Six Birds Foundations II: Admissibility, Meta-Theory, and the Exact-Six Program

Six Birds Foundations I [1] introduced a stratified closure calculus with six primitive closure roles. The present paper extends that calculus into a theorem-level framework in three parts: a primitive role architecture fixing the meanings, levels, local boundaries, and selected lifts of the six typed closure roles \(P_{1},\ldots,P_{6}\); an admissibility and meta-theory regime in which honest bookkeeping, typed non-collapse, level profiles, forgetting with selected lifts, activation thresholds, instrument-relative visibility, and empirical-bridge admissibility are stated as theorem schemas; and a scoped exact-six theorem chain over the domain \(\mathsf{F}\mathsf{A}\mathsf{T}\mathsf{C}\mathsf{D}\) of finite audited typed closure descriptions. Within this domain we prove six role-lower-bound witnesses, an upper-bound classification with corrected role-projection alignments (closing the probability, causality, and agency threats in the covered scope), a decomposition/exhaustion theorem, and an integrated stress certification. The terminal theorem asserts that every \(D \in {\mathsf{F}\mathsf{A}\mathsf{T}\mathsf{C}\mathsf{D}}\) admits a well-formed decomposition/exhaustion record with exactly six role channels, each in one of five recorded statuses, with residual features classified under an explicit scheme and, under an explicit recognition hypothesis (the three principal threat classes — probability, causality, and agency — closed unconditionally), no seventh irreducible role required in the covered scope. The claim concerns role channels rather than active projections, and the decomposition is existential, not unique. We do not claim unrestricted exact-six, all-emergence coverage, a full six-primitives algebra, global primitive independence, empirical realization, internal instrument-family completeness, or recognition-completeness of the six roles.