Daily Papers

Agent skills today are hand-crafted, generated one-shot, or evolved through loosely controlled self-revision, none of which behaves like a deep-learning optimizer for the skill, and none of which reliably improves over its starting point under feedback. We argue the skill should instead be trained as the external state of a frozen agent, with the same discipline that makes weight-space optimization reproducible. SkillOpt is, to our knowledge, the first systematic controllable text-space optimizer for agent skills: a separate optimizer model turns scored rollouts into bounded add/delete/replace edits on a single skill document, and an edit is accepted only when it strictly improves a held-out validation score. A textual learning-rate budget, rejected-edit buffer, and epoch-wise slow/meta update make skill training stable while adding zero inference-time model calls at deployment. Across six benchmarks, seven target models, and three execution harnesses (direct chat, Codex, Claude Code), SkillOpt is best or tied on all 52 evaluated (model, benchmark, harness) cells and beats every per-cell competitor among human, one-shot LLM, Trace2Skill, TextGrad, GEPA, and EvoSkill skills. On GPT-5.5 it lifts the average no-skill accuracy by +23.5 points in direct chat, by +24.8 inside the Codex agentic loop, and by +19.1 inside Claude Code. Transfer experiments further show that optimized skill artifacts retain value when moved across model scales, between Codex and Claude Code execution environments, and to a nearby math benchmark without further optimization.

Large language model agents are increasingly envisioned as always-on personal assistants with access to anything relevant in the user's digital world. Yet current systems operate over only narrow slices of that world, limiting context-sensitive reasoning and effective assistance. Existing benchmarks similarly provide only partial user state and therefore fail to capture performance in such a broad, always-on setting. To address this gap, we introduce Claw-Anything, a benchmark that expands agent context along three dimensions: long-horizon activity histories, interdependent backend services, and integrated GUI and CLI interaction across multiple devices. To instantiate this setting, we simulate months of user activity through multi-round event injection, producing complex world states and realistic noise, including irrelevant events and conflicting signals. Agents must reason over rich contextual environments while remaining robust to such noise. This expanded scope also enables the evaluation of proactive assistance, requiring agents to anticipate user needs and deliver timely recommendations. Experiments show that GPT-5.5 achieves only 34.5% pass@1, substantially below prior benchmarks, underscoring a gap between current agent capabilities and the demands of always-on personal assistance. Alongside the benchmark, we release an automated data-generation pipeline that yields 2,000 training environments and improves the base model by 23.7%, demonstrating its utility of scalable data infrastructure.

Computer-aided design (CAD) is the backbone of modern industrial design, yet learned CAD generators still fall short of real engineering pipelines: they neither iterate like engineers nor evaluate what engineering requires. Prior work has treated CAD generation as two disjoint steps, part synthesis and assembly, where the former is graded by proximity to a gold reference and the latter, when handled at all, is reduced to a separate constraint solving step. In this work, we introduce a more industry-native task formulation that requires a model to produce a fully assembled multi-part STEP file from a free-form engineering brief, which is then validated via finite element analysis (FEA). FEA validation reveals that Codex (GPT-5.5) and Claude Code (Opus-4.7) agents do not produce a single strict-passing artifact in the main first-attempt sweep, with the best configuration meeting only about 20% of typed requirements on average. Moreover, we introduce two additional supervision signals, a novel text-only blueprint schema and a 21-view image renderer that aids the agent's visual inspection, that better align the generation loop with how engineers iterate in practice. On S2O and Fusion360, the same feedback tools improve geometric reconstruction, with GPT-5.5/xhigh rising from 0.444 to 0.592 Box-IoU on S2O and from 0.397 to 0.505 on Fusion360. Together these signals move CAD programs toward artifacts that are not only visually plausible but also checked against physical and structural requirements.

Recent LLM-based agents have closed substantial portions of the scientific discovery loop in software-only machine-learning research, in chemistry, and in biology. Extending the same loop to high-fidelity physical simulators is harder, because solver completion does not imply physical validity and many failure modes appear only in field-level imagery rather than in solver logs. We present AI CFD Scientist, an open-source AI scientist for computational fluid dynamics (CFD) that, to our knowledge, is the first to span literature-grounded ideation, validated execution, vision-based physics verification, source-code modification, and figure-grounded writing within a single inspectable workflow. Three coupled pathways cover parameter sweeps within a fixed solver, case-local C++ library compilation for new physical models, and open-ended hypothesis search against a reference comparator, all running on OpenFOAM through Foam-Agent. At the center of the framework is a vision-language physics-verification gate that inspects rendered flow fields before any result is accepted, rerun, or written into a manuscript. On five tasks under a shared GPT-5.5 backbone, AI CFD Scientist autonomously discovers a Spalart-Allmaras runtime correction that reduces lower-wall Cf RMSE against DNS by 7.89% on the periodic hill at Reh=5600; under matched LLM cost, two strong general AI-scientist baselines (ARIS, DeepScientist) execute partial CFD workflows but lack the domain-specific validity gates needed to convert runs into defensible scientific claims; and a controlled planted-failure ablation shows that the vision-language gate detects 14 of 16 silent failures missed by solver-level checks. Code, prompts, and run artifacts are released at https://github.com/csml-rpi/cfd-scientist.

Does a lexical retriever suffice as large language models (LLMs) become more capable in an agentic loop? This question naturally arises when building deep research systems. We revisit it by pairing BM25 with frontier LLMs that have better reasoning and tool-use abilities. To support researchers asking the same question, we introduce Pi-Serini, a search agent equipped with three tools for retrieving, browsing, and reading documents. Our results show that, on BrowseComp-Plus, a well-configured lexical retriever with sufficient retrieval depth can support effective deep research when paired with more capable LLMs. Specifically, Pi-Serini with gpt-5.5 achieves 83.1% answer accuracy and 94.7% surfaced evidence recall, outperforming released search agents that use dense retrievers. Controlled ablations further show that BM25 tuning improves answer accuracy by 18.0% and surfaced evidence recall by 11.1% over the default BM25 setting, while increasing retrieval depth further improves surfaced evidence recall by 25.3% over the shallow-retrieval setting. Source code is available at https://github.com/justram/pi-serini.

Large language models (LLMs) are increasingly proposed as clinical agents, yet static, single-turn benchmarks cannot capture how a model dynamically delivers care across an encounter: gathering information, planning treatment, and adapting longitudinal management across successive patient states. Medical education has long addressed an analogous challenge through standardized patients (SPs): trained actors who consistently portray clinical cases, enabling realistic practice and objective, scripted assessment. Here we introduce MedSP1000, an SP-derived interactive benchmark for clinical-agent evaluation, including 1,638 SP cases with 24,602 trajectory-level peer-reviewed rubrics. MedSP1000 converts peer-reviewed SP teaching cases into executable scenarios with defined SP case scripts, clinical environment contexts, and human-validated structured rubric. In each simulation evaluation run, a clinical agent interacts in closed loop with a patient agent and an environment controller, and its behaviour is scored throughout the encounter against expert criteria specified in the original materials. Applying MedSP1000 to a range of general-purpose and medically specialized LLMs, we find that performance on static benchmarks does not reliably translate to such educational scenarios. The best-performing model, GPT-5.5, completes only 60.4% of expert-defined rubric items, whereas the strongest medically specialized model reaches 40.0%; increasing test-time compute produces no measurable gain. These results suggest that current LLMs, including agentic systems tuned for medicine, are not yet reliable enough to be safely integrated into actual clinical practice. More broadly, MedSP1000 shows how process-level, SP-style evaluation can reveal clinically relevant failure modes that single-turn benchmarks miss.

LLM agents benefit from reusable skills, yet test-time tasks often require guidance more specific than a static skill library can provide. We propose SkillTTA, a Test-Time Adaptive Skill Synthesis method that retrieves a small set of training trajectories relevant to the current task and synthesizes them into a temporary, task-specific textual skill. The solver model is kept fixed, so adaptation happens entirely through generated context rather than parameter updates. We evaluate the method on SpreadsheetBench, ALFWorld, and BigCodeBench. Compared with static trajectory-to-skill synthesis using GPT-5.5, task-specific skills improve SpreadsheetBench Pass@1 from 0.397 to 0.505 and BigCodeBench Pass@1 from 0.517 to 0.651. On ALFWorld, the method matches a heavier memory-learning baseline within four points of success rate while producing the shortest successful trajectories among reported methods. Ablations on SpreadsheetBench further show that synthesized skills outperform raw trajectory prompting, that top-k retrieval should stay small, and that failed trajectories are especially useful because they expose recurring evaluator-facing mistakes.

Autoresearch offers a flexible paradigm for automating scientific tasks, in which an AI agent proposes, implements, evaluates, and refines candidate solutions against a quantitative objective. Here, we use composition-based materials-property prediction to test whether such agents can perform a task beyond model selection and hyperparameter optimization: the design of input descriptors. We introduce Automat, an autoresearch framework where a coding agent based on a large language model generates composition-only descriptors for chemical compounds and evaluates them using a random forest workflow. The agent is restricted to information derivable from chemical formulas and iteratively proposes, implements, and tests chemically motivated descriptor strategies. We apply Automat, with OpenAI Codex using GPT-5.5 as the coding agent, to the prediction of experimental band gaps in inorganic materials and Curie temperatures in ferromagnetic compounds. In both tasks, Automat improves over fractional-composition, Magpie, and combined fractional-composition/Magpie baselines, while producing descriptor families that are chemically interpretable. These results provide a demonstration that autoresearch agents can generate competitive, task-specific materials descriptors without manual feature engineering during the run. They also reveal current limitations, including descriptor redundancy, sensitivity to greedy feature expansion, and the need for explicit complexity control, descriptor pruning, and more sophisticated search strategies.

Frontier model evaluations are shifting from foundational capabilities (e.g., instruction following and reasoning) toward compositional, agentic ones, but Korean agentic benchmarks remain scarce. We introduce K-BrowseComp, a web-browsing agent benchmark grounded in Korean contexts, consisting of 400 problems. The 300-problem K-BrowseComp-Verified subset is manually constructed and validated by native Korean speakers. On this subset, frontier LLMs, including GPT-5.5, DeepSeek-V4-Pro, and GLM-5.1, reach only 30.00--45.67\%, a substantial drop from BrowseComp, while Korean LLMs released through Korea's Proprietary AI Foundation Model program obtain only 0.00--10.33\%. We further construct a 100-problem synthetic split using hard few-shot exemplars and failure-mode-targeted generation to exploit the asymmetry between solving and creating web browsing problems. On the adversarially filtered synthetic diagnostic split, the strongest model reaches only 26.00\%, and we report this split separately as a targeted stress test. We publicly release our data and code.

Structured financial audit verification is difficult for language-model agents because correctness depends on structured evidence rather than text alone. A model must link reported facts to taxonomy concepts, traverse calculation or dimensional relations, and recompute expected values before applying an audit rule. We propose AuditFlow, a graph-grounded multi-agent framework that separates adaptive search from deterministic verification. AuditFlow builds a symbolic environment from a static US-GAAP taxonomy graph and a dynamic XBRL filing graph, and exposes it through typed tools for fact retrieval, taxonomy traversal, numerical checking, and rule evaluation. Two junior auditors inspect each case from regulatory and evidentiary views, while a senior auditor resolves disagreements and can request further investigation. The final reports are fused through evidential aggregation to produce an audit verdict, expected value, evidence trail, and trustworthiness score. On a FinAuditing-derived FinMR sample, AuditFlow reaches 82.09% joint audit accuracy under GPT-5.5, outperforming the strongest baseline by 14.93 points. Removing deterministic checks drops accuracy to 17.91%, showing that the symbolic environment performs the verification step that the model cannot reliably replace.

Large language models (LLMs) are becoming increasingly capable mathematical collaborators, but static benchmarks are no longer sufficient for evaluating progress: they are often narrow in scope, quickly saturated, and rarely updated. This makes it hard to compare models reliably and track progress over time. Instead, we need evaluation platforms: continuously maintained systems that run, aggregate, and analyze evaluations across many benchmarks to give a comprehensive picture of model performance within a broad domain. In this work, we build on the original MathArena benchmark by substantially broadening its scope from final-answer olympiad problems to a continuously maintained evaluation platform for mathematical reasoning with LLMs. MathArena now covers a much wider range of tasks, including proof-based competitions, research-level arXiv problems, and formal proof generation in Lean. Additionally, we maintain a clear evaluation protocol for all models and regularly design new benchmarks as model capabilities improve to ensure that MathArena remains challenging. Notably, the strongest model, GPT-5.5, now reaches 98% on the 2026 USA Math Olympiad and 74% on research-level questions, showing that frontier models can now comfortably solve extremely challenging mathematical problems. This highlights the importance of continuously maintained evaluation platforms like MathArena to track the rapid progress of LLMs in mathematical reasoning.

Evaluating embodied systems on real dexterous hardware requires more than isolated primitive skills: an agent must perceive a changing tabletop scene, choose a context-appropriate action, execute it with a dexterous hand, and leave the scene usable for later decisions. We introduce DexHoldem, a real-world system-level benchmark built around Texas Hold'em dexterous manipulation with a ShadowHand. DexHoldem provides 1,470 teleoperated demonstrations across 14 Texas Hold'em manipulation primitives, a standardized physical policy benchmark, and an agentic perception benchmark that tests whether agents can recover the structured game state needed for embodied decision making. On primitive execution, π_{0.5} obtains the highest task completion rate (61.2%), while π_{0.5} and π_0 tie on scene-preserving success rate (47.5%). On agentic perception, Opus 4.7 obtains the best strict problem-level accuracy (34.3%), while GPT 5.5 obtains the best average field-wise accuracy (66.8%), exposing a gap between isolated visual sub-capabilities and complete routing-relevant state recovery. Finally, we instantiate the full embodied-agent loop in three case studies, where waiting, recovery dispatches, human-help requests, and repeated primitive execution reveal how perception and policy errors accumulate during closed-loop deployment. DexHoldem therefore evaluates dexterous tabletop execution, agentic perception, and embodied decision routing in a shared physical setting. Project page: https://dexholdem.github.io/Dexholdem/.

We present Metal-Sci, a 10-task benchmark of scientific Apple Silicon Metal compute kernels spanning six optimization regimes (stencils, all-pairs in n-body problems, multi-field Boltzmann, neighbor-list molecular dynamics, multi-kernel PDE, FFT). Each task ships a CPU reference, a roofline-anchored fitness function, and a held-out generalization size. We pair the benchmark with a lightweight harness for automatic kernel search that runtime-compiles each candidate, scores it against the roofline across multiple sizes, and feeds structured compile and per-size correctness diagnostics back to a frozen LLM driving a (1{+}1) evolutionary loop. We report matched single-model sweeps of Claude Opus 4.7, Gemini 3.1 Pro, and GPT 5.5 on M1 Pro: in-distribution self-speedups span 1.00times to 10.7times. Beyond raw speedup, our central methodological claim is structural: the held-out gate scoring function Φ_T (evaluated once at end-of-run on a configuration the agent never sees during search) functions as a cheap mechanical oversight primitive on this automatic search loop, catching e.g. an Opus template <uint D> HMC win that returns wrong samples at unseen dimensions, and a GPT FFT3D best that wins in-distribution at 2.95times speedup but collapses to 0.23times on a 256^3 held-out cube, a silent regression that the in-distribution score alone cannot see. Code at https://github.com/vicgalle/metal-sci-kernels

We introduce ShapeCodeBench, a synthetic benchmark for perception-to-program reconstruction: given a rendered raster image, a model must emit an executable drawing program that a deterministic evaluator re-renders and compares with the target. The v1 DSL has four primitives on a 512 x 512 black-on-white canvas, but every instance is generated from a seeded RNG, so fresh held-out sets can be created to reduce exact-instance contamination. We release a frozen eval_v1 split with 150 samples across easy, medium, and hard tiers, scored by exact match, pixel accuracy, foreground IoU, parse success, and execution success. We evaluate an empty-program floor, a classical computer-vision heuristic, Claude Opus 4.7 at high and max effort, and GPT-5.5 at medium and extra_high reasoning effort. The heuristic is competitive on easy scenes but collapses when overlaps fuse components; the strongest multimodal configuration preserves much of the foreground structure but still misses exact match because of small parameter errors. Best overall exact match remains low, so ShapeCodeBench is far from saturated. The benchmark code, frozen dataset, run artifacts, and paper sources are released to support independent replication and extension.

We present E3, an automated review assistant that augments reviewers and engineering teams by identifying decision-relevant technical concerns in research papers. For each concern, E3 reports its nature, its location, its bearing on the contribution, and the analysis or evidence that would resolve it, covering unsupported claims, missing ablations, weak baselines, hidden assumptions, threats to validity, and leakage risks. To evaluate E3 without contamination confounds we adopt an issue-level backtesting protocol: the corpus is restricted to papers postdating the training cutoff of every automated source, and for each paper a meta-judge that observes only anonymised reviews labels every issue-source pair as Caught, Partial, or Missed. Applied to 100 ICLR 2026 papers and 4598 judged issue rows, comparing E3 against the ICLR human reviews and two prompt-matched LLM baselines built on gpt-5.4 from OpenAI and claude-opus-4-6 from Anthropic, with meta-judge gpt-5.5, E3 attains the highest recall on every aggregate metric. Partial-inclusive recall reaches 90.2 percent, which is 15.5 points over GPT, 17.1 points over Claude, and 29.2 points over the human reviews, and strict recall preserves the ordering at 65.8 percent. On concerns raised by the human reviewers, E3 recovers 89.6 percent; on concerns the human reviewers missed it surfaces 1635 additional rows admitted into the judged union, 406 above the next-best source. Corpus, baseline prompts, judge prompt template, and evaluation code are released.

Despite significant advancements in general artificial intelligence, such as GPT-4, their effectiveness in the medical domain (general medical AI, GMAI) remains constrained due to the absence of specialized medical knowledge. To address this challenge, we present GMAI-VL-5.5M, a comprehensive multimodal medical dataset created by converting hundreds of specialized medical datasets into meticulously constructed image-text pairs. This dataset features comprehensive task coverage, diverse modalities, and high-quality image-text data. Building upon this multimodal dataset, we propose GMAI-VL, a general medical vision-language model with a progressively three-stage training strategy. This approach significantly enhances the model's ability by integrating visual and textual information, thereby improving its ability to process multimodal data and support accurate diagnosis and clinical decision-making. Experimental evaluations demonstrate that GMAI-VL achieves state-of-the-art results across a wide range of multimodal medical tasks, such as visual question answering and medical image diagnosis. Our contributions include the development of the GMAI-VL-5.5M dataset, the introduction of the GMAI-VL model, and the establishment of new benchmarks in multiple medical domains. Code and dataset will be released at https://github.com/uni-medical/GMAI-VL.

Evaluating LLMs' instruction-following ability in multi-topic dialogues is essential yet challenging. Existing benchmarks are limited to a fixed number of turns, susceptible to saturation and failing to account for users' interactive experience. In this work, we propose a novel framework featuring a three-layer tracking mechanism and a query synthesis agent to mimic sequential user behaviors. Grounded in Flow Theory, we introduce process-centric metrics and terminate a conversational evaluation only upon exhausting user patience. Leveraging this framework, we present EvolIF, an evolving benchmark covering 12 constraint groups. Our analysis reveals deficiencies in failure recovery and fine-grained instruction following, with performance stratification becoming evident as conversational depth increases. GPT-5 demonstrates the most sustained resilience, maintaining a 66.40% robustness score, outperforming Gemini-3-Pro by 5.59%, while other models lag behind. Data and code will be released at https://github.com/JiaQiSJTU/EvolIF.

Current Multimodal Large Language Models (MLLMs) may struggle with understanding long or complex videos due to computational demands at test time, lack of robustness, and limited accuracy, primarily stemming from their feed-forward processing nature. These limitations could be more severe for models with fewer parameters. To address these limitations, we propose a novel framework inspired by cybernetic principles, redesigning video MLLMs as adaptive systems capable of self-monitoring, self-correction, and dynamic resource allocation during inference. Our approach, CyberV, introduces a cybernetic loop consisting of an MLLM Inference System, a Sensor, and a Controller. Specifically, the sensor monitors forward processes of the MLLM and collects intermediate interpretations, such as attention drift, then the controller determines when and how to trigger self-correction and generate feedback to guide the next round. This test-time adaptive scaling framework enhances frozen MLLMs without requiring retraining or additional components. Experiments demonstrate significant improvements: CyberV boosts Qwen2.5-VL-7B by 8.3% and InternVL3-8B by 5.5% on VideoMMMU, surpassing the competitive proprietary model GPT-4o. When applied to Qwen2.5-VL-72B, it yields a 10.0% improvement, achieving performance even comparable to human experts. Furthermore, our method demonstrates consistent gains on general-purpose benchmarks, such as VideoMME and WorldSense, highlighting its effectiveness and generalization capabilities in making MLLMs more robust and accurate for dynamic video understanding. The code is released at https://github.com/marinero4972/CyberV.

Automatic toxic language detection is critical for creating safe, inclusive online spaces. However, it is a highly subjective task, with perceptions of toxic language shaped by community norms and lived experience. Existing toxicity detection models are typically trained on annotations that collapse diverse annotator perspectives into a single ground truth, erasing important context-specific notions of toxicity such as reclaimed language. To address this, we introduce MODELCITIZENS, a dataset of 6.8K social media posts and 40K toxicity annotations across diverse identity groups. To capture the role of conversational context on toxicity, typical of social media posts, we augment MODELCITIZENS posts with LLM-generated conversational scenarios. State-of-the-art toxicity detection tools (e.g. OpenAI Moderation API, GPT-o4-mini) underperform on MODELCITIZENS, with further degradation on context-augmented posts. Finally, we release LLAMACITIZEN-8B and GEMMACITIZEN-12B, LLaMA- and Gemma-based models finetuned on MODELCITIZENS, which outperform GPT-o4-mini by 5.5% on in-distribution evaluations. Our findings highlight the importance of community-informed annotation and modeling for inclusive content moderation. The data, models and code are available at https://github.com/asuvarna31/modelcitizens.

Existing large vision-language models (LVLMs) are largely limited to processing short, seconds-long videos and struggle with generating coherent descriptions for extended video spanning minutes or more. Long video description introduces new challenges, such as plot-level consistency across descriptions. To address these, we figure out audio-visual character identification, matching character names to each dialogue, as a key factor. We propose StoryTeller, a system for generating dense descriptions of long videos, incorporating both low-level visual concepts and high-level plot information. StoryTeller uses a multimodal large language model that integrates visual, audio, and text modalities to perform audio-visual character identification on minute-long video clips. The results are then fed into a LVLM to enhance consistency of video description. We validate our approach on movie description tasks and introduce MovieStory101, a dataset with dense descriptions for three-minute movie clips. To evaluate long video descriptions, we create MovieQA, a large set of multiple-choice questions for the MovieStory101 test set. We assess descriptions by inputting them into GPT-4 to answer these questions, using accuracy as an automatic evaluation metric. Experiments show that StoryTeller outperforms all open and closed-source baselines on MovieQA, achieving 9.5% higher accuracy than the strongest baseline, Gemini-1.5-pro, and demonstrating a +15.56% advantage in human side-by-side evaluations. Additionally, incorporating audio-visual character identification from StoryTeller improves the performance of all video description models, with Gemini-1.5-pro and GPT-4o showing relative improvement of 5.5% and 13.0%, respectively, in accuracy on MovieQA.

Block diffusion speculative decoding accelerates LLM inference by predicting all tokens within a block simultaneously for the target model to verify in parallel. Predicting an entire block at once requires a sufficiently capable draft model and effective utilization of the target model's internal knowledge. However, the state-of-the-art method DFlash constrains all draft layers to share a single fused representation derived from only a few target layers, limiting per-layer expressiveness and hindering further scaling of draft capacity. In this paper, we present \modelname, which flares out the narrow conditioning bottleneck of DFlash through a lightweight layer-wise fusion mechanism: each draft layer attends to its own learnable combination of a broad set of target layers at negligible overhead, simultaneously injecting richer target knowledge and providing every draft layer with a distinct input. This enhanced per-layer expressiveness enables scaling the draft model to deeper architectures with consistent gains. We further scale training data from 800K to 2.4M samples to fully exploit the enlarged capacity. On six benchmarks spanning mathematical reasoning, code generation, and conversation, \modelname attains average wall-clock speedups of 5.52x on Qwen3-4B, 5.46x on Qwen3-8B, and 3.91x on GPT-OSS-20B, improving over DFlash by roughly 11\%, 8\%, and 5\% respectively. Our code is available at https://github.com/Tencent/AngelSlim.

Large Language Models (LLMs) achieve strong performance on natural language tasks but remain unreliable in mathematical reasoning, frequently generating fluent yet logically inconsistent solutions. We present NeuroProlog, a neurosymbolic framework that ensures verifiable reasoning by compiling math word problems into executable Prolog programs with formal verification guarantees. We propose a multi-task Cocktail training strategy that jointly optimizes three synergistic objectives in a unified symbolic representation space: (i) mathematical formula-to-rule translation (KB), (ii) natural language-to-program synthesis (SOLVE), and (iii) program-answer alignment. This joint supervision enables positive transfer, where symbolic grounding in formula translation directly improves compositional reasoning capabilities. At inference, we introduce an execution-guided decoding pipeline with fine-grained error taxonomy that enables iterative program repair and quantifies model self-debugging capacity. Comprehensive evaluation on GSM8K across four model scales (3B--32B parameters) demonstrates consistent improvements: cocktail training achieves significant accuracy gains of +5.23\% (Qwen-32B, p < 0.01), +3.43\% (GPT-OSS-20B, p < 0.01), and +5.54\% (Llama-3B, p < 0.05) over single-task baselines. Systematic error analysis reveals scale-dependent learning dynamics: at 32B scale, cocktail training transforms unfixable type errors (12\% repair rate) into correctable domain errors (96\% repair rate), achieving 92.7\% overall correction; at 8B scale, the same training eliminates syntactic errors but introduces semantic failures, revealing a critical capacity threshold for type-safe symbolic reasoning.