10.5281/zenodo.18845666 A universal geometric structure governing matter across 45 orders of magnitude fromatomic nuclei to cosmic voids is presented. The theory is based on a single axiom: theprojection operator P : O → R3,1 from 8-dimensional octonion space to 4-dimensionalspacetime. Its spectrum yields three fundamental constants: κ1 = 8/9 (synchronization),κ2 = 1/3 (structural hierarchy), κ3 = 1/7 (cyclic periodicity). From these, the universalseries an = 3 + n(n + 9)/2 emerges.A unified formula for all isotopes is derived:n = −9 + √81 + 8(M − 3)2where M is the atomic mass. This reveals three distinct types of isotopes: 21 with integern forming the geometric skeleton, 9 with rational n forming the geometric fine structure,and the remaining ∼ 2970 with irrational n forming the continuous background. Statisticalsignificance p 11σ).The kernel of the projection operator has dimension dim(ker P) = 8 − 4 = 4. Thisgeometric number appears in the fundamental identity:4(kernel) + 14G2 + 21a3 + 98M o = 137matching the fine-structure constant 1/α = 137.035999084 with 0.026% accuracy (p 18σ). New neutron clusters and the superheavy element 304120Ubn (Androsium) are predicted.Twenty-five parameters of the Standard Model and cosmology reduce to one geometricaxiom.