The Common Support Interval of Coordinate Factorization and Term Presentation in Finite Boolean Functions.

This paper studies the relation between coordinate factorization and term presentation for finite typed Boolean functions. For a function f: 2^N → 2 with determinational support R_f, two families of coordinate sets are considered. The factorization-support family F_N(f) consists of those subsets M ⊆ N through which f factors. The presentation-support family S_N(f) consists of the exact syntactic supports of term presentations of the same typed function in a fixed Boolean term language. The main result is the common support interval F_N(f) = S_N(f) = [R_f, N]_⊆. Thus a coordinate set supports a factorization of f if and only if it occurs as the exact syntactic support of a presentation of the same typed function on the same frame. In particular, the determinational support R_f is the common least element of both families. The result identifies the admissible support sets on the two sides while preserving the structural distinction between coordinate factorization and syntactic presentation. The paper provides a direct bridge between the canonical factorization through determinational support developed in Minimal Reality, Volume I: The Minimal Structure of Change and the presentation-support structure established in Formal Frame, Syntactic Support, and Real Determination in Finite Boolean Functions.