Admissibility Precosheaves and Conditional AQFT Nets in Modal Triplet Theory

An admissibility-indexed precosheaf is a pregeometric covariant functor only after its algebras and extension morphisms are supplied; overlap or failure of joint representability does not create commutators or imply locality. A physical Haag–Kastler net instead requires a selected Lorentzian base, an upper local operator net, and a localization-preserving coherent reduction. For a decomposable orthogonal projector P, we prove that compression of the P-compatible upper subalgebras preserves isotony and spacelike commutation. This is the rigorous MTT-to-AQFT bridge currently available. We also show why absence of a global admissible chart does not forbid an abstract quasilocal algebra or categorical colimit. The current q79 source now also supplies a selected free twisted-Dirac construction whose even CAR observable net has locality, covariance, the time-slice property, and nonempty positive Hadamard state space through standard AQFT results. The chart-to-region natural equivalence and the nonperturbative interacting C-star completion remain open.