Generalized Mapping Theory — Used to Describe Phenomena That Cannot Be Characterized by Generalized Functions

Traditional theories of generalized functions (e.g., Schwartz's distribution theory and Colombeau algebras), while demonstrating excellent performance in static distribution problems, inherently fail to characterize the generative relationships between dynamic behaviors and their outcomes. As exemplified by college students' enlistment decisions where existing methods can only employ Dirac measures to represent final state distributions without capturing the influence of dynamic processes like medical examinations and qualification reviews, similar limitations manifest in cases ranging from blacksmith forging to graduate admissions, where conventional approaches merely reflect terminal states or statistical probabilities while discarding crucial causal mechanisms such as "hammering intensity-deformation response" or "exam preparation-admission outcome". To overcome these deficiencies, this paper proposes Generalized Mapping Theory (GMT) that establishes a quadruple framework consisting of object set A, operation set F, result set B and generative relation ⊢ to mathematically model dynamic behavior-outcome correlations, with its theoretical innovations embodied in three aspects: (1) explicit mathematization of physical/social behaviors as formal objects, (2) complete retention of dynamic generative pathways from operations to results, and (3) native support for multi-branch outcome scenarios. Theoretical analysis confirms that GMT not only fully subsumes all functionalities of traditional generalized functions (e.g., representing Dirac delta as identity-operation-generated mappings) but also solves their unsolvable dynamic behavior modeling problems, thereby providing innovative mathematical tools for quantitative research in materials science and social sciences.