Binary presentations of center-valued quantification on ortholattices

We study associative binary presentations of center-valued quantifiers on ortholattices. The fixed elements of such a quantifier form a Boolean subalgebra of the center. We construct an adaptive product whose multiplication agrees with lattice meet at precisely the left operands in this Boolean algebra. Within a family obtained by modifying the quantifier on fixed central components, associativity and this recovery property determine the product uniquely. On orthomodular lattices, twelve identities characterize the product without the family assumption. The product and quantifier define each other by terms, and the product together with its reverse De Morgan dual forms a distributive skew lattice. In the orthomodular expansions, we give explicit descriptions of congruences and factor congruences in terms of ideals of the fixed Boolean algebra, and express a residual through the Sasaki implication. An equivalent presentation with separate lattice and Boolean sorts yields a weak Boolean product representation on ortholattices. In the orthomodular case, the representation is a Boolean product of simple algebras, with an elementary product on each stalk. We thereby obtain a concrete realization of the representation theory of the associated discriminator variety.