The Global Maximum Point Invariance Theorem: A Statistical Invariant Theory of Structural Break Detection — Unified Shape Function, Standardized Limits, and Multi-Regime Generalization

Abstract

This paper establishes a mathematical theorem system for between-group regression difference statistics in structural break detection. The central result is the Global Maximum Point Invariance Theorem: in a two-regime linear switching model with homogeneous second-moment design, with fixed noise variance σ² > 0, under the standardized limit as n → ∞, the standardized limits of Chow F/n, Wald/n, and LR/n all attain their global unique maximum at the true regime boundary. On both sides of the true boundary, the standardized limits of the three statistics are strictly increasing functions of a common unified shape function h(τ)—Wald/n shares the same form as Chow F/n up to a factor of the constant p, while LR/n is another nonlinear monotone transformation; if the distribution of the regime variable T is symmetric about the true boundary, the three can be merged into a globally single-valued strictly monotone function of h(τ). This invariance property is derived rigorously from two independent facts: convex combination geometry and statistical algebra. Under fixed σ² > 0, the paper establishes a non-asymptotic shrinkage bound: the estimated coefficients at an incorrectly split point converge to the mixed coefficient at rate n^(−1/2), and the upper bound holds for every fixed candidate threshold; a corollary further demonstrates that in finite samples, the statistic value at any incorrectly split point is suppressed below the value at the true boundary with high probability. The paper proves a multi-regime global optimality theorem: BIC attains its global minimum at the true set of regime boundaries, with complete proofs covering underfitting, overfitting, and mislocation. It also establishes a switching intensity monotonicity theorem and an initial parameter–position–difference-vector identifiability proposition. Furthermore, the paper formulates the relaxation of A3 as a generalized regime mixing lemma, proving that when the discrepancy between the second moments of the design matrices in the two regimes tends to zero, the peak location offset is O(‖Σ₁ − Σ₂‖); through a slow-variation robustness proposition for the cusp structure, it further proves that when the in-regime parameter vector varies slowly at a rate below a critical threshold, the maximum point of the statistic does not shift. The Onsager–Yang exact solution of the 2D Ising model verifies the slow-variation robustness proposition—the Chow F peak precisely locks onto T_c = 2.269185; the intercept/slope ratio is strictly monotone in the distance from T_c (ρ = 1.0000), providing cross-disciplinary support for the switching intensity monotonicity theorem. This paper establishes a unified mathematical framework for the consistency of the maximum points of between-group regression difference statistics in 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.