Strange Region Discovery, Precursor Cascade, and the Transition Zone Width Theorem— Four-Tool Cross-Validation, Cascade Asymmetry Confirmation, and Mathematical Proof of the Switching Type Dichotomy(II)

Abstract

Building upon the discovery of the charm and bottom regime-switching centers in Paper 14, this paper further discovers the strange region (~1.4–1.6 GeV) as the third confirmed regime center. Four tools — ICSS, MOSUM, Chow, and Bai-Perron — provide cross-validation of the strange region's existence. The Bai-Perron double-breakpoint blind test automatically locks T₁ = 2.0200 GeV and T₂ = 2.1052 GeV (F = 16.6, significantly superior to the single-breakpoint F = 9.6), with a transition zone width of 0.0852 GeV. Integrating the three regime centers — strange, charm, and bottom — we discover that the fluctuation transformation point (T₂) of each center is precisely the precursor boundary of the next center: strange T₂ ≈ charm T₁ (deviation 0.0052 GeV), charm T₂ = bottom precursor (deviation 0.0000 GeV) — forming a complete precursor cascade. The transition zone width rigorously follows two regularities: non-overlapping transition zones satisfy the inter-center transition zone width equaling the sum of the two internal transition zone widths plus the gap between the centers; overlapping transition zones have width equal to the smaller of the two internal transition zone widths (minimum rule, verified by strange→charm). The switching type spectrum is fully determined: all four thresholds of the strange region are rule-resetting type (intercept/slope ratio 1.14–1.95), the charm region is rule-resetting type (0.77), and all three thresholds of the bottom region are direction-reversal type (5.48–6.27) — nine data points across six centers, with no exceptions. Based on the above empirical findings, cascade asymmetry is confirmed as a new form of regime factor not covered in the original classification of Paper 7 — a processual regime factor that modulates the exposure coefficients of all three executive factors and produces a significant Chow rupture in the strange→charm transmission (F = 12.63, p = 3.57×10⁻⁹). This paper formally proves three mathematical theorems. Theorem 1 (Inter-Center Transition Zone Width Theorem): The inter-center transition zone width equals the sum of the two internal transition zone widths plus the gap between the two centers. Theorem 2 (Switching Type Dichotomy Theorem): In finite samples, the intercept/slope ratio must be a dichotomy (r < 2 or r > 5). Theorem 3 (Switching Type Intrinsic Property Theorem): All thresholds within the same regime center must have the same switching type. The proofs of the three theorems require no physical assumptions, being entirely based on the mathematical properties of structural break detection and the empirical axioms of the precursor cascade, providing a rigorous mathematical foundation for the derivation of the gauge group structure theorem and direct product theorem in Paper 3. 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.