Constructible Witt theory of schemes

We study the constructible Witt theory of \'etale sheaves of $\Lambda$-modules on a scheme $X$ for coefficient rings $\Lambda$ having finite characteristic not equal to 2 and prime to the residue characteristics of the scheme $X$. Our construction is based on the recent advances by Cisinski and D\'eglise on six-functor formalism for derived categories of \'etale motives and offers a background for the study of constructible Witt theory as a cohomological invariant for schemes. In the case of smooth complex algebraic varieties and finite coefficient rings, we show that the algebraic constructible Witt theory studied in this paper can be identified with the topological constructible Witt theory.