Computer Science (arXiv)

We establish a central limit theorem in Rényi divergence for independent and identically distributed lattice random variables $X_1, \cdots, X_n$ with zero mean, unit variance, and maximal span $h>0$. Let $S_n=(X_1+\cdots+X_n)/\sqrt n$. Let $Z_n$ denote the standard Gaussian distribution quantized on the support lattice of $S_n$. For every $α>1$, with $β=α/(α-1)$, we prove that the Rényi divergence $D_α(S_n\|Z_n)\to 0$ if and only if the divergence is finite at some convolution level and the strict sub-Gaussian condition $$ \mathbb E e^{tX}<e^{βt^2/2},\quad t\in\mathbb R,~ t\ne0 $$ holds. Under these conditions, we further derive an Edgeworth-type asymptotic expansion of the divergence to arbitrary order. These results provide a lattice counterpart of the Rényi entropic central limit theorem for continuous random variables due to Bobkov, Chisyakov and Götze (\emph{Ann. Probab.} \textbf{47} (2019), 270--323).

We present AnyTalk, a novel method for generating 3D speech animations for arbitrary characters without requiring any animation data. While existing audio-driven 3D speech animation methods rely on character-specific training data or laborious rigging/re-meshing, AnyTalk circumvents these limitations by leveraging recent video diffusion models trained on extensive video datasets. We first adapt a pre-trained video diffusion model to a target character through our Character-specific Fine-tuning (\textit{CsF}) technique. By fine-tuning on rendered images of the 3D character paired with zeroed-out audio embeddings (representing "no motion"), we eliminate the need for animation data while preserving the motion prior of large-scale video diffusion model. We then uplift the resulting talking-head video into a 3D speech animation by estimating blendshape parameters through a proposed optimization process. AnyTalk enables lip-synced animations across diverse face meshes and blendshape configurations, significantly reducing manual effort and data requirements. We further enhance usability by distilling AnyTalk into a streamlined network, $\text{AnyTalk}_{RT}$, thereby enabling real-time performance. By leveraging talking-head video generation, our method broadens access to audio-driven speech animation technology for arbitrary characters. The code is publicly available at https://serin-yoon.github.io/projects/anytalk/.

UAV on-board vision systems are widely used for different activities, including monitoring in no-fly zones. In this case, the vision-equipped UAV streams a video to a ground server where an operator assists its activities. The latency of video transmission has a profound impact on the effectiveness of the operator assistance. However, most techniques available for video transmission still incur significant latency costs. In this paper, we propose a graph convolutional neural network-assisted (GCN-Assisted A2C) deep reinforcement learning (DRL) system model to find the optimal pixel-correlated area of a suspicious object. We combine the Lagrangian dual form with gradient descent to prevent lack of convergence and over- and under-penalization constraint violation during latency optimization. The proposed system model sends a sub-group pixel-correlated area of the frame from the UAV to the server rather than the transmission of the whole video frame. The proposed framework utilizes the GCN model to explore hidden representations of feature-correlated groups of pixels. Moreover, the GCN supervises the A2C model, which selects a subgroup to enhance transmission latency, thus supervising the training of UAV actions in A2C. Experimental results show that GCN-assisted A2C reduces video frame transmission latency together with false detection rate in UAV vision systems over other DRL and state-of-the-art models.

By Tarski's undefinability theorem, no consistent classical formal system that includes arithmetic can define its own truth predicate. Reflective Grounded Arithmetic (RGA) is a powerful arithmetic whose universal quantifier is grounded in its own reflected proof search, and whose paracompleteness circumvents Tarski's theorem. This paper presents a machine-checked Isabelle/HOL development that defines a truth predicate for RGA's full language, quantifiers included, as an internal term of RGA itself. This term is compiled from a primitive-recursive decider for its operational semantics, and proven adequate in both directions. Around this predicate the development closes a square of metatheorems: for every formula RGA proves, RGA derives the formula's internal truth; every grounded-true formula is internally provable; internal truth implies internal provability; and the consistency of RGA follows. The two directions run on disjoint internal machines---a certified decider and a certified proof-checker, both RGA terms. Reaching these results involved substantial ordinary reasoning carried out within RGA: coded syntax and substitution, compiled primitive-recursive functions with symbolic unfolding laws, internal strong induction, and a verified proof-checker for the system written in the system's own formal language. The development thus demonstrates along the way that RGA is a workable formal system supporting nontrivial mathematical reasoning.

In clinical motor imagery brain-computer interface (MI-BCI) decoding, cross-day transferability and online operation remain two critical challenges. Hypergraphs can improve transferability by capturing higher-order sample relationships, yet existing hypergraph-based methods for online emotion recognition neglect the cross-day benefits of Riemannian geometry widely adopted in EEG transfer learning. To bridge this gap, we propose the Multi-feature Riemannian Hypergraph (MRieHy), a framework tailored for online test-time adaptation in MI-BCI decoding that leverages Riemannian geometry to strengthen cross-day transferability. MRieHy first computes Riemannian means of covariance matrices from cross-day training data to align multi-day distributions. It then constructs a hypergraph over covariance matrices using Riemannian distance, complemented by a second hypergraph over deep features built with cosine similarity. The two hypergraphs are fused via adaptively learned combination weights, jointly optimized with the label projection matrices. During online testing, MRieHy maintains a first-in-first-out buffer of recent samples, performs Riemannian alignment on the buffered data, and decodes with the learned hypergraph. Extensive experiments on a private four-class ECoG dataset and two public four-class EEG datasets validate that MRieHy achieves notable performance gains over state-of-the-art baselines.

Electric vehicles (EVs) represent a critical intelligent terminal within the Internet of Things (IoT). Despite the year-on-year growth in EV penetration, the highway driving experience still requires improvement. Accurate prediction of EV highway charging behavior is crucial to addressing this issue. This paper introduces a novel bounded rationality framework to analyze highway charging decisions. Specifically, we utilize prospect theory to capture the tendency of drivers to reserve more electricity than theoretically necessary. We then propose a Bayesian game in which EV drivers, unaware of others' decisions, aim to minimize costs, including range anxiety, charging fees, and queuing time. To gain insights into the game, we prove the existence and uniqueness of the Bayesian Nash Equilibrium in two practical scenarios. Our numerical experiments, based on real-life data, demonstrate that drivers' risk aversion tendency significantly influence EV charging decisions, charging demand, queuing lengths at charging stations, and the departure rate on the highway network. Furthermore, our strategy reduces cumulative EV cost and CSs' charging costs compared to other benchmarks.

This study evaluates the effectiveness of utilising large language models (LLMs) to retrieve targeted segments from delivered video recordings to answer student questions in introductory programming environments. By restricting AI to identifying existing, educator-verified media rather than generating open-ended text, this approach aims to mitigate common pedagogical risks such as generative hallucinations and cognitive bypassing. We benchmarked three distinct models, two proprietary (Gemini 3.1 Pro and GPT 5.4 Pro) and one open-weight (Qwen3.5 397B), against a human lecturer's manual video selections. An automated judging framework subsequently assessed the outputs for relevance, sufficiency, redundancy, and the presence of extraneous material. While the AI-retrieved timestamps rarely shared exact overlaps with the human baseline, the proprietary models achieved near-parity with the expert in delivering sufficient and highly relevant answers. Furthermore, a pilot deployment of this retrieval system in a large C programming cohort (n~=900) demonstrated strong user engagement, with students primarily utilising the tool to review foundational concepts. By leveraging AI to retrieve established lecture material, this approach shows potential for a reliable, high-fidelity pathway for safely integrating LLMs into novice computing courses.

Proportionality (PROP) is one of the simplest fairness criteria for allocating items among agents with additive preferences. With indivisible chores, however, PROP is not always satisfiable. We study proportionality up to any item (PROPX), which requires every agent to satisfy proportionality after any chore is removed from her bundle. Under strictly positive costs, we settle the weighted compatibility question negatively: weighted PROPX and Pareto optimality are incompatible already for two agents and four chores. Moreover, for every $n\geq3$, we give an $n$-agent, $(n+1)$-chore counterexample whose shares can be arbitrarily close to equal. These counterexamples are item-minimal: under strictly positive costs, weighted PROPX and Pareto optimality are always compatible when the number of chores is at most the number of agents, and they are compatible for two agents with at most three chores. Our impossibility result contrasts with the compatibility theorem of Mahara (2026) for weighted envy-freeness up to one item (EF1) and Pareto optimality .

Identification of dominant polynomial-chaos modes is usually formulated as a sparse-regression problem on a sampled multivariate polynomial dictionary. We develop coded Hankel polynomial chaos (CH-PC), a complementary spectral formulation for dominant-mode identification. A finite generating transform converts PCE coefficients into a coefficient-generating polynomial, and evaluation along a geometric phase orbit produces a finite exponential sum. Its model order and spectral nodes are encoded by low-rank Hankel matrices, while coordinate phase shifts attach root-of-unity labels from which the full polynomial multi-indices are recovered. Coordinate-shifted probes are combined as common-node snapshots, and independent phase encodings provide redundant representations when a single spectral encoding is poorly conditioned. For finite observations, population, finite-data, and observed probes are kept distinct: sampling or quadrature error and observation error enter as separate Hankel perturbations, which are then connected to spectral stability, discrete decoding, and phase voting. For tensor-product candidate sets, the generating kernel factorizes into one-dimensional sums and can be evaluated without assembling the full multivariate PCE design matrix. Numerical experiments on sparse Legendre benchmarks and a stochastic Darcy problem illustrate exact recovery, noise stabilization, unknown-order identification by phase persistence, and dominant-mode recovery for a PDE-generated quantity of interest.

Let $A, B \in \mathbb{Z}_{\ge 0}^n$ be nonnegative vectors and let $t = |\operatorname{supp}(A \star B)|$. We give a Las Vegas algorithm that computes $A \star B$ in $O(t \log t)$ expected time. More generally, for every $0 < δ\le \frac{1}{2}$, the algorithm terminates within $O(t \log t \log \frac{1}δ)$ time with probability at least $1 - δ$. The algorithm uses dense convolution, linear hashing, and the length reduction of \cite{BFN22}. Its main ingredient is a carry-free representation of the indices as vectors of constant dimension $d$ whose coordinates have size $O(t / \log t)$. We can then take our hash function to be the inner product with a random element of $\mathbb{F}_p^d$ for a prime $p$ of size $Ω(t / \log t)$: this preserves addition and gives collision probability exactly $1/p$, while identities regarding the moments of the vectors identify and recover the isolated terms as in \cite{BFN22}. Our expected running time matches that of Jin and Xu~\cite{JX24} while using substantially different tools and yielding a simpler algorithm. Note that their algorithm also terminates within $O(t \log t)$ time with probability at least $1 - \frac{1}{t}$, while our tail bound is weaker.

Adolescent Idiopathic Scoliosis (AIS) is a prevalent spinal deformity in adolescents that, if left untreated, can result in severe health outcomes. Traditional screening methods are limited by subjective interpretation, reliance on professional expertise and low scalability. To address these challenges, we present ScoliGait dataset, which comprises 1,516 gait video clips paired with corresponding X-ray records. We also introduce TokenSTFormer, a novel model that tokenizes spatial and temporal semantics to enhance feature representation and convergence. Our model achieves state-of-the-art performance, surpassing vanilla Vision Transformer encoder across key metrics, including accuracy of 0.79. This study highlights the potential of leveraging holistic motion features derived from gait video and attention-based models for scalable, cost-effective AIS screening, paving the way for future clinical applications in scoliosis detection.

How should we assess whether large language models can perform mathematical invention? I argue that this question is currently underspecified: mathematical creativity is not one capacity but several mechanistically distinct modes of meaning-making - reflexive introspection on mathematical practice, analogical import from the sciences, problem-driven construction, and the bridging of distant domains - together with a further, cross-cutting distinction between meaning pursued because a pattern was observed and meaning pursued because it is strategically wanted, a distinction I develop through the case of conjecture-formation. These mechanisms are likely non-substitutable, so that competence in one does not transfer to the others. Grounding each in a historical case study and in an architecture-level account of current transformer-based systems, I suggest that today's models concentrate their competence in modes shaped by recombination and search over existing building blocks; if that description holds, the remaining modes are out of reach in principle, not just slower - though whether it holds is itself the open, empirical part. Because proof is getting cheaper as AI improves at generating it - a shift the field's own leading voices are now diagnosing - mathematical value is migrating toward the modes current systems cannot yet perform, and evaluations of AI mathematical ability should be organized around this taxonomy rather than around aggregate benchmarks that conflate it.

As agentic tasks grow in complexity, LLM agents increasingly rely on experiential memory to reuse procedural knowledge across tasks. Effective memory design must jointly address what to store, how memory is structured and retrieved, and how memory evolves. Existing systems tackle each only partially: they store trajectories, insights, or workflows as isolated entries, discarding compositional relationships among subtasks and reusable skills; retrieve by flat embedding similarity that ignores relational signals; and maintain memory without leveraging its relational structure. We propose HyperSkill, a hypergraph-based memory framework that jointly improves all three. HyperSkill represents memory as a hypergraph with two node types, subtask steps and reusable skills, where each hyperedge links the subtasks and skills from a single trajectory. Dual-path retrieval queries both subtask and trajectory levels, ranking skills by co-occurrence across retrieved trajectories. Periodic structure-informed maintenance prunes low-utility nodes and merges redundant skills via quality-weighted propagation. Across xBench, GAIA, and WebWalkerQA with GPT-4o and Qwen3-30B-A3B, HyperSkill outperforms ten memory baselines, yielding gains of up to +11.51 on GAIA and +11.18 on WebWalkerQA.

Technical debt is treated almost universally as an engineering pathology. This paper argues that under the conditions defining early-stage software work (high hypothesis uncertainty, cheap experiments, and the freedom to abandon), deliberately incurred technical debt is a rationally priced financial instrument: a call option on the validated product, purchased at a discount that is largest exactly when uncertainty is highest. We make three contributions. First, a demarcation: debt is strategic when its expected cost loads on the success branch of the venture (repaid only if the hypothesis validates) and toxic when it imposes unconditional cost while held (security exposure, data loss, corrupted experimental signal), a boundary stated formally that renders the popular "prudent vs. reckless" intuition testable. Second, a sequential model: a finite-horizon dynamic program over belief and debt stock yielding four results: a shadow price of debt equal to the risk-discounted probability of repayment; a technical-debt overhang (the belief threshold for scaling rises with the debt stock, proved via an envelope Lipschitz bound); a refactoring-pivot theorem (optimal repayment concentrates at the commitment boundary, predicting the practitioner-reported refactoring burst at product-market fit, registered here as a falsifiable prediction); and a volatility result under risk-neutral valuation. A pivot-salvage correction shows the folk rule "maximum debt at maximum uncertainty" fails whenever failure redirects rather than terminates the venture and the salvage differential clears the discounted cost premium. Third, a two-test primary empirical program (validation-event refactoring timing; the first repository measurement of pivot salvage), pre-registered with frozen analyzers before any data contact. Calibrations are illustrative, not estimates.

Retrosynthesis is a cornerstone of drug discovery and organic synthesis. While data-driven deep learning models have shown remarkable progress, they autonomously learn reaction patterns from extensive datasets with limited integration of established chemical knowledge as priors. To address this limitation, we introduce RetroMPA, a molecular property-aware, post-hoc enhancement module that injects chemical knowledge into the retrosynthesis pipeline. Rather than functioning as an independent SMILES sequence generator, RetroMPA is a broadly applicable, model-agnostic chemical filter designed to recalibrate and optimize the predictive pathways of existing algorithms. This plug-and-play framework integrates seamlessly with a range of data-driven retrosynthesis methods, enhancing outputs without modifying model architecture or requiring resource-intensive retraining. By leveraging a property-aware latent embedding space, RetroMPA consistently improves top-1 accuracy across eight representative retrosynthesis models by an average of 5.50% on USPTO-50K. Furthermore, we validate its scalability on the large-scale USPTO-Full dataset, achieving an average improvement of about 2.03% across both template-based and template-free architectures. Wet-lab experiments provide preliminary support for the practical utility of the framework. These syntheses confirmed viable, previously unreported substrate combinations for classic reaction paradigms---specifically, Suzuki-Miyaura coupling, Bucherer reaction, and Friedel-Crafts acylation---suggesting that RetroMPA can operate beyond mere data fitting. The code is open-sourced at https://github.com/MengzhouLu/RetroMPA.

Cold Start Active Learning (CSAL) is important in improving the performance of a medical image segmentation model with low annotation budget by querying a small subset for annotation from an unlabeled training set. Existing CSAL methods typically rely on inefficient dataset-specific Self-Supervised Learning (SSL) to map the unlabeled images into a feature space for sample selection. Recently, the advent of foundation models such as the Segment Anything Model (SAM) offer a promising alternative as the pre-trained model can provide strong generalizable feature embeddings, and allow high performance in downstream tasks after fine-tuning (adaptation). However, how to systematically exploit SAM's inherent embeddings for cold-start sample selection during adaptation with low annotation budget remains underexplored. To address this, we propose an extended SAM-based Uncertainty-guided Feature Weighting (SUGFW+) framework for CSAL and adaptation of SAM. Specifically, it leverages the SAM for Patch-level Feature and Uncertainty Calculation (PFUC), and introduces a Patch-based Global Distinct Representation (PGDR) module that aggregates patch-level embeddings into highly discriminative, uncertainty-aware image-level features. These features are then utilized by a Greedy Selection with Cluster and Uncertainty (GSCU) strategy to combine diversity and uncertainty during sample selection. Unlike prior CSAL methods that decouple sample selection from model training, SUGFW+ tightly integrates these two stages via an Uncertainty-Prompted Fine-Tuning (UPFT) process of SAM in model training. Extensive experiments on four public datasets demonstrate that SUGFW+ achieves state-of-the-art performance against existing CSAL methods. Code is available at https://github.com/HiLab-git/SUGFW-plus.

We study fair division of indivisible goods when agents' valuations are accessed only through ordinal comparisons between bundles, with arbitrary tie-breaking. In this model, even deciding whether a given allocation is envy-free up to one good (EF1) can be impossible. This suggests explicit fairness certificates as a natural algorithmic object. Our main contribution is a certificate-preserving scaling framework, which recursively contracts goods, solves a smaller instance, and expands the solution while repairing an explicit envy-eliminating witness certificate. For arbitrary identical monotone valuations, this yields a certified EF1 allocation for $n$ agents and $m$ goods using $O(n \log n \log(m/n))$ comparison queries, within an $O(\log n)$ factor of the $Ω(n \log (m/n))$ communication lower bound. For identical additive valuations, we additionally obtain a $1/2$-MMS guarantee within the same query complexity. For non-identical additive valuations, exploiting our EF1+$1/2$-MMS algorithm, we accelerate the existing matching-based PROP1+$1/2$-MMS framework, improving the query complexity from $O(n^4\log m)$ to $O(n^3\log m)$. Finally, we study the structure of such certificates through $k$-witness EF1, a hierarchy between EF1 and EFX.

One-Time Passwords (OTPs) have become a common option for multi-factor authentication in several applications. For instance, during website login processes, OTPs are often used in conjunction with traditional text-based usernames and passwords to verify whether the access request originates from a legitimate human user rather than an automated agent. However, in scenarios involving automated connections and system-to-system interoperability, Time-Based One-Time Passwords (TOTPs) may be required to establish secure connections and access Web Services (WSs). Therefore, this study focuses on exploring the development of a quantum-safe web service architecture. The proposed approach achieves transmission security management by implementing Transport Layer Security (TLS) and HyperText Transfer Protocol Secure (HTTPS) based on Post-Quantum Cryptography (PQC). Furthermore, web service security management is realized through the construction of keyed-Hash Message Authentication Code (HMAC)-driven TOTPs. Within the experimental environment, this study evaluates and compares the computational performance of the Secure Hash Algorithm-2 (SHA-2), SHA-3, Ascon-Hash256, and SM3. The required computation time under different hardware resource conditions is analyzed for future web service deployment.

Diffusion and flow matching models have made significant progress in text-to-image generation, yet high computation, quadratic complexity, and large memory footprint hinder high-resolution synthesis and edge deployment. We propose Nexus, which integrates sparse architecture, linear complexity, and low-bit quantization. It combines MoE feed-forward layers, gated DeltaNet attention, and per-expert low-bit training to reduce computation and memory. Their joint optimization allows Nexus to achieve generation quality comparable to mainstream models such as SDXL and SD3 while delivering markedly higher inference efficiency. Experiments on COCO and LAION validate its effectiveness.

Open-vocabulary referring segmentation in 3D Gaussian Splatting (3DGS) requires a neural model to select Gaussian primitives according to free-form language expressions. Existing 3DGS-based methods usually rely on global text-region similarity, which is weak for queries involving attributes, reference objects, spatial relations, and fine-grained parts. This often causes target-reference confusion, granularity mismatch, part-whole leakage, and relation violations. We propose QAGaussian, a query-adaptive neural reasoning framework for language-guided Gaussian primitive selection. QAGaussian first learns query-conditioned multi-scale Gaussian slots as differentiable candidates whose receptive fields are shaped by the input expression. It then builds a relation-aware slot graph with language-conditioned edge weighting to propagate target-reference, attribute, part-whole, and contextual evidence. A granularity-adaptive router softly combines region-level, object-level, part-level, attribute-aware, and relation-aware mask branches, followed by relation-constrained refinement for spatial, part-whole, attribute, and geometric consistency. QAGaussian is pretrained only on Mosaic3D-5.6M for Gaussian-text alignment and evaluated on independent benchmarks without target-dataset fine-tuning. It achieves 47.2 Avg. mIoU and 63.2 Avg. F1, outperforming the strongest 3DGS referring baseline by 2.7 mIoU points and 2.9 F1 points. It also improves Part-mIoU from 38.6 to 43.4, Rel-mIoU from 44.4 to 50.8, and reduces target-reference confusion from 10.8 to 7.4. These results demonstrate that query-conditioned slot learning, relation-aware graph reasoning, and adaptive routing provide an effective neural modeling strategy for open-vocabulary referring segmentation in 3DGS. The code is available at https://github.com/zqeslwyz/QAGaussian.

Coresets distill large datasets into small, representative subsets for efficient downstream learning. Yet Optimal Transport (OT)-based selection typically requires intensive computation of transport plans, limiting scalability. We introduce a scalable Sinkhorn coreset method that permits closed-form updates of the entropically regularized OT coupling by allowing non-uniform coreset weights. This produces centroids that generalize k-means via soft assignments. We establish asymptotic consistency of the selected measure and Lipschitz stability to data perturbations, providing accuracy and robustness guarantees. Across synthetic and real-world benchmarks, the proposed method achieves competitive or improved approximation quality while substantially reducing runtime compared to Wasserstein- and standard Sinkhorn-based coreset selection, especially at large scale.

Generative image semantic communication converts an image into a text description and then performs text-to-image reconstruction at the receiver via diffusion-based generative models. This paradigm has attracted broad attention due to its extremely low bandwidth cost. However, existing methods still face two critical bottlenecks across image-to-text (I2T) semantic extraction at the transmitter and text-to-image (T2I) semantic reconstruction at the receiver: (i) semantic loss and distortion in I2T, where holistic image descriptions may omit fine-grained object attributes and spatial-position information, causing the generated text to deviate from the original image semantics; and (ii) insufficient semantic faithfulness in T2I, where even with the same semantically faithful text description, different initial noise settings may lead diffusion-based reconstruction to produce images with different levels of semantic consistency with the original image. These issues jointly limit the semantic faithfulness of image reconstruction. To address them, we propose TISC, a text-driven image semantic communication framework tailored for faithful reconstruction. TISC incorporates two key designs: (1) Tree-Structured Attribute Semantic Extraction (TSASE), which decomposes semantic extraction into global scene, background, and object-level attribute descriptions, covering spatial position, shape/pose, color, material, and other physical attributes for each detected object; and (2) an Initial Noise Optimization (INO) mechanism, which selects an initial noise seed at the transmitter according to a comprehensive similarity score that jointly considers visual and semantic consistency. Experiments on multiple datasets show that TSASE improves object-position recovery and semantic description faithfulness, while the INO parameter study supports the adopted configuration for noise selection.

Multivariate time-series forecasting faces a structural dilemma: sharing one temporal predictor across variables is parameter-efficient but forces heterogeneous variables through an identical history-to-future map, whereas learning an independent predictor per variable restores flexibility at a cost that grows with the product of variable count, context length, and horizon. We argue that this dilemma dissolves once the object being compressed is the forecasting operator rather than the observed series. Auditing per-variable linear history-to-future maps across standard benchmarks, we find that a phase-locked seasonal component paired with a compact residual operator outperforms a dense phase-blind reference in most audited settings. The residual transport is also directional: lag-invariant alternatives consistently underperform asymmetric history-to-future maps. Guided by this structure, we propose AsyTO, an Asymmetric Temporal Operator that factorizes the tensor of per-variable operators into shared but distinct history-reading and future-writing temporal modes with per-variable mode-wise gains, complemented by a low-rank periodic prototype and a cycle-separable factorization of the temporal modes. Each forecast reads only its own variable's history, so parameters and compute grow linearly in the number of variables. Across eleven benchmarks and multiple forecast horizons, AsyTO attains the best lightweight error in 30 of 44 dataset-horizon settings, locating at the accuracy-compute Pareto frontier.

Graph neural networks are commonly described through family-specific equations whose notation obscures shared computations and structural differences. We introduce a common layer equation that represents covered architectures through seven components: an update domain, channel set, propagation bank, per-channel message maps, channel-fusion operator, ego/residual map, and update map. The central factorization separates where information moves, encoded by the propagation bank, from what moves, encoded by the message maps. Function-valued fillings extend the same equation across local message passing, attention, spectral filtering, global communication, relation-specific channels, higher-order domains, and geometric messages. We make this unification explicit and checkable through worked reductions of canonical layers and component assignments spanning seven nonexclusive architectural families. A fixed slot discipline assigns operations by computational role and defines the framework's coverage boundary. The decomposition also yields component-level theoretical insights: under endpoint-local messages and node-local updates, operator support bounds one-layer dependencies, and one-layer global mixing requires a full effective operator row under the stated hypotheses. The resulting framework organizes more than 200 architectures in a common design space, enables component-wise comparison and generation of structurally consistent architectures, and connects propagation choices to oversmoothing, oversquashing, heterophily, and expressivity. It further exposes the empirical inverse problem of mapping measurable graph and task properties to validated component choices.

Enterprises connect language models to their own data through retrieval. The benchmarks that rank multi-hop retrieval systems leave out two facts a buyer needs before a published number can be used: whether the retrieval backbone may be deployed commercially, and what it costs to build. On licensing: the field's dense-retrieval anchor, NV-Embed-v2, is licensed cc-by-nc-4.0. Of the four leading MuSiQue systems we audit (HippoRAG-2, PropRAG, SAG, KET-RAG), three depend on it for their best numbers and none says so. On performance: we measure thirteen embedders from eight makers on one identical MuSiQue harness with bootstrap confidence intervals throughout. Until mid-2026 there was a real commercial tax: the best commercially-licensed embedder trailed the anchor by 2.31 Recall@5 points (95% CI [0.91, 3.71], p=0.001). NVIDIA's Nemotron-3-Embed-8B, released 2026-07-16, has closed it: +0.24 at Recall@5 (95% CI [-0.94, +1.43], p=0.69), -0.58 at Recall@10 (p=0.28). It matches the anchor, does not beat it, and is the only entrant that is commercially licensed, free to self-host, and indistinguishable from the anchor; every other entrant meeting the first two conditions sits 5.2 to 14.6 points below. The durable finding is the paid-versus-free divide: API embedders charge per token on every re-index, self-hosted ones charge nothing. On cost: three of five audited systems (adding Microsoft's GraphRAG) do not disclose indexing cost, and the only published GraphRAG dollar figures span 11x inside one third-party paper (USD 2.30 vs USD 24.94 to index a 5.64 MB corpus once); extrapolated to 1 TB that undisclosed choice separates roughly USD 428K from $4.6M. Our cost model keeps one-time embedding apart from recurring answering: at 1 TB, embedding sits 7.5x-900x below graph construction, and a year of answering at 10,000 queries/day sits 350x or more below it.