Geometric Necessity of Structural Breaks:Cusps, Slope Jumps, Cascade Transmission Efficiency, and Detection Power of the Unified Shape Function

In the k-regime linear switching model, the switching between adjacent regimes is fully characterized by the parameter difference Δθj = θ_{j+1} − θj. This paper establishes five core results. Theorem 1 (Geometric Necessity of Rupture): under the switching existence assumption (θj ≠ θ_{j+1}), the unified shape function h(τ) exhibits a cusp at the true boundary τj, with slope jump J(τ) > 0. Theorem 2 (Local Uniqueness of the Critical Value): h(τ) attains a locally unique maximum at each true boundary τj, and the standardized limits of Chow F/n, Wald/n, and LR/n share the same locally unique maximum point. Theorem 3 (Rupture Strength–Slope Jump Correspondence Theorem): the slope jump of the unified shape function h(τ) at τ* is proportional to ||Δθ||_Σ² and the boundary density f_T(τ):J(τ) = |h′(τ−)| + |h′(τ+)| = 2||Δθ||_Σ² · f_T(τ) / [F_T(τ)(1 − F_T(τ*))]. Theorem 4 (Cascade Transmission Efficiency Theorem): in a cascade regime system, if ||Δθj||Σ² + ||Δθ{j+1}||_Σ² is held fixed and the regime variable densities at adjacent boundaries are equal, the root mean square error is minimized when ||Δθj||Σ = ||Δθ{j+1}||_Σ. Theorem 5 (Optimal Effect Size Allocation under Unequal Densities): if the densities at adjacent boundaries are unequal, the optimal effect size ratio is given by the cube root of the density ratio, ||Δθj||Σ / ||Δθ{j+1}||Σ = (f{j+1}/f_j)^{1/3}; when the cumulative distribution values at adjacent boundaries are equal, this optimal ratio is equivalently the cube root of the cusp sharpness ratio: (R_{j+1}/R_j)^{1/3}. Proposition 1 (Geometric Strength–Detection Power Relation) establishes a bridge between geometric rupture strength and statistical detection power: in the two-regime model, the noncentrality parameter of the Chow test is proportional to the slope jump J(τ*), and this relation is completely invariant to the choice of the reference vector θ_ref. Definition (Cusp Sharpness) introduces a geometric index R(τ) = J(τ)/h(τ) that depends only on the boundary location and density. Proposition 2 (Cusp Sharpness and Decay Rate) further establishes an explicit relation between R(τ) and the first-order decay rate of the noncentrality parameter, and discusses the left–right asymmetric decay. Corollary 1 generalizes Theorems 1–3 boundary by boundary to the k-regime model and points out that the cusp sharpness is a pointwise invariant. Corollary 2 gives a lower bound on the efficiency loss when deviating from the optimal effect size allocation, and discusses its tightness. The paper derives the explicit piecewise form of h(τ) and its left and right derivatives within the main text; the proofs are fully self-contained. Numerical verification based on real simulation data (MC = 500, n = 2000) confirms the theoretical predictions of the slope jump and the left/right derivatives, and provides an explicit account of the relation between the Chow F mean and the noncentrality parameter. 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.