  We build on previous work on multirings (\cite{roberto2021quadratic}) that
provides generalizations of the available abstract quadratic forms theories
(special groups and real semigroups) to the context of multirings
(\cite{marshall2006real}, \cite{ribeiro2016functorial}). Here we raise one step
in this generalization, introducing the concept of pre-special hyperfields and
expand a fundamental tool in quadratic forms theory to the more general
multivalued setting: the K-theory. We introduce and develop the K-theory of
hyperbolic hyperfields that generalize simultaneously Milnor's K-theory
(\cite{milnor1970algebraick}) and Special Groups K-theory, developed by
Dickmann-Miraglia (\cite{dickmann2006algebraic}). We develop some properties of
this generalized K-theory, that can be seen as a free inductive graded ring, a
concept introduced in \cite{dickmann1998quadratic} in order to provide a
solution of Marshall's Signature Conjecture.
