A Survey of the Mathematization of the Factor Hierarchy Law : Formalization of the Three Axioms, the Cross-Disciplinary Verification Matrix, and a Preliminary Establishment Statement

The Factor Hierarchy Law (Tang, 2026g) consists of three axioms: Axiom 1 (Factor Dichotomy), Axiom 2 (Regime Switching/Structural Break), and Axiom 3 (Interaction Effect). This law has been systematically verified in six completely independent disciplines, but the mathematical formalization of the three axioms has been missing until now. This paper surveys the three papers of the Factor Hierarchy Law mathematization series (Tang, 2026am–2026ao) and formally declares that the three axioms have acquired a complete mathematical foundation within the k-regime linear switching model. Formalization of Axiom 1 (Tang, 2026am): the Switching Type Classification Theorem proves that the parameter difference between any adjacent regimes must be one of three types—pure slope switching (corresponding to a pure regime factor), full-parameter switching (corresponding to a dual-identity factor), or pure intercept switching (corresponding to a pure execution factor)—and the fourth combination is excluded by switching existence. The Exhaustiveness Theorem for Six Hierarchical Structures proves that the hierarchical structure within the standard model belongs to one and only one of four types (single-layer system, single-switching system, multi-switching system, dual-switching system); after introducing additional moderating variables, hierarchical switching systems and cyclic switching systems may arise, and the six types jointly exhaust all possibilities. The Completeness Theorem for the Switching Type Sequence proves that the sequence is a bijective invariant of the hierarchical partial order. Numerical verification discovers two original phenomena: the discretization plateau effect and the sample-size sensitivity of weak-effect-size classification. Formalization of Axiom 2 (Tang, 2026an): the Geometric Necessity of Rupture Theorem proves that the unified shape function exhibits a cusp at the true boundary, with a strictly positive slope jump. The Local Uniqueness of the Critical Value Theorem proves that each true boundary is a locally unique maximum point. The Rupture Strength–Slope Jump Correspondence Theorem establishes the explicit relation between J(τ*) and ‖Δθ‖_Σ². The cusp sharpness index R(τ*) = J/h provides a geometric measure depending only on the boundary location and density. The Cascade Transmission Efficiency Theorem reveals the monotone relation between the effect size ratio and the cascade alignment precision. Formalization of Axiom 3 (Tang, 2026ao): the η Divergence Theorem proves that the limiting behavior of η must be one of 0, a finite positive number, or ∞. The η Limit Theorem for Three Morphologies precisely characterizes the η limiting behavior of the three switching types. The proposition on the η trajectory of a processual variable proves that η exhibits a plateau (rather than a peak) in a neighborhood of the boundary along the regime variable, with the plateau value proportional to the square of the slope Switching Degree. The Geometric Interpretation Theorem for η interprets the finite limit of η as the squared tangent of the direction angle of the switching difference vector: η* = tan²φ. This paper presents a Preliminary Establishment Statement: within the k-regime linear switching model (A0–A8), the three axioms of the Factor Hierarchy Law have all been mathematically formalized; the exhaustiveness of the three-way switching type classification and the six hierarchical structures has been proved; and testable definitions of the extended configurations have been provided. This paper also gives a cross-disciplinary verification matrix, mapping the three axioms to empirical findings in six disciplines; consolidates open problems FHL-T1 through FHL-T23; and provides a concrete list of tools for the subsequent Information Isomorphism Law mathematization series. 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.