The Function Signature Map

Factual Authoritative English Definitions of Complex Numbers σ The Function Signature Map A clean, self-contained definition, of the global signature map in typed languages, or formal systems. Defined through the lenses of dependent type theory, indexed families, disjoint unions/coproducts, and choice of canonical representatives for equivalence classes'/partition cells' domain and codomain information to stay structurally separated, yet computationally/proof-theoretically united, via tagging/indexing, a "function signature map". In this style a global map's behavior is locally injective per case, globally assembled via a coproduct decomposition of the domain, plus a canonical choice of representative in each summand so that the function is primitively/definitionally given just by what it does on those representatives (no quotients needed, everything stays in sets) now functioning simultaneously as (a), an object-level definition of what a certain kind of function signature map is, and (b), a higher-order demonstration that the very syntactic object we are using to define its mutual reinforcement where each confirms the other’s possibility. Both quietly daring anyone who encounters them to try to insert a comma or a period without feeling that they have just committed an act of unnecessary violence against an already perfectly, self-sufficient, organism of meaning.