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·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. 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.