Formal Frame, Syntactic Support, and Real Determination in Finite Boolean Functions.

This paper studies finite typed Boolean functions f:2N→2f:\\mathbf{2}^N\\to\\mathbf{2} by separating three distinct structural levels: the formal frame NN, the determinational support RfR_f, and the syntactic support SτS_\\tau of a chosen Boolean-term presentation τ\\tau. The determinational support coincides formally with the classical essential support of a Boolean function, while the syntactic support records the variables that occur literally in a particular representation. Every term representation satisfies Rf⊆Sτ⊆N.R_f\\subseteq S_\\tau\\subseteq N. For a fixed typed function ff, the paper introduces the family SN(f)\\mathfrak S_N(f) of syntactic supports of all term presentations of ff on the frame NN, and proves the presentation-support interval theorem SN(f)=[Rf,N]⊆.\\mathfrak S_N(f)=[R_f,N]_{\\subseteq}. Thus every set of coordinates between the determinational support and the formal frame occurs as the exact syntactic support of some representation. Equivalently, SN(f)≅P(N∖Rf),\\mathfrak S_N(f)\\cong\\mathcal P(N\\setminus R_f), so the presentation-support family forms a finite Boolean lattice. Variant 11, f11(p,q)=¬q,f_{11}(p,q)=\\neg q, serves as the minimal-arity Boolean model containing one determinationally real coordinate and one determinationally fictitious coordinate. Alternative presentations of the same typed function realize the two endpoint supports Rf11R_{f_{11}} and NN. The paper also establishes the unique determinational-core factorization f=Cyl⁡RfN(fRf),f=\\operatorname{Cyl}_{R_f}^{N}(f_{R_f}), showing that the core contains all information required to determine the values of ff, while not determining the ambient frame or a chosen syntactic presentation. The resulting framework isolates, within a static Boolean setting, the distinction between formal argument membership, syntactic visibility, and determinational power.