Switch Measure Theory: Definitions, Theorem System, and Assessment of Mathematical Foundations — A Survey of the Unified Shape Function, Switching Degree, Spectral Weighting Family, and Information Lower Bound

Science describes the world through measurement. Measure theory measures the size of objects; information theory measures uncertainty. The third fundamental category—structure—has long lacked a standardized measurement theory. This paper formally names Switch Measure Theory and establishes it as a mathematical branch for measuring regime switching intensity. It should be emphasized that the parallel relationship between Switch Measure Theory and measure theory/information theory rests on a threefold division of measurement objects—size, uncertainty, and structural switching intensity—rather than on a claim of equal theoretical maturity; the theorem foundation of Switch Measure Theory remains in its early stages. Switch Measure Theory comprises six mathematical papers (Tang, 2026af–2026ak). The six papers form a complete logical chain: localization (the unified shape function guarantees that switching can be consistently estimated) → measurement (axiomatization and inference of the Switching Degree) → generalization (nonlinear, post-selection, weak dependence) → limit (the n²-scale information lower bound). This paper systematically surveys the core definitions and notation of these six papers, unifying and consolidating them into standard notation; presents the theorem system by logical hierarchy; and strictly distinguishes three categories of completion status: fully proved, derived but condition-dependent, and conjectural. Within the explicitly delimited model classes (two-regime and k-regime linear switching models, A0–A8), the core theorem system of Switch Measure Theory has been established: five core results—the Global Maximum Point Invariance Theorem, the axiomatic characterization and asymptotic normality of the Switching Degree, post-selection asymptotic equivalence, the n²-scale information lower bound, and the direction-dependent asymptotic variance under weak dependence—have all been completely proved. The generalization to broader model classes is constrained by a set of explicit open problems. Switch Measure Theory provides the measurement-tool foundation for the two subsequent series of the Factor Hierarchy framework—the mathematization of the Factor Hierarchy Law and the mathematization of the Information Isomorphism Law. This paper explicitly lists the concrete inventory of these tools in Section 5. Research Paradigm Statement: The core methodology, research direction, and final decisions were independently directed by the author. DeepSeek assisted with code implementation, data presentation, and text drafting. The author takes full academic responsibility for the final content.