From Diaz's Enriques Product to an $n$-Fold Cup-Product Bockstein Family of Integral Hodge Counterexamples

Abstract: We reinterpret Diaz's construction of Chow-trivial smooth projective varieties violating the integral Hodge conjecture as the level-two case of an (n)-fold cup-product Bockstein mechanism. Diaz's dimension-four example is (V=S_1\times S_2), where (S_1,S_2) are Enriques surfaces, and its obstruction is the Bockstein of $$π_1^{<em>α_1\cupπ_2^</em>β_2\in} H^3(V,\mathbb Z/2(2))$$. Here (α_1) is the K3 double-cover class and (β_2) is an Enriques Brauer-detecting class. We extend the finite-coefficient source construction to (X_n=S_1\times\cdots\times S_n) by forming $$Θ_n=π_1^{<em>α_1\cupπ_2^</em>β_2\cup\cdots\cupπ_n^*β_n} \in H^{{2n-1}(X_n,\mathbb} Z/2(n))$$,with Bockstein $Δ_n=δ(Θ_n)\in H^{{2n}(X_n,\mathbb} Z(n))$. Using external products of perverse sheaves, categorical Bockstein compatibility, and a Leibniz rule for the MacPherson--Vilonen boundary, we prove unconditionally that (Δ_n) has nonzero image in a distinguished Enriques--Brauer component of the MV obstruction channel. Under the Brauer-separation hypothesis, which asserts that algebraic codimension-(n) cycle classes have zero image in this same component, the class (Δ_n) is a non-algebraic (2)-torsion integral Hodge class. We verify this separation for decomposable algebraic cycles and reduce the remaining non-decomposable case, via integral even Chow--Künneth projectors on the Enriques factors, to a single coefficient-level algebraic-control problem involving the (H^1(S_1,\mathbb Z/2(1))) Enriques double-cover direction. We also record a motivic finite-coefficient lift of the tower via the finite-coefficient cone (\mathbf 1_X(n)/2:=\operatorname{Cone}(\mathbf 1_X(n)\xrightarrow{\times 2} \mathbf 1_X(n))), and explain which formal part of the MacPherson--Vilonen zig-zag construction lifts motivically under Betti realization.