Flavor-Changing Neutral Currents from Moduli Curvature: The Joint Hierarchy–Flavor Landscape on the Moduli Torus

Paper VI established that the fermion mass hierarchy is a topological feature of the moduli torus T2 of three-kink configurations on a compact orbifold S1/Z2. This paper computes the flavor-changing neutral current (FCNC) couplings as geometric fields on the same torus, by evaluating the off-diagonal components of the mass matrix derivative ∂M/∂θi in the mass eigenbasis at each point (θ1, θ2). The resulting ΔS=2 Wilson coefficient Csd defines a flavor landscape on T2 whose intersection with the SM-like hierarchy region determines the viable region of parameter space. We find three principal results. First, the FCNC constraint imposes a smooth, approximately linear reduction in viable volume, reflecting a stable unimodal distribution of flavor couplings across the SM-like region with no emergent fine-tuning scale: at the median threshold, the viable area is 50% of the SM-like region (±0.5% across all tested α), and the total naturalness cost is approximately 1 in 14 at the optimal α* ≈ 3.0, with no additional fine-tuning from FCNC physics beyond the hierarchy cost itself. Second, the hierarchy and flavor optimal compactness coincide: α*(hierarchy) = α*(flavor) ≈ 3.0, meaning the FCNC constraint does not shift the preferred compactification scale. Third, the inverted FCNC hierarchy predicted in Paper V (|Csd| > |Cbd|, |Cbs|) is confirmed at the optimal compactness, with the s→d channel dominating by a factor of 1.1 over b→d and by six orders of magnitude over b→s in the SM-like region. The hierarchy–FCNC correlation is positive but moderate (Pearson r ≈ 0.35–0.44), reflecting the geometric fact that both hierarchy and flavor violation originate from proximity to eigenvalue collision lines on T2.