The Finite Cell as the Origin of the MAT Universe: Rigid arithmetic topology, finite quantum registration, cell-written RG transport, and global vacuum selection

Fundamental physics usually treats microscopic topology, internal state spaces, renormalization-group thresholds, and infrared boundary conditions as separate layers of description. This paper develops the opposite architecture within Mittermeier Attractor Theory (MAT): a single finite arithmetic–spectral origin structure is tested as the common source of all four. The central result is an explicit finite-to-continuum chain. A rigid discriminant -23 hyperbolic topology supplies a finite torsion sector space; a finite phase-space quantization route compresses its twenty-five sectors to a five-dimensional quantum-register target; an eight-dimensional interface then generates the active forty-dimensional MAT transmission space; the finite support radius writes both thresholds of the exact three-channel Mittermeier RG attractor; and the back-integrated monotone vacuum chart is finally required to return the same boundary coordinate as the finite trace, selecting a unique compatible vacuum scale on the active branch. The work separates established mathematics, exact MAT identities, verified SSOT readouts, and open microscopic hypotheses at every stage. Its scientific novelty is therefore not a new isolated numerical coincidence, but a sharply constrained mechanism by which one finite object may determine the scales, channel structure, global flow shape, and boundary condition of a continuum theory. If this architecture survives independent derivation and experiment, constants and cosmological readouts that are normally adjusted in different sectors become correlated outputs of one falsifiable microscopic source.