Principle Emerges Function: Structure-Independent Synchronization Admissibility Boundary

We discover a synchronization admissibility boundary defined solely by the states of oscillators. The boundary is independent of structure and determines whether any two oscillators share a cluster in real time, unifying global synchronization, cluster partition, and the real-time onset of synchronization loss. This uniformity has been validated through dozens of adversarial tests. Mathematical proofs show that this boundary is mathematically equivalent to the constraint that the synchronous frequency must be a real number. This constraint is a direct corollary of a cornerstone of physics long taken for granted: all measurable physical quantities are real numbers. This equivalence reveals that the synchronous admissibility boundary (a key function) emerges directly from the principle that is logically prior to any specific structure.