Paper CCX: The Pion as a Kaon Returned from the Bulk: Handle-Crossing Geodesics on the Klein Quartic, Kaon Degeneracy from Fano Crossing Modes

Within the Klein Quartic meson mass framework of the One-Octonion Brane- Bulk series (Papers CXC, CCVIII, CCIX), the pion and kaon do not correspond to distinct geodesic paths on the brane surface. Instead, we show that the pion is the kaon's brane projection: the kaon is the pion after its geodesic has transited through a handle of the genus-3 Klein Quartic surface into the AdS5 bulk and returned. The relevant geodesic for the pion is not a surface geodesic but a bulk-mediated handle-crossing geodesic that passes through one of the three topological handles of M = PSL (2, 7) 2. The transit probability per oscillation cycle is exactly fbulk = 3. 6917%, the universal brane-bulk transit fraction. The crossing degeneracy — the number of topologically distinct ways to traverse the handle while preserving Fano incidence — is three, corresponding precisely to the three observable kaon states (K±, K0, ¯K0). The mass splitting between K± and K0 is the degeneracy-lifting from the asymmetric Fano incidence structure at the crossing node. The pion-kaon mass ratio mK/mπ = 493. 68/139. 57 = 3. 537 is therefore not a free ratio: it is determined by the handle geometry of the Klein Quartic combined with the brane-bulk transit fraction fbulk and the Fano crossing degeneracy. This paper records the theorem and its immediate corollaries; the full numerical computation of mK/mπ from first principles is deferred to a companion paper. 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.