The actual first EM stage is the rank-zero K space on all qcqs schemes

The presheaf $L(V)$ of locally constant integer functions on $\mathrm{\mid }V\mathrm{\mid }$ is a cdh sheaf on the site in [F1]. Indeed, for a finite Nisnevich cover its disjoint-union map is a surjective open quotient map by [F2]; for an abstract blowup covering its disjoint-union map is proper surjective and hence closed quotient. Matching functions are constant on each set-theoretic fiber: two points over the same base point have a common point in their scheme fiber product since the tensor product of their residue fields over that base field is nonzero. A matching function therefore descends uniquely, and the quotient topology makes the descended function continuous to the discrete set $\mathbb{Z}$. This proves descent for the generators and hence the generated topology.

The map from the constant presheaf to $L$ is locally an isomorphism: locally constant functions are constant on an open cover (finite after quasi-compactness), and equality of constant values is detected there. The empty object uses its empty cover and the zero group. Thus $L$ is the ordinary cdh constant sheaf. Consequently $H_{\mathrm{c}\mathrm{d}\mathrm{h}}^0(X,\mathbb{Z})=L(X)$ and negative cohomology is zero, as for the derived sections of any ordinary sheaf. Applying Ω∞ to gr⁰ therefore gives precisely the discrete sheaf $L$. This is computation on the actual big site, not a small-site substitution.

For every point $x$ of $X$, pull back $q_0$ to $\mathrm{Spec}\kappa (x)$. Its $K_0$ and weight-zero target are $\mathbb{Z}$, and the map is unital by [F1], hence identity. Its value at $x$ on any perfect class is therefore its Euler rank. Values at all points determine the function in $L(X)$, so $q_0$ is the full rank function. Finally Ω∞ preserves fibers, and the discrete target has no positive homotopy groups; the fiber sequence in [F2] gives the asserted rank-zero space and the injective identification on ${\pi }_0$.