arXiv subjects

Wenlin Zhang

Publications and source records attributed to Wenlin Zhang.

At least 19 recordsLinked to original sources

arXiv2026

Block-exchange descent for insertion: commuting projections and minimum masks

For fixed nonempty words $u,v$, the minimum insertion degree of a shuffle is the least number of inserted-source runs among its representations. We prove that, after a block exchange from a minimum mask of degree $r$, the minimum over masks with commuting source projections is exactly $r-1$; a larger decrease occurs precisely when a noncommuting mask has fewer runs. If either source is unary, every such exchange is exact. We prove an explicit repair for an infinite periodic degree-three family. Computer-assisted results establish first-mask descent through combined length seventeen over every alphabet, degree-three first-mask descent through length thirty, and a uniform repair of all degree-three failures through length thirty, including all further letter identifications. Counterexamples to arbitrary, first, and last minimum-mask choices explain the need for alternative masks. Finally, we prove that minimum iteration depths have no gaps for arbitrary languages. General block-exchange descent and Singleton No-Gap remain open.

cs.FL↗

arXiv2026

Proactive Memory for Event-Driven LLM Agents

Proactive LLM agents anticipate service opportunities from ongoing observations and accumulated history. Existing approaches emphasize recent context or retrieval of related memories, providing limited support for validating recurring behavioral regularities and assessing their applicability to the current event. We propose Proactive Memory (ProactMem), which organizes history into recent events, temporal trajectories, and behavioral regularities. Trajectory facts are checked against source events, and regularities are assessed through historical replay before being matched to current conditions. We also construct ProactiveStream, a longitudinal benchmark covering three application scopes, where agents process isolated event streams chronologically under episode-aware opportunity labels. On ProactiveStream, ProactMem achieves an overall effectiveness (F1) of 52.6%, a 21.4% relative improvement over the strongest of five external baselines. Code is available at: https://anonymous.4open.science/r/ProactMem-5351/.

cs.AI↗

arXiv2026

Residue Sums Modulo Products of Two Distinct Odd Primes

Let $S_m(a)$ denote the sum of the least positive residues in one multiplicative period of $a$ modulo $m$, of length $k$. We study the equality $S_m(a)=mk/2$ for $m=pq$, where $p,q$ are distinct odd primes and $k$ is even. Complementary pairing settles subgroups containing $-1$; the remaining cases depend on their embedding in the two local unit groups. At index four, we exclude mixed cyclic subgroups with biquadratic annihilator and classify every eligible imprimitive quartic kernel using two odd integer moments at its prime conductor. For each fixed conductor, the proportion of admissible second primes giving equality is $1/4$ or $1/2$; full coset balance has proportion $1/4$. For primitive quartic characters, classical triangular character-sum identities yield explicit short-sum criteria, and the relative class-number formula gives an obstruction to a centered coset. At index six, we give structural counts and construct an infinite family modulo $19q$ in which nonzero sextic and quadratic moments cancel. These applications use a standard odd-character expansion, which also recovers the known index-two balance classification and determines the sign of its deviation. A reproducible census for $pq\le10000$ counts $75\,090$ eligible cyclic subgroups and identifies $101$ mixed index-four subgroups whose own sum is centered although some coset sums are not.

math.NT↗

arXiv2026

Generalized Fidelity, the Data Processing Inequality, and Convexity

In this paper, we show that the generalized Bures--Wasserstein distance induced by the generalized fidelity fails the data processing inequality (DPI) in every dimension $n\geq2$. In dimension two, we derive three sufficient conditions guaranteeing the DPI by using an explicit formula for the real part of the generalized fidelity. Furthermore, we find a necessary and sufficient condition under which the generalized fidelity reduces to the Uhlmann fidelity. We also show that neither the generalized Bures--Wasserstein distance nor its square is convex in one state, even with the other state and the reference fixed, in every dimension $n\geq2$.

quant-ph↗

arXiv2026

TrajEvolve: Trajectory Evolution for Reinforcement Learning with Hindsight Credit Assignment

The Reinforcement Learning (RL) paradigm has achieved remarkable success in enhancing the reasoning capabilities of Large Language Models (LLMs). However, when applied to long-horizon tasks, it faces severe challenges: i) Due to the high task complexity, the model struggles to sample enough successful trajectories, resulting in slow convergence. ii) Relying solely on sparse final rewards causes all steps to be updated indiscriminately, encouraging redundant step generation. When both the quantity and quality of successful trajectories become difficult to guarantee, the performance is severely constrained. This dilemma stems from the limitations: existing RL paradigms rely entirely on the reasoning capability of base models, lacking the ability to actively optimize trajectories. To address the above issues, we propose a novel trajectory evolution paradigm with hindsight credit assignment, termed TrajEvolve. TrajEvolve not only relies on the trajectory inferred by the model but also generates ``evolutionary trajectories'' by refining low-quality steps in historical trajectories, thereby exposing the policy to high-quality compositions that lie within the base model's support but are rarely sampled within practical training budgets. Furthermore, TrajEvolve reformulates step-level continuous credit assignment as a binary classification task, whether each step is necessary for task completion. This simplification not only reduces the learning difficulty but also enables step labels to be obtained through counterfactual verification, achieving precise identification of redundant steps. Extensive experiments on two datasets demonstrate the superiority of the proposed method.

cs.AI↗

arXiv2026

Sharp zero forcing numbers for the generalized Petersen graphs $P(n,3)$ and $P(n,4)$

We prove that $Z(P(n,3))=8$ for every $n\ge13$ and $Z(P(n,4))=10$ for every $n\ge18$, with both stabilization thresholds sharp, and give the exact values for all smaller admissible $n$. The first equality proves Krishnan's Conjecture 5; his Proposition 4 supplies its upper bound. The lower bounds use the external boundary of the first forcing sources: ten vertices have boundary at least eight in $P(n,3)$ for $n\ge14$, and sixteen vertices have boundary at least ten in $P(n,4)$ for $n\ge20$. Deletion within an empty column interval reduces these assertions to finite checks, and forts settle the remaining cases. The $k=3$ stabilization theorem has a Lean 4 formalization; the $k=4$ proof uses 89 boundary formulas and 11 fort formulas with separately checked DRAT certificates, together with explicit forcing sequences.

math.CO↗

arXiv2026

On the coincidence of discrete and continuous definitions of Entanglement of Formation

It is proved that every Borel ensemble of pure states on a separable bipartite Hilbert space with finite average entanglement admits a countable ensemble with the same barycenter and arbitrarily small additional average entanglement. Consequently, the discrete and continuous versions of the Entanglement of Formation coincide on the whole space of states of infinite-dimensional bipartite quantum system (with infinite values allowed). The main idea of the proof (realized with the AI-assistance) is the lifting a Borel ensemble of pure bipartite states to a Borel probability measure on the unit sphere in the space of Hilbert--Schmidt operators followed by dividing the support of this measure into appropriate cells and replacing each cell by its conditional barycenter. The resulting countable ensemble is obtained by a recursive construction.

quant-ph↗

arXiv2026

Generation by lazy and invertible cellular automata over the Klein four-group

Let $G=C_2\times C_2$ and $A=\{0,1\}$. We characterize the submonoid of $\text{CA}(G;A)$ generated by lazy and invertible cellular automata. A cellular automaton belongs to this submonoid if and only if, whenever it maps the two free $G$-orbits onto the same free orbit, it maps both constant configurations and some two-point orbit to the same constant. The proof gives a factorization using elementary orbit collapses and three two-cell lazy maps. The criterion yields the cardinality $59\,712$ and identifies the maps missed by a weaker constant-separation invariant. Adjoining one idempotent generates the full monoid, and the generated submonoid is contained in exactly one maximal proper submonoid, of cardinality $65\,024$. The submonoid generated by lazy maps alone has $4\,007$ elements, and for both generating sets the maximum shortest-word length is six. The criterion and the cardinalities $59\,712$ and $4\,007$ are formally verified in Lean.

math.GR↗

arXiv2026

Memory Type Varies: Empowering LLM Agents for Long-Term Memory with Diverse Strategies

The memory capabilities of Large Language Models (LLMs) have garnered increasing attention recently. Despite great success achieved, existing retrieval-based memory approaches typically overlook the differences between memories and employ a unified strategy to process all memories, leading to suboptimal performance. Thus, an intuitive question arises: can we categorize memory into different types and select appropriate strategies? However, given the topic-rich, scenario-complex, and boundary-blurred nature of memory scenarios, achieving precise classification of memories is not easy. To address this challenge, we propose a memory multi-class dataset in this paper, termed TriMEM, which provides precise annotations for memory types across diverse scenarios. Building upon this foundation, we propose a novel memory framework, named MemoType, which can adaptively recognize each memory and query type with the learned router model. With the memory and query routing, MemoType can retrieve the memory with corresponding query types and design tailored retrieval strategies, thereby enhancing the retrieval performance. Moreover, we theoretically prove that any single retrieval strategy is subject to a fundamental upper bound on its expected retrieval precision in multi-class corpora, leading to systematic precision degradation. Extensive experiments on three datasets demonstrate that MemoType consistently outperforms existing methods, achieving up to 16.18% improvement in Recall@1.

cs.AI↗

arXiv2026

EvoSkillRec: Skill-Genome Evolution for Recommender Architecture Discovery

Modern recommender systems advance not only by scaling data and parameters, but also by encoding task-specific inductive biases through architecture, including sparse feature interactions for click-through rate (CTR) prediction, temporal attention for sequential recommendation, and expert routing for multi-task learning. However, these biases are typically human expert designed or searched within predefined operator spaces. Although Recent LLM-driven code evolution expands this space, unconstrained edits often produce invalid or ineffective architectures, underuse established architecture design knowledge, and fail to preserve successful innovations for reuse. We introduce EvoSkillRec, a promotion-and-reuse framework for cumulative recommender architecture evolution. It first decomposes recommenders into atomic executable skills and represents architectures as typed skill genomes, with each skill equipped with input--output types, semantic annotations, and implementation code. We then evolve models with different tasks through two coupled spaces: a constrained skill--space that mutates, recombines, specializes, and reuses validated skills, and an open-ended code--space in which LLM planners and synthesizers invent new skill modules using prior evolution traces and accumulated experience. An autoresearch controller evaluates candidates, diagnoses failures, retrieves relevant skills, promotes validated innovations into the skill library, and adaptively allocates the proposal budget between the two spaces. Extensive experiments on CTR prediction, multi-task learning, and multi-domain learning, including resource-constrained co-optimization of predictive quality and model FLOPs utilization in generative ranking models, consistently demonstrate the effectiveness of our proposed EvoSkillRec.

cs.IR↗

arXiv2026

Persona Following Is Not Selective Control: The Neutrality Gap in LLM User Simulation

Persona prompting is widely used to construct user simulations with large language models (LLMs), yet it relies on a largely untested assumption: specifying one user attribute should change that attribute alone. We test this assumption and identify a systematic failure of selective control: across all eight black-box LLMs we audit, changing a target attribute also shifts responses on unspecified, non-target attributes. For example, describing a user as more risk-seeking shifts color choices, even though the prompt never mentions color; we term this cross-attribute influence. Semantic, contextual, and internal analyses collectively suggest that models treat a persona prompt as evidence about the user and extend the inferred profile to unspecified preferences, a process we call trait-conditioned completion. We next ask whether explicitly specifying non-target attributes restores selective control. When a non-target attribute is assigned a clear direction, models generally follow the declaration and suppress the target attribute's influence. However, when the same attribute is declared neutral, the target continues to affect choices across all five open-weight checkpoints, even when the model correctly reports the declared state. This disparity, the neutrality gap, demonstrates that successful persona following does not imply selective persona control, which additionally requires keeping non-target attributes stable. We operationalize this distinction with a three-state diagnostic that leaves the non-target attribute unspecified or declares it directional or neutral; because directional tests can be passed by simply following the stated persona, the neutral state reveals failures they miss. In a post hoc analysis of independent items, neutral declarations leave 51-81% of items target-sensitive, against at most 1 of 320 item-pole comparisons under directional ones.

cs.AI↗

arXiv2026

A Padovan-automatic description of a nested recurrence

We study the sequence $a(0)=0$, $a(1)=1$ and $a(n)=n-a(n-a(n-a(n-1)))$ for $n\ge 2$, listed as A076502 in the On-Line Encyclopedia of Integer Sequences. We identify $a(n)$ as a two-position shift in the greedy Padovan numeration system, with a finite-state correction. The proof constructs an addition automaton from an exact integer-carry invariant and certifies its completeness by finite-language inclusion; a synchronized automaton then verifies the nested recurrence. We establish bounded discrepancy from the line of slope $c$, where $c^3-c^2+2c-1=0$, and show that the exact set of offsets from $\lfloor cn\rfloor$ is $\{-1,0,1,2\}$. We construct an explicit 26-letter non-erasing morphic presentation of the first-difference word, prove that its least balance constant is 4, and give an effective procedure for enclosing the global discrepancy extrema to arbitrary accuracy. We formalize the recurrence identification, six-decimal discrepancy bound, exact offset set, concrete morphic identity, least balance constant, and an effective extrema algorithm in Lean. Separate exact computations refine the numerical enclosures.

math.NT↗

arXiv2026

TAD: Token-Adaptive Contrastive Decoding with Confidence-Guided Gating for Hallucination Mitigation in Large Audio-Language Models

Large audio-language models (LALMs) can hallucinate audio objects, answering "yes" to absent sound events, thus undermining reliability in audio question answering. We propose Token-Adaptive Decoding (TAD), a training-free strategy for hallucination mitigation that grounds the initial yes/no decision by contrasting logits under real audio with a matched silent reference. TAD introduces a token-adaptive, confidence-guided gate that is decision-critical at the first decoding step and class-conditional on affirmative tokens, using the audio-silent margin to avoid overcorrection when evidence is weak or already sufficient. Experiments on AudioCaps-Hallucination show that, relative to Audio-Aware Decoding (AAD), a contrastive baseline with fixed contrast strength, TAD improves F1 for Qwen2 by 0.059 to 0.117 across Popular, Adversarial, and Random splits, and for Gemma by 0.025 to 0.064, while on Clotho-AQA it raises F1 from 0.810 to 0.816 on Qwen2 and remains comparable to AAD on Gemma.

cs.SD↗

arXiv2026

A Certificate-Producing Cascade for Equational Implication: The SAIR EQT2 Stage 2 Solver

The SAIR Mathematics Distillation Challenge on Equational Theories asks a solver to classify whether one magma identity implies another and, for either verdict, to return a certificate accepted by a deterministic Lean judge. We present a single-file solver organized as a cheapest-first cascade. Its false branch combines coefficient tests over structured algebra families, bounded finite-model search, an explicit central-groupoid witness, and several infinite-carrier witnesses. Its true branch is a proof-producing ordered unit superposition procedure with Knuth-Bendix ordering, bidirectional demodulation, indexing, memoised substitution, and anytime size deepening. Search results remain outside the trusted base: successful derivations are replayed as small Lean terms, and countermodels are rechecked by the competition judge. The frozen solver is a 189,504-byte Python file with SHA-256 f2392533c9f4c03b.... In local runs through official judge revision 2848228, it produced accepted certificates for all 1,889 rows of the six public sets with no language-model calls. Separate measurements recorded full agreement on the 800 published Stage 1 evaluation-distribution problems, 100 accepted rows in the canonical Marathon manifest without tokens, and 200 accepted rows in the hosted playground. These are regression and playground measurements, not a leaderboard result and not evidence about a hidden set. All quantitative claims are tied to immutable result ledgers; the paper makes no completeness or comparative-superiority claim.

cs.CL↗

arXiv2026

GUI-Lens: Coarse-to-Fine Cropping for GUI Grounding with General-Purpose VLMs

GUI grounding maps natural-language instructions to click locations and is essential for reliable GUI agents. The task remains difficult on high-resolution, densely populated interfaces because a vision-language model (VLM) may recognize a requested control without locating it precisely enough for interaction. Most existing methods provide various forms of localization assistance, but still rely on a direct click prediction, allowing visual ambiguity or an inaccurate initial estimate to propagate to the final result. In this paper, we introduce GUI-Lens, a coarse-to-fine grounding framework that allows a general-purpose VLM to determine the target through active visual observations. Specifically, GUI-Lens extracts OCR text and detected UI components from the screenshot and presents their positions as coordinate references. Using the instruction, the current view, and these references, the VLM selects the region and scale of the next view, which is cropped and enlarged to provide finer visual details. This process continues over successively focused views until the target is determined. Proposed crops and clicks are checked against the instruction throughout the process, and the final local position is mapped back to the original screen coordinates. Experiments on four GUI grounding benchmarks and three general-purpose VLM backends show that GUI-Lens improves overall grounding accuracy by up to 24.9 percentage points and achieves state-of-the-art performance with GPT-5.5.

cs.CV↗

arXiv2026

Mechanism-level routing failure in LLMs over Lean-verified algebraic structures

We present an empirical study of structural routing failure in large language models (LLMs) over a formally verified algebraic corpus. The task requires selecting the correct proof-mechanism label from a fixed closed template set for compact mathematical objects drawn from the FiberRing formalization in Lean 4, where each item is anchored to a Lean-verified artifact and assigned a label from the corresponding certificate family. Our central finding is a mechanism-level routing ceiling: under blind conditions, gpt-oss-120b achieves 80.3% template accuracy on 22 FiberRing items (n=66; temperature=0, seed=0), while Llama 3.3 70B reaches 68.2%. Exposing a mechanism-bearing Lean verdict/witness cue (Condition A2) raises accuracy to 90.9% and 81.8% -- gaps of +10.6 and +13.6 pp termed cue-induced routing uplift. The dominant failure is a CRT-to-ring-equivalence misroute: gpt-oss-120b misroutes 7 of 12 CRT items (58.3%) blind, zero under A2. A cross-model dissociation in Llama is notable: verdict accuracy is identical in both conditions (95.5%), while template accuracy improves 13.6 pp -- confirming that truth inference and proof-mechanism classification are separable capacities. A cross-corpus extension (Set B; 6 POM/CollisionKernel items, 72 evaluations) provides a small cross-module check: CRT-granularity compression reappears with different labels, and an inverse cross-model dissociation emerges. These findings extend the router hypothesis (Cazares 2026) to formal algebraic structures. The full pipeline, manifest, and results are at https://github.com/bytepro-ai/fiber-routing-eval.

cs.CL↗

arXiv2026

R$^2$-Searcher: Calibrating Retrieval and Reasoning Boundaries for Agentic Search

Recent search agents for multi-hop reasoning often fail by either retrieving incomplete evidence or reasoning over irrelevant portions of the retrieved content, leading to a retrieval-reasoning boundary shift. We propose R$^2$-Searcher, a novel framework that explicitly explores and calibrates the retrieval and reasoning boundaries via fine-grained, query-token-guided evidence modeling and post-retrieval reflection. Specifically, R$^2$-Searcher: (1) constructs fine-grained reasoning contexts by extracting precise facts from retrieved content based on query token semantics (e.g., subjects, actions, temporal markers, and degree modifiers), thereby guiding the attention of search agent; (2) introduces a retrieval reflection mechanism that evaluates and corrects boundary deviations after each retrieval step, guiding the generation of improved queries grounded in the extracted reasoning contexts; and (3) employs an end-to-end reasoning-reflection-guided reinforcement learning algorithm, R$^2$PO, which jointly optimizes both boundaries through a tree-based exploration of reasoning regions and reflections. Our method significantly enhances the quality of both retrieval and reasoning, establishing an iterative loop where retrieval and reasoning mutually enhance each other. Extensive experiments on seven complex multi-hop QA benchmarks demonstrate that R$^2$-Searcher significantly outperforms state-of-the-art agentic search methods in answer accuracy and retrieval-reasoning quality. Ablation studies further confirm the critical role of retrieval-reasoning boundary calibration.

cs.IR↗

arXiv2026

LLM-as-Code: Agentic Programming for Agent Harness

Every major LLM agent framework gives the LLM the role of orchestrator; the model decides what to do next, when to call tools, and when to stop. We argue that token explosion, control-flow hallucination, and unreliable completion are not implementation bugs but architectural consequences of assigning the deterministic work of looping, branching, and sequencing to a probabilistic system. A better prompt or a stronger model cannot guarantee the reliability of the LLM agent. We therefore propose Agentic Programming, in which the program governs all control flow, and the LLM is itself part of it, an adaptive component we call LLM-as-Code and invoke only where a task calls for reasoning or generation. Within each call the model keeps full flexibility, but it cannot alter the program's execution path. With control in the program, the LLM's context is built from the execution history's call tree and forms a directed acyclic graph (DAG). Each call's context length is then determined by its call depth rather than by accumulation over steps. A case study of computer-use agents shows that the design is practical, not just a theoretical stance, substantially improving the stability of long visual operation sequences.

cs.AI↗