The physical universe is a fundamental continuum, devoid of absolute voids or inherent demarcations. Yet, the architectures of intelligence—from geopolitical borders drawn on maps to the segmentation of time in calendars—rely entirely on the imposition of artificial, discrete lines. What is the ontological nature of these boundaries, and why must they exist? This paper introduces the General Theory of Boundaries to formalize the inescapable tension between continuous reality and discrete representation. We model the world as an omnipresent, continuous flux governed by a time-evolution operator (hat (F) ), upon which systems apply discrete boundary operators (hat (B) ) to extract computable and manageable states. We establish that this interaction is fundamentally non-commutative: hat (B), hat (F) != 0. This non-commutativity generates systemic impedance, forcing a continuous phase transition between the fluid nature of the universe and the discrete architectures of cognition and society. By categorizing boundary dynamics into intensity-threshold (self-organizing) and order-dependent (epistemological) types, this framework unifies phenomena across macroscopic geopolitics, microscopic ecosystems, and cognitive discretization costs. Ultimately, we demonstrate that intelligence itself—both biological and artificial—emerges not from the continuum alone, but as a boundary operator continuously negotiating the friction of discretizing a universe that refuses to be divided.
Whas O (Sun,) studied this question.
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