<![CDATA[The transition from differential calculus to real analysis represents a significant ontological shift for undergraduate mathematics students, often accompanied by difficulties in constructing formal proofs. While existing frameworks broadly categorize imperfect proofs as procedural failures, they lack the diagnostic resolution to capture transitional cognitive states. To address this gap, this study integrates the SOLO taxonomy with APOS theory to analyze the cognitive mechanisms of students' proof construction. Employing a qualitative instrumental case study, data from written tests and task-based interviews were collected from 35 undergraduate students. These participants were purposefully selected because they had completed foundational calculus and logic courses, placing them squarely in the crucial transition stage from computational calculus to axiomatic analysis. The data were analyzed using the Constant Comparative Method. Moving beyond purely narrative descriptions, the empirical findings reveal that a majority of students (63%) operate within transition zones. Specifically, the study introduces two refined transitional states: 34% of the students were identified at the Semi-Relational level, characterized by logical gaps stemming from unstable process internalization, and 29% at the Semi-Extended Abstract level, where students display strong intuition but lack formal rigor due to rigid concept encapsulation. Ultimately, identifying these specific cognitive barriers provides a practical roadmap for educators to design targeted instructional interventions, actively helping students overcome their logical or intuitive hurdles to master formal mathematical proving successfully.]]>
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