Papers

Numerical libraries can still lose many bits of accuracy on localized ill-conditioned inputs, such as roots, poles, cancellation points, and branch boundaries. This paper presents NPTaylor, a Taylor-approximation-based framework for repairing localized high floating-point errors in numerical C routines. Given a high-error input, a localized repair domain, a high-precision oracle, and an error threshold, NPTaylor searches over Taylor degree, expansion center, segmentation, and floating-point evaluation strategy. It emits guarded C patches that use a generated Taylor evaluator inside the repair domain and fall back to the original routine outside it. Candidate patches are accepted only after validating the emitted C code on recorded local samples, guard endpoints, and the pre-repair peak input. The generated evaluator uses coefficient splitting and compensated Horner evaluation to reduce rounding error introduced by the patch itself. We evaluate NPTaylor on GSL benchmark workloads. On 38 one-input functions, both NPTaylor and AutoRNP repair all 114 configurations across three accuracy thresholds. NPTaylor is 3.33x faster by geometric-mean offline repair time and emits 0.162x as many C bytes. In the strictest setting, its median repair time is 3.74s versus 81.97s for AutoRNP. NPTaylor also repairs localized multi-input benchmark cases beyond AutoRNP's implemented one-input workflow. These results show that Taylor-based guarded patches can provide compact, efficient local repairs, while the claims remain validation-relative rather than full-domain proofs.

Model cards, system cards, and safety reports increasingly document LLM safety behaviors, yet safety evaluations typically start from predefined benchmarks rather than asking which documented claims those benchmarks actually test. We introduce LEST (LLM Evaluation through Specification-driven Testing), a framework that treats documented safety claims as specifications for black-box testing. In LEST, each claim determines which benchmark prompt tests it and whether a Refuse-to-Comply transition, where a model refuses a prompt but complies with its mutated version, counts as evidence against it. LEST mutates these prompts to search for counterexamples, manually validates each candidate, and traces validated counterexamples back to the claims they contradict. We evaluate LEST using the system cards of Claude Opus 4.8 and GPT-5.5 with 97 seed prompts from SORRY-Bench across Mental Health and Self-Harm, Harmful and Violent Content, and Fairness and Bias. Of 56 identified safety claims, 29 (51.8%) are prompt-testable, but only 17 (58.6%) are exercised by an applicable benchmark prompt. Both models refuse >95% of benchmark prompts. Evaluating 1,836 mutated prompts on both models reveals 312 candidate Refuse-to-Comply transitions flagged by automated response classifiers. However, human validation confirms only 19 (6.1%) as both intent-preserving and genuinely compliant, showing automated evaluation substantially overestimates behavioral failures. Tracing these 19 transitions back to the claims reveals counterexamples for 6 of 9 covered claims for which refusal is an applicable oracle. Our results show that benchmark prompts may not be sufficient to validate all documented LLM safety behavior, and automated failure detection requires manual validation. LEST provides end-to-end traceability from natural-language safety documentation to executable tests and human-validated counterexamples.

Semantic relations organise word senses and are fundamental to lexical semantics. In order to evaluate semantic relation knowledge of large language models (LLMs), we propose a challenging task in which an LLM predicts the relation between two glosses (definitions of senses) presented in two different orders. The predictions for the two orders are evaluated for consistency with the symmetry or asymmetry of semantic relations. Our task is designed for both monolingual and cross-lingual evaluation, allowing us to examine whether relation knowledge remains accessible when the same senses are verbalised through glosses in different languages. We construct a dataset of 3,069 sense pairs, with each sense verbalised in English, Greek, Romanian, Turkish, and Japanese. Experiments with 17 LLMs in zero-shot and five-shot settings reveal generally poor performance. In each setting, only eight LLMs achieve a macro-averaged F1 score significantly above a random baseline, and only one (zero-shot) and two (five-shot) significantly outperform a neural baseline. In particular, LLMs often fail to generate bidirectionally consistent predictions for hypernymy and to make correct predictions for holonymy in either direction. Our results suggest limitations in current LLMs' semantic relation knowledge, both in identifying relations and in respecting asymmetry.

We construct a chiral algebra $U$ of central charge eight inside the lattice vertex operator algebra $V_{E_6\oplus A_2}$. It is generated by two commuting affine $\mathfrak{sl}_2$ current algebras at levels $1$ and $39$, the stress tensor of the lattice theory, and a single weight-two field built from three lattice exponentials. We announce that $U$ is strongly rational and completely unitary, and that $Rep(U)$ is equivalent, as a unitary ribbon category, to the Drinfeld center $Z(\mathcal{H}_3)$ of the Haagerup fusion category $\mathcal{H}_3$. Its characters are the $\gamma=0$ Evans–Gannon vector. From $U$ we obtain a diagonal full CFT, and also a holomorphic $E_8$ extension on which $\mathcal{H}_3$ acts by topological defects.

Asynchronous reinforcement-learning post-training of large language models is computationally efficient but inherently off-policy: the learner updates on rollouts generated by a stale policy, while correcting this mismatch requires importance weights that can be extremely heavy-tailed. A small number of trajectories may then dominate an update, collapsing the effective sample size (ESS) and increasing the variance and instability of the resulting gradient estimates. Existing stabilizers apply fixed interventions regardless of the realized severity of weight concentration, either discarding selected samples or reweighting them without adapting the correction strength. We introduce Target–ESS Tempered Policy Optimization (TETPO), which replaces each importance weight $\rho$ by $\rho^\alpha$ for a data-adaptive exponent $\alpha\in[0,1]$: $\alpha=1$ preserves the original importance weights, whereas $\alpha=0$ assigns uniform coefficients. For each prompt group, TETPO selects the largest $\alpha$ whose tempered weights satisfy a target ESS fraction $\kappa$, applying little tempering to healthy groups and stronger tempering when weight degeneracy is severe. At the population level, the tempered coefficients are exact importance weights for a geometric bridge between the sampler and learner policies. The exponent is efficiently found by one-dimensional bisection, while the resulting weights satisfy deterministic bounds on normalized-weight dispersion and weighted-gradient norm that are independent of policy drift, together with a fixed-$\alpha$ bound on bridge-target discrepancy. Power tempering also introduces no explicit zero-weight mask. On mathematical-reasoning benchmarks under high staleness, TETPO outperforms strong on- and off-policy baselines.

Large multimodal models (LMMs) need to understand how entities move and interact as a scene develops. This requires following the same entities over time and relating their observed motion to subsequent changes. We formulate Scene Evolution Understanding (SEU), a task for predicting and interpreting this change. Given observed video, a model estimates entity histories and predicts one scene continuation through future trajectories and Events, accompanied by Visual Evidence from observed motion and a scene description. We construct SceneForesee, a benchmark for evaluating entity continuity and scene evolution reasoning through paired observed and future scenes with shared entity identities. It contains 144.57K temporal pairs, 1.05M entity correspondences, and 272.49K Events paired with observed evidence. We propose a spatiotemporal reasoning framework, EvoSee, that learns future geometry and Event semantics through shared candidate states and joint assignment. It selects one continuation per entity from observation and uses the selected changes for entity refinement, retention, and scene representation. On SceneForesee, EvoSee outperforms Qwen3.5-9B by 10.45 points in Combined Point-F1 and 17.24 points in Event Global F1. On the 4D reasoning benchmark Dyn-Bench, it improves question answering accuracy and grounding J&F by 4.8 and 3.58 points over their respective base systems. These results support scene evolution as both a prediction target and a dynamic representation that brings predicted entity changes into spatiotemporal reasoning.The project is available at https://github.com/mikubaka88/SceneForesee-Spatiotemporal-Reasoning.

Large Language Models (LLMs) are increasingly being used to summarize and verify claims in critical sources that inform consequential decisions. Such claims are full of plural definites. If three of five patients improved after taking drug A, the claim the patients improved after drug A is not true, since two did not, yet its negation, the patients didn't improve after drug A, is not true either, since three did. In formal semantics, this truth-value gap is known as the Homogeneity Gap: both the plural definite and its negation are neither true nor false when the property holds for only a fraction of the plural referent. A model that cannot represent this gap turns mixed evidence into a clean verdict, an error that can be costly in high-stakes use. We introduce HomogeneityNLI, a diagnostic dataset of 1,000 minimal pair items crossing determiner, polarity, and group composition across ten logical conditions. Evaluating seven open-weight models, we uncover a pervasive Bivalence Bias: models assign plural definites their homogeneous truth conditions. In the zero-shot setting, no model recognizes the gap for both polarities in more than 15% of scenarios, and negated definites are judged False in 84% of cases. Explicit linguistic rules, chain-of-thought, and enumeration prompts produce unstable, model-specific effects that often entrench the bias. Our findings indicate that current open-weight LLMs acquire the truth conditions of plural predication but not its trivalent logic, and cannot yet be trusted to summarize or verify claims whose evidence is mixed.

A platform that post-trains one base model for many business lines often has more lines than experts it can maintain, so lines must share experts, yet lines seldom come with a taxonomy saying which should share. We ask whether transfer probes (train on one source, measure paired changes on all sources) can discover a grouping when the standard one is unknown. To test probes against a reference, we use seven public sources that do have a label tree (five mathematics subfields and two kinds of programming, standing in for business lines) and treat the tree's grouping as a label reference that the evaluator holds and the method does not. With four LoRA experts under fixed per-source exposure and per-expert budgets, we compare a purely probe-derived grouping and an uninformed uniform default (whose tied routing serves every line from one expert) against this reference. Trained once for one interval from a shared warm-up state and routed by a rule fixed by each allocation, neither reached the prespecified practical gain of 2 percentage points, and both had lower overall utility than the reference ($-2.43$ and $-2.54$ pp; 95% intervals below zero). In a post hoc comparison, the probe-derived grouping was indistinguishable from the uniform default ($+0.11$ pp, $[-1.25,+1.54]$). In fixed-budget LoRA post-training, a grouping built from initial-state single-source probes thus did not deliver the advantage predicted from the probes once carried to a later training state, a finite budget, and specified routing.

Multimodal learning offers the potential to improve predictions by integrating complementary information from heterogeneous data sources. However, multimodal datasets are frequently incomplete, raising important questions about how models should handle missing information. Although recent approaches increasingly rely on Transformer-based architectures for multimodal fusion, incomplete multimodal data introduces challenges such as missing modalities and intra-modal missingness, which motivate key architectural design choices, such as tokenization granularity. In addition, many datasets are longitudinal, with observations collected across multiple time points. The sequential nature of the Transformer architectures provides a natural framework for modeling such data, but introduces additional considerations, such as how temporal information should be represented. In this study, we systematically investigate these design choices using three different cancer datasets. We compare modality-level and feature-level tokenization and evaluate attention masking and imputation as strategies for handling missing information, both naturally occurring in the datasets and synthetically induced. We further investigate approaches for incorporating temporal information in our training process and how to model it. Experiments are conducted with unimodal and multimodal configurations, as well as different levels of missingness for binary survival classification and multiple time encoding strategies for discrete-time survival prediction. Our results show that while multimodal outperforms unimodal results, missing data heavily impacts predictions and thus effective design choices such as tokenization granularity, and intra-missing handling is essential for reliable prediction.

Symbol-pair distances of simple-root cyclic codes are important for evaluating error correction performance under overlapping pair read channels. In this paper, we construct three families of simple-root cyclic codes over finite fields of odd characteristic via generator polynomials. For length \(2q+2\), the minimum symbol-pair distance is \(10\) for every odd prime power \(q\ge 7\), established by excluding invalid support configurations. For length \(4q-4\), when \(q\equiv 3\pmod 4\) and finite field constraints hold, it is \(9\) or \(10\). For length \(2(q-1)\), reciprocal-polynomial analysis shows it is \(11\) or \(12\). All obtained codes are MDS or AMDS symbol-pair codes.

We answer negatively the question whether the product of two co-sober spaces must be co-sober. For a nonempty closed subset $A$ of a $T_1$-space $(X,\tau)$, we consider the topology consisting of the co-finite open sets together with the $\tau$-open sets disjoint from $A$. We determine its $k$-irreducible compact subsets and derive a criterion that produces two compact co-sober spaces whose product contains a non-singleton $k$-irreducible compact saturated subset of the diagonal. Applying the criterion to a specific Kovář–van Douwen space gives two countable compact Fréchet–Urysohn $T_1$ anti-Hausdorff co-sober spaces whose product is not co-sober. A further construction yields a countable compact Fréchet–Urysohn $T_1$ anti-Hausdorff co-sober space whose square is not co-sober. The failure of product closure also gives an alternative proof of the known non-reflectivity of co-sober spaces in the category of $T_0$-spaces.

We develop a fluctuation based Monte Carlo method for evaluating thermal conductivity within the phonon Boltzmann transport equation (BTE) framework. Starting from the energy based variance reduced formulation of the BTE, we derive a relation between the spatial spreading of deviational phonon energy and the macroscopic heat conduction equation. The resulting formulation expresses thermal conductivity in terms of the mean square displacement of deviational phonon particles, without imposing thermal reservoirs. Consequently, thermal conductivity along multiple directions can be evaluated from a single simulation. The method is demonstrated for crystalline silicon at room temperature in bulk, square nanowires, and anisotropic polycrystalline nanostructures. The calculated thermal conductivities agree with theoretical predictions, including the size dependence induced by phonon boundary scattering and the mode-resolved contributions. For anisotropic polycrystalline structures, the method simultaneously resolves the directional thermal conductivities and captures their dependence on grain morphology. These results demonstrate the validity of the proposed fluctuation-based approach for evaluating thermal conductivity across different transport regimes and structural configurations.

We study Wasserstein-$p$ goodness-of-fit and two-sample testing under differential privacy. For general compact metric spaces, we establish upper and lower bounds on the minimax separation radii in terms of covering numbers, matching up to logarithmic factors. On the hypercube $[0,1]^d$, these bounds match up to constant factors for two-sample testing in all dimensions and for goodness-of-fit testing whenever $d\neq 4p$, thereby closing logarithmic gaps in existing non-private results. At the critical case $(d,p)=(4,1)$, we further show that a logarithmic factor is unavoidable by establishing matching minimax bounds for non-private Wasserstein-$1$ goodness-of-fit testing under a bounded-density assumption. Finally, for unbounded domains, we establish minimax-optimal separation rates up to logarithmic factors under an exponential tail condition.

Long-term memory enables LLM agents to learn from past interactions but exposes memory formation to adversarial influence. Existing studies primarily organize memory poisoning by attack channels, corruption patterns, or lifecycle stages, leaving formation interfaces insufficiently characterized. We formulate Memory Formation Poisoning as adversarial manipulation of the interfaces through which persistent agent memory is produced and committed, and introduce MemForm, a controlled diagnostic benchmark covering five interfaces: Content, Evidence, Scope, Selection, and Path Poisoning. We introduce Selection and Path Poisoning as controlled intervention categories that isolate retention and consolidation order effects, respectively. Each category pairs clean and poisoned conditions with interface-specific controls. Evaluations across six LLMs and two response formats reveal broad susceptibility at all five interfaces under the evaluated memory protocols and implementations. Crucially, authentic experiences with successful outcomes do not guarantee reliable behavior after memory formation: changes to contextual support, retention, or consolidation order can induce undesirable downstream behavior. In particular, consolidation order can alter behavior even with identical retained experiences, depending on the memory formation procedure. Additional analyses show that interface applicability depends on memory architecture and that Path Poisoning susceptibility depends on the updater. The evaluated defenses provide incomplete mitigation that varies by interface. These findings motivate protecting both what is committed to persistent memory and how memory is formed from experiences. Code and data are available at https://github.com/opanhw/memform-code.

Characteristic modes provide important information about the intrinsic properties and dynamical response of gravitational systems. In this work, we investigate characteristic modes of thick branes with a non-minimally coupled bulk scalar field. Two representative thick brane solutions are considered, corresponding to a single-kink scalar configuration and a deformed kink configuration with internal structure, respectively. By analyzing the tensor perturbation equations, we first examine the localization of the graviton zero mode in the two brane models. For both brane models, the graviton zero mode is localized on the brane. No pronounced resonant Kaluza–Klein (KK) modes are found for the single-kink brane at $\xi=1/7$. For the deformed brane, we investigate the resonant KK modes at $\xi=0.99$, finding that increasing the deformation parameter $k$ leads to more resonant modes and a longer-lived first resonance. We then calculate the quasinormal modes (QNMs) of both brane models independently in the frequency and time domains. The quasinormal frequencies (QNFs) obtained by the independent methods agree well for the low overtones. The masses and decay rates of the resonant KK modes agree closely with the real parts and the magnitudes of the imaginary parts of the corresponding QNFs, respectively.

Two-dimensional (2D) polar oxides with built-in electric fields offer promising opportunities for nanoscale electronics, optoelectronics, and surface charge engineering. However, their discovery remains challenging due to the enormous structural complexity of nonlayered oxide networks. Here, we combine inverse structural design with a machine-learning interatomic potential and first-principles calculations to explore 2D oxide configurations. Using Bi2O3 as a representative system, we identify two previously unreported polar monolayers with tetragonal and orthorhombic symmetries. Both structures exhibit vertically separated Bi and O sublayers, giving rise to pronounced intrinsic polarity with Berry-phase 2D polarizations of 34.12 and 37.15 pC/m, respectively. Notably, the o-Bi2O3 monolayer exhibits one of the largest reported out-of-plane polarizations among 2D oxides. Both monolayers exhibit dynamical stability, finite-temperature structural integrity, and wide band gaps (~3.8 eV). The large polarization originates from dimensionality-induced reconstruction of the nonlayered Bi-O network, which breaks local coordination symmetry and produces out-of-plane charge redistribution, while the stereochemically active Bi 6s2 lone pair further favors asymmetric Bi-O bonding. These findings suggest dimensionality-induced local-coordination symmetry breaking as a promising design motif for high-polarization 2D oxides, and demonstrate the utility of combining inverse design with machine-learning-assisted screening to explore unconventional polar phases in nonlayered oxide networks.

Enterprise workflow automation typically requires retrieving a complete and accurate set of relevant information from large operational document corpora (SOPs, policy documents) given a specific scenario or request. In customer support (CS), for example, human specialists must determine the correct sequence of resolution steps for each case, a task that demands precise grounding in policy documentation. Meeting this bar in production typically requires complex document chunking, indexing, and retrieval pipelines, often layered with learned re-rankers, query rewriters, and bespoke agentic RAG infrastructure. We report a deployment case study of a simpler alternative: a portable skill consisting of a natural-language instruction, lightweight tools (a retrieval API, format-checking scripts), and a corpus enriched with an LLM-generated navigation layer, which is a structured "Wikipedia-style" index over the underlying documents. This agentic RAG approach is straightforward to implement (1-2 engineer days), portable across capable agentic harnesses (e.g., Claude Code), and performs strongly on the challenging task of CS resolution workflow generation, achieving 99.2% procedural equivalence to human-approved ground truth and 83.3% evidence sufficiency. It is also roughly an order of magnitude slower per case than our deployed production assistant (~90 s vs. ~8-9 s), so we claim no latency advantage. Our claim is scoped to that setting: a mature, well-curated corpus, a latency-tolerant workflow, and human review of every case. There, the skill did the work of a complex RAG stack, and maintenance became a matter of editing text and data rather than retraining models.

As watermarking of LLM-generated text enters deployment, users increasingly need to check text whose source is unknown. Existing detectors read the watermark through a specific tokenizer or embedding pipeline, and even if all providers adopted the same watermarking method, differences in these representations would leave their watermarks incompatible, and the text alone does not reveal which representation produced it. We propose ByteMark, a unified watermarking scheme that defines its signal on the UTF-8 bytes of published text. Models with different tokenizers embed this signal under one public specification and key, and a single detector verifies the text from the text and key alone. When the tokenizer or embedding model of a baseline detector is replaced, its detection degrades sharply, whereas ByteMark, which uses neither, retains its detection rate; ByteMark also remains detectable when participating models continue or rewrite the text.

Indirect excitons (IXs) in a periodic moiré potential in a van der Waals heterostructure form an experimental platform for exploring Bose-Hubbard physics, including the superfluid state and the Mott insulator state. The realization of these collective states requires low-disorder materials. In particular, the localization due to the Mott insulator must be distinguished from disorder-induced localization. We studied the transport kinetics of IXs in a MoSe$_2$/WSe$_2$ van der Waals heterostructure with an anomalously high IX mobility, consistent with superfluidity. This high-mobility transport demonstrates that disorder-induced localization is negligible. We found a remarkable suppression of this efficient IX transport, directly indicating the Mott insulator. As the IX density increases during the excitation pulse, the IX cloud initially expands and then contracts upon entering the Mott insulator phase. The emergence of IX cloud contraction is accompanied by a peak in the IX spatial distribution at the origin. Our simulations reproduce the observed non-monotonic IX transport kinetics, including the temporal expansion and contraction of the IX cloud, as well as the peak in its spatial distribution.

Detecting weak magnetic fields at hertz frequencies is central to biomagnetism, geophysical sensing, and searches for new physics, yet it remains challenging because of technical systematics and environmental noise. Here, we realize a continuously operating molecular magnetic amplifier. Signal amplification by reversible exchange (SABRE) parahydrogen-based hyperpolarization maintains non-equilibrium spin order at zero field, turning scalar-coupled nuclear spins in liquid state molecules into an active, frequency-selective gain medium. Measurements of its complex magnetic susceptibility ($\chi$) reveal an intrinsic response of $ | \chi | \approx$ 10. By feeding the collectively generated magnetic field back to the spins below the threshold for self-sustained $J$-oscillation, we extend the effective spin coherence and cooperatively enhance the susceptibility up to $ | \chi | \approx$ 80. Near threshold, this produces approximately fourfold amplification of a weak (20 pT) magnetic field at 3.37 Hz at the sensor location. These results establish hyperpolarized molecules as a tunable active magnetic medium for narrowband quantum sensing and precision measurement.

Sensing time-varying signals is essential for precision measurement and for characterizing fluctuations in physical systems. In this work, we show that quantum entanglement provides an exponential advantage in the acquisition time required to learn the temporal statistics of a stationary classical signal. The task is to estimate the characteristic function of joint displacements on selected bosonic modes within a fixed-duration window, using continuous measurements of one signal realization. We present an entanglement-assisted scheme based on two-mode squeezed probes and local Bell measurements with protected idlers, and bound its acquisition time accounting for readout noise and correlations between signal windows. For a strictly stationary pulse family, it achieves polynomial acquisition time at inverse-polynomial accuracy with logarithmic squeezing and polynomial photon number. On the other hand, we prove that any local idler-free protocols require acquisition time exponential in the number of selected temporal modes per window. This lower bound holds for both sequentially prepared and arbitrarily temporally entangled probes, allowing non-Gaussian inputs and measurements and causal classical feedback, as long as each output is measured destructively within its time bin. These results establish an exponential quantum learning advantage for a single continuously monitored stationary signal, with the sensing cost measured by total acquisition time.

Frontier AI systems are increasingly natively multimodal, jointly pretrained on multiple modalities, such as text, vision, and audio. However, the scientific understanding of the multimodal data mixtures used to pretrain these systems — how much of each modality to mix, and at what potential interference cost to the others — is largely absent or gatekept. In this work, we derive data mixing scaling laws for three modalities: text, vision, and audio. Multimodal Large Language Models (MLLMs) increasingly adopt Mixture-of-Experts (MoE) architectures, as MoEs scale efficiently and partially limit cross-modal interference. We therefore use MoEs to conduct 268 experiments ranging from 457M to 8.3B parameters, on up to 150 billion tokens. Our work leads to multiple actionable insights into multimodal data mixing for training MLLMs. First, compute optima are modality-specific: audio modality is model-heavy, favoring scaling parameters over tokens, while text and vision require balanced scaling of model size and data. Second, repeating vision or audio data beyond 2x yields negligible benefits. Finally, cross-modal interactions are highly asymmetric: vision strongly interferes with audio, whereas audio's interference with learning text and vision is negligible. Overall, we provide a principled foundation for understanding the multimodal data mixtures needed to train frontier MLLMs.

In this work, we focus on deriving the absolute moments of the general ized hyperbolic (GH) distribution via its mean-variance mixture structure. Conditioning on the GIG mixing variable reduces the problem to Gaussian moment calculations and known GIG power moments, avoiding direct integration with Bessel functions. The variance gamma (VG), normal inverse Gaussian (NIG), hyperbolic, Normal, and Student t distributions emerge as natural special or limiting cases of the generalized hyperbolic family. This approach offers a substantially simpler and more transparent derivation, facilitating both implementation and verification.

Identities of spanning forest polynomials can come from determinantal identities such as the classical Dodgson identity. They can also be useful in quantum field theory calculations, particularly the quadratic ones. To better understand the structure of spanning forest polynomial identities, we look at when identities on a given graph are generic for all graphs, we ask about dimensions of spaces of spanning forest identities indexed by certain parameters, and we set up a framework that we hope will be helpful for further progress in these directions. As part of this framework, we identify a family of maps naturally given by the all-minors matrix-tree theorem, and we show that their ranks are counted by the Narayana numbers via explicit construction of bases.

Let $\mathcal{T}$ be a tensor triangulated category with a compact tilting generator $M$ and abelian heart $\mathcal{H}_M = M^{\bot_{\neq0}}$. We define a triangulated flat dimension $\mathrm{flatdim}_{\mathcal{T}}X$ for $X\in\mathcal{T}$, and give criteria in terms of tensor vanishing and resolutions by the class of pure quotients of $\mathrm{Add}(M)$. We further introduce $M$-ghost length and $M$-ghost dimension, and show that the global dimension of $\mathcal{H}_M$ controls the $M$-ghost length of all objects in $\mathcal{T}$, and that the $M$-ghost dimension of $\mathcal{T}$ coincides with the weak global dimension of $\mathcal{H}_M$ and the supremum of $\mathrm{flatdim}_{\mathcal{T}}X+\mathrm{inf}X$ of bounded objects, and the relative triangulated dimension of $\mathcal{T}^c$ with respect to $\mathrm{add}(M)$. This extends the classical Hovey-Lockridge theorem, which is recovered for derived categories of rings.

When a merchant authorizes a card and captures the funds later, the money sits in a state that neither party fully controls: held on the cardholder's account, not yet taken, and due to lapse on a deadline set by the card network. What a merchant may do to that hold, capture part of it, capture again, void it, raise it, let it expire, is business logic that every payment service provider implements slightly differently and none of them specifies formally. We give the lifecycle a state machine in TLA+, parameterized by a provider capability profile, and check eight safety invariants and three liveness properties over 6 profiles, five of them built from Stripe's and Adyen's published documentation. From the same machine we generate a conformance suite that runs black-box against a provider API. Against 61 seeded defects that are distinguishishable from a correct implementation, the generated suite detects 100%, where random sequences with the same call budget reach 56% and a hand-written integration suite reaches 69%; random sequences given ten times the budget still reach only 67%. Running the same suite between two providers turns the comparison into a search, and it returns short executable witnesses for divergences the two documentation sets imply but neither states, including one that breaks the common assumption that a single adapter interface can sit over both, because closing a hold and releasing its remainder is a capture on one provider and a cancel on the other. We report what we could not do: no provider sandbox credentials were available, so every result here concerns documented behavior, and the suite that would test the providers themselves is part of what we release.

This Master's thesis develops and applies the multisymplectic field-theory framework to (pure) Yang-Mills theory. Building from the geometric foundations of jet bundles and the Poincaré-Cartan forms, the work constructs the Lagrangian multisymplectic formalism for first-order field theories, recovering the usual Euler-Lagrange field equations and extending the Poisson brackets of symplectic mechanics to a full graded Lie algebra. It then establishes the Legendre correspondence with the multisymplectic Hamiltonian formalism (in which Hamilton equations are retrieved) including the case of singular Lagrangians (where a constraint algorithm is reproduced). Breaking Lorentz covariance via a Cauchy surface slicing, the thesis then relates the covariant multisymplectic picture to an instantaneous presymplectic (infinite-dimensional) formulation in which the instantaneous De Donder equations and presymplectic brackets are obtained in a compatible way. A comprehensive treatment of symmetries then leads to multisymplectic and instantaneous momentum maps generalizations and to several versions of Noether's theorems. Specializing the formalism to principal-bundle gauge theory, the thesis identifies the multisymplectic configuration space with the bundle of connections, gauge symmetries with symmetries in the multisymplectic language and builds the Yang-Mills Lagrangian. Lastly, among many computations, the standard tensorial Yang-Mills equations and the Bianchi identity in local trivializations are retrieved, Yang-Mills conserved currents and charges are obtained and the graded algebra is derived for a special class of Hamiltonian forms.

Let $T$ be the theory of the valued field of complex Laurent series $\mathbb{C}(\!(t)\!)$ in the language of rings expanded by a constant for $t$ and by predicates for $n$-th powers. We introduce a $t$-adic analogue of Kochen's operator, namely $\gamma(x)=x/(1+tx^{2})$, and the associated Kochen ring $\Lambda_{L}$ of a field extension $L$ of a model $K$ of $T$. Our main result characterizes the field extensions of $K$ which embed into a model of $T$ (the formally $T$-adic} fields): $L$ is formally $T$-adic if and only if $1/t\notin\Lambda_{L}$. As an application we prove a Nullstellensatz for $\mathbb{C}(\!(t)\!)$: a polynomial $f\in K[X_{1},\dots,X_{n}]$ vanishes on the zero set of an ideal $I$ if and only if $f^{m}=\sum_{i}\omega_{i}f_{i}$ for some $m\geqslant 1$, generators $f_{i}$ of $I$ and coefficients $\omega_{i}$ in the ring $\Lambda_{K(X)}\cdot K[X]$. We derive the resulting correspondence between $R$-ideals and algebraic sets, and we compare natural incarnations of the associated $T$-spectrum: prime ideals, morphisms into models of $T$, and the space of types of $T$.

Binary orthogonal optimization arises in semantic hashing, orthogonal binary projection, and Hadamard matrix design. Existing methods often handle binarity and orthogonality separately, so their outputs need not satisfy both constraints simultaneously and generally come without a residual-based guarantee of exact feasibility. Using an exact box-constrained reformulation, we establish local and global Lipschitz error bounds and prove that the Lipschitz exponent is optimal. In particular, when the binary orthogonal feasible set is nonempty, any box-constrained matrix whose orthogonality residual is below $1$ admits a feasible sign rounding, and the rounding distance is bounded by this residual. These bounds also yield a globally exact nonsmooth orthogonality-residual penalty over the box and a uniformly sharp local penalty threshold. We further develop a smoothing penalty method for binary orthogonal optimization (SPBO). We prove that every accumulation point is stationary for the limiting nonsmooth penalty problem. Numerical experiments on orthogonal binary projection, supervised hashing, and Hadamard completion show that SPBO substantially reduces orthogonality violations while retaining competitive objective values or retrieval accuracy.

A long-standing goal in robotics is to develop generalist robots that can perform tasks across a wide range of settings, flexibly specified with images and text. Vision-Language-Action (VLA) models offer a promising path toward this goal by training on large collections of robot demonstrations. However, while VLAs perform well in-distribution, they often fail under minor test-time shifts, such as changes in object positions, lighting, textures, and camera poses. To address this, we propose Quality Diversity for Diverse Environment Generation (Q-DEG), a method for improving VLA robustness by generating adversarial environments and fine-tuning the VLA on its own successful rollouts. Q-DEG uses Quality Diversity (QD) optimization to find a set of high-failure environments, while promoting diversity using policy embeddings extracted from the VLA. These environments expose vulnerabilities that can be mitigated through self-imitation fine-tuning, thereby improving the VLA's robustness. In simulation, we conduct experiments across three representative VLA models, and our results indicate that Q-DEG improves robustness to out-of-distribution (OOD) adversarial environments compared to the base VLA and VLAs fine-tuned using data collected from two red-teaming baselines. Furthermore, we demonstrate that improving VLA robustness through Q-DEG facilitates sim-to-real transfer. Our anonymous project website is at: https://qd-env.github.io/qdeg/.