Cover letter

The author introduces as an extension to the field of topology a sub-field Analytic Gauge Theory. The concepts of analytic numbers, the analytic field and analytic gauge functions are introduced and defined. The sub-field analytic gauge theory has an enormous application to the fields of topology, number theory, QFTs, amongst others, some of which are introduced. A rigorous examination and presentation will be contained in later works. *Note*: This and all subsequent related papers are highly technical. Any reader should have a relatively advanced understanding of current mathematics, specifically the study of elliptic curves, topology, and the strictly mathematical applications of gauge theories. Definition of terms: Gauge: By the term gauge the author means to represent either a) the normal definition or b) the representation of the quantity: z α β = + l , where l is a number in an analytic field, α and β are the sets of automorphisms of connective geometries, and z is the metric quaternion structure. Analytic field: The field of analytic numbers. Analytic number: A number z α β = + l , where l is a number in an analytic field, α and β are the sets of automorphisms of connective geometries, and z is the metric quaternion structure. Gauge function: ( ) s l , a function whose range is in the analytic numbers is a gauge function. 1) Table of