PGT4: The CKM and PMNS matrices are parameterised by τ = arccos (1/√7) for the CP phases and by combinations of θ₆䃒 for the mixing angles. Specifically: the Cabibbo angle θC = arcsin (sin (θ₆䃒) × Fanofactor), derivable from the same formula. This is Paper CXXX.

Since Paper I, the One-Octonion Brane-Bulk Framework has used the principle of least action as an assumed variational principle governing brane dynamics. Paper CXXX establishes that this principle is not an assumption: it is the statement that physical brane evolution is G₂-gauge-invariant. The identification rests on three observations: (1) G₂ = Aut( ) is the gauge group of the brane — G₂ 𝕆 automorphisms cost zero action because they are internal symmetries that leave all physical content unchanged; (2) Mostow rigidity (Paper CXVIII) makes the physical moduli space of brane configurations a single point, so the 'least-action path' is uniquely determined; and (3) the sump focal zone is the G₂- invariant vacuum — the region of maximum G₂ symmetry (κ → −1) where the gauge group acts with full compact structure. The principle of least action is not preferred among other formulations of mechanics. It is forced: it is the unique formulation compatible with G₂ gauge invariance and Mostow rigidity. This is the same reason it is 'preferred compared to other energy–momentum transformations with the same beginning and end points' (as the MRI analogy makes precise): the on-resonance path is the gauge-invariant one; all other paths dephase. 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.