Talagran's correlation inequality provides quantitative lower bounds on the covariance of two increasing Boolean functions in terms of their coordinate influences, but, in general, a logarithmic loss is necessary. Motivated by a question of Kalai, Keller and Mossel, we identify a natural log-free regime. We prove that if two increasing Boolean functions on $\{0,1\}^n$ are either both submodular or both supermodular, then $$ \mathbb{E}[fg]-\mathbb{E}[f]\mathbb{E}[g]\ge \frac{1}{4}\cdot\sum\limits_{i=1}^n\mathrm{Inf}_i[f]\mathrm{Inf}_i[g], $$ where the constant $1/4$ is optimal. We also prove a real-valued extension: for two functions with the same second-difference sign, the covariance is bounded below by the sum of products of their Level-1 Fourier coefficients. As a consequence, we verify the Friedgut--Kahn--Kalai--Keller spectral conjecture in this structured setting. The proofs combine a heat-semigroup representation based on second-order discrete derivatives with an independent induction argument for the Boolean case.