  We present here a product between vectors and scalars that {\it mixes} them
within their own space, using imaginaries to describe geometric products
between vectors as complex vectors, rather than introducing higher
order/dimensional vector objects. This is done by means of a {\it mixture
tensor} that captures the rich geometries of geometric algebras, and
simultaneously lends itself naturally to tensor calculus. We use this to
develop a notion of analyticity in higher dimensions based on the idea that a
function can be made differentiable -- in a certain strong sense -- by
permitting curvature of the underlying space, and we call this {\it analytic
curvature}. To explore these ideas we use them to derive a few fundamental laws
and operators of physics which, while considered somewhat lightly, have
compelling features. The mixture, for instance, produces rich symmetries
without adding dimensions beyond the familiar space-time, and its derivative
produces familiar quantum field relations in which the field potentials are
just derivatives of the coordinate basis. The symmetries of physical laws are
seen to arise not directly from the bases, but from their mixtures.
