Discrete Mass Spectrum and Horizon Quantization of Schwarzschild Black Holes: Convergent Kinematic and Dynamic Derivations

We derive a discrete mass spectrum for Schwarzschild black holes through two independent, convergent analytical paths, establishing that Planck-area quantization is an inescapable consequence of fundamental physical principles rather than an ad hoc postulate. In the kinematic formulation, applying the Heisenberg uncertainty principle at the event horizon boundary determines a fundamental minimum length scale l_P. Projecting this constraint onto the differential geometry of a spatial slice (dV/dA = R/2) naturally yields Planck-area discretization without invoking internal microstate dynamics. In the dynamic formulation, classical accretion onto an evolving horizon under the continuous volume-mass scaling condition generates an identical M·dM structural coupling across both momentum exchange and black hole entropy production rates (dS_BH/dt). The convergence of these decoupled frameworks demonstrates that horizon quantization constitutes an invariant geometric property of the horizon manifold. We obtain the discrete mass spectrum M_n = (m_P / 4√π)√n and microstate entropy S_n = (k_B / 4)n, resolving full consistency with Bekenstein-Mukhanov area quantization. Furthermore, we establish a thermodynamic cosmic duality wherein black hole evaporation represents the localized, time-reversed mirror of cosmological horizon expansion under the same invariant M·dM engine. We outline three testable signatures: (i) discrete emission lines in the high-frequency Hawking radiation spectrum, (ii) characteristic picket-fence quasi-normal mode frequencies in gravitational wave ringdowns, and (iii) discrete cosmological horizon entropy units.