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

Abstract

This paper is the mathematical middle layer between Single-Chain Persistence Theory I (SCPT-I) (v3.4, under journal review, frozen) and Single-Chain Persistence Theory III (SCPT-III) (full manuscript, v1.2 Ultimate Edition). SCPT-I bears the burden of proof; the present paper formalizes the four criteria and answers the question “why the four criteria can be assigned values”; SCPT-III answers “how to use the four criteria to interpret history, diagnose the present, and conditionally predict the future.” This paper legislates no new definitions—all definitions are quoted from SCPT-I/III; what is newly given consists only of formulas, theorems, phenomenological models, and boundary declarations. Revision ledger of this edition: v1.0 implemented the six-member first-round 8 required items + 4 recommendations; v1.1 implemented the second-round 3 required items + 6 recommendations (archived as item No. 04); v2.0, per the six-member resolution “push to citable status,” upgrades T4-5 and T7-8 from skeletons to conditional theorems (with complete derivations and condition lists), and adds the quantitative Proposition T5-6a to T5-6—from this point on, the theorems of this paper can be cited directly by SCPT-III and subsequent work.v2.0 (2026-08-04): theorem candidates T4-5 and T7-8 upgraded to conditional theorems with full derivations and condition lists; quantitative proposition T5-6a added. Theorems herein are citable by SCPT-III and subsequent work.