One of the primary problems in the theory of quadratic forms over a field of characteristic different from two is the determination of its graded Witt ring GWF, where GWF is the graded ring associated to the Witt ring relative to the fundamental ideal of even dimensional forms. In particular, a relationship between this ring and the cohomology ring \(H^*(Gal(F_ q/F),{\mathbb{Z}}/2{\mathbb{Z}}),\) where \(F_ q\) denotes the quadratic closure of F, has been sought. In this paper, an isomorphism between these two rings is established for a class of realizable abstract Witt rings which include the so-called elementary Witt rings, i.e., those abstract Witt rings which are generated from the Witt rings of local, real, complex, and finite fields via the fiber product and group extension operation in the category of abstract Witt rings. The method used is to suitably generalize the notion of (double) rigidity and lift such a property up certain two extensions. Analogous relationships with the Milnor k-theory of fields is also established. Indications are also given which allows one to avoid the use of Merkur'ev's theorem.