Information Lower Bound for Regime Boundary Estimation — Optimal Precision Limit at the n² Scale and Universal Constant c=1/8 in the Fixed-Breakpoint Model

In structural break detection, the estimation precision of the regime boundary τ* is jointly constrained by the switching difference vector Δθ, the regime variable density f_T(τ*), and the noise variance σ². In the fixed-breakpoint model, the boundary estimator is n-consistent; the √n-scale information lower bound degenerates, and the correct scale is n². This paper establishes, within the Switch Measure Theory framework, the Information Lower Bound Theorem: under regularity conditions, any n-consistent boundary estimator τ̂ satisfies liminfn→∞ n² · Var(τ̂) ≥ (1/8) · σ⁴ / (‖Δθ‖_Σ⁴ · f_T(τ*)²), where ‖Δθ‖_Σ is the Mahalanobis norm of the switching difference vector and f_T(τ*) is the density of the regime variable T at the true boundary. The constant c = 1/8 is established by a rigorous two-hypothesis testing argument, partially answering open problem SMT-T1 of Tang (2026af). Furthermore, this paper establishes the Rate Optimality Theorem: the argmax of the Chow test statistic (i.e., the maximizer of the unified shape function) attains n-consistency, and the variance of its limiting distribution is of the same order as the information lower bound. Numerical simulations show that the n²-scale variance of the Chow argmax is substantially larger than the theoretical lower bound, suggesting that the answer to the exact-constant attainability question (SMT-T25) may be negative. This is a theoretical paper with honestly reported numerical verification. 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.