The “Not” Function—Paradox’s Mechanism in Linguistics, Mathematics, and Physics

The 1-D geometric model studies the structure of paradox found in states containing only self-reference for their property. The mechanism of absolute closure is the logical “not” function. Two correlated fundamental reference frames and the boundary that applies are identified. In the first framework, the discrete identification of the state's internal structure is prohibited from enumeration within its boundary. In the second, two segments share property for the same infinity but are excluded from common membership across the boundary that divides them. The geometric model is applied to analyze the role of the “not” function in linguistics, mathematics, physics, and the generic structure of emergence. Logical formalisms necessarily discount paradoxes as anomalies open to more advanced understanding, worked around by restrictions to logic or ignored as nonsensical. The 1-D geometric model takes an opposing analysis perspective, considering paradox a fundamental mechanism. The geometric proof examines two paradoxical constructions of the right triangle within the unit circle. Although the two formats are paradoxical, with the second having no rational basis, the cosine squared calculations agree. The study concludes that two paradoxical frameworks cohabit within a universal state, and the paradox mechanism is validated as the basis of the relationship between them.