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

v2 (2026-06-14) — ERRATUM appended. The headline cubic-QNM amplitude prediction η₃ = |C(2,2,0)³|/|C(2,2,0)|³ = n·f_bulk = 1.255×10⁻² is FALSIFIED by direct measurement on the public Dyer & Moore CCE catalog (SXS:BBH_ExtCCE:0010/0011/0012, superrest frame): η₃ ≈ 0.6 (0/21 fits inside the stated factor-3 band), ~48× high. The original “confirmation” conflated the raw cubic amplitude (~10⁻²) with the dimensionless ratio. Corrected, derived prediction: η₃ = f_III = 1 − 4/π² ≈ 0.595 (matches +0% to +2.3%) — the cubic is the coherent brane→bulk projection of the 3 spatial brane dimensions (power-3 = spatial dimensionality; single power of f_III = coherence, the same Born-once principle that fixes f_III in the neutrino sector; bulk fraction via the H⁻ associator [e4,e5,e6]=−2e7 → e7). The cubic is the unique framework nonlinear mode; quadratics are incomplete (2-of-3-dimension) projections = ordinary GR mode-coupling, carrying no framework prediction. The cubic mode’s existence and (6,6) localization, the Fano quadratic selection rule, the retrograde-spin rule, and n_max stand. See appended erratum + correction note. v2 (2026-06-14) — ERRATUM appended. The headline cubic-QNM amplitude prediction η₃ = |C(2,2,0)³|/|C(2,2,0)|³ = n·f_bulk = 1.255×10⁻² is FALSIFIED by direct measurement on the public Dyer & Moore CCE catalog (SXS:BBH_ExtCCE:0010/0011/0012, superrest frame): η₃ ≈ 0.6 (0.55–0.69, 0/21 fit start-times inside the stated factor-3 band), ~48× high. The original “confirmation” conflated the raw cubic amplitude (~10⁻²) with the dimensionless ratio. A post-hoc, parameter-free corrected target η₃ = f_III = 1−4/π² ≈ 0.595 (H⁻ associator [e4,e5,e6]=−2e7 → e7 bulk) matches at +2.3%; an attempted “all quadratics = n·f_III” generalization was tested and failed. The cubic mode’s existence and (6,6) localization, the Fano quadratic selection rule, the retrograde-spin rule, and n_max stand. See appended erratum + correction note. Pipeline validated against the known quadratic (2,2,0)²→(4,4) ratio. 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.