Sharp constants for Fourier factorization defects: universal ceilings, strong mixing and interaction measures

Sharp dependence inequalities have two complementary aspects: the largest possible factorization defect and the cost of imposing a mixing or a multilinear structure. Starting from the universal characteristic-function bound \(C_d\), we identify sharp covariance constants \(4K_G(k)\) for real Hilbert-space-valued observables, including \(4√2\) for complex scalars; we establish the small-mixing slope for several factors and quantify restricted-phase and centred-interaction variants. For \(n\) \(σ\)-fields, the sharp real and complex multilinear factors are \(2^n\) and \(2^nκ_n\), respectively, where \(κ_n\) is the inverse planar GHZ local-model visibility. A square-wave/Möbius construction gives the matching formula \(κ_n=(π/2)^nζ(n)/(2(1+2⁻ⁿ)ζ(2n))\) for every \(n≥4\); in particular \(κ_4=105/34\). For three parties a finite-harmonic Bell functional built from the divisors of \(105\) and the primes \(11,13,17,19,23,29,31\) yields the strengthened bound \(κ_3≥2.0349681535…\), whereas \(κ_3≤105ζ(3)/(2π^3)\) and exact equality remains open. The three-party lower bound is computer-assisted: the analytic reduction is completed by 272 explicitly specified inequalities verified using exact rational arithmetic and directed bounds, documented in a reproducible proof supplement. We state precisely where this exceptional finite verification enters the argument and separate it from the independent results on mixing and interaction measures.