Sum and product totalisations on an infinite diagonal

Only total degree zero has nonzero components. Consequently ${\mathrm{Tot}}_0^{\oplus }C=S$ and ${\mathrm{Tot}}_0^{\mathrm{\Pi }}C=P$ by [F1, F2]. All other total degrees and every total differential vanish. In either complex every degree-zero element is a cycle and the boundary subgroup is zero. Thus the homology groups in degree zero are $S$ and $P$ respectively, and all other homology groups are zero.

The canonical comparison has degree-zero component the finite-support inclusion. The tuple $(1,1,\dots )$ has infinite support, so it is not in the image; its homology class is unchanged because there are no boundaries. Even an abstract homology isomorphism is impossible: $S$ has a bijection with $\mathbb{N}$, while every purported enumeration $e:\mathbb{N}\to P$ misses the sequence $y_j=1-e(j{)}_j$. Therefore a bijection $S\to P$ would contradict [F2].

The support includes $(j,-j)$ for every positive $j$, so is not first quadrant and is infinite on its sole nonzero diagonal. Hence neither first-quadrant finite-diagonal comparison nor finite-filtration convergence is contradicted. The index $j=0$ contributes one copy of $k$, the omitted total degrees are genuinely zero, and all constructions and diagonalization are explicit and choice-free.