Beta Hypothesis: A Unified Geometric Origin of Mass, Charge, and Spin from Wave Propagation in a Hopf-Fibered Space

We present the second revision of a geometric framework in which mass, electric charge, and spin emerge as mode properties of a single monochromatic wave propagating in the Hopf-fibered space M = D³ × S³. The theory rests on a single physical postulate: the total phase velocity of the wave is constant and equal to V throughout M. The principal change with respect to the previous revision [2] is structural rather than physical: the action of SU(2) on the internal factor — implicit in [2] in the choice of the Hopf fibration — is here derived as a lemma from the requirement that the unique postulate be observer-independent under spatial rotations of D³. From this single derived structure follow the relativistic dispersion relation E² = p²c² + m²c⁴, the Klein–Gordon equation for each spin-j mode, the discrete mass spectrum m_j = ℏ√(j(j+1))/(R₃c), the spinorial sign (−1)^(2j) acquired by half-integer modes under a 2π rotation of D³ — including its explicit identification with the geometric construction of Penrose — and the topological precondition for the spin–statistics theorem. The Dirac equation is identified as the unique first-order form compatible with the j = 1/2 spinorial mode. Electric charge is identified with the integer Hopf-winding number; the chirality convention introduced in the lemma is shown to coincide with the orientation convention of the Hopf fiber, yielding a single global chirality convention for the entire theory. The framework has one free parameter, R₃ = (√3/2)λ̄_e ≈ 3.34 × 10⁻¹³ m, fixed by the electron mass. Several results are geometrically motivated rather than rigorously derived; gravity is not addressed in the present revision. The distinctions are made explicit throughout.