Paper CXVI · One-Octonion Brane-Bulk Framework · Prediction P193 The Final Parsec Problem as Brane Geometric Scattering, and the NANOGrav Amplitude Excess

The Final Parsec Problem — the apparent stalling of supermassive black hole (SMBH) binaries at ~1 pc separation where dynamical friction becomes inefficient — is naturally explained in the One- Octonion Brane-Bulk Framework as a geometric scattering effect at the brane curvature transition radius r_T2* = √(GM/a₀). For SMBH masses of 10⁸–10⁹ M , r_T2* ≈ 300–1600 pc. At this radius the ☉ brane transitions from flat (κ = 0) to fully hyperbolic (κ = −1). Stellar orbits entering this region acquire the character of hyperbolic geodesics, which carry more angular momentum per unit energy than flat- space orbits. Stars therefore pass farther from the binary centre, reducing loss-cone refilling. This is not a depletion of stars — it is a geometric rerouting of their orbits. The binary hardens more slowly below r_T2* than standard models predict, extending its residence time at sub-parsec separations where it radiates in the NANOGrav band. This naturally explains the NANOGrav 15-year result that the GWB amplitude is 2–4.5σ above standard SMBHB model predictions: more binaries are present in the band at any given time because the geometric scattering delays their final inspiral. We also explain the low- frequency deficit at 2 nHz and the excess at 16 nHz as signatures of the transition from stochastic background to discrete-source regime, modified by the geometry-extended residence time. One clean prediction follows: the binary hardening rate should show a break at r ~ r_T2* √M in N-body ∝ simulations, scaling as √M with SMBH mass. Mathematical note: An earlier draft of this paper incorrectly computed a GW power enhancement of order r_T2*/r ~ 10^4-10^5 from the hyperbolic brane geometry. This was wrong: at parsec separations (r << r_T2*), the brane correction to the orbital metric is of order (r/r_T2*)^2 ~ 10^-10, entirely negligible. The correct mechanism is kinematic, not energetic: hyperbolic geodesics route stars away from the loss cone at r ~ r_T2*, extending binary residence time. This paper presents the corrected analysis. 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.