评测智能厨房助手的营养管理能力，填补动态环境下的持续状态追踪空白。

An embodied kitchen assistant must do more than recognize food in isolated frames. It must track ingredient states over time and integrate visual observations with recipe and nutritional knowledge to support constraint-aware decision-making. We formalize this capability as \emph{Embodied Nutrition Management}: perceiving nutrition-relevant events, maintaining a persistent food state, and using it for knowledge-grounded planning. Existing benchmarks evaluate static food understanding or embodied cooking actions, but do not measure whether an agent can continuously update and use nutrition-relevant states in dynamic kitchens. To fill this gap, we introduce \textbf{NutriBench-Kitchen}, a benchmark containing 1,500 manually verified question--answer pairs from 160 cooking videos. It covers five task families: Ingredient Entry, Memory Management, Recipe Query, Long-Term Planning, and Short-Term Planning, spanning food-state construction, maintenance, knowledge retrieval, and decision-making across different planning horizons. Evaluations of proprietary and open-source large vision-language models reveal a substantial gap from human performance, particularly in quantitative ingredient estimation, long-term state tracking, and reasoning under interacting constraints. We further introduce \textbf{Nutri-Vgent}, a diagnostic long-video agent with separate episodic, food-state, and recipe memories. Its consistent improvements demonstrate the value of explicit state representations and structured memory for nutrition management. Together, NutriBench-Kitchen and Nutri-Vgent provide a testbed for studying persistent state tracking and knowledge-grounded reasoning in dynamic kitchens. Code is available at https://github.com/V1ol1n/NutriBench-Kitchen.