1 The geŶeralized Stokes theoreŵ By J.A.J. van Leunen Last modified: 4 januari 2016 Abstract When applied to a quaternionic manifold, the generalized Stokes theorem can provide an elucidating space-progression model in which elementary objects float on top of symmetry centers that act as their living domain. The paper indicates that integration depends on the ordering of the involved parameter spaces. In this way, it elucidates the origin of the electric charges and color charges of elementary particles. 1 Introduction This paper uses the fact that separable Hilbert spaces can only cope with number systems that are division rings. We use the most elaborate version of these division rings and that is the quaternionic number system. Quaternionic number systems exist in multiple versions, that differ in the way they are ordered. Ordering influences the arithmetic properties of the number system and it appears that it influences the behavior of quaternionic functions under integration. Another important fact is that every infinite dimensional separable Hilbert system owns a companion Gelfand triple, which is a non- separable Hilbert space. We will use these Hilbert spaces as structured storage media for discrete quaternionic data and for quaternionic manifolds. We use the reverse bra-ket method in order to relate operators and their eigenspaces to pairs of functions and their parameter spaces. Subspaces act as Hilbert space domains in relation to which manifolds are defined. The existence of Hilbert spaces are a corollary of the existence of a deeper foundation, which is an orthomodular lattice. The set of closed subspaces of a separable Hilbert space has the relational structure of an orthomodular lattice. This deeper foundation does not use number systems. Thus notions, such as space and progression do not exist at this level. The number systems are introduced by extending the deeper foundation to a separable Hilbert space. This extension enforces the restriction to number systems that are division rings. With other words, space and progression must be expressible by elements of a division ring.