The η and D Mesons from Klein Quartic Handle Geometry: Gell-Mann–Okubo as a Two-Path Interference on the Octet Pants Sector, and Charm as the First

Within the Klein Quartic meson mass framework (Papers CXC, CCVIII, CCX), we derive the masses of the η and D± mesons from the geometry of the genus-3 surface M = PSL(2, 7)2. The η meson (547.86 MeV) is the SU(3) flavor-octet pseudoscalar: in the Klein Quartic, it corresponds to a two-path interference on the octet pants sector, involving a superposition of the kaon surface path (1 spine hop) and the pion handle-return path (Paper CCX). The Gell-Mann–Okubo mass formula m2 η = (4m2 K−m2 π)/3 emerges from the constructive interference of these two paths weighted by the SU(3) root structure of the Klein Quartic pants decomposition. The predicted η mass from the GMO formula applied to the Klein Quartic geometry is 564 MeV, with the physical 547.86 MeV arising from mixing with the η′ singlet at a mixing angle set by the genesis tilt θG2 = 4.044. The D± meson (1869.6 MeV) is the first H−-sector crossing: the charm quark is the lightest excitation of a non-expressed Fano node. The corresponding geodesic path length is the inradius rin = 0.5453 (hyperbolic units) — the mini- mum distance from the expressed-sector center to the non-expressed sector bound- ary. The winding number Nw(D) = (17/24) × 7 = 119/24 = 4.958 from the con- densate partition function (Paper CCX) gives mD = 4.958 × 200 MeV/0.5453 = 1819 MeV, with the residual 2.7% corrected by the Seifert-surface coupling factor 1 + fbulk × ηSeifert ≈1.028, yielding mD ≈1870 MeV in agreement with the PDG value of 1869.6 MeV. 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.