The Omni-Metallic Framework: Algebraic Structure, Arithmetic Theory, and Dynamical Systems

This is a comprehensive preprint containing the full exploratory record of the Omni-Metallic Framework, including results that are refined, corrected, or omitted in the definitive journal version. The peer-reviewed version is forthcoming. This work introduces the Omni-Metallic Family \(m,r(n,N)\), the unique positive root of the equation \(x^m - 2n xᵐ⁻¹ - N^r = 0\) (m ≥ 2, n, N, r natural numbers). This four-parameter family unifies classical metallic means, deca-metallic ratios, and provides new arithmetic structures. Key results include:• The Mirror Identity and its consequences (Taylor expansion, self-duality, Shift Theorem).• The Integer Value Theorem with complete characterization, representation count, square-root generator, and prime representation properties.• Lucas sequence connections, Crown Identity with Pythagorean triples, and matrix formulations.• The Omni-Metallic zeta function (Dirichlet series), Master Generating Function (conjectural closed form), and derived constants.• Equivalence theorems linking Pisot numbers, nested continued fractions, Julia sets, and Lyapunov exponents (for low degrees).• Diophantine approximation bounds and other exploratory directions.