A mathematical framework for a standard theory using extended representations of paths and world lines

An analysis using a composition of currently-accepted theories is given. Starting with a synthesis of what may be generically termed paths'', analysis of representations for these paths’’ is developed. Foreground and background interactions are explicitly treated by using a local representation that treats the two representations equally and symmetrically. A restriction coupling from the global space-time representation to local interaction source terms is treated in terms of mass and charge couplings. Rewriting the connection in terms of the global manifold and the coupled terms yields compatibility with Dirac and Klein-Gordan equations for electro-weak coupled particles and fields. Compatibility with currently-accepted theories that includes standard charge assignments, SU(3) confinement, and a definition for particle flavor generations is used to constrain and validate the composition and the analysis.