Paper CCXXXIV of the One-Octonion Brane-Bulk Series Mode-by-Mode Projection of the Akivis–Mal'tsev BCH Tower onto the Kerr Spin-(−2) Spheroidal Basis:

Paper CCXXXIII stated, but did not prove, the BCH↔QNM correspondence theorem mapping Akivis–Mal'tsev nested-commutator depth to ringdown radial overtone index and associator depth to cubic mode content. Here we close that derivation by explicit projection of the BCH series onto the spin-(−2) spheroidal harmonic basis on the Kerr remnant background. Three results follow. (i) The mode-by-mode identification of CCXXXIII Theorem 1 is established: linear X1+X2 sources (I,m,0,+); each additional Lie bracket adds one to n; the Akivis–Mal'tsev associator at cubic BCH order sources (2,2,0,+)3 in (I,m)=(6,6) via the unique G2-equivariant projection of e4,e5,e6=−2e7 onto spin-(−2) tensors. (ii) The cubic coupling η3=K/J=n·fbulk=1.255% is shown to be a G2 Lie-algebraic ratio, not an energy-dependent running coupling: both n=0.34 and fbulk=3.6917% are topological/structural invariants of the framework, so no QCD-scale-to-merger-frequency re-normalization is required and the headline prediction of Paper CCXXXIII stands at full strength at the BBH ringdown frequency. (iii) The explicit amplitude ratios A(I,I,0)/A(2,2,0) for I=3,4,5,6 are obtained from the G2 Gram-matrix eigenvalue decomposition of Paper CXLVII applied to the spheroidal-source projection, yielding a six-row table of zero-parameter predictions directly comparable to the Dyer & Moore (PRL 136, 191403, 2026) catalog. The previously "deferred" status of CCXXXIII Theorem 1 is now closed. 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.