Switch Measure Theory: Axiomatic Foundations of the Third Fundamental Measure Theory—Three-Dimensional Unification of Intensity, Type, and Information, and Embedding in Information Geometry

This paper establishes the axiomatic foundations of Switch Measure Theory (SMT) and argues, on the basis of five structural criteria, that it is a candidate for the third fundamental measure theory alongside measure theory and information theory. Switch Measure Theory unifies three mathematical dimensions: intensity measurement (SMT) measures the intensity of regime switching; type measurement (FHL) classifies the type of regime switching; information measurement (IIL) proves that statistical structure carries information about regime switching. The three dimensions share the unified shape function h(τ) as a common mathematical object, and share the regime boundary τ* as the localization benchmark. The core contributions of this paper are sevenfold. First, definitional completion and five criteria for a fundamental measure theory: the narrow positioning of Switch Measure Theory is completed into a broad positioning—Switch Measure Theory = SMT + FHL + IIL, the complete mathematical structure of measuring switching—and a formalized standard is given for each criterion: representation theorem (characterizing the functional form), independent axiom system (not reducible to existing measure theories), broad application domains (at least five independent disciplines), structural discontinuity and singularity difference (derivative jump at the true boundary), and rich theorem system and open problems (29 core theorems, 81 open problems). Second, two-level axiom structure and three equivalent forms of the switching degree (Theorem 1): core axioms (B structural measurability, C information carrying) are distinguished from the entry condition (A switching existence); axiom B contains six conditions (B1–B6), among which B6 (covariant naturality) is the key to uniqueness. It is proved that the Mahalanobis norm ratio form, the Fisher information ratio form, and the covariant naturality form are equivalent—thereby Switch Measure Theory is strictly embedded in information geometry (Rao, 1945; Amari, 1985; Amari & Nagaoka, 2000). Third, unification of the three dimensions and the master formula lemma (Proposition 1): SMT, FHL, and IIL share h(τ); the three dimensions are homologous to the master formula h(τ) = C · r(τ)^2, where C = ||Δθ||_Σ^2 and r(τ) are decoupled—C determines the peak height (intensity), and r(τ) determines the shape (type and position). This decoupling is the mathematical root of why the three dimensions can be independently classified yet mutually coupled. Fourth, cross-dimensional theorems and embedding in information geometry: detection-estimation duality (Theorem 2), statistical resolution fiber entropy and the information asymmetry theorem (Theorem 3), Fisher metric characterization (Theorem 4), Fisher-Rao geometric characterization (Theorem 5), multi-regime Fisher geodesic network (Proposition 2), the KL identity of h (Proposition 3) and the L² identity (Proposition 4), Wasserstein distance separation (Theorem 6), Wishart representation (Proposition 5), and FWER control for multiple comparisons (Theorem 7). Fifth, axiom coupling structure and false-peak immunity under model misspecification (Propositions 6–7): it is proved that axioms B and C are formally independent but coupled through A under the correct model; a complete characterization is given for the emergence of a false peak if and only if both conditions—misspecification of the conditional mean and T not independent of X—hold simultaneously, elevating A0 from a technical assumption to a structural condition. Sixth, comparison with effect sizes in change-point detection and FWER control for multiple comparisons: the relations between S and the Chow statistic (including the 1/(4σ²) factor correction), Andrews sup-F, Bai-Perron breakpoint scores, Cohen’s d, and Wasserstein distance are clarified. Seventh, structured presentation of 81 open problems (T numbering: SMT-T1–T36, FHL-T1–T23, IIL-T1–T22): classified and summarized by the three dimensions, including Level distribution (L1=5, L2=38, L3=37, L4=1). This paper strictly distinguishes three completion statuses: fully proved, condition-dependent, and conjectural. Section 1.6 proactively responds to three possible objections. 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.