Axiomatic Emergence AD; Why Does Mathematics Work? The Physical Foundation of Information, Computability, and Description Boundaries

This article starts from two fundamental physical principles—‘‘Information Conservation’’and ‘‘Finite-Step Computability’’—to explore the effective boundaries ofmathematical structures in describing physical reality. The study indicates that anymathematical framework (such as set theory, category theory, or geometry) is effective only within a ‘‘description window’’determined by the information dynamics ofthe system. The boundary of this window is regulated by the ratio η between therate of information reorganization and the rate of entropy capacity expansion (theη criterion). When the accumulated ‘‘information torsion’’of the system reachesa critical threshold, the ‘‘closure’’of the current level of mathematical descriptionwill experience ‘‘leakage,’’manifesting as phase transitions in logical rules or geometric structures, thereby forcing the descriptive language to leap to higher-ordermathematical structures. This article interprets Gödel’s incompleteness theoremsas the metamathematical counterpart of this physical process: the undecidability offormal systems arises from the inevitable constraints of finite generability and information conservation when a physical system engages in self-referential description.By analyzing the Langlands program as a case study for bridging the ‘‘incompleteness gaps’’between levels, this paper argues that the hierarchical tower structure ofmathematics and its dynamic evolution are essentially emergent phenomena of thephysical universe seeking self-consistent description under information constraints.This article provides a unified framework based on physical principles for empiricallyinvestigating ‘‘why mathematics works’’and ‘‘where its boundaries of effectivenesslie.’’