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 ≥ r_min ≈ 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 ).