Paper CCXXXIII: One-Octonion Ringdown Mode Content as the Akivis–Mal'tsev BCH Commutator Tower of Coalescing G2 Infundibula:

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. Dyer & Moore (PRL 136, 191403, 2026) present a fully Bayesian, data-driven identification of quasinormal mode (QNM) content in high-fidelity SpECTRE Cauchy characteristic evolution (CCE) ringdown waveforms. They confirm overtones to n=6, quadratic nonlinear modes, retrograde modes prominently in component-spin anti-aligned configurations, and a stable cubic QNM (2,2,0,+)3 in the (I,m)=(6,6) harmonic. We show that this entire hierarchy is the Akivis–Mal'tsev extension of the Baker–Campbell–Hausdorff (BCH) composition of two coalescing κ=−1 hyperbolic infundibula (Papers CXXIX, CXLVIII, CL, CLXII), each described by a G2-valued brane connection. The mapping is: BCH commutator depth k ↔ QNM radial overtone index n; nested Lie-bracket structure ↔ quadratic and higher multi-mode sourcings; octonion associator [e4,e5,e6]=−2e7 (Paper CVIII) ↔ the cubic mode (2,2,0,+)3. Five zero-parameter predictions follow: (CCXXXIII-1) the cubic-to-fundamental amplitude ratio is the non-associative three-body coupling K/J = n·fbulk = 1.255% already derived in Paper CXCVIII; (CCXXXIII-2) quadratic mode amplitude ratios follow the Fano-line incidence structure constants 2εijk; (CCXXXIII-3) the overtone tower truncates at nmax=5 from the G2 Cartan depth, with an n=6 recurrence corresponding to the free e7 radial breathing mode of Paper CCXIII; (CCXXXIII-4) retrograde-mode amplitude in anti-aligned-spin binaries scales as sin2(θmisalign/2) from H+/H− ordering reversal (Paper CXLIX); (CCXXXIII-5) the framework predicts a late-time power-law tail of amplitude ∝ a0·M·t in Cauchy-characteristic matching simulations, pre-registered before any such catalog publishes such a measurement. The paper closes a triangle between Paper CCIX (BCH Taylor expansion of spacetime), Paper CCXIII (Onsager {6,3} disk lock-in), and the GW birefringence prediction of Paper CCXXXII into a unified Lie-algebraic description of ringdown. 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.