This work introduces a formal admissibility framework for a Theory of Everything based on recursive logical closure rather than force-level unification. The proposed criterion defines a 60-glyph quaternionic inter-domain syllogistic system (SEXA Mathematics) in which physical laws, dimensional regimes, and observer constraints are admitted only if they satisfy global closure, invariance preservation, and recursive self-consistency across scales. At the core of the framework is the SEXA Unified F