To investigate how task complexity cognitive factors influence the lexical complexity in essay writing, this study was guided by Robinson's Cognition Hypothesis (CH) and Skehan's Limited Attentional Capacity Model (LACM), and examined the effects of task complexity on lexical complexity in undergraduates' essay writing. Using Lu's Lexical Complexity Analyser (2010, 2012), the study analysed essays written by 60 undergraduates across 25 dimensions of lexical complexity, following the SSARC grouping method by Robinson and Gilabert (2007). The Wilcoxon signed-rank test evaluated the relationship between pre-existing standardised writing scores (PS) and the essay outputs under the experimental condition (QM), focusing on lexical density, lexical sophistication, and lexical variation dimensions. Findings indicate that verb sophistication and the Uber index are significantly associated with higher proficiency, supporting Robinson's CH, which links cognitive resource allocation in complex tasks to improved language proficiency. Noun variation also emerged as significant, aligning with recent studies emphasising lexical diversity in writing. However, measurements like lexical density and type/token ratio showed no significant differences, suggesting a limited role. The study highlights the importance of sophisticated lexical use; however, it also acknowledges that cognitive resource limitations, as posited by Skehan's LACM, may constrain the deployment of more complex lexical structures.