Rigorous Constraints on Multi-Sector Spectral Triples: The Disjointness Criterion, Tree-Level Sequestering, and Obstructions for Dark Sectors in Noncommutative Geometry

We present a mathematically rigorous analysis of multi-sector extensions of finite real spectral triples within noncommutative geometry. For an algebra satisfying the first-order condition, we prove that off-diagonal Dirac blocks can be non-zero only if the left or right indices coincide. From this “Vanishing Theorem” we establish the Disjointness Criterion: if the right-representation spectra of the visible and dark sectors are completely disjoint and the real structure is block-diagonal, the FOC forces the Dirac operator to be strictly block-diagonal. This block-diagonality is stable under inner fluctuations, guaranteeing tree-level factorization of the spectral action and the geometric absence of Higgs and kinetic mixing portals.

For the minimal Standard Model algebra we prove that a completely neutral algebraic singlet intersection cannot exist. We then analyse the simplest dark-sector candidates and prove an obstruction theorem: the minimal Abelian dark algebra cannot simultaneously satisfy anomaly cancellation and a non-degenerate intersection pairing (Poincaré duality). This no-go result is circumvented when the dark algebra contains no U(1) factors. We explicitly construct a finite real spectral triple for the purely non-Abelian algebra Adark=H⊕HAdark=H⊕H that satisfies all NCG axioms, is anomaly-free, has a non-degenerate pairing, and keeps the right-spectrum disjoint from the visible sector – a mathematically consistent, geometrically sequestered dark sector.