Single-Chain Persistence Theory II: The Mathematical Foundations of Persistence

This paper is the mathematical middle layer between Single-ChainPersistence Theory I (SCPT-I) and Single-Chain Persistence Theory III (SCPT-III) . SCPT-I bears the burden ofproof; the present paper formalizes the four criteria and answers the question “why the fourcriteria can be assigned values”; SCPT-III answers “how to use the four criteria to interprethistory, diagnose the present, and conditionally predict the future.” This paper legislates no newdefinitions—all definitions are quoted from SCPT-I/III; what is newly given consists only offormulas, theorems, phenomenological models, and boundary declarations. In this edition (v2.0),T4-5 and T7-8 are upgraded from skeletons to conditional theorems (with complete derivationsand condition lists), and the quantitative Proposition T5-6a is added to T5-6—from this point on,the theorems of this paper can be cited directly by SCPT-III and subsequent work.