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

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 ${\pi }_1^{\ast }{\alpha }_1\cup {\pi }_2^{\ast }{\beta }_2\in H^3(V,\mathbb{Z}\mathrm{/}2(2))$. Here ${\alpha }_1$ is the K3 double-cover class and ${\beta }_2$ 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 ${\mathrm{\Theta }}_n={\pi }_1^{\ast }{\alpha }_1\cup {\pi }_2^{\ast }{\beta }_2\cup \cdots \cup {\pi }_n^{\ast }{\beta }_n\in H^{2n-1}(X_n,\mathbb{Z}\mathrm{/}2(n))$,with Bockstein ${\mathrm{\Delta }}_n=\delta ({\mathrm{\Theta }}_n)\in H^{2n}(X_n,\mathbb{Z}(n))$. Using external products of perverse sheaves, categorical Bockstein compatibility, and a Leibniz rule for the MacPherson–Vilonen boundary, we prove unconditionally that ${\mathrm{\Delta }}_n$ 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 ${\mathrm{\Delta }}_n$ 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 $H^1(S_1,\mathbb{Z}\mathrm{/}2(1))$ Enriques double-cover direction. We also record a motivic finite-coefficient lift of the tower via the finite-coefficient cone ${\mathbf{1}}_X(n)\mathrm{/}2:=\mathrm{Cone}({\mathbf{1}}_X(n)\stackrel{\times 2}{\to }{\mathbf{1}}_X(n))$, and explain which formal part of the MacPherson–Vilonen zig-zag construction lifts motivically under Betti realization.