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arXiv2026

The Rate of the Delsarte Bound for Binary Codes and Spheres

We give the asymptotic rate of Delsarte's linear programming bound for binary codes and spherical codes. This extends previous work of the author which formulated and proved an analogue of the conjecture of Afkhami-Jeddi, Cohn, Hartman, de Laat, and Tajdini for binary codes, obtaining the rate of the Delsarte linear program at distance $1/2-1/π$. The lower bounds introduce ideas from potential theory into the study of Delsarte's bounds, whereby the measures underlying a system of orthogonal polynomials give an obstacle problem. We solve the resulting obstacle problem in the case of binary codes and spherical codes, obtaining a closed form for the lower bound in terms of elliptic integrals. For the upper bounds, we relate the $N$-qubit construction for binary codes used in previous work of the author to the constructions in OpenAI's Chapter 2 for spherical codes and observe that they both involve interlacing spectra. This leads to a variational problem involving measures constructed from these spectra under the Markov-Krein correspondence. This variational problem admits an exact solution and matches the lower bound.

math.CO↗
arXiv2026

AdaBool: Sparse Boolean Matrix Processing with Adaptive Data Representation

Many data processing tasks revolve around determining whether logical relationships exist between two sets of entities, a pattern that can be naturally modeled as Sparse Boolean Matrix Multiplication (SpBMM). Under Boolean semantics, matrix representations no longer need to store numerical values and instead focus on encoding the positions of non-zero elements, making it highly amenable to data compression for substantially lower memory access overhead. The key challenge, however, is that the most effective compression depends heavily on sparsity distribution: different methods perform best at different sparsity levels, while real-world matrices often contain sharply varying sparsity across regions. Consequently, forcing the entire matrix into any fixed, monolithic representation leaves considerable optimization opportunities untapped. We present AdaBool, a SpBMM framework designed to reconcile this tension through an adaptive data representation. AdaBool partitions matrices into fine-grained element blocks and represents each block with the encoding best matched to its local sparsity. On top of this adaptive encoding, AdaBool introduces a heterogeneous GPU execution model that groups blocks by encoding type and maps them to matching processing paths, enabling efficient execution and scheduling across GPU resources. We implement AdaBool end to end with low-level system optimizations, evaluate it on both production and public workloads, and show that it outperforms representative baselines by over an order of magnitude.

cs.DB↗
arXiv2026

Local Extrema and Infinite Derivatives of the Delange Function of Base 3

In the ternary Hamming cube $Q_n^3$, the smallest number $ξ_n(m)$ of edges leaving a set of $m$ vertices is $2(nm-S_3(m))$, where $S_3(m)=\sum_{j<m}s_3(j)$ and $s_3$ is the ternary digit sum. After rescaling, these numbers are the values at the points $m/3^n$ of a continuous function $T$ of Takagi type, the Delange function of base $3$. Both facts are known. In this paper, we study the local structure of $T$. The local minima of $T$ are exactly the ternary rationals, that is, the numbers $m/3^n$ with integers $m$ and $n\ge0$. The local maxima are exactly the points $(m+1/2)/3^n+k$, where $k$, $m$ and $n$ are integers with $n\ge0$, $0\le m<3^n$ and $s_3(m)=n$. We also characterize the points $x$ where $T$ has an infinite derivative from one or both sides, in terms of their ternary digits $ξ_1,ξ_2,\ldots$. At a ternary rational the derivative is $+\infty$ from the right and $-\infty$ from the left. At every other point, the derivative from the right is $+\infty$ exactly when the digit slopes $2\sum_{j\le n}(1-ξ_j)$ tend to $+\infty$. The derivative from the left is $+\infty$ exactly when a stronger condition holds, which also involves the lengths of the maximal strings of consecutive zeros among the digits $ξ_j$. For the value $-\infty$ the two sides are exchanged, and twos replace zeros. In particular, $T$ has an infinite derivative at every rational point that is not a ternary rational and whose period has different numbers of zeros and twos. For the finite cubes, we show that over every block of sizes $a3^k\le m\le(a+1)3^k$, the maximum of $ξ_n(m)$ exceeds the larger of its values at the two ends of the block by exactly $\max\{3^{k-|d|}-1,0\}$, where $d=n-k-s_3(a)$. Finally, the variation $V_n$ of $T$ over the partition of $[0,1]$ into the $3^n$ intervals $[m/3^n,(m+1)/3^n]$ satisfies $V_n/\sqrt n\to4/\sqrt{3π}$, and we compute the Fourier coefficients of $T$.

math.CO↗
arXiv2026

TRACER: A Multimodal Harness for Patient Care in Hospitals

Continuous video and clinical records provide complementary information for understanding how patient behavior evolves over a hospital stay. A central challenge is how to represent behavior across the entire stay while preserving access to fine-grained visual and clinical evidence. We introduce TRACER, a training-free multimodal harness that constructs minute-level action records from video and combines them with clinical records into hourly reports. The harness equips an LLM agent with tools to query this hierarchy, inspect relevant video, and retrieve clinical records. To evaluate TRACER, we construct two benchmarks. The first contains 397 annotated one-minute clips spanning 15 atomic actions; the second poses 144 clinician-informed multiple-choice questions, grounded in 264 hours of video and clinical records from two ICU patients. TRACER achieves up to 84.8% accuracy for action detection and 61.0% QA accuracy versus 49.3% for the strongest baseline using the same backbone. Across 280 patients spanning approximately 60,000 camera-hours, TRACER identifies significantly less purposeful activity in patients with a positive delirium assessment during their stay than in those without (p < 0.0001). Our results show that combining continuous video with clinical records can support understanding of patient behavior over time and across patient cohorts without training on protected health data.

cs.CV↗
arXiv2026

TRACER: Trajectory-Aware Tool Retrieval for LLM Agents

Modern agents select tools from registries, but exposing entire registries incurs high token costs and can impair selection as they grow. We present TRACER, a retrieval front-end that indexes observed tool-call sequences in a trie and ranks continuations using a learned blend of empirical and contextual priors. On a mixed retail-and-telecom $τ^2$ environment, TRACER improves GPT-5.4's tool-selection accuracy by 4.7 points while reducing input tokens per step by 80%, exposing just 5 of 128 tools. In autonomous full-turn execution, it matches GPT-5.4's full-registry task success with 61% fewer input tokens per step. Across six environments and four models, TRACER improves tool-selection accuracy over full-registry selection in 21 of 24 model-environment pairs, with average gains of 6 to 10 points; the three exceptions trail by at most 2.2. We analyze failures in both regimes to identify directions for future work.

cs.CL↗
arXiv2026

Poisson statistics for one-dimensional Anderson--Bernoulli Schrödinger operators

We prove Poisson local eigenvalue statistics for one-dimensional Schrödinger operators with nondegenerate Bernoulli potentials at any fixed nonzero coupling. After unfolding by the integrated density of states (IDS), the local eigenvalue process converges to a Poisson point process of unit intensity at every energy in the interior of the almost-sure spectrum. The theorem also holds for bounded nonconstant iid potentials, provided the IDS is continuous and strictly increasing near the reference energy. If the IDS also has a finite positive derivative at that energy, the usual linear normalization gives the same limit. The proof uses a phase coordinate with exact IDS drift to obtain a uniform exponential crossing law and joint Poisson counts in disjoint energy windows.

math-ph↗
arXiv2026

Trigger Value Selection in Clean-Label Malware Backdoors: An Effectiveness-Detectability Analysis

Clean-label backdoor poisoning is a threat to machine learning-based malware classifiers because poisoned samples can be inserted into training data without changing their labels. Previous explanation-guided attacks mainly focus on selecting important trigger features, but the values assigned to these features can also affect the attack behavior. In this paper, we study trigger value selection as an independent design factor in clean-label backdoor attacks against feature-based malware classifiers. We evaluate several value-selection strategies on EMBER 2018 and a Win64 PE subset of EMBER 2024 using LightGBM classifiers. We compare rare-value, SHAP-based, frequency-bounded, co-occurrence-aware, and quantile-based selectors under random and distribution-based malware-similar benign sampling. The results show that aggressive selectors such as MinPopulation and CountAbsSHAP can achieve high attack success rate, especially under distribution-based sampling, but they are also easier to detect by sanitization methods such as Isolation Forest. In contrast, selectors such as CorrCountAbsSHAP and Quantile95 can reduce poison recall while keeping useful attack effectiveness. These findings show that trigger value selection is not merely an implementation detail, but an important design choice that shapes the effectiveness and detectability profile of clean-label malware backdoor attacks.

cs.CR↗
arXiv2026

Auditing Escalation Reliability in Healthcare LLM Agents: A Perturbation-Controlled Benchmark of Workflow-State Failures

Trustworthy biomedical AI requires reliable action selection when evidence is missing, stale, conflicting, or late, not only accurate answers on static inputs. We instantiate an episode-level evaluation protocol as 400 synthetic healthcare workflow episodes: 40 hand-authored semantic seeds across four task classes, two surface parameterizations, and five controlled variants, scored on ANSWER/ASK CLARIFICATION/ABSTAIN/ESCALATE under a pre-declared 5:1 missed-versus-unnecessary escalation cost. Reference actions are single-author study-policy labels, not clinician-adjudicated ground truth. Across five open-weight configurations (1.7B--9.7B), clean decision accuracy reaches 0.97, yet uncertainty-prompted models miss 32--97% of required escalations. The stronger 9.7B configuration narrows but does not remove the gap, falling from 0.97 clean accuracy to 0.78 under conflict. Verbalized confidence is inconsistent (ECE 0.10--0.54; failure AUROC 0.48--0.77), and a development-tuned confidence-only policy yields no utility gain. A transparent workflow-aware controller improves 5:1 utility for every model and reduces missed escalations to 0.12--0.24, at specificity 0.83--0.93. However, a 30-concept lexical stress test reduces each rule detector's recall from 1.00 on trigger phrases to 0.00 on unseen paraphrases, limiting the controller claim to this benchmark. Seed-clustered statistics, ablations, cross-task transfer, raw outputs, and manifests are packaged for audit. The results motivate explicit workflow-state evaluation while showing that apparently effective routing controls need their own generalization tests.

cs.AI↗
arXiv2026

Beyond 11/16: dense Kuramoto networks that resist synchrony

How densely must a network of identical Kuramoto oscillators be wired before synchrony is its only stable state? The answer is a critical connectivity $μ_c$ (the minimum degree divided by $n-1$), known to satisfy $11/16\leμ_c<3/4$. We raise the lower bound from $11/16=0.6875$ to $μ_c\ge(4+2q^*)/(7+q^*)=0.6929077\ldots$, where $q^*\approx0.6506$ is a root of an explicit polynomial of degree 8. The networks approaching this bound carry stable states that hug the four-cluster state: three neighboring clusters each split into two slightly separated halves, and the four crossed links between neighboring halves (those whose phase gap deviates most from a quarter turn) have densities approaching $q^*$. We build these networks from a coarse-grained cluster model, verify stability with interval arithmetic, and blow the model up into large networks with an exact minimum degree. The states are fragile and easily missed by numerical searches that impose stability margins.

nlin.AO↗
arXiv2026

A refinement of the strange duality conjecture on K3 surfaces

Let $v,w$ be primitive Mukai vectors of positive rank on a polarized K3 surface with $χ(v\otimes w)=0$. Marian and Oprea conjectured that the strange duality map $H^0(M(v),Θ_w)^\vee\to H^0(M(w),Θ_v)$ is either zero or an isomorphism. We give three families of counterexamples to this conjecture. In each of them the map is nonzero but fails to be an isomorphism: in the first (a quartic surface) it is not surjective; in the second (Picard rank one, including very general polarized K3 surfaces) the source has strictly larger dimension than the target; and in the third (an elliptic surface) source and target both have dimension six, yet the map has rank five. The three examples share a common mechanism: one of the moduli spaces carries an effective divisor of square $-2$ on which a theta line bundle has negative degree, so that this theta line bundle is not nef and has nonvanishing higher cohomology, and the theta sections produced by the strange duality map vanish to excess order along this divisor. We therefore propose a refined conjecture, in which both theta line bundles are assumed to be big and nef, for instance ample.

math.AG↗
arXiv2026

Regret Lower Bound for Online Learning of Quantum States with Logarithmic Loss, with Applications to Quantum State Tomography and Confidence Sequences for Matrix Means

We study the problem of online learning of quantum states with logarithmic loss, which is a quantum generalization of online portfolio selection and has applications to quantum state tomography, the task of estimating an unknown quantum state from measurement outcomes. We first establish a regret lower bound, showing that the regret rates of Exponential Weights and the algorithm of Tseng et al. (2025) have optimal polynomial dependence on the dimension. Our analysis leads to two further results. First, our lower-bound proof leads to a quantum state tomography algorithm that is sample optimal up to polylogarithmic factors in the sample size. Second, we develop an algorithm for constructing confidence sequences for the mean of a matrix-valued random variable. A numerical experiment shows that the latter algorithm is competitive with existing methods.

cs.LG↗
arXiv2026

TimeCap: A Scalable Open-Source Pipeline for Captioning Wearable Health Time-Series Data

Time-series data generated continuously across industrial, environmental, and especially health domains produce multimodal signals whose scale and heterogeneity make their processing and interpretation particularly challenging. Time-Series Language Models (TSLMs) promise scalable interpretation of these signals, yet remain bottlenecked by a fundamental lack of text-annotated time series pairs that are signal-grounded, reproducible, and extensible across modalities and schemas. Existing paired corpora are either closed or limited in scale and modality coverage. Annotation with large language models could scale them up, but its outputs cannot be regenerated exactly, which prevents others from re-deriving, extending, or auditing the resulting corpus. To address this, we release TimeCap, an open deterministic captioning pipeline with three reusable extractor tiers and a cross-channel synthesis layer that operates directly on the signal with no language model in the loop, extensible to new schemas by writing channel definitions alone. This positions TimeCap as a reproducible mechanism for future dataset generation rather than a one-off release. Applying TimeCap to OpenMHC, the MyHeart Counts wearable dataset, yields MHCap, the largest open caption layer for multimodal wearable time-series data. MHCap spans 19 Apple HealthKit channels and 7,005,763 captions over 908,944 person-days. To showcase the impact of the corpus, we train OpenTSLM, a time-series language model, on MHCap and benchmark on health-outcome prediction. Caption-augmented training (OpenTSLM-Cap) outperforms a label-only baseline on 29 of 32 per-user binary evaluations across 8 tasks.

cs.LG↗
arXiv2026

Broadband radio-frequency TEM specimen holders for in situ biasing excitation and 100-ps-scale electron beam control

Broadband electrical excitation inside a transmission electron microscope (TEM) is required to investigate electrically driven nanodevices at their actual operating frequencies and to develop fully electrical time-resolved experiments at a sub-nanosecond range. Here, we report the performances of homemade two- and four-channel radio-frequency (RF) TEM specimen holders based on a hybrid architecture combining coaxial transmission lines with miniaturized impedance-controlled printed circuit boards (PCB). Removable mechanical contacts provide direct electrical connection to interchangeable TEM chips without wire bonding. Vector network analyzer measurements demonstrate broadband transmission through the complete electrical path up to the specimen. With a representative TEM chip, the two-channel holder exhibits a -3 dB bandwidth of 7.9 GHz, while insertion losses remain below 12 dB up to 18 GHz. The same RF holder architecture was used to drive a miniaturized electrostatic electron-beam deflector. The measured temporal response indicates that the holder deflector assembly remains compatible with electrical pulse durations approaching 100 ps, with the shortest accessible times presently limited by the available waveform generator. High-resolution TEM imaging with a 200 ps probe window does not show detectable degradation of spatial resolution, while off-axis electron holography is successfully performed for probe durations down to 0.5 ns under conditions representative of operando measurements. These results establish a versatile broadband platform combining GHz electrical excitation and sub-nanosecond electron-beam control. The dual-stage configuration of our Hitachi HF3300 microscope further provides the basis for a future fully electrical pump-probe TEM approach using independently driven RF holders for specimen excitation and electron-probe control.

cond-mat.mtrl-sci↗
arXiv2026

When Rotation Bounds Nothing: Streaming Image Classifiers to Field Nodes Under Physical Capture

Field nodes run image classifiers locally, exposing a first-class asset, the trained model, to theft. Trusted-execution enclaves, collaborative-inference splits, and encrypted storage assume the device owner is the adversary from deployment. Here the operator owns the node, and an adversary seizing it reads only what cold-boot capture finds resident. One answer stores nothing: a chunkable multi-exit model streams into volatile memory, rotating to bound residency. We characterise the capture envelope on real LTE traces across two victims of one architecture, against passive and adaptive adversaries. Neither survives a capable adversary; recovery cost alone differs. Against a from-scratch victim the uncaptured stages must be learned: the adaptive adversary gains most at shallow seizures, ends the CIFAR-10 envelope at every rotation, and stays rotation-bounded on CIFAR100, from 42.6% of usable seizures at 5 s to 86.8% at 30 s. Against the same architecture sliced from a public pretrained backbone, the adversary downloads those stages, recovering at least 90% of maximum accuracy at all depths, taking usable CIFAR-100 seizures from 0.0% to 97.3% at 5 s, given the victim's full labelled training set. Delivery-side secrecy raises a seizure's price; a public copy of the withheld stages erases it.

cs.CR↗
arXiv2026

Optical observations of supernova candidates with the ILMT

The International Liquid Mirror Telescope (ILMT) is a 4-m optical ({\em gri} spectral bands) zenithal survey telescope with a field of view of 22.3$\times$22.3 arcmin, suitable for discovery and follow-up of faint transients. With the \texttt{PyLMT} pipeline, several new transients, including supernova candidates, are being discovered and detected routinely. We report the discovery of six supernova candidates, AT~2023yjc, AT~2024agkc, AT~2024aifv, AT~2024aiha, AT~2024fxn, and AT~2024zsm, using the ILMT. Ten additional supernova candidates were discovered in other surveys but were also detected with the ILMT. Here, we present the light curves of all supernova candidates observed with the ILMT in the {\em gri} bands. The basic light curve parameters, such as the rise time, decline rate, and peak magnitude, are estimated and reported.

astro-ph.HE↗
arXiv2026

Non-vanishing mod $p$ of derived Hecke algebras of multiplicative groups over number fields, II

We investigate the derived Hecke action on the cohomology of an arithmetic manifold associated to the multiplicative group over a number field. Under mild assumptions, we prove that the derived Hecke action is non-vanishing modulo $p$ in every degree. Moreover, the total cohomology ring with $\mathbb F_p$-coefficients is a free graded module of rank one over the derived Hecke algebra, generated in degree zero.

math.NT↗
arXiv2026

Iterative Bounded Distance Decoding With 2-Bit Reliability Marker For Product-Like Codes

We propose a novel soft-aided hard-decision decoding scheme for product codes based on two-bit quantized reliability markers. In particular, one bit is used for a two-level tolerance miscorrection check to validate the bounded distance decoding (BDD) result, while the other identifies bits to be randomly flipped upon BDD failure and miscorrection. List decoding is incorporated to further facilitate the error performance. Density evolution is performed for the design of the quantization thresholds. Compared to iterative BDD, the proposed scheme requires only moderate increases in memory and computational overhead. Simulation results verify that our scheme outperforms several unquantized soft-aided decoders, while maintaining small memory costs and low BDD attempts.

cs.IT↗
arXiv2026

Unbounded Nilpotency Gaps on Rationally Elliptic Manifolds

We construct simply connected, closed smooth manifolds whose rational homotopy types are elliptic and whose homology nilpotency exceeds rational category by arbitrarily large amounts. This separates short models from short quotients of minimal models. The construction starts from an explicit family of minimal Sullivan algebras with finite-dimensional cohomology. A finite quotient establishes Poincaré duality and gives an upper bound for homology nilpotency, while a closed orientation functional computes rational category. Two forced primitives yield the lower bound. An even regrading separates the relevant weights, extending this obstruction to every graded acyclic differential ideal. Rational surgery realizes the resulting models in their formal dimensions, with vanishing rational Pontryagin classes and arbitrarily large rational connectivity. The ratio of homology nilpotency to rational category tends to four thirds. Finally, under an external spherical stabilization statement, attaching a single cone can increase homology nilpotency by an arbitrarily large amount while leaving rational category unchanged.

math.AT↗