Presentization Number System (PNS): A Mathematical Framework for Boundary-Based Number Emergence

This paper introduces the Presentization Number System (PNS), a theoretical framework in which numbers emerge from persistent boundary events detected within a continuous state space under an observational frame. Unlike classical set-theoretic definitions where numbers represent the cardinality of predefined objects, PNS defines numbers as the cardinality of stable boundary events that arise through frame-dependent segmentation of continuous systems. The framework connects topology, information theory, and physical field dynamics, suggesting that counting can be interpreted as the detection of persistent structural distinctions within continuous fields.