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

1. Exact planar GHZ nonlocality threshold for all n ≥ 4. We determine the sharp supremum of violations of full-correlation Bell inequalities by the n-qubit GHZ state under planar (XY) measurements, over arbitrary finite numbers of measurement settings. The result is the exact formula κₙ = (π/2)ⁿ ζ(n) / [2(1 + 2⁻ⁿ) ζ(2n)], n ≥ 4. In particular, κ₄ = 105/34, corresponding to the critical visibility v₄ = 34/105. The result replaces bounds for particular finite measurement configurations with an exact analytic expression for the stated class. 2. Substantially improved rigorous bounds for the open three-party GHZ problem. We establish 2.0349681535… ≤ κ₃ ≤ 105 ζ(3)/(2π³) = 2.0353294291… . Both bounds improve the previously published 2024 interval. The lower bound comes from a finite-harmonic Bell witness supported by 272 reproducible exact inequalities; the upper bound follows from an explicit square-wave/Möbius local model. The conjecture that the upper bound is attained remains open. The paper discusses an obstruction involving the Möbius function and zeros of the Riemann zeta function. 3. Sharp covariance inequalities under strong mixing. We establish the optimal coefficient 4√2 for bounded complex-valued observations, improving the classical bound 2π. More generally, the optimal Hilbert-space coefficient is 4K_G(k), where K_G(k) is the real Grothendieck constant of order k. For real scalar observations, the sharp coefficient is 4. 4. Sharp multilinear dependence inequalities. For n-linear forms that are real-valued on real inputs, we identify the optimal real and complex comparison factors 2ⁿ and 2ⁿ κₙ. These results strengthen classical estimates in a natural restricted class. They do not settle the more general problem for arbitrary complex multilinear forms considered by Bradley and Bryc (1985). 5. Asymptotically sharp growth of higher-order interaction measures. For the centred Fourier–Lancaster interaction functional, we derive explicit finite-dimensional bounds and establish L_d = 2ᵈ/(ed) · (1 + O(d⁻¹)). Thus, unlike the universal characteristic-function factorization defect, this higher-order interaction functional admits no dimension-independent bound. 6. Further structural results. The paper also develops restrictions to discrete phase groups, centred-interaction comparisons and quantitative links between universal Fourier factorization ceilings, strong mixing and multilinear dependence.