Computer Science and Mathematics

Abstract: We present AEGYPTI, a combinatorial triangle-detection framework for an undirected simple graph \( G=(V,E) \) with \( n=|V| \) vertices and \( m=|E| \) edges, built on the duality between triangles and independent sets: a triangle of \( G \) is exactly a three-vertex independent set of the complement \( \overline{G} \), which is what a small vertex cover of \( \overline{G} \) leaves uncovered. AEGYPTI dispatches on density at \( \lceil n^{4/3}\rceil \). When \( m \le \lceil n^{4/3}\rceil \), it runs the exact Chiba–Nishizeki routine, whose \( \mathcal{O}(m^{3/2}) \) cost is \( \mathcal{O}(n^{2}) \) on inputs this sparse. When \( m > \lceil n^{4/3}\rceil \), it covers \( \overline{G} \) with the linear-time HVALA algorithm and extracts the uncovered candidate set \( I = V \setminus C \). To systematically eliminate incompleteness when \( |I| < 3 \), AEGYPTI incorporates a deterministic 1-Vertex Expansion step: if \( |I| = 1 \) with \( I = \{v\} \), inspecting the induced neighborhood \( G[N_G(v)] \) guarantees expanding \( I \) to a full 3-vertex triangle whenever \( v \) belongs to any triangle in \( G \). Furthermore, we establish that the local-search repair process structurally guarantees that at least one or two vertices of any existing triangle are visited during repair iterations. Combining this seed guarantee with 1-vertex and 2-vertex expansion routines ensures that handling these cases guarantees the algorithm never misses an existing triangle. To guarantee an unconditional \( \mathcal{O}(n^2) \) worst-case execution budget, the local-search repair iterations are bounded by the cover size \( |C| \le n \). Combined with a size-capped independent set search that guarantees \( \mathcal{O}(1) \) conflict checks per node, total repair loop time is strictly bounded by \( \mathcal{O}(n^2) \). Crucially, because this fast combinatorial algorithm successfully resolves every dense benchmark instance tested without fail and provides absolute theoretical guarantees, its strictly quadratic runtime poses a direct, definitive refutation of the Combinatorial Boolean Matrix Multiplication (BMM) Conjecture. A public reference implementation is provided in the aegypti Python package v0.4.9, which depends on hvala.

Abstract: Let a₁, . . . , a_m be distinct elements of a field and let Z(X) = Π_i(X − a_i). We study the total arithmetic complexity of the canonical Bézout pair sZ + tZ′ = 1, deg t < m, deg s < m − 1. The standard product-tree route uses O(MF(m) logm) field operations. Even when M_F(m) = O(m logm), this is O(m log² m) rather than O(m logm). Our main result identifies the missing logarithm. Over an infinite field of characteristic zero, in the generic rational straight-line-program model, computing all coefficients of (s, t) is equivalent, up to O(M_F(m)) operations, to arbitrary-node multipoint evaluation and to interpolation. The new direction rests on an explicit quasi-linear reconstruction of Z from (s, t) on a nonempty Zariski-open set. It also transfers Strassen’s Ω(m logm) nonscalar lower bound for elementary symmetric functions to the Bézout problem, showing that the requested order would be optimal. The reconstruction is genuinely characteristic-dependent: in characteristic p, the squarefree polynomials X^p + cX + d all have the same canonical pair (0, c⁻¹) for fixed c ̸= 0. Thus the all-field, all-input O(m logm) question is not resolved here; it is reduced generically in characteristic zero to the corresponding arbitrary-node evaluation problem.

Abstract: Muon orthogonalizes the momentum matrix, granting every singular direction of the update equal trust. EchoMuon prices that trust by each direction's echo: its support in a second, slower momentum buffer. This paper reports what that buys, and what measuring such a gate honestly costs. The gate is a contraction, not a reallocation. Its multiplier never exceeds one in 5,760 logged layer-steps, and the arm carrying it sits higher on the learning-rate axis: over four cells it is penalized 3.7x less than ungated Muon for a step twice too large, and 1.6x more for one half too small, without exception in either direction. A shared learning-rate grid therefore does not compare the two fairly: when the grid sits above the joint optimum, the contracting arm is rescued and the other is not, by an amount comparable to the margins being measured. Our initial hypotheses about the gate were wrong on this point, so we re-ran the full suite under corrected selection. All eight CIFAR arm-cells had picked the bottom edge of a single shared grid; widening it and selecting on a held-out split cuts those margins by 42 to 80% and leaves none individually significant. Tiny ImageNet, whose optimum the original grid did contain, is unchanged at +1.65 pp (t = +7.4) pooled over its two cells, and on the clean cell a norm-matched scalar control reproduces only 30% of the gain, so what remains is directional rather than a step-size effect. Three further boundaries are measured. The FineWeb advantage falls from -0.031 nats at 0.3 tokens per parameter to -0.003 at 0.9, verified not to be a learning-rate artifact. Across ten vision cells the margin scales with the baseline's error rate and crosses zero near 90% accuracy. On byte-level text EchoMuon wins on clean enwik8 (t=-10.1) and loses only under input corruption, which replaces the tokenization boundary we first proposed. We release 1,815 runs, the re-run suite behind these numbers, and the reporting practice we recommend: publish the margin by which each learning rate beat its runner-up. Here a pick that won by 0.0026 in validation loss reversed under an equally valid split, with a 1.45 pp consequence on test accuracy.

Abstract: This study examines dynamic volatility transmission among WTI crude oil, gold, the U.S. Dollar Index (DXY), and the Stock Exchange of Thailand (SET) using a Bayesian TVP-VAR-SV framework. Using 4,206 daily observations from 2008 to 2025, the study combines generalized forecast-error variance decomposition, dynamic connectedness, directional measures, generalized impulse responses, and crisis-regime analysis. Volatility connectedness intensifies markedly during financial stress, with the Global Financial Crisis (GFC) recording the highest mean TCI (3.0522), compared with 1.6598 during Normal periods. Gold emerges as the dominant net transmitter during the GFC, while the USD becomes the strongest receiver, with Gold-to-Oil, Gold-to-USD, and Gold-to-SET transmission also increasing substantially. GIRF results further show that Gold shocks generate immediate responses across markets but attenuate rapidly, indicating strong yet short-lived transmission. Importantly, market roles are state dependent: during the GFC, gold becomes a major transmitter within a financial-system crisis, whereas during the Russia–Ukraine episode, gold becomes a receiver and the USD a transmitter amid geopolitical and real-economy shocks. These findings highlight that both the direction and persistence of volatility transmission depend on the nature of the underlying shock.

Abstract: Wi-Fi 7 (IEEE 802.11be) introduces key physical and MAC layer advancements including wider channel bandwidths up to 320 MHz, 4096-QAM modulation, preamble puncturing, and tri-band operation across 2.4 GHz, 5 GHz, and 6 GHz to support high bandwidth and low latency wireless applications. Despite these advances, the application-level streaming behavior of Wi-Fi 7 networks under realistic multi-factor conditions remains insufficiently characterized in the literature. This paper presents an empirical and simulation-based evaluation of Wi-Fi 7 streaming stability, assessing the combined effect of received signal strength (RSS), codec bitrate, frequency band, network layer protocol, media type, and network contention on playback delay for MP3 audio and MP4 video streams. A physical testbed comprising a Wi-Fi 7 access point and client devices is used to conduct controlled experiments across varying signal and network conditions, complemented by OMNeT++ simulation to assess MAC layer scalability under higher client densities. Results show that RSS degradation non-linearly increases playback delay and reduces link stability, particularly in the 5 GHz and 6 GHz bands. The 6 GHz band achieves the lowest average playback delay under strong signal conditions, while the 2.4 GHz band offers greater reliability at extended range. Higher codec bitrates are found to reduce playback delay due to improved protocol efficiency and more effective buffer utilization. Network layer protocol (IPv4 vs. IPv6) has negligible impact on streaming performance. Network contention, modelled through increasing concurrent client counts, introduces substantial AP side airtime partitioning effects that degrade per-client streaming quality. These findings establish a reproducible, multi-parameter performance baseline for Wi-Fi 7 audio and video streaming, providing practical guidance for network deployment and serving as a reference for future benchmarking of next-generation Wi-Fi 8 (IEEE 802.11bn) networks.

Abstract: We settle the complexity of reaching a globally Pareto-efficient allocation by mutually improving swaps along the edges of a tree. The problem is NP-complete, even though reachability of a specified full allocation is polynomial-time decidable on trees. The proof is a linear-size reduction from the top-target restriction of object reachability on trees. Its two devices are a fixed sentinel, which certifies global Pareto inefficiency whenever the target object has not arrived, and private cleanup leaves, which turn every successful target-reaching state into a globally Pareto-efficient allocation. The construction doubles the number of agents, preserves the tree property, and increases maximum degree by at most one. We also separate the standard global notion used here from Pareto maximality within the reachable set, which is the notion used in the earlier social-network allocation literature.

Abstract: We prove that the optimal sphere packing density in \( \mathbb{R}^7 \) is \( \pi^3/105 \), achieved uniquely by the scaled root lattice \( \sqrt{2}\,E_7 \). The argument is local: it suffices to show that every Voronoi cell of any unit-ball packing in \( \mathbb{R}^7 \) has volume at least~\( 16 \), with equality characterised by the \( E_7 \) kissing configuration. The Weyl group \( W(E_7) \) of order~\( 2{,}903{,}040 \) partitions the configuration space into \( 115 \) chamber types; for each, positivity of the free-volume surplus \( F_\Omega(\theta) \)on \( (0,\pi/2) \) is certified by an exact rational \( 7\times7 \) \( LDL^\top \) decomposition (minimum pivot \( 823/3360 \)) together with a complete polynomial positivity argument covering the full angular range. The large-angle range follows from a geometric analysis of the freed-root body associated to each cap-cutting direction. The proof is unconditional and self-contained, carried out in exact rational arithmetic throughout.

Abstract: Validation-based machine-learning model selection in temporally dependent, weak-signal data can produce a clear numerical winner without sufficient evidence that the choice will remain reliable out of sample. This study develops PC-Audit (Prequential-Calibration Audit), a decision-reliability framework for auditing the evidence behind validation-selected model choices. The framework combines target-date partitioning, prequential non-negative calibration, Model Confidence Sets (MCS), transfer diagnostics, dependence-aware inference, cost and execution checks, temporal stress testing, and fallback simulation. It is evaluated on nine Chinese index ETFs in 45 walk-forward cases from 2020 to 2024 under a controlled CPU-only setting. For signed returns, mean validation-to-test rank correlation is 0.090, PC-Select identifies the lowest-test-MAE candidate in 15 of 45 cases, the label-permutation value is p=0.0521, and the prequential MCS retains 4.80 of five candidates. No Holm-adjusted MAE reduction is found relative to naive references or individual-model baselines. Volatility-like targets show stronger main-period ranking transfer, but rolling/EWMA references remain competitive and 2025 stress weakens the evidence. Audit-triggered fallback does not improve mean MAE but modestly reduces worst-year MAE. PC-Audit therefore provides an auditable and reproducible reliability layer for validation-based time-series model selection rather than a forecasting-accuracy enhancement method.

Abstract: Autonomous, goal-directed AI agents are increasingly deployed on consumer edge devices — smart-home hubs, wearables, and ambient environments — where they observe context, reason over objectives, plan multi-step actions, and invoke tools with reduced human involvement. This shift exposes assurance gaps that periodic, organization-level governance frameworks were not designed to close: goal misgeneralization, unsafe or irreversible actuation, prompt injection via untrusted content, unauthorized tool invocation, inferential privacy exposure, and insufficient runtime human oversight. We present TOAIAF, a runtime AI Assurance Control Plane comprising seven components — a Policy Enforcement Engine, an Agent Runtime Governance Layer, a Trust Evaluation Engine, an Explainability Engine, a Privacy and Security Assurance Layer, a Human Oversight Boundary, and a Continuous Learning Assurance Loop — that mediates every proposed agent action through a formally specified, three-stage decision procedure (Algorithm 1): a hard capability/scope check, a non-compensatory safety gate, and a risk-weighted composite trust score. We operationalize six trust metrics (Agent Reliability, Goal Alignment, Privacy Risk, Explainability Confidence, Safety Compliance, and a newly introduced Human Oversight Effectiveness Score). We evaluate the control plane's decision logic through a synthetic, template-generated benchmark of 100 scenarios spanning smart-home, wearable, and ambient-intelligence contexts across six risk categories (benign, unsafe-policy, privacy-risk, goal-misaligned, prompt-injection, tool-misuse), against an ungoverned baseline and a keyword-based guardrail baseline, in seven experiments. TOAIAF's decision procedure achieved 100% flagging on safety-policy-violation, prompt-injection, and tool-authorization scenarios and 92.3% on privacy-risk and goal-misalignment scenarios, versus 100%/0%/75%/27.3% and 0%/0%/0%/0% for the guardrail and ungoverned baselines, respectively, with zero false restrictions across 40 benign scenarios. We report two ablation results transparently rather than selectively: removing the safety gate produced no measurable change in this dataset (23/24 flagged either way), traced to a confound in the experimental design rather than a finding about the gate's value; and including the Human Oversight Effectiveness term produced a small, mathematically-traceable negative effect on sensitivity (47/49 vs. 48/49 flagged), a concrete metric-calibration finding reported transparently. A companion reference-implementation prototype further surfaced a genuine design gap: the WARN verdict currently allows an action to execute, which for physically irreversible actions is a disclosed, unresolved limitation. We map each control-plane component to NIST AI RMF, ISO/IEC 42001, IEEE 7001–7003, and the EU AI Act, and provide a forward mapping to five external agent-safety benchmarks (AgentHarm, AgentDojo, R-Judge, SmartBench, SMH-Bench) for the real-agent evaluation this package does not yet perform. The benchmark, source code, simulation artifacts, and supplementary materials will be made publicly available through a permanent repository upon publication.

Abstract: Execution-based verification has been shown to be effective in enhancing the mathematical reasoning abilities of large language models due to its computational soundness guarantees and dependency-aware filtering. Previous works involving preference optimization often include reward models that utilize Bradley-Terry assumptions, which fail to capture the logical dependencies and execution consistency requirements essential for scientific and computational reasoning tasks. In this paper, we introduce a novel method for generating computationally sound solutions accompanied with corresponding dependency graphs for execution-consistent preference optimization. Our approach begins with the construction of a high-quality scientific reasoning dataset by incorporating UltraFeedback prompts, base model generations, computational verification, and execution consistency results. Next, we construct dependency graphs by extracting reasoning step expressions, the computational prerequisites needed for the expressions, and the derivability relationships of the expressions from the previously collected dataset. Based on this extracted information, we generate corresponding execution consistency scores to accurately capture the mathematical verification process. Appending the generated execution consistency scores to each reasoning step results in data consisting of paired filtered reasoning steps and their corresponding execution consistency scores. Training Llama-3-8B and DeepSeekMath-7B with this corpus achieves substantial improvements across scientific reasoning domains: +17.0\% on MATH, +15.1\% on GSM8K, while extending our Scientific Feasibility Control framework to achieve 50.1\% accuracy on PhyX multimodal physics reasoning—outperforming DeepSeek-R1 (49.8\%) and OpenAI o3-mini (48.2\%)—with 91.7\% scientific validity coverage at \(\alpha = 0.10\) confidence level and 73\% reduction in scientific law violations across architectures, leading to the creation of the CCPO family of models.

Abstract: This work introduces a unified mathematical framework within the Zero Pairs Interaction Functional (ZPIF) framework for understanding the distribution of prime numbers through three novel concepts: spectral orbits, zero mirrors, and the imaginary complement. We demonstrate that the non-trivial zeros of the Riemann zeta function follow closed spectral orbits with defined phase angles, revealing an organized structure behind prime distribution. A remarkable symmetry is uncovered: for every zero γn, there exists a mirror zero γN+1-n such that the sum of their fractional parts equals approximately 1. This symmetry is further confirmed through logarithmic and inverse transformations. The framework provides a natural explanation for the stability of prime gaps and the coherence of zero interactions without requiring external regulation. Numerical simulations using the first 10,000 non-trivial zeros confirm the convergence and stability of the proposed model with high precision.

Abstract: This paper provides a presentation and structural overview of the book Evolving Intelligence. The Generative Path from Information and Computation to Cognition and Intelligence. The monograph is forthcoming with World Scientific in 2026. Evolving Intelligence develops a naturalistic, generative framework for understanding how intelligence emerges from the organization of nature. Rather than beginning with human reasoning or with artificial intelligence, the book traces a continuous path from physical and chemical organization in nature through life and cognition to human symbolic culture, artificial intelligence, and hybrid human-AI systems. Its central framework, ICON (Information-Computation-Cognition Naturalism), connects these domains through morphology, interaction, observer-agency, information, computation, cognition, consciousness, and intelligence.The guiding question is not simply “What is intelligence?” but “How can intelligence arise at all within nature?” The book therefore shifts attention from intelligence as an isolated human faculty to the generative processes through which increasingly complex forms of organization, sensing, memory, anticipation, agency, meaning, and problem solving emerge.

Abstract: In our previous works we developed a neuromorphic decoder of intended movements of tetraplegic patients using ECoG recordings from brain motor cortex composed by a Motor Control Decoder (MCD) and a Neural Response Decoder (NRD). It was an actor-critic structure able to adapt via reinforcement learning the MCD (actor) based on NRD (critic) predictions. In this paper, we continue the development of novel neuromorphic methods for BMI aiming at further improvement of their functionality. First of all, feature extraction from ECoG data was improved by introduction of additional filtration of raw data. Second, the auto adaptive ability of NRD was upgraded using Intrinsic Plasticity (IP) tuning mechanism. Third, the mechanism of MCD decision improvement using NRD predictions was changed considering labeling approach of NRD training data. The MCD-NRD training and testing was done on a bigger data base and its improved accuracy was demonstrated on new test data sets as well.

Abstract: Assessing scale mismatch, visual obstruction, and circulation conflict from two-dimensional drawings remains difficult in public art design. Finally, a VR simulation framework is developed, which integrates real scale calibration, space drawing, synchronous behavior trajectory, and dynamic scene analysis. Continuous interaction records were converted into behavioral, semantic, and pedestrian envelope. In a 10-week controlled trial, 180 students were enrolled in the study, with class level stratification, standardized familiarisation, matching tasks, and blind scores. About 4.3 million interaction records supported comparisons with single-view, multi-view, and distance-aware baselines. The proposed approach achieves 4.4% visual area error, 0.903 occlusion-recognition F1, and 0.872 passage-collision F1, and 28.6 ms for each key point. Ablative analysis has demonstrated the contribution of trajectory correction, adaptive weighting, semantic occlusion, and passage-envelope modeling. Compared with the conventional design, the VR group had lower elevation, better visual field visibility, fewer circulation conflicts, and higher site suitability scores, while the viewpoint coverage mediated 26.1% of the overall effect.

Abstract: The paper presents a parallel implementation of an algorithm for modeling a single dislocation source based on an ensemble of physics‑informed neural networks (Physics‑Informed Neural Networks, PINN). The task of approximating the Berlage pulse, which describes the displacement of a dislocation source under rock deformation, is considered; this is a key element in the study of high‑frequency geoacoustic emission arising during the failure of geomaterials under the influence of mechanical stresses. A method for decomposing the time domain into an arbitrary number of subdomains (\( n=1,\dots,8 \)) is proposed, each of which is trained on a separate computational process. To combine the predictions of the subdomains, the MultiDomainPINN ensemble architecture is used, which ensures smooth coordination of solutions at the boundaries of the overlapping regions. The quality of the approximation is assessed using statistical metrics: mean squared error (MSE), root mean squared error (RMSE), normalized root mean square error (NRMSE), coefficient of determination (\( R^2 \)), and maximum absolute error, which allows for a quantitative comparison of the PINN solution with the analytical Berlage impulse. To quantitatively assess the computational efficiency of the parallel implementation, TAECO metrics are used, which allow for evaluating the algorithm’s efficiency. The assessment was carried out at the inference stage of trained models for parameters that were not involved in the training, by comparing them with successive runs of the 4th‑order Rosenbrock method. MultiDomainPINN provides high performance, with a 70--76‑fold speedup compared to the Rosenbrock method while maintaining comparable accuracy (MSE at the level of \( 10^{-10} \)—\( 10^{-8} \)). The developed software is scalable and can be adapted for a wide range of tasks related to modeling seismic signals and wave processes, including machine learning tasks in geophysics, seismology, and solid mechanics.

Abstract: A pure geometric construction is presented that recovers a substantial portion of classical mathematics from a single primitive figure: the non-proper Archimedean conical helix. Starting from three geometric axioms that govern continuous coiling, radial growth under maximal cohesive spread, and indecomposable flux conservation, the basic objects of Euclidean geometry are reconstructed—points, lines, planes, congruence, circles, triangles, parallels, and ruler-and-compass constructions—strictly as geometric shadows of the helix. From the same structure, obtaining the fundamental constants π, the golden ratio, the Fibonacci sequence, the trigonometric ratios, the imaginary unit, and the prime numbers (as irreducible cycle lengths). Geometric arithmetic operations, the successor function, and an induction principle are then introduced, followed by a complete pure-geometric development of the calculus (limits, derivatives, integrals, series, and vector calculus), real and complex analysis, measure theory, and the core structures of differential geometry (tangent bundle, Riemannian metric from flux, covariant derivative, curvature as holonomy, geodesics, and the Laplace–Beltrami operator). The construction concludes with geometric sets, geometric functions, and a pure geometric form of wave–particle duality embodied by the helix itself. Throughout, no external arithmetic, analytic, or set-theoretic primitives are assumed; every notion is derived from the helix and its intrinsic geometric operations.

Abstract: Let p > 3, and let A be a finite pre-Lie ring whose additive group has p-power order and which satisfies Property 3 of Smoktunowicz. We give a two-part answer to Kourovka Notebook Problem 21.123, distinguishing the typographically literal formulas from an explicit factorial-compatible corrected prescription.First, the formulas displayed around the proposed modified group-of-flows construction do not, when read typographically literally, define a total operation. Several local indices do not align, an arbitrary set-theoretic right inverse of multiplication by p need not be iterable, and the movement identity printed in Remark 63 of the source need not hold with its stated shielding exponent. Second, after the factorial-compatible local adjustments and replacement of that identity by the block-shielding law proved here, the corrected partial prescription has a unique conservative totalization. Here conservative means that the totalized value agrees with every corrected literal section iterate at every point where that iterate is defined. The result covers both natural readings of the pullback notation: a direct set-theoretic \( p^{s_j} \)-pullback and a repeated p-section wherever the latter is defined.We construct this totalization through protected divided operator words. Every p-pullback is separated from the next pullback and from both endpoints by enough operator factors for Property 3 to annihilate all root-choice and additivity defects. This gives a choice-independent noncommutative functional calculus on finite abelian p-groups, stable under products and substitution and supporting protected exponential, logarithm, and Baker-Campbell-Hausdorff series.A faithful affine representation of the pre-Lie ring yields the decisive closure statement: the protected BCH series returns to the affine image, so the protected exponentials are closed under actual composition. Their orbit map is bijective, and transport of composition defines a group law on A whose \( \lambda \)-maps are additive automorphisms. Thus the conservative totalization is a left brace. If one fixed section has all required literal iterates defined, its literal construction is exactly this group law. Consequently, the typographically literal reading does not yield a total operation, whereas the explicitly corrected reading formalized in this paper gives an affirmative answer to the mathematical question underlying Problem 21.123.

Abstract: Can AI improve AI? AI agents can now run experiments, modify code and training pipelines, and iteratively improve AI artifacts—bringing a long-standing idea closer to an empirical research question. Yet the rapidly expanding literature remains fragmented across long-horizon agents, AI for AI (AI4AI), self-improvement, and recursive self-improvement, making it difficult to tell how much progress has actually been made. This survey synthesizes evidence from hundreds of studies to answer this question. We organize the emerging AI4AI landscape around a simple question: how far can an AI system reliably carry an improvement process from idea to verified result? Across model design, agent harnesses, benchmarks, automated research, and self-modifying systems, we examine what current systems can do, how progress should be evaluated, and where claims of self-improvement remain unsupported. A striking pattern emerges in the information flow within the taxonomy– benchmark-model-harness structure of this survey. AI systems increasingly excel at the work of improvement—planning, coding, experimentation, optimization, and repair—but humans still largely determine the goals, evaluation criteria, and what ultimately counts as progress. Moreover, strong performance on individual components rarely translates into reliable end-to-end AI improvement. We call this composition gap. Today’s systems can already produce impressive improvements under bounded conditions, but evidence for reliable research judgment, causal experimentation, persistent gains, and compounding improvement remains limited. By separating demonstrated capability from extrapolated autonomy, this survey maps what AI4AI can do today—and what must change before AI can reliably improve AI itself. The eve ends the moment a system, for the first time, reliably strengthens its successors without humans specifying the goals or evaluation criteria. We hope this taxonomy and survey will help that moment arrive a little sooner.

Abstract: AI scientists generate hypotheses, propose experiments, and analyze datasets, and several have produced findings that were confirmed in the laboratory. Although they are opening a path to autonomous discovery, the loop from hypothesis to experiment to revised hypothesis is still often anecdotal. Here, we consider how that loop could be closed to enable AI-driven biomedical discovery. Closing the loop requires AI scientists that reason over biomedical knowledge and data across horizons of days to months. It also requires verifying hypotheses with computational tools and experiments in biological laboratories and clinical environments. Agentic and reasoning methods need to learn over time from sparse, delayed, and noisy feedback that verifiers return. In mathematics and program search, where candidate solutions can be verified at negligible cost and often as soon as they are generated, methods that generate and score large numbers of candidates have improved bounds on problems that had been open for decades. In biomedical sciences, comparable progress requires closing the loop where each test is slow, costly, and partially informative. Closed-loop AI-driven biomedical discovery can enable a mode of research in which AI continuously reviews hypotheses against accumulating experimental results. Scientists choose which experiments to pursue, and AI maintains a record that links each result to the hypothesis that produced it.

Abstract: Suppose that R is a commutative multiplicative hyperring with identity and \( S\subseteq R \) is a multiplicatively closed subset. In this paper, we introduce and study the notions of weakly 2- prime hyperideals and weakly S-2 prime hyperideals, which extend the concepts of prime, 2- prime , weakly prime, weakly S - prime, and S-2 prime hyperideals in multiplicative hyperrings. A proper I of R is called a weakly 2- prime hyperideal if whenever \( \left\{0\right\}\neq x\circ y\subseteq I \) for \( x,y\in R \), then either \( x^2\subseteq I \) or \( y^2\subseteq I \). Furhermore, a proper hyperideal I with \( I\cap S=\emptyset \) is said to be a weakly S-2 prime hyperideal if there exists an element \( s\in S \) such that for all \( x,y\in R \) with \( \left\{0\right\}\neq x\circ y\subseteq I \), we have \( s\circ x^2\subseteq I \) or \( s\circ y^2\subseteq I \). We examine various basic properties of these hyperideals and investigate their characterizations under good hyperring homomorphisms, quotient hyperrings, and fraction hyperrings \( S^{-1}R \). In addition, we generalize the classical 2- prime avodience theorem to weakly 2-prime and weakly S-2 prime hyperideals, and we delineate the structural relationships among prime, 2- prime, weakly 2- prime, weakly S- prime, S-2 prime, weakly \( S-2 \) prime hyperideals.