Paper LVI: The CKM Matrix from δM/M: Three Generations as A3 Colour Orderings, the Cabibbo Angle as the Gatto-Sartori-Tonin Relation from Odd-Ordering Vacuum Mixing, and CP Violation from Z3 Chirality

The three generations of quarks and leptons, the CKM quark mixing matrix, the Cabibbo angle, and CP violation are derived from the single baryogenesis parameter δM/M ∼6 × 10−10 of Paper I through four steps. (1) Three generations: the six orderings of the three colour imaginary units {e4, e5, e6} that generate e7 via the octonionic associator split into three even permutations (+e7, matter) and three odd permutations (−e7, antimatter). The three even orderings are the three generations; their count is the order of the alternating group A3 ∼= Z3, a theorem with zero free parameters. (2) Generation mass hierarchy: the δM asymmetry tilts the Fano colour line {e4, e5, e6} relative to the e7 equatorial plane, ordering the three cyclic permutations by proximity to the H+ matter brane. Generation 3 is heaviest, generation 1 lightest xed by the tilt direction. (3) The Cabibbo angle: at QCD connement, the three odd orderings are conned but not annihilated. They form a mixed vacuum state whose intermixing with the even-ordering quark states produces the CKM o- diagonal elements. The mixing amplitude between generation-1 and generation-2 odd- ordering vacuum dressings is proportional to the geometric mean of the respective quark condensate contributions, giving the Gatto-Sartori-Tonin relation sin θC = p md/ms ≈ 0.224, in agreement with the measured 0.2257 to 0.7%. This is the rst derivation of the Gatto-Sartori-Tonin relation from a microscopic geometric model. (4) CP violation: the Z3 generation space is chiral even permutations map to odd under colour unit exchange, which is a CP transformation. The CP-violating phase exists because the three even orderings cannot be simultaneously eigenstates of this parity. The Jarlskog invariant is proportional to the product of all six quark masses normalised to Λ6 QCD, giving J ∼10−5 consistent with observation. The derivation requires no free parameters beyond the quark masses, which are themselves expected to emerge from the G2 overlap integrals of Paper LII. Four new predictions follow (Predictions 9295).

Part of the One-Octonion Brane-Bulk Framework series. Anchor DOI: 10.5281/zenodo.19120873. Community: one-octonion-brane-bulk. Author: Bharathi Dasan Jagadeesan, M.D., University of Minnesota. ORCID: 0000-0002-1143-941X.