No-False-Peak Theorem and Extreme Value Limit under the Null Hypothesis of Regime Switching: Necessity Direction, Exact Control of Permutation Tests, and Function-Space Generalization

This paper proves the necessity direction of structural break detection: if a regime switch does not exist, then the statistical structure produces no false signal. Under the null hypothesis of the k-regime linear switching model (the required assumptions for each theorem are specified in the main text), three core results are established. Theorem 1 (No-False-Peak Theorem): the sample unified shape function ĥ(τ) tends to zero in probability at every candidate threshold τ, sup_{τ ∈ [π, 1-π]} ĥ(τ) = O_p(n^{-1} log n). No “false peak” exists—the statistical structure carries no spurious regime-structure information when no switch is present. Theorem 2 (Finite-Sample False-Peak Control): under the null hypothesis and A0 (T ⊥ X), the permutation test exactly controls the false positive rate at any significance level α, P(permutation test rejects H_0 | H_0 holds) ≤ α, and this control is non-asymptotic—it holds strictly for any finite sample size n. Theorem 3 (Extreme Value Limit Theorem): define the standardized statistic M_n = max_{τ ∈ [π,1-π]} [n ĥ(τ) F̂_n(τ)(1 - F̂_n(τ))] / σ̂², then under the null hypothesis M_n converges to a non-degenerate limit distribution G, M_n →_d G, where G = sup_{u ∈ [π,1-π]} ||B̃(u)||²/[u(1-u)], B̃(·) is a p-dimensional standard Brownian bridge, and the limit distribution is completely identical to that of the sup-Wald statistic of Andrews (1993); the permutation version M_n* converges to the same G in the permutation-probability sense. Proposition (non-asymptotic bound) gives a non-asymptotic upper bound on sup_τ ĥ(τ). Proposition (power) proves that the permutation test has power tending to 1 under a fixed alternative. Proposition (local power) gives the non-degenerate power limit under a local alternative as the supremum of a drifted Brownian bridge (which degenerates to a noncentral χ²_p if the boundary is known), where the drift function is the product of the contraction weight w(τ) and the local parameter μ, and w(τ) is continuous at τ* with w(τ*)=1. Proposition (residual permutation) establishes the asymptotic false-positive control framework of the residual permutation test under the conditional independence assumption A0′ (which allows T and X to be correlated). Corollary (L²) generalizes Theorems 1–3 to the null hypothesis of the L² functional regime switching model (f_1 = ⋯ = f_k), where the no-false-peak property and the exact control of the permutation test hold only under A0, while under A0′ only the residual permutation test is asymptotically valid; for B-spline bases and Fourier bases, the Donsker-class condition of the extreme value limit is explicitly verified, and when d_n is fixed and the series basis consists of linear regressors, the limit distribution degenerates to G. The proofs of this paper directly cite the regime mixing lemma of Tang (2026af), the L² sufficiency framework of Tang (2026aq), and empirical process theory. Together with the sufficiency direction of Tang (2026aq), this paper constitutes the complete mathematical formalization of the correspondence level of structural break detection. 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.