Axiom of Valuation and Hierarchical Uncertainty: A Unified Theory of Global Envelope-Projection-Mathematical-Physical Distribution

Existing theories generally regard valuation as an approximate method, and attribute uncertainty to observation limitations, incomplete information or systematic constraints. This paper proposes a minimalist, self-consistent and globally closed axiom system. It proves that valuation is not an optional technique, but a logical necessity of hierarchical structure; uncertainty is not a defect, but an inherent hierarchical difference between the high-order global total envelope and the low-order mathematical-physical distribution package. This paper defines core concepts including global total envelope, mathematical-physical distribution package, hierarchical difference, downward projection and projection number. It establishes the axioms of global completeness, hierarchical uncertainty, valuation, projection uniqueness, and evolutionary distribution of mathematical-physical entities. It demonstrates that high-order states can only be expressed by interval, range and steady-state domain valuation, while a single downward projection inevitably generates a unique accurate projection number. This paper unifies the origin of all valuation phenomena in various fields, and points out that the persistent pursuit of precision by low-order mathematical-physical systems precisely reflects that contemporary mathematical and physical systems have not yet broken through the limitations of the low-order framework.