By Jones's theorem, the negative cyclic homology $\HC^-_\ast(C^\ast(Y))$ of a simply connected space $Y$ computes the $S^1$-equivariant cohomology of its free loop space $LY$. Rationally, the right-hand side is computable via Sullivan minimal models (Burghelea--Vigu\'e-Poirrier), but integrally HKR fails for exterior algebras over $\Z$ and explicit cyclic torsion enters.
I describe an integral computation of $\HC^-_\ast(\Lambda_\Z(e_1, \ldots, e_r))$, equivalently the integral $S^1$-equivariant cohomology of $LX$ for $X = \prod_i S^{2k_i - 1}$ a finite product of odd spheres. Eilenberg--Zilber reduces the negative cyclic complex to a tensor-product mixed complex $(W, B^\otimes) = \bigotimes_i (V_i, B_i)$ with single-factor input $H_j(V_i, B_i) = \Z/j$ for $j \ge 1$. Iterated K\"unneth produces gcd-indexed cyclic torsion with binomial multiplicities, and the kernel of $B^\otimes$ has rank generating function
\[
K(t) \;=\; \frac{(1 - t)^r + 2^r t}{(1 + t)(1 - t)^r}.
\]
The Borel spectral sequence for $LX$ collapses at $E_3$ integrally with no extensions. Time permitting, I will indicate a degree-$D_G$ naturality identity transporting this to compact Lie groups via the Hurewicz-scaled map $\varphi_G: X_G \to G$.
