L² Sufficiency Theorem under Functional Regime Switching: X-Dependent Convex Combination, Detection Information Loss, Spectral Weighted Sufficiency, and Detection Resolution–Information Lower Bound Duality

This paper proves the sufficiency direction under the L² functional regime switching model: if a regime switch exists, then the statistical structure can detect it. Three core results are established. Theorem 1 (L² Sufficiency Theorem): under the local continuity assumption A0′ and the conditional non-degeneracy assumption A0″, the L² version of the unified shape function hL²(τ) = ||Δ̄f(τ)||²L²(P_X) attains a global unique maximum at the true regime boundary τ*. Theorem 2 (Multi-Regime Sufficiency Theorem): the functional-space BIC attains a global minimum at the true boundary set B*, with complete proofs covering underfitting, overfitting, and mislocation, where the effective parameter dimension is d_n. Theorem 3 (Spectral Weighted Sufficiency Theorem): for any α ∈ [0,1], when the fractional power kernel K_α satisfies the multiplier algebra condition and Δf ∈ HK_α, the spectral weighted unified shape function h_α(τ) = ∑ₖ₌₁∞ a_k(τ)²/λ_k^α attains a global maximum at τ*; strict uniqueness holds for α=0 under A0″, and for α ∈ (0,1] under the multiplier contraction condition. Lemma 2 provides a verifiable sufficient condition for the multiplier contraction condition under the Gaussian kernel, and Appendix A gives the complete rigorous proof. Lemma 1 (X-Dependent Convex Combination Lemma) is the core technical contribution of this paper: under A0′, A0″(a), and the overlap condition, the conditional mean function of the mixed subsample is an X-dependent convex combination. Corollary 1 (Detection Information Loss) defines the information loss quantity I(τ) = C - hL²(τ), explicitly I(τ) = ∫_X (1 - w(τ,x)²)(Δf(x))² dP_X(x), and proves that I(τ) > 0 holds for any τ ≠ τ*. Theorem 4 (Uniform Convergence of ĥL²) gives the finite-sample uniform convergence rate of ĥL²(τ). Theorem 5 (Argmax Consistency) proves τ̂_n → τ* under the cusp separation condition. Theorem 6 (Convergence Rate) gives the O_p(r_n) convergence rate of τ̂_n. Theorem 7 (BIC Selection Consistency) proves that the boundary set selected by multi-regime BIC converges in probability to B*. Proposition (Density Sufficient Condition for A0″(a)) gives a verifiable sufficient condition for non-degeneracy of the conditional Gram matrix. Proposition (Detection Resolution–Information Lower Bound Duality) quantitatively connects detection geometry with the estimation information limit for the first time. Proposition (Positive Semi-Definiteness of the Information Loss Matrix) further enriches the geometric structure of information loss. This paper explicitly states that A0 (T ⊥ X) has been replaced by A0′. 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.