To each pair, R, T , consisting of a unitary commutative von Neumann-regular ring, R, where 2 is a unit and T is a preorder on R, we associate a reduced special group, G T (R), which faithfully reflects quadratic form theory, modulo T , over free R-modules and then show, using the representation of R as the ring of global sections of its affine scheme, together with results from [M. Dickmann, F. Miraglia, On quadratic forms whose total signature is zero mod 2 n . Solution to a problem of M. Marshall, Invent. Math. 133 (1998) 243-278; M. Dickmann, F. Miraglia, Lam's Conjecture, Algebra Colloq. 10 (2003) 149-176; M. Dickmann, F. Miraglia, Algebraic K-theory of special groups, J. Pure Appl. Algebra 204 (2006) 195-234], that G T (R) satisfies a powerful K-theoretic property, the [SMC]-property. From this we conclude that quadratic form theory modulo T over free R-modules verifies Marshall's signature conjecture, Lam's conjecture, as well as a reduced version of Milnor's Witt ring conjecture.The main result of this paper is that, if R is a commutative von Neumann-regular ring in which 2 is a unit, then the reduced theory of quadratic forms with invertible coefficients in R,