The particle mass spectrum from K(7) geometry: Klein Ladder, Selberg trace formula, and exact triangle identification The One-Octonion Brane-Bulk Framework derives the particle mass spectrum from the geometry of the Klein

The One-Octonion Brane-Bulk Framework derives the particle mass spectrum from the geometry of the Klein quartic K(7) via three connected structures. RESULT 1 (GI permutohedron): The W(GI)=DI orbit of weight (2,1) generates a 12-cycle CII whose graph Laplacian eigenvalues are exactly the Klein Ladder {2-√3, 1, 2, 3, 2+√3, 4} with sum 14 = dim(GI), zero free parameters. RESULT 2 (Triple identity): The 6 Weyl reflection planes of DI are identical to the 6 systolic seams of K(7) (length lsys = 2b = 1.0905, = 2arccosh(1/2sin(π/7))) and to the 6 Jarlskog geodesic cuts that partition K(7) into 4 pairs of pants separating the fermion generation sectors. RESULT 3 (Exact triangles): Each of the 7 expressed heptagons heI contains exactly one expressed triangle, identified as the GI root generator Eα at Klein Ladder position k(i), with 13 suppressed triangles. RESULT 4 (Mass formula): The electron occupies a half-heptagon superposition (area π/6, the topological minimum), giving E_unit = 6me/π = 0.976 MeV; the mass formula m = E_unit × N_eff then gives mπ = 140.5 MeV (∆=0.7%), mµ = 105.4 MeV (∆=0.24%), mp = 936.9 MeV (∆=0.15%), all from N_eff = (3/7 × 336), (3/4 × Nπ), (20/7 × 336) respectively. 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.