Abstract

We present a rigorous and purely mathematical white paper formalizing functional classes through behavioral, algebraic, topological, and categorical lenses. Functions are classified by predicates invariant under composition, limits, and homotopy.

A minimal calculus based on lambda abstraction and Lisp-style combinators generates broad families of functions. We prove core theorems using classical implication, modal necessity/possibility (□, ◇), and topological interior/closure arguments.

Extensions to functionals, functors, computability, and speculative quantum analogy are delegated to appendices.