Quantitative towers in finite difference calculus approximating the continuum

Multivector fields and differential forms at the continuum level have respectively two commutative associative products, a third composition product between them and various operators like $\mathrm{\partial }$, $d$ and $\ast$ which are used to describe many nonlinear problems. The point of this paper is to construct consistent direct and inverse systems of finite dimensional approximations to these structures and to calculate combinatorially how these finite dimensional models differ from their continuum idealizations. In a Euclidean background there is an explicit answer which is natural statistically.