PRIMBIT toward Tube(Fib): Algebraic Framework, Prism Realization, and a Falsifiable Certification Program

This preprint develops a falsifiable mathematical and computational route from PRIMBIT toward a finite realization of the Fibonacci tube algebra. The abstract target is Tube(Fib) ≅ C³ ⊕ M₂(C). An explicit triangular-prism annulus is specified, with a raw Hilbert space of dimension 512, a closed admissible subspace of dimension 50, and a cut-open formulation of dimension 170. The work derives a redundant family of exact algebraic certificates based on two-projector geometry, central-support reconstruction, synthesized unitary presentations, Gram-determinant tests, real forms, and alternative Wedderburn reconstructions. A real target form A_R ≅ R ⊕ C ⊕ M₂(R) is also identified under a real Fibonacci gauge. Version 1.0 does not yet contain the directly evaluated prism matrices e_diag, W_loop^diag, and U_diag, and therefore does not claim a completed microscopic realization of Tube(Fib), nor a physical demonstration of non-Abelian anyons. The document explicitly separates algebraic tube-data certification from the further requirements of a scalable local many-body model, a thermodynamic gap, deconfined localized excitations, physical transport, and emergent noncommuting adiabatic braid holonomy.