Performing deliberate mathematical reasoning in visual contexts is a hallmark of expert-level Multimodal Large Language Models (MLLMs) and requires a sophisticated synthesis of perceptual grounding and symbolic logic. However, in the realm of mathematical functions, our investigation reveals a critical *modality interference phenomenon*: even advanced models, while performing textual computational reasoning, tend to disregard or misinterpret essential visual cues. To address this challenge, we propose **Func-R1**, which synergistically harmonizes precise visual perception and rigorous logical reasoning. Concretely, built upon an explicitly decoupled architecture, we employ a hierarchical post-training framework to progressively identify critical visual evidence and conduct in-depth theoretical reasoning. Furthermore, the *Perception-Aligned Theoretic Optimization (PATO)* strategy is proposed to steer policy updating towards internalizing fundamental theoretical properties while dynamically rectifying heterogeneous visual information throughout the reasoning process. Extensive experiments across diverse benchmarks demonstrate that Func-R1 delivers the optimal performance among open-source MLLMs, even surpassing GPT-5 with an 8.4% improvement on MathVerse’s function-oriented tasks.

Open Domain Multi-Hop Question Answering (ODMHQA) plays a crucial role in Natural Language Processing (NLP) by aiming to answer complex questions through multi-step reasoning over retrieved information from external knowledge sources. Recently, Large Language Models (LLMs) have demonstrated remarkable performance in solving ODMHQA owing to their capabilities including planning, reasoning, and utilizing tools. However, LLMs may generate off-topic answers when attempting to solve ODMHQA, namely the generated answers are irrelevant to the original questions. This issue of off-topic answers accounts for approximately one-third of incorrect answers, yet remains underexplored despite its significance. To alleviate this issue, we propose the Discriminate→Re-Compose→Re- Solve→Re-Decompose (Dr3) mechanism. Specifically, the Discriminator leverages the intrinsic capabilities of LLMs to judge whether the generated answers are off-topic. In cases where an off-topic answer is detected, the Corrector performs step-wise revisions along the reversed reasoning chain (Re-Compose→Re-Solve→Re-Decompose) until the final answer becomes on-topic. Experimental results on the HotpotQA and 2WikiMultiHopQA datasets demonstrate that our Dr3 mechanism considerably reduces the occurrence of off-topic answers in ODMHQA by nearly 13%, improving the performance in Exact Match (EM) by nearly 3% compared to the baseline method without the Dr3 mechanism.