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·fbulk = 1. 255×10⁻² is FALSIFIED by direct measurement on the public Dyer single power of fIII = coherence, the same Born-once principle that fixes fIII 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 nₘax 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·fbulk = 1. 255×10⁻² is FALSIFIED by direct measurement on the public Dyer an attempted “all quadratics = n·fIII” generalization was tested and failed. The cubic mode’s existence and (6, 6) localization, the Fano quadratic selection rule, the retrograde-spin rule, and nₘax stand. See appended erratum + correction note. Pipeline validated against the known quadratic (2, 2, 0) ²→ (4, 4) ratio. Dyer 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.