The Spectral Action Information Hierarchy: What the Bosonic Spectral Action Can and Cannot See About Flavor

Abstract We prove that the bosonic spectral action of the minimal Chamseddine–Connes–Marcolli (CCM) Standard Model spectral triple is blind to all flavor information beyond fermion mass eigenvalues. Specifically, we show that (i) the spectral action determines the Yukawa eigenvalues (fermion masses) through single-sector power-sum invariants; (ii) the spectral action contains no cross-sector traces coupling up-type and down-type Yukawa matrices, and is therefore independent of CKM and PMNS mixing angles; and (iii) the spectral action, being a real-valued trace functional of D2, is structurally incapable of producing CP-odd invariants such as the Jarlskog invariant. We further prove that even in non-minimal extensions where cross-sector traces arise, the bosonic spectral action can access mixing angles only through mass-weighted sums of |Vij |2 and remains permanently blind to CP-violating phases. These results establish a precise boundary for the predictive power of the spectral action in flavor physics: masses are within its reach; mixing and CP violation are not.