PFUSRC-13: A Century of the Kakeya Problem and the Equivalent Closure

This paper revisits the century-old Kakeya problem, arguing that its mathematical abstraction over the last hundred years has progressively stripped away essential physical realities, such as continuity and structural constraints. By introducing the PFUSRC (Unified Theory of the Universe) framework, we demonstrate that rotation is not merely a geometric transformation but a fundamental process of cosmic growth and entropy generation. We propose that a sustainable universe requires a rigid topological boundary to sustain persistent entropy growth, identifying the 45° bicone as the optimal structure for this mechanism. Furthermore, we introduce a new physical constant, \pi_1 = 12/11, derived from the geometric boundary ratio of the bicone, and establish a linear relationship between rotation angle and entropy residual. This study concludes that the "Equivalent Closure" of a 360° rotation represents a topological update rather than a reset, offering a falsifiable physical interpretation of the Kakeya problem that bridges geometry, thermodynamics, and cosmology.