Science & Mathematics

Mass formulas for prescribed hull dimensions have mainly been studied for linear codes viewed as subspaces of the whole space. We study internal enumeration for an arbitrary fixed parent code. Let $C$ be an $[n,k]_q$ linear code with an $\ell$-dimensional hull, and let $s$ be the subcode dimension. We determine the number of subcodes with prescribed dimension, hull dimension, and type over finite fields of both even and odd characteristic. We decompose the parent code into an LCD complement and its hull, and use projection to reduce enumeration to the corresponding problem inside the LCD complement. This yields explicit formulas refined by complete type and shows that the counts depend only on the relevant parameters, not on the particular parent code or its length. The known whole-space mass formulas and the enumeration of LCD subcodes of LCD codes arise as special cases. Motivated by code extension, we also study the dual containment problem: for a fixed code, we enumerate larger codes containing it with prescribed hull dimension and type. By duality, this problem is converted into the subcode enumeration problem already solved above, leading to corresponding conditional mass formulas. Finally, we investigate the asymptotic distributions arising from the exact enumeration formulas. When both the subcode dimension $s$ and $k-\ell-s$ tend to infinity, the hull distribution is governed by the LCD complement, and the limiting proportions are explicit constants determined by the field, the prescribed hull dimension, and the relevant type. When $s$ approaches or exceeds the dimension $k-\ell$ of the LCD complement, we determine the asymptotic proportion of subcodes attaining the minimum possible hull dimension and establish the associated concentration behavior. Analogous asymptotic results are obtained for the containment problem.

Coding agents can adapt their harnesses, the software, configuration, and instructions used to run them, based on experience from earlier tasks, but whether these adaptations improve their behavior on later tasks remains unclear. We study \retro, a skill that asks an agent to review a completed session and propose harness changes, and ask what changes it proposes and whether implemented changes reduce repeated work on later tasks. We ran Claude Code on 30 SWE-bench Verified issues, qualitatively coded the 125 proposals from \retro, implemented 24, and evaluated 12 harness changes on later issues from the same repositories in fresh sessions with and without each change. Most proposals referred to an event in the session, but 18 of 43 lint-related proposals recommended a practice regardless of what had happened. Among the seven evaluations in which sessions without the change encountered the targeted problem, two changes eliminated the targeted action, one reduced its occurrence, four showed no reduction, and some changes added work on unrelated issues. Harness self-evolution requires more than generating and implementing changes: it needs ways to establish whether a change helps on later tasks and to revise or remove changes that do not.

RGB-X tracking complements RGB appearance with thermal, depth, or event observations, but centralized training requires collecting multimodal videos at a single location. Federated training keeps raw videos local, but sensor heterogeneity can create a mismatch between parameter-sharing domains and the clients contributing to them. This mismatch is particularly relevant when RGB-only clients coexist with sensor-equipped clients: their RGB observations remain useful, but they provide no auxiliary measurements for learning sensor-dependent interactions. We propose Fed-MAPTrack, a modality-aware federated prompt tuning framework for RGB-X tracking. The pretrained tracking backbone and prediction head are frozen, and clients optimize and exchange only prompt-related parameters. The server aggregates RGB-shared parameters globally and maintains separate thermal, event, and depth banks for auxiliary-specific, fusion, and normalization parameters. Selective participation retains RGB-only clients' contributions to the global RGB bank while excluding their fusion and normalization updates from modality-specific aggregation. Across three matched training repetitions under a fixed client partition, Fed-MAPTrack improves the mean FedAvg scores by 2.35 AUC points on LasHeR, 3.45 SR points on VisEvent, and 2.11 F-score points on DepthTrack. Among the nine evaluated methods, it achieves the highest mean thermal AUC and depth F-score, while its mean event SR is 0.07 points below that of Independent FL. Controlled ablations show that extending the bucketing scope without a compatible participation rule can reduce performance. These results highlight the need to jointly define the sharing domain and eligible contributors for each parameter group.

Fact-verification benchmarks score the final label, while an interactive verifier must first decide whether the available evidence is sufficient. We focus on that intermediate Stop-or-Clarify Decision. Uncertainty alone is insufficient, because the benefit of another evidence-acquisition step is unknown before the step is taken. Sequential Experience-Adaptive Clarification (SEAC) wraps a fixed verifier with a lightweight evidence-acquisition policy. For each claim, SEAC retrieves evidence, runs the verifier, forms a label-free decision state, and combines stop risk with an observational clarification-utility estimate from earlier executed trajectories. Clarification is requested only when this score exceeds the action cost and decision threshold. On SciFact, LIAR, and FEVER, SEAC improves mean accuracy and macro-F1 over the compared fact-verification and active-retrieval baselines. Equal-budget and formula-level diagnostics support selective clarification as a contributor to the gain, while the small SciFact split leaves its paired improvement inconclusive.

Hyperdimensional computing (HDC) offers efficient learning and inference for resource-constrained systems, yet ownership verification of its models remains underexplored. We present the first systematic investigation of model watermarking in HDC by adapting backdoor-based embedding to three classifiers with different prototype-learning mechanisms. Across four benchmarks, with conventional neural networks as references, we examine whether watermarks can be learned without compromising task accuracy and whether they survive subsequent model modification and replication. The error-driven classifiers, AdaptHD and OnlineHD, support verification at a nominal trigger exposure of 1% with at most 0.58 percentage points of mean accuracy loss, whereas single-pass Vanilla HDC requires greater exposure and can incur substantial accuracy costs. The error-driven models retain verification under native continued training, but extraction produces surrogates that reproduce much of the task behavior without verifiable watermarks. Watermark overwriting readily embeds attacker watermarks into HDC prototypes and erodes the owner's watermark more strongly than in the neural network references, although the strongest erosion coincides with large accuracy losses. We further deploy watermarked AdaptHD on an FPGA and stress its execution through undervolting and overclocking. Verification survives substantial computation errors, whereas more severe stress either causes verification failure together with accuracy collapse or results in inference timeouts. These findings establish the feasibility of watermarking in HDC while identifying limitations in retention under extraction and sensitivity to overwriting.

Large Reasoning Models (LRMs) generate long reasoning traces that expose intermediate inference behavior, offering opportunities to study and control reasoning in AI coding. However, the underlying patterns in these traces and their potential for analyzing and guiding coding models remain poorly understood, limiting systematic characterization and inference-time control of LRM reasoning. We conduct an empirical study of reasoning patterns across four representative coding tasks. We transform reasoning traces into structured action sequences and mine recurring fragments using within-task support, cross-task contrasts, and valid--invalid support contrasts. We characterize four general reasoning patterns by coding agents: design-to-implementation for code generation, state tracing for execution reasoning, evidence--fix--recheck for debugging, and semantic mapping for translation. We further identify positive and negative patterns associated with valid and invalid reasoning traces, respectively. Positive patterns reflect verification-driven convergence, whereas negative patterns reflect repeated design revision, reflection, and knowledge recall without closure. To demonstrate their practical utility, we evaluate three pattern-based applications: task identification, pattern-guided prompting, and pattern-aware early stopping. Features capturing general reasoning patterns enable accurate task identification from reasoning traces without accessing task descriptions or final answers. Prompting guided by positive and negative patterns improves the overall accuracy of downstream tasks, while negative pattern-aware early stopping substantially reduces API-reported token usage with only a modest decrease in pass@1. All three applications also generalize to previously unseen models, showing that reasoning patterns can support both the analysis and control of LRM behavior in AI coding.

A dot product means something only when both vectors share a coordinate system. Content words have one; a fresh entity does not: rotating its coordinates alone changes nothing. We take this per-entity frame freedom as a symmetry and ask what it forces in attention. For orthogonal transports, invariant scores depend only on norms and the transported inner product $\langle x_i,T_{ij}x_j\rangle$; equivariant messages lie in the span of $x_i$ and $T_{ij}x_j$. Endpoint channel count bounds transport rank, so with one vector per entity relations must be supplied; standard attention is the flat sector of a shared frame. Representation theory attaches an integer to each mechanism; the readout decides which. With orthogonal transitions, a convex-cell readout composes a finite group exactly only at or above its smallest faithful real representation dimension. At this minimum, exactness forces a faithful representation. An even readout can decode a cover at smaller width, and the gap is unbounded. Below its threshold, the part of the group a mechanism cannot see sets an accuracy ceiling computable before training. 17/17 fitting seeds at minimal width passed tests through length 1000, versus 7/18 at twice the width; 12 models carry correctness certificates to explicit lengths, from the weights alone. On $C_3\times A_4$ (faithful width 5), descent at width 4 learned the 72-state double cover under sign-blind attention (6/48 seeds; monotone 0/48); on a frozen cover the cross-entropy-optimal affine head is uniform, a quadratic head exact. On parsed CLUTRR, transported composition beats a matched transformer by 41 points at 4-10 hops, but forcing isometries costs 65 points; on StepGame's commutative algebra, isometries help. Static embeddings gave each word one vector; contextual models gave each occurrence its own; per-entity frames make each relation an operator, the unit in which relations compose.

Reasoning models trained with Group Relative Policy Optimization (GRPO) now produce chains of thought thousands of tokens long, and recent studies reveal that such training is prone to policy collapse. Entropy regularization is the standard remedy, but it works by pushing probability mass away from the few tokens that carry it and toward the vast low-probability tail of the vocabulary, and it applies this force once for every token of the reasoning trace. This puts the bonus in direct conflict with the entropy decay that convergence requires, and makes its coefficient difficult to set in the long-trace regime. To resolve this tension, we propose Info-GRPO, an information-theoretic framework that cultivates correlation between the policy and a latent prior. Rather than raising entropy everywhere, Info-GRPO builds on the mutual information between a latent variable and the trajectory it induces: augmenting prompts with latent seeds during training lets the model explore diverse policies correlated with the prior, while the seed-conditioned entropy is guided toward convergence. Diversity is therefore carried across whole reasoning traces rather than by per-token randomness, and exploration no longer fights entropy reduction. Experiments show that Info-GRPO outperforms vanilla GRPO and entropy-regularized GRPO across diverse reasoning benchmarks. On AIME24, it improves $Avg@8$ over GRPO by 3.75%, 1.66%, 4.16%, and 4.17% on four backbones, and it reaches the same majority-vote accuracy with $8\times$ fewer samples. Code and models will be publicly released.

Illicit account detection is a critical task for financial risk control systems, aiming to pinpoint malicious actors within transaction networks. Existing efforts typically recast transaction networks into directed multigraphs and leverage graph neural networks (GNNs) to detect anomalies via topological structures under the homophily assumption. However, directly grafting GNNs onto transaction graphs is faced with at least two pivotal challenges, i.e., structural inconsistency and label scarcity. The former arises from deceptive connections forged by malicious actors, while the latter stems from the lack of annotated data. In light of these, we introduce MINT, a novel graph neural framework that mitigates structural inconsistency via an attentive message-passing scheme on transaction graphs. Specifically, MINT introduces a discrepancy-aware feature propagation, which tracks malicious actors by quantifying deviations between nodes across bi-directional transaction flows. A transaction salience module is proposed to mitigate structural inconsistency by adaptively regularizing the interaction strength between nodes. Furthermore, to alleviate label scarcity, MINT is optimized via Positive-Unlabeled (PU) learning, which exploits only a minority of illicit nodes coupled with a majority of unlabeled ones. Extensive experiments conducted on three financial transaction benchmarks, including two large-scale public and one in-house real-world datasets, validate the superior effectiveness of MINT over several cutting-edge baselines.

Recent methods for Reinforcement Learning with Verifiable Rewards (RLVR) use rollout diversity to improve reasoning performance or training efficiency. Despite these gains, the extent to which differences among correct-answer rollouts reflect useful alternative reasoning or unresolved errors remains poorly understood. We investigate this question through a blind paired analysis of a model trained with Group Relative Policy Optimization (GRPO), comparing the highest- and lowest-agreement correct-answer traces for the same problem. Across 126 pairs, low-agreement traces have more spurious reasoning than high-agreement traces (35% vs. 22%), including internal inconsistencies and misuse of problem conditions. Based on these findings, we propose RoAR (Rollout-Agreement Reward), which adds reasoning-level agreement to GRPO's rewards for verified-correct rollouts. RoAR computes this signal from existing on-policy rollout groups, requiring no human-written rationales, additional sampling, or separately trained process evaluator. On five mathematical reasoning benchmarks, the 1.5B GRPO-RoAR model reaches 55.5% mean pass@1 under zero-shot evaluation using 7K training examples, exceeding GRPO and GRPO-CARE by 6.0 and 5.5 percentage points. A separate blind audit of 100 problems answered correctly by all four 1.5B models finds spurious rates of 18% for GRPO-RoAR, 32% for GRPO, 27% for GRPO-CARE, and 37% for DRA-GRPO. Our findings identify reasoning validity as an important consideration when using rollout differences to guide learning. Our code is available at https://github.com/ldi-kyunghee/RoAR.

Dynamic Gaussian reconstruction requires compact scene features to support coordinated changes in geometry and appearance. A factorized grid provides efficient local queries, but accurate interpolation alone does not ensure that the resulting dynamic attributes remain compatible across space and time. We introduce DimFlow, a framework that learns these features through bidirectional interaction between a multi-resolution grid and a coordinate-based expert branch. Feature-level adversarial alignment connects their distributions, adaptive fusion transfers complementary information, and activation-masked residual feedback refines the grid during training. Four attribute experts organize deformation, rotation, opacity, and temporal information before Gaussian rendering. In the aggregate comparison across D-NeRF, HyperNeRF, and DAVIS, DimFlow achieves 39.73 dB PSNR, improving on 4D-GS by 1.92 dB while matching its reported 85 FPS rendering speed. The model occupies 18.9 MB, compared with 18.3 MB for 4D-GS. Component and attribute-expert ablations reveal complementary contributions from feature alignment, adaptive exchange, and temporal processing.

Decision needs vary with each application, language, and deployment environment. The Decision Model Factory turns those needs into specialized, sovereign models on demand. A brief recipe guides the training, calibration, verification, and evaluation of a selected backbone, delivering a deployable decision model under the operator's control. The same workflow creates a new specialist whenever the requirements change. Across Japanese decision tasks, compact deployment, and long-context passage ranking, five factory-built models improve upon their backbones; three surpass the strongest results in a fixed public Japanese benchmark comparison. A dedicated runner computes only permitted-answer probabilities and reuses shared context, more than doubling serving speed on matched multi-question workloads with essentially unchanged decision scores. Together, the factory and runner make fast, task-specific decision models practical to build and deploy whenever needed.

A converged topology optimization can solve the wrong problem when loads, supports, or the domain are underspecified. Assuming defaults, or asking the problem's author about every detail, ignores how much each answer changes the design. We propose consequence-guided clarification (CGC), which bases acceptance on computed mechanical consequences. CGC computes a reference design for each of a finite list of complete specifications, called readings. A design is accepted only if its compliance is at most 5% above the reference at every reading consistent with the answers. Otherwise, CGC asks the question whose worst answer leaves the smallest worst-case excess, and withholds acceptance at its question limit. To save optimizations, the screening variant CGC-S postpones some references until a candidate must be checked. We developed and evaluated both on 36 arXiv-derived tasks, treating each of 134 readings in turn as intended and simulating correct answers. In 26 tasks, the default-reading design failed another reading. Apart from always-asked questions on load direction, sign, and material budget, asking everything took 188 questions; CGC and CGC-S each took 105. The acceptance test supplied most of this saving: applying it to the default design alone took 117. Relative to asking everything, CGC-S requested 4.6 extra reference designs per avoided question, each from two optimization starts, counting optimization time only. References are not certified optima: better ones pushed three accepted cases above tolerance, and finer meshes changed some decisions. Within these limits, the test shows which readings a design covers and which details block acceptance.

GUI agents built on vision-language models interact with continuously evolving interfaces, yet conventional inference repeatedly recomputes representations of largely overlapping observations and histories. This mismatch makes multi-step interaction costly, while changes in context, interface content, and action arguments prevent straightforward cache reuse. We introduce GUIFlow, a framework for full-pipeline incremental inference in multi-image GUI interaction. Our central insight is that advancing a decision need not require rebuilding all supporting representations: each interaction step can instead selectively update computational states maintained across the trajectory. To support new decisions as the context evolves, GUIFlow adapts how past computation is reused: historical KV states are position-aligned and injected into the current context; change signals guide selective updates of current-observation states in the vision encoder and language model; and earlier action responses serve as speculative drafts verified by the target model. Together, these mechanisms enable cross-step reuse throughout visual encoding, prefill, and decoding without additional training, a separate draft model, or changes to the model-visible inputs and action interface. Experiments demonstrate up to 3.39x geometric-mean speedup across offline multi-image benchmarks and up to 3.49x in online, closed-loop interaction on OSWorld, while maintaining task performance broadly comparable to the dense baseline.

Despite the remarkable capabilities of large language models (LLMs) across diverse applications, they remain vulnerable to generating content that violates safety regulations and policies. To mitigate these risks, LLMs undergo safety alignment; however, they can still be effectively jailbroken. Off-the-shelf guardrail models are commonly deployed to monitor generations, but these models primarily focus on detection rather than ensuring safe decoding of LLM outputs. Moreover, existing efforts lack rigorous safety guarantees, which are crucial for the universal deployment of LLMs and certifiable compliance with regulatory standards. In this paper, we propose a Claim-based Stream Decoding (CSD) algorithm coupled with a statistical risk guarantee framework using conformal analysis. Specifically, our CSD algorithm integrates a stream guardrail model to safeguard sequential claims generated by LLMs and incorporates a backtracking mechanism to revise claims flagged with high safety risks. We provide theoretical guarantees demonstrating that the CSD algorithm achieves the desired generation distribution subject to safety constraints. Furthermore, we introduce a generation risk certification framework and derive a high-probability upper bound on the safety risk of the proposed CSD algorithm. We prove that our method can asymptotically control safety risk to any desired level. Empirical evaluations demonstrate the effectiveness and efficiency of the CSD algorithm compared to state-of-the-art safety decoding approaches. Additionally, we validate the soundness and tightness of the derived safety risk upper bound using realistic data.