From Modal Triplet Theory to Noncommutative Geometry: A Typed Almost-Commutative Encoding and Its Current Finite-Triple Frontier

Corrected fourth edition. This paper gives a typed relation between Modal Triplet Theory (MTT) and noncommutative geometry. It proves the available compression theorem for a projector reducing an already specified real even spectral triple and records the selected finite MTT construction: three chiral families, the faithful gauge group (SU(3) x SU(2) x U(1)Y)/Z6, a profile-tier real-even finite triple, and an explicit reduction of a rank-twelve raw scalar space to one complex Higgs doublet. Scope boundary. The finite Dirac entries remain profile data; the spacetime triple, Wick dictionary, cutoff function, spectral moments, and a no-knob Standard Model derivation are not supplied by the fixed-point construction.