharing one 32-bit identifier. The study provides practitioners with an artifact catalogue, interpretation rules and acquisition recommendations.

In this paper the generic behavior of the Jordan structure of symplectic matrices under low-rank structure-preserving transformations is considered. So far, only the case of rank-one perturbations was covered in the literature, and this result is now extended to the case of arbitrary rank $k$. As a key step toward the main result, it is shown that all symplectic matrices that differ from the identity by rank $k$ can be represented as a product of $k$ symplectic reflectors unless they are similar to a diagonalizable matrix having only the eigenvalues $\pm 1$.

Let $\mathcal{C}(\Phi)$ denote a subclass of class $ \mathcal{C}$ of convex functions in the unit disc $\mathbb{D}=\{z:|z|<1, z\in \mathbb{C}\}$ defined by \begin{equation*} {Re}\left \{1+ \frac{ zf^{\prime \prime}(z)}{f^{\prime}(z)}\right \} \prec \Phi(z), \end{equation*} where $\Phi$ represents the Ma–Minda function in $\mathbb{D}$. For $f\in\mathcal{C}(\Phi)$, we find the sharp estimates of $||\gamma_{2}|-|\gamma_{1}||$, where $\gamma_{1}$ and $\gamma_{2}$ are the first two logarithmic coefficients of $f$. The bound is also obtained for the inverse logarithmic coefficients $\Gamma_{1}$ and $\Gamma_{2}$. We then show that, under suitable conditions, the bounds for $||\gamma_{2}|-|\gamma_{1}||$ and $||\Gamma_{2}|-|\Gamma_{1}||$ are identical, which gives the invariance result between these functionals. We also present bounds for the differences for various classes of univalent functions by choosing particular function $\Phi$. Thus our results unify several results studied in the literature.

A proper coloring of Euclidean space is a coloring that assigns different colors to points at distance one. Using the planar character-law rigidity theorem of OpenAI, we prove that every finite proper coloring of $\R^3$ can be replaced by a proper Lebesgue-measurable coloring with the same number of colors. As a corollary we obtain that the ordinary and measurable chromatic numbers of $\R^3$ are equal, and both lie between $7$ and $15$.

In this paper we provide an introduction to some of the efficient features of the Raspberry Pi Pico (RP2) family of microconroller boards (RP2040 and RP2350). More exactly, we show how to use PIO (Programmable Input/Output) and DMA (Direct Memory Access) to program the RP2 boards for efficient DAC (Digital to Analog) and ADC (Analog to Digital) conversions. Using OP-AMPS (Operational Amplifiers) we also show how to build a minimal electronic frontend interface for signal input/output. Finally, we integrate all these components into a product, the O-SCOPE, which is a practical and easy to use oscilloscope and wave function generator. The RP2 component of the O-SCOPE is built using Micropython, while the user interface is developed using vanilla Javascript and it runs on the web browser of any computer, interacting with the RP2 board via the Web Serial API. The O-SCOPE web interface is hosted on GitHub Pages such that it is directly accessible everywhere using a web browser, such that no local installation is required. Also, the RP2 software component is freely distributed on GitHub using the UF2 (USB Flashing Format) mass storage file, and it can be installed by simply dragging the file on the RP2 boards. These features make the O-SCOPE easily accessible, transforming it into a very low cost and useful tool for laboratories everywhere in the world.

Entanglement entropies of multipartite pure states satisfy linear inequalities; for stabilizer states, the backbone of quantum error correction, the achievable entropy vectors generate a convex cone. We show that, after a linear lift, the closure of this cone is exactly the self-dual slice of a classical object on one more element than there are parties: the closed cone generated by the rank functions of arrangements of subspaces over $\mathrm{GF}(d)$ for a prime local dimension $d$, and of subgroups of abelian groups of exponent dividing $d$ in general. This correspondence brings two features of rank inequalities into play. First, the purifier acts as an extra variable: already for five parties the stabilizer cone is strictly smaller than the quantum linear rank cone, with which earlier evidence suggested it might coincide; an explicit six-variable inequality witnesses the gap. Second, the characteristic of the field enters, so the cone depends on the local dimension. It does not for up to four parties. With enough parties, the cone of $\mathbb{Z}_d$ qudits is contained in that of $\mathbb{Z}_{d'}$ qudits if and only if $d$ divides $d'$, and already for six parties the logical zero state of Steane's code lies outside the stabilizer cone of every odd local dimension. For this last result we show that the seven-variable characteristic-dependent inequalities we checked all follow from Shannon inequalities on the self-dual slice, and instead transfer one of them through a family of subgroups derived from the Fano plane.

A reading departs from what a monitoring system expects for one of three reasons: the physical process changed, the instrument is failing, or the value was corrupted between the instrument and the software. The three call for different responses and look alike in a single sample. We state the three-cause problem, prove which pairs of causes one channel cannot separate, and propose a bank of online detectors - a Kalman tracker with CUSUM and EWMA charts on its innovation, a Hampel median with generalized-ESD confirmation, a robust noise statistic with ADWIN, exact-value checks and an optional two-channel complementary filter - combined by a deterministic, inspectable attribution rule. The detectors were merged into a public algorithm library. On 2,942 injected events of thirteen fault types the method detects 96.3% (95% interval 95.7-97.0) at 0.22 false alarms per hour and attributes 91.2% (90.6-91.8) of detected events correctly from one channel and 99.0% (98.6-99.4) with a second, rate channel; from one channel only 2 of 207 detected sensor drifts are attributed to the instrument, as the problem statement predicts. Injected into recorded industrial signals, the same catalogue is detected at 80.8% and attributed correctly at 84.5%, at 3.6 false alarms per hour. On the Skoltech Anomaly Benchmark the method detects 32 of 33 experiments with a median delay of 15 s, at 46 false alarms per hour against 11 for a rolling z-score.

We demonstrate the well-known fact that the diagonal of a unit regular pentagon has length the golden ratio. While similar to other proofs, we provide a novel figure that we believe is particularly well-suited to achieve full understanding at a glance.

We study the LR2 one-parameter lifetime distribution obtained by mixing an exponential component of rate $\theta/2$ and a gamma component with shape four and rate $\theta^2$. The exponential mixing probability is $\theta^2/(1+\theta^2)$ for $\theta>0$. We derive the density, survival function, moments, transforms, truncated moments, quantiles, order-statistic distributions, mean residual life, and stress–strength reliability. Identifiability follows from the density at the origin, and strict monotonicity of the mean yields a unique moment estimator. A complete density-turning-point criterion and analytic initial and tail hazard expansions clarify the possible shapes. The component rates cross at one half. The hazard decreases initially for every positive parameter and eventually increases in the gamma-dominated regime; it eventually decreases in the exponential-dominated regime. We develop likelihood estimation, observed information, constrained expectation–maximization, spacing estimation, and inference under regularity conditions, and state the likelihood for independent right censoring. A reproducible Monte Carlo study with eight settings and 4,000 generated samples compares likelihood and moment estimation and evaluates log-scale confidence intervals. The analysis distinguishes mathematical properties from empirical validation and explains the restrictions associated with a fixed working time unit.

We construct a fixed triangular family of noncommutative polynomials that spans the joint cyclic module of a real symmetric pair from any prescribed vector. In dimension at most $N$, when the second matrix has spectral diameter at most $N-1$, the family has $N(N+1)/2$ members, degree at most $N-1$, and a shared arithmetic circuit of size $O(N^2)$. No rank-dependent choice of coefficients is made. The proof uses a rational differential connection with a small self-adjoint residue: a primitive polynomial exterior vector for an invariant subspace must have an integer degree smaller than one. For connected trees, the probes and the pointed adjacency–degree reconstruction theorem give one explicit self-adjoint polynomial $p_N\in\mathbb{Z}\langle a,d\rangle$ whose trace separates all trees of order at most $N$. Its circuit has $O(N^2)$ gates, formal degree at most $3N-2$, and polynomial bit complexity. Only the original adjacency and degree matrices, the identity, and ordinary matrix arithmetic are used. A separate integer sum-of-squares construction gives a fixed positive semidefinite representative on which the all-one vector is cyclic; its degree is $O(N^2)$. A linear trace-degree lower bound shows that the first construction has the correct asymptotic degree scale.

For an odd prime p, we study the change in matrix order under lifting from Z/p^k Z to Z/p^(k+1) Z. The first variation of the order defect is an affine translate of a cyclic adjoint transfer on sl_2(F_p). We classify the full-order transfer for every q in GL_2(F_p): its rank is 3 for scalar elements, 1 for nonscalar semisimple elements, 1 for nonscalar Jordan elements when p = 3, and 0 for nonscalar Jordan elements when p > 3. For a Jordan element q = lambda(I + U), U^2 = 0 and U != 0, the vanishing first-order transfer for p > 3 admits an exact higher-order refinement. Writing d = (p - 1)/2, the full-order perturbation has depth d when tr(UX) != 0, depth p - 1 when tr(UX) = 0 and X is not in the span of U, and vanishes identically for X in the span of U. For p = 3, among the 27 lifts of a nonscalar order-preserving parent with determinant fixed to +1 or -1, exactly 9 preserve its order and 18 triple it; after the first increase, every further lift triples the order. We also give a scalar obstruction criterion for order preservation among unrestricted Jordan lifts.

Thompson sampling (TS) achieves logarithmic and asymptotically optimal regret in classical stochastic bandits, but its behavior in multi-agent learning remains less understood. We study TS in an unknown finite two-player zero-sum matrix game with Gaussian bandit feedback. Under centralized self-play, TS samples a single payoff matrix from the posterior and plays a Nash equilibrium of the sampled game. For every fixed zero-sum game with a unique Nash equilibrium, we show that the expected cumulative duality gap is $O(\sqrt T)$. The rate improves to $O(\log T)$ for pure equilibria. We then identify the exact coefficient of the $\sqrt T$ term, which depends only on the equilibrium strategies and the noise level. We also establish a minimax lower bound matching the square-root dependence on $T$ and an instance-dependent lower bound matching the logarithmic dependence on $T$ for a class of games with a unique pure equilibrium. Experiments support the predicted rates and coefficient.

Knowledge distillation transfers representational knowledge from a large teacher network to a compact student. Existing relational and contrastive distillation methods treat all samples uniformly, spending equal gradient on well-matched and poorly-matched regions of the representation space alike. We propose Representational Difference Distillation (RDD), which diagnoses where teacher and student representations structurally disagree and concentrates the distillation signal on those regions. RDD uses two asymmetric affinity matrices that capture opposing directions of misalignment: one identifies samples the student incorrectly considers similar, the other identifies teacher relationships the student has yet to learn. These affinities determine which sample pairs the distillation loss aligns and how strongly it corrects each misaligned sample. On CIFAR-100 with resnet32x4 to resnet8x4, RDD reaches 75.58% test accuracy in our runs, on par with CRD (75.45%) and above KD and RKD. We additionally explore curriculum-based and sampling-based variants, finding that weighting the loss is more effective than filtering the data.