The Primal Energy Scale: Derivation and Physical Significance of E₀ and ℓ₀ from Riemann Zeta Zeros

We present a rigorous derivation of two fundamental physical scales: the primal energy E0 = 2.916601 × 10^{-16} J = 1820.469 eV and the primal length l0 = lP = 1.616255 × 10^{-35} m (the Planck length). These quantities emerge uniquely from the arithmetic-geometric structure encoded in the zeros of the Riemann zeta function ζ(s). We demonstrate that E0 serves as the natural energy unit connecting quantum mechanics, gravitational physics, and number theory, while l0 establishes the fundamental length scale of spacetime geometry. The derivation employs: (1) the exact conformal transformation Φ(z) = α arcsinh(βz) + γ with αβγ = 2π connecting quantum spectra to zeta zeros; (2) combinatorial relations among the first four nontrivial zeros γ1, γ2, γ3, γ4; and (3) consistency conditions with established physical constants (CODATA 2018). The resulting framework provides a unified basis for understanding fundamental constants, predicts testable modifications to quantum and gravitational phenomena, and offers new insights into the geometric structure of reality at the Planck scale.