This paper presents a rhetorical and philosophical analysis of the public discourse surrounding the P vs NP problem, using the Clay Mathematics Institute’s description of the problem as a case study. The analysis does not challenge the formal validity of computational complexity theory itself. Instead, it examines how institutional and public-facing explanations extend formal asymptotic statements into broader narratives concerning impossibility, technological limitation, epistemic boundaries, and the limits of rationality. Drawing on the rhetoric and sociology of science, philosophy of modeling, and epistemology of formal systems, the paper identifies three recurrent mechanisms of rhetorical expansion: technical ambivalence, asymptotic extrapolation, ontological reification of computational difficulty. The paper argues that these mechanisms transform model-relative computational abstractions into culturally powerful narratives about intelligence, civilization, and the structure of reality itself. Additional sections analyze: the semantic ambiguity of terms such as “easy” and “hard,” the epistemology of verification, the transformation of formal models into ontological frameworks, hardness results as cultural symbols, and the institutional role of impossibility narratives in constructing scientific authority and disciplinary prestige. The broader thesis is that contemporary mathematical and computational discourse increasingly functions not only as formal analysis, but also as a producer of metaphysical and civilizational imaginaries.