The Proton–Electron Mass Ratio from Quaternion Phase Space

A first-principles derivation of the proton-electron mass ratio m_p/m_e = 6π⁵ ≈ 1836.12 from quaternion phase space. The derivation yields m_p/m_e = (N/d) · |S³|^N / |S¹| = (3/2) · (2π²)³ / (2π) = 6π⁵ agreeing with the measured value 1836.153 to 0.002% with no fitted parameters. Each factor has independent justification: |S³| = 2π² from the unit-quaternion wave axiom, |S¹| = 2π from the standard U(1)_em gauge redundancy, N = 3 from the proton's quark content, and d = 2 from the wave-persistence theorem (which forces exactly two perpendicular wave components related by i² = −1). Combined with the previously established proton mass formula m_p = M_Pl · 3^(1/4) · α⁹, this derivation closes the electron mass — previously listed as an open derivation in the framework's inventory — at 0.5% from one mass scale (M_Pl) and one dimensionless coupling (α). The construction provides a geometric origin for the Lenz coincidence (1951), which has been an open puzzle in mathematical physics for over seven decades despite prior derivation attempts by Wyler (1969, 1971) and others. The formula generalizes to M_N = (N/d) · 2^(N−1) · π^(2N−1), with the power of π tracking the mode count as 2N − 1. The N = 3 entry recovers the proton-electron ratio; higher-N predictions are falsifiable.