Let $X$ be a Stein manifold with the density property. We present methods of constructing two kinds of non-Runge Fatou-Bieberbach domains in $X$, which by definition are proper open subsets of $X$ biholomorphic either to $\mathbb{C}^n$ or to $X$. For both kinds, we provide examples where our methods apply.

The Pathfinder of the Type-A satellites in the Chasing All Transients Constellation Hunters (CATCH) space mission is equipped with Micro-Pore Optics (MPOs) and four single-pixel Silicon Drift Detectors (SDDs). Due to the lack of position resolution in an individual SDD, we propose a new method based on the cross-arms in the point spread function (PSF) of MPOs to enhance the satellite's capability in identifying contaminating sources and locating target sources. By placing one detector on each of the horizontal and vertical cross-arms on the focal plane, we can use the changes in the relative counts on the cross-arms detectors to deduce the location of the source. Simulated observations demonstrate that, for a target source with a flux of 1 Crab and an exposure time of 200 s, the cross-arms detectors can identify contaminating source with the same flux level at an off-axis angle larger than 8', and improve positioning accuracy to 6'. Furthermore, we extend the simulation study to CATCH Type-A, which plans to use an SDD array. In situations where sources exhibit the same flux of 1 Crab and the exposure time is merely 1 s, a 16x16 SDD array is capable of identifying contaminating source with an off-axis angle greater than 2.4' and can achieve a positioning precision of 1.8'.

We extend the single source approach of Valencia et al in order to synchronize the clocks of an N user start network, connected both through fiber and in free space. Entangled photon pairs from a centralized SPDC source are distributed through a 1 by N splitter to four remote users arranged in a star topology. Using commercially available single photon detectors and time taggers, we achieve median time precision of 50 ps for atomic oscillators and 20 ps for GPS displayed oscillators in our Kalman models. Thus, we achieve three order of magnitude improvement over GPS alone. By monitoring the drift fo the correlation peaks over time, we also extract the frequency skew between users's local clocks to 35ps/s precision. From these measurements, e3ach user can compute its offset and drift relative to every other user, achieving full network synchronization without a central clock.

We investigate preservation results for the independent fusion of one-variable first-order modal logics. We show that, without equality, Kripke completeness and decidability of the global and local consequence relation are preserved, under both expanding and constant domain semantics. By contrast, Kripke completeness and decidability are not preserved for fusions with equality and non-rigid constants (or, equivalently, counting up to one), again for the global and local consequence and under both expanding and constant domain semantics. This result is shown by encoding Diophantine equations. Even without equality, the finite model property is only preserved in the local case. Finally, we view fusions of one-variable modal logics as fusions of propositional modal logics sharing an S5 modality and provide a general sufficient condition for transfer of Kripke completeness and decidability (but not of finite model property).

Enterprise knowledge graphs (EKGa) are a novel paradigm for consolidating and semantically integrating large numbers of heterogeneous data sources into a comprehensive dataspace. The main goal of an EKG is to provide a data layer that is semantically connected to enterprise data, so that applications can have integrated access to enterprise data sources through that semantic layer. To make legacy relational data sources accessible through the organization's knowledge graph, it is necessary to create an RDF view of the underlying relational data (RDB2RDF view). An RDB2RDF view can be materialized to improve query performance and data availability. However, a materialized RDB2RDF view must be continuously maintained to reflect updates over the relational database. This article proposes a formal framework for constructing the materialized data graph for an RDB2RDF view and for incrementally maintaining the view's data graph. The article also presents an architecture and algorithms for implementing the proposed framework.

Cell migration plays a fundamental role in numerous physiological processes, including embryonic development, wound healing, and cancer metastasis. While cell-cell adhesion is known to regulate motion by shaping cell morphology and intercellular force balance, its dynamic, rate-dependent contributions to tissue behavior remain poorly understood. In this study, we examine how the dissipative nature of cell-cell adhesion influences tissue dynamics and collective migration using an extended vertex model with explicit junctional viscosity. Our findings reveal a nontrivial interplay between two distinct components of adhesion: an interfacial adhesion energy (energetic, rate-independent) contribution, which sets the effective junctional tension, and a dissipative (rate-dependent) contribution, which controls resistance to relative motion during cell rearrangements. We show that increasing the energetic component promotes migration by modifying cell shape and lowering the barrier to neighbor exchanges, whereas strengthening the dissipative component induces jamming and suppresses cell motion. Linear rheological analysis further demonstrates that, in the unjammed regime, vertex-model tissues exhibit power-law viscoelastic behavior, with adhesion modulating the power-law exponent and thereby controlling the spread of relaxation timescales. Together, these findings clarify the dual role of adhesion in governing tissue mechanics and rheology and provide a mechanistic framework for understanding the balance between fluidity and rigidity in epithelial monolayers.

Query rewriting is essential for database performance optimization, but existing automated rule enumeration methods suffer from exponential search spaces, severe redundancy, and poor scalability, especially when handling complex query plans with five or more nodes, where a node represents an operator in the plan tree. We present SLER, a scalable system that enables efficient and effective rewrite rule discovery by combining standardized template enumeration with a learning to rank approach. SLER uses standardized templates, abstractions of query plans with operator structures preserved but data specific details removed, to eliminate structural redundancies and drastically reduce the search space. A learn to rank model guides enumeration by pre filtering the most promising template pairs, enabling scalable rule generation for large node templates. Evaluated on over 11000 real world SQL queries from both open source and commercial workloads, SLER has automatically constructed a rewrite rule repository exceeding 1 million rules - the largest empirically validated rewrite rule library to date. Notably, at the scale of one million rules, SLER supports query plan templates with complexity up to channel level depth. This unprecedented scale opens the door to discovering highly intricate transformations across diverse query patterns. Critically, SLER's template driven design and learned ranking mechanism are inherently extensible, allowing seamless integration of new and complex operators, paving the way for next generation optimizers powered by comprehensive, adaptive rule spaces.

Edema is a potential indicator of underlying pathological changes. However, its low-contrast signature is often masked in conventional B-mode imaging by strong scatterers, making reliable detection challenging. Ultrasound (US) provides a non-invasive, non-ionizing, and cost-efficient imaging option that is widely used. Conventional techniques, which rely on beamforming, often lack sufficient physical interpretability. Quantitative US (QUS) can estimate physical properties such as the speed of sound (SoS) and density by solving a physics-based inverse problem directly on the measured US wavefields, i.e., the raw per-element channel data (CD), to recover their spatial distribution. However, state-of-the-art physics-based inversion methods, including full waveform inversion (FWI) and model-based quantitative radar and US (MB-QRUS), are computationally intensive and susceptible to local minima, which constrains their clinical utility. We introduce deep unfolded FWI (DUFWI), a physics-faithful unfolded iterative inversion method that exhibits FWI-like refinement behavior while learning the update rule from data, requiring only a small number of iterations for real-time SoS reconstruction. Across both simulated datasets and hardware measurements acquired with a Verasonics US system, the DUFWI significantly outperforms classical FWI and MB-QRUS in reconstruction quality while maintaining high computational efficiency. These results demonstrate real-time edema diagnosis in both simulation and hardware experiments, with phantom-based validation using cylindrical rods, supporting practical deployment under typical US imaging setting.

In the ALP model, lepton flavor violation (LFV) can arise from off-diagonal ALP-lepton couplings ($g_{a\ell_i\ell_j}$), which are absent in the Standard Model. We focus on ALP production via gluon fusion ($pp \to a$) at the Large Hadron Collider (LHC), which dominates the ALP mass range of 5-1000 GeV due to its high cross section and manageable backgrounds. We are interested in the decay of an axion into an electron and a muon, two oppositely charged leptons of different flavors. After background suppression, we obtain the sensitivity to ALP in this mass range, finding significantly improved limits for $5 < m_a < 1000$ GeV, where SM backgrounds are suppressed. Our research is complementary with relevant studies.

We propose Scrambler, and e-graph-based MBA obfuscation tool using Equality Expansion to efficiently generate complex and diverse expressions with equivalence guaranteed by construction. Experiments show Scrambler improves existing tools in expressiveness and complexity.

Many leading AI researchers expect AI development to exceed the transformative impact of all previous technological revolutions. This belief is based on the idea that AI will be able to automate the process of AI research itself, leading to a positive feedback loop. In August and September of 2025, we interviewed 25 leading researchers from frontier AI labs and academia, including participants from Google DeepMind, OpenAI, Anthropic, Meta, UC Berkeley, Princeton, and Stanford to understand researcher perspectives on these scenarios. Though AI systems have not yet been able to recursively improve, 20 of the 25 researchers interviewed identified automating AI research as one of the most severe and urgent AI risks. Participants converged on predictions that AI agents will become more capable at coding, math and eventually AI development, gradually transitioning from `assistants' or `tools' to `autonomous AI developers,' after which point, predictions diverge. While researchers agreed upon the possibility of recursive improvement, they disagreed on basic questions of timelines or appropriate governance mechanisms. For example, an epistemic divide emerged between frontier lab researchers and academic researchers, the latter of which expressed more skepticism about explosive growth scenarios. Additionally, 17/25 participants expected AI systems with advanced coding or R&D capabilities to be increasingly reserved for internal use at AI companies or governments, unseen by the public. Participants were split as to whether setting regulatory ``red lines" was a good idea, though almost all favored transparency-based mitigations.

In recent work, the notion of $m$-rigidity was introduced as a sufficient condition for the existence of infinite antichains of $1$-degrees inside many-one degrees. Motivated by a recent preprint of Richter, Stephan, and Zhang on finite-one degrees inside many-one degrees, we study the finite-one structure of the many-one degree of an $m$-rigid set. First, combining bi-immunity of $m$-rigid sets with a theorem of Richter, Stephan, and Zhang, we show that for Lebesgue-almost every set $A$, and for a comeager class of sets $A$, the many-one degree $°_m(A)$ contains a least finite-one degree. Second, we prove that if $A$ is $m$-rigid, then $°_m(A)$ contains infinitely many pairwise incomparable finite-one degrees. More precisely, we construct representatives $B_S \equiv_m A$, indexed by computable sets $S$, such that $T \setminus S$ infinite implies $B_T \not\leq_{fin} B_S$. Third, inside a single finite-one degree we build a strict ascending chain \[ A_{(1)} <_1 A_{(2)} <_1 \cdots \] of $1$-degrees. These results yield almost-sure and comeager partial answers to the first two open problems posed by Richter, Stephan, and Zhang.

We demonstrate how the beyond-mean-field Lee-Huang-Yang (LHY) corrections and its related physics can be naturally incorporated into the representation of an ultracold Bose gas using the truncated Wigner approach without invoking effective energy terms or local density assumptions. By generating a Bogoliubov ground-state representation with appropriately tailored bare interaction strength $g_0$ and condensate density $n_0$, the expected initial energy and densities are obtained while retaining access to quantum effects beyond the reach of the extended Gross-Pitaevskii equation (EGPE) formulation. This approach enables the study of correlations, coherence decay, single realisations, and the onset of quantum fluctuation effects with growing interaction strength. Numerical demonstrations for a weakly interacting single-component Bose gas show that observables deviate significantly from both the plain GPE and the EGPE incorporating LHY corrections. In regimes of strong interaction, many of the interference effects predicted by the GPE and EGPE suppressed, and the EGPE offers no improvement over the plain GPE compared to the full Wigner model. In the weakly interaction limit, the EGPE appears accurate but resolving its deviation from mean-field results requires extensive ensemble averaging.

We demonstrate the novel effect of gain-induced spectral shifts in the type-II parametric down-conversion (PDC) process, which results in a transition from degenerate to non-degenerate PDC with increasing parametric gain. This effect, originating from the second-order dispersion terms, significantly alters the properties of PDC in the high-gain regime, where it leads to increased distinguishability of the generated photon pairs. The effect is established by evaluating a rigorous theoretical model, which is based on solving a system of coupled integro-differential equations for monochromatic operators. The widely used spatially-averaged approximate model fails to reproduce this important effect.

Given a labelled tournament on $[n]$, \emph{inverting} a vertex subset $X$ means reversing every edge with both endpoints in $X$. Alon, Powierski, Savery, Scott, and Wilmer~\cite{AlonPowierskiSaveryScottWilmer2024} asked for the mixing time of the Markov chain that repeatedly inverts a uniformly random subset of $[n]$. We show that this \emph{inversion walk} undergoes total-variation cutoff at time $n$. More precisely, there is a universal constant $C>0$ such that for all $c\ge 0$, $d_n(n+c)\le C\,2^{-c}$, while for all $s\in\{0,1,\dots,n\}$, $d_n(n-s)\ge 1-2^{\,n+s\logtwo n-\binom{s}{2}}$. In particular, the lower tail threshold lies within $O(\sqrt{n})$ below $n$, while the upper tail decays within $O(1)$ above $n$. As a second result, we characterise the state space of the \emph{$k$-restricted inversion walk}, which inverts a uniformly random $k$-subset at each step. For $n\ge 4$ and $2\le k\le n-2$, the reachable states form a coset of a subgroup $H_k\le\F_2^{\binom{n}{2}}$ whose codimension is determined solely by $k\bmod 4$.

This paper is concerned with fault/disturbance compensation control for fully actuated systems. In particular, we explore observer-based control, incorporating an active compensation mechanism. First, we propose a novel observer with enhanced design flexibility for the fully actuated system model, enabling simultaneous estimation of system states and exogenous unknown signals, such as faults or disturbances. Then, a nonlinear controller is developed with an active fault or disturbance compensation term, leveraging the fully actuated system approach. The asymptotic stability of both the state estimation error and the closed-loop control system is systematically established. Finally, the feasibility and merits of the proposed method are validated through comparative simulations and experiments.

Optical Character Recognition (OCR) is a critical but error-prone stage in digital humanities text pipelines. While OCR correction improves usability for downstream NLP tasks, common workflows often overwrite intermediate decisions, obscuring how textual transformations affect scholarly interpretation. We present a provenance-aware framework for OCR-corrected humanities corpora that records correction lineage at the span level, including edit type, correction source, confidence, and revision status. Using a pilot corpus of historical texts, we compare downstream named entity extraction across raw OCR, fully corrected text, and provenance-filtered corrections. Our results show that correction pathways can substantially alter extracted entities and document-level interpretations, while provenance signals help identify unstable outputs and prioritize human review. We argue that provenance should be treated as a first-class analytical layer in NLP for digital humanities, supporting reproducibility, source criticism, and uncertainty-aware interpretation.

Aeroacoustic emissions from intense evaporation are widely measured yet often treated as noisy byproducts and used mainly in empirical monitoring. Here, we show that airborne sound encodes physics-governed sub-millisecond fingerprints of vapor-jet dynamics in excessive vaporization, exemplified by vapor keyholes in laser metal processing. From first principles, we develop a vapor-jet-cavity oscillation framework and incorporate it into an aeroacoustic formulation, thereby coupling measured sound to transient cavity depth and oscillation frequency. Reconciled with synchronized multimodal in-situ data, airborne acoustics enable accurate tracking of vapor-cavity properties within tens to hundreds of microseconds. Combined with newly discovered correlations, cavity-jet-acoustic theory recasts the transition from steady, pore-free to pore-shedding vaporizations as a critical-frequency event. Aeroacoustic emissions thus become scalable, physics-guided, and cost-efficient probes of rapidly evolving liquid-vapor systems.

We propose a correction to one of the elliptic blocks in the NS sector of 2d $\mathcal N = 1$ superconformal field theories. We analyze the 4-point block in the pillow geometry to demonstrate the necessity of the correction and verify the formula by numerically checking the crossing symmetries in the $\mathcal N =1 $ super Liouville theory, as well as directly comparing the $c$-recursion and $h$-recursion results.

Photoacoustic tomography (PAT) and thermoacoustic tomography (TAT) both leverage acoustic signals generated by electromagnetic absorption to noninvasively image deep tissues. PAT operates by detecting optical absorption, whereas TAT targets radiofrequency (RF) absorption, providing complementary information on tissue composition and structure. Combining these modalities into a single system promises richer contrast but remains difficult due to the expense and complexity of the RF source. Here, we show that PAT can be integrated with a low-cost RF heater and used to image both optical and RF absorption in tissue phantoms. RF Heating-Enhanced Photoacoustic Tomography (HEPAT) maps RF absorption via temperature-dependent changes in thermomechanical properties, which enables the use of slow, inexpensive RF subsystems and provides an additional layer of contrast. HEPAT therefore provides distinct, complementary contrast relative to existing photoacoustic imaging systems, expanding specificity and diagnostic power while opening new avenues for studying temperature-related tissue phenomena.

Large-scale e-commerce search must surface a broad set of items from a vast catalog, ranging from bestselling products to new, trending, or seasonal items. Modern systems therefore rely on multiple specialized retrieval channels to surface products, each designed to satisfy a specific objective. A key challenge is how to effectively merge documents from these heterogeneous channels into a single ranked list under strict latency constraints while optimizing for business KPIs such as user conversion. Rank-based fusion methods such as Reciprocal Rank Fusion (RRF) and Weighted Interleaving rely on fixed global channel weights and treat channels independently, failing to account for query-specific channel utility and cross-channel interactions. We observe that multi-channel fusion can be reformulated as a query-dependent learning-to-rank problem over heterogeneous candidate sources. In this paper, we propose a unified ranking model that learns to merge and rank documents from multiple retrieval channels. We formulate the problem as a channel-aware learning-to-rank task that jointly optimizes clicks, add-to-carts, and purchases while incorporating channel-specific objectives. We further incorporate recent user behavioral signals to capture short-term intent shifts that are critical for improving conversion in multi-channel ranking. Our online A/B experiments show that the proposed approach outperforms rank-based fusion methods, leading to a +2.85\% improvement in user conversion. The model satisfies production latency requirements, achieving a p95 latency of under 50\,ms, and is deployed on Target.com.

A finite set $ S \subset \mathbb{R} $ is called a Sidon set if all sums $ x+y $ with $ x,y \in S $ and $ x \le y $ are distinct, and a weak Sidon set if all sums $ x+y $ with $ x,y \in S $ and $ x < y $ are distinct. For a finite set $ A \subset \mathbb{R} $, let $ h(A) $ denote the maximum size of a Sidon subset of $ A $, and define $$ g(n) := \min\{\, h(A) : A \subset \mathbb{R},\ |A| = n,\ A \text{ is a weak Sidon set} \,\}. $$ Sárközy and Sós asked whether the limit $ \lim_{n\to\infty} g(n)/n $ exists and, if so, to determine its value. We resolve this problem completely by determining $g(n)$ exactly: $$ g(n)=\left\lceil \frac{n+1}{2}\right\rceil \qquad\text{for all } n\ge 1. $$ In particular, $\lim_{n\to\infty} g(n)/n=\frac12$. We also investigate a related problem of Erdős concerning a local difference condition. A finite set $ A \subset \mathbb{R} $ is called a $(4,5)$-set if every $4$-element subset of $A$ determines at least five distinct values among its six pairwise absolute differences. Erdős asked for the optimal constant $ c_* > 0 $ such that every $(4,5)$-set of size $ n $ contains a Sidon subset of size at least $ c_* n $. Gyárfás and Lehel reduced this to an extremal problem of $3$-uniform hypergraphs and proved $\frac{1}{2} + \frac{1}{141 \cdot 76} \le c_* \le \frac{3}{5}$. We improve both bounds by establishing $$ \frac{9}{17} \le c_* \le \frac{4}{7}, $$ where the lower bound uses a reformulation of the extremal problem, and the upper bound follows from an explicit construction together with a convenient characterization of $c_*$.

We investigate variational problems in quantum thermodynamics at positive temperature, in which admissible states are constrained by prescribed outcomes of a finite set of measurements. We solve a problem raised by the recent work [Liu, Minervini, Patel, Wilde; arXiv:2505.04514 - Section C] and develop a general mathematical setup which allows a broad class of possible regularizations. Employing methods inspired by non-commutative optimal transport, we analyze the dual formulation of the problem, study the existence and characterization of maximizers, and investigate the qualitative behavior of the model in the zero-temperature limit. In the second part, we tailor this framework to quantum state tomography and quantum optimal transport. Finally, we address computational aspects, with particular attention to the convergence of algorithms in selected cases.

The GW190814 event, involving a black hole of mass $22.2$--$24.3 M_{\odot}$ and a compact object of mass $2.50$--$2.67 M_{\odot}$, challenges our understanding of the mass gap between the heaviest neutron stars and the lightest black holes. If the secondary is a neutron star exceeding $2.5 M_{\odot}$, hyperons are likely to appear in its core, softening the equation of state. Rapid rotation can offset some of this softening, enabling higher maximum masses, but it may simultaneously excite the Chandrasekhar--Friedman--Schutz $r$-mode instability. Bulk viscosity arising from nonleptonic weak interactions in hyperonic matter provides an efficient damping mechanism that can stabilize such configurations. In this work, we investigate the combined effects of rotation, thermal evolution, and hyperon-induced bulk viscosity on the stability of massive neutron stars. We demonstrate a direct connection between the suppression of $r$-mode instabilities and the long-term dynamical stability of hyperon-rich stars, offering a plausible interpretation of the GW190814 secondary as a rapidly rotating, hyperon-rich neutron star rather than a low-mass black hole. Our unified framework extends beyond previous studies restricted to static equations of state or extreme viscous damping assumptions, providing new insights into the stability of massive, exotic neutron star configurations.

We study the total cut complexes and their Alexander duals. The homotopy type of these complexes is calculated for de $p$th power of a cycle with at least $2rn$ vertices where $p\leq r$, solving part of a conjecture of Bayer, Denker, Milutinović, Rowlands, Sundaram and Xue. The homotopy type of the $2$-total cut complex for any $r$th power of a cycle with $r\geq3$ also is calculated, solving a conjecture of Chauhan, Shukla and Vinayak. We give some results about the connectivity. The homotopy type of the complexes for complete multipartite graph is determined. We also study the complexes of cartesian products of paths and of cartesian products of complete graphs for the total $2$-cut complex.

The putative dusty torus is a key ingredient of the unification scheme of active galactic nuclei (AGN), but its origin remains a mystery. Here we put forward a new physical model to explain how a large number of small dusty gas clumps form and they collectively appear as a geometrically thick dynamic dusty torus. The circumnuclear hot gas flows towards the central black hole (BH) and forms a rotating disk on sub-pc scales. A fraction of inflowing hot gas condenses to form small cold clumps due to thermal instabilities, when the accretion rate is sufficiently high. These cold dusty gas clumps are irradiated by the central accretion disk and re-radiate as dust emission mostly in the infrared. We propose that the dusty torus in AGN consists of such cold clumps vertically supported by the radiation force against gravity. For clumps with suitable column density, the vertical component of the BH gravity is in quasi-static equilibrium with the infrared radiation force together with the vertical component of the disk radiation force. Our model is robust in the sense that for any reasonable range of parameters concerning clump vertical dynamical equilibrium a torus exists. We further show that the hot gas in the rotating flow condenses to cold clumps only if its accretion rate is higher than about one percent of the Eddington rate. The radiation force is unable to lift the cold gas clumps up away from the mid-plane when the luminosity of the disk surrounding the BH is lower than 0.1 percent of the Eddington luminosity. These two features of our model may provide a physical explanation for the lack of evidence of dusty tori in low-luminosity AGNs.

We present StochasticBarrier.jl, an open-source Julia-based toolbox for generating Stochastic Barrier Functions (SBFs) for safety verification of discrete-time stochastic systems with additive Gaussian noise. StochasticBarrier.jl certifies linear, polynomial, and piecewise affine (PWA) systems. The latter enables verification for a wide range of system dynamics, including general nonlinear types. The toolbox implements a Sum-of-Squares (SOS) optimization approach, as well as methods based on piecewise constant (PWC) functions. For SOS-based SBFs, StochasticBarrier.jl leverages semi-definite programming solvers, while for PWC SBFs, it offers three engines: two using linear programming (LP) and one based on gradient descent (GD). Benchmarking StochasticBarrier.jl against the state-of-the-art shows that the tool outperforms existing tools in computation time, safety probability bounds, and scalability across over 30 case studies. Compared to its closest competitor, StochasticBarrier.jl is up to four orders of magnitude faster, achieves significant safety probability improvements, and supports higher-dimensional systems.

We approach the 3-SAT satisfiability problem with the quantum-inspired method of imaginary time propagation (ITP) applied to matrix product states (MPS) on a classical computer. This ansatz is fundamentally limited by a quantum entanglement barrier that emerges in imaginary time, reflecting the exponential hardness expected for this NP-complete problem. Strikingly, we argue based on careful analysis of the structure imprinted onto the MPS by the 3-SAT instances that this barrier arises from classical computational complexity. To reveal this connection, we elucidate with stochastic models the specific relationship between the classical hardness of the $\sharp$P $\supseteq$ NP-complete counting problem $\sharp$3-SAT and the entanglement properties of the quantum state. Our findings illuminate the limitations of this quantum-inspired approach and demonstrate how purely classical computational complexity can manifest in quantum entanglement. Furthermore, we present estimates of the non-stabilizerness required by the protocol, finding a similar resource barrier. Specifically, the necessary amount of non-Clifford operations scales superlinearly in system size, thus implying extensive resource requirements of ITP on different architectures such as Clifford circuits or gate-based quantum computers.

Given an automorphism $x$ of order bigger than $2$ of a sporadic simple group $S$, we show that there are at most $3$ conjugates of $x$ required to generate a subgroup of order divisible by a fixed prime divisor $r$ of $|S|$. The only exception is the case where $S=Suz$, $x$ is in class $3A$, $r=11$, and then the required number of generators is $4$.

Modern artificial intelligence (AI) systems act with a high degree of independence yet lack legal personhood-a paradox that fractures doctrines grounded in human-centric notions of mens rea and actus reus. This Article introduces Operational Agency (OA)-a permeable legal fiction structured as an ex post evidentiary framework-and Operational Agency Graph (OAG), a tool for mapping causal interactions among human actors, organizations, and AI systems. OA evaluates an AI's observable operational characteristics: its goal-directedness (as a proxy for intent), predictive processing (as a proxy for foresight), and safety architecture (as a proxy for a standard of care). OAG operationalizes that analysis by embedding these characteristics in a causal graph to trace and apportion culpability among developers, fine-tuners, deployers, and users. Drawing on corporate criminal liability, the innocent-agent doctrine, and secondary and vicarious liability frameworks, the Article shows how OA and OAG strengthen existing doctrines. Across five real-world case studies spanning tort, civil rights, constitutional law, and antitrust, it demonstrates how the framework addresses challenges ranging from autonomous vehicle collisions to algorithmic price-fixing, offering courts a principled evidentiary method-and legislatures and industry a conceptual foundation-to ensure human accountability keeps pace with technological autonomy, without conferring personhood on AI.