Locally Weighted Unified Shape Function and the One-to-One Correspondence with Regime Boundaries: Multi-Regime Local Peak Correspondence, Population Limit Geometry, and the Pseudo-Peak Phenomenon of Global Splitting

In multi-regime structural break detection, the global unified shape function produces “convex-combination pseudo-peaks” because the two groups on the left and right each mix multiple regimes, causing the local peaks to lose the one-to-one correspondence with the true boundaries. This paper proposes the one-sided kernel locally weighted unified shape function: within the left and right windows of a candidate threshold τ, locally weighted least squares are performed separately, and the squared Mahalanobis norm of the coefficient difference is taken. Under the k-regime linear switching model (A0–A13), three core results are established. Theorem 1 (Two-Regime Bijection): a regime switch exists if and only if the population unified shape function attains a global unique maximum at the true boundary. Theorem 2 (Local Bijection Theorem, the core result of this paper): under the bandwidth condition b_n → 0 with n b_n / log n → ∞, the set of local maxima of h_{b_n} corresponds pointwise to the set of true boundaries in probability, with error O_p(b_n). Proposition (Pseudo-Peak Phenomenon of Global Splitting): there exists an explicit three-regime counterexample in which the global unified shape function produces a pseudo-peak in the inter-boundary region that is strictly greater than the peak of the weakest true boundary, whereas the locally weighted version is identically o_p(1) in the same region. This counterexample establishes the necessity of localization. In addition, Theorem 3 proves that under the null hypothesis sup_τ h_{b_n} = O_p(log n / (n b_n)) and the permutation test exactly controls the false positive rate; Lemma (population shape) gives a deterministic geometric characterization of the population limit shape (single peak plus zero band); Lemma (uniform convergence) proves the uniform convergence of the sample function to the population limit. Corollaries 1 and 2 generalize the results to the L² function space and give the consistency of the permutation test. Numerical verification includes a multi-regime counterexample (global pseudo-peak h(0.5) = 46.23 > h(τ₁*) = 30.71, local two-boundary correct rate 100%) and a bandwidth plateau. 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.