Critical Correlation in Non-Centered Gaussian Vectors: The ρ*(r) Formula

For non-centred Gaussian vectors (X, Y) ~ N (μ, Σ_ρ) with opposite-sign means and symmetric strip events A = B = -a, a, we study the signed corner sum C (ρ): = ∂∆/∂ρ where ∆ (ρ) = Pr (A ∩ B) − Pr (A) Pr (B). We prove the exact identity ∂∆/∂ρ = C (ρ) (Theorem 1) and characterise the critical correlation ρ* (r), where r: = a/|m|, at which the GCI violation ∆ (ρ) is deepest. For r ≥ rₘin ≈ 2. 08 we identify an empirical formula for ρ* (r) supported by a dominant-corner heuristic (Remark 5) and a rigorous error bound (Remark 7): ρ* (r) = r (r − 1) − sqrt (r² (r − 1) ² − ln (2) (2r − ln (2) ) ) / ln (2) + O (e^-4r). The Regime-2 approximation (r − (ln 2) /2) / (r (r − 1) ) is the leading asymptotic term of this formula. This is the companion paper to "On the Failure of the Gaussian Correlation Inequality for Non-Centered Distributions" (Paper #1 https: //zenodo. org/records/20078486).