The Omni-Metallic Family: A Complete Research Program

The Omni-Metallic Family constitutes a long-term research program centered on the four-parameter algebraic object α_{m,r}(n,N), defined as the unique positive real root of the equation x^m − 2n x^{m−1} − N^r = 0 (m ≥ 2, n, N, r ∈ ℕ). This program originated with the discovery of deep mathematical relationships between classical metallic means and primitive Pythagorean triples in 2021, which were published in 2024 in Publications on Number Theory and Discrete Mathematics. The research continued through successive generalizations: the Deca-Metallic Ratios, the Decimal Hierarchy, Decimal-Metallic Family, the Universal Metallic Family, and reached its current form with the Omni-Metallic Family — a unified framework that contains all previously studied metallic families as exact special cases. This preprint presents the complete research program developed to date, including the Integer Value Theorem, Mirror Identity, representation theory, analytic framework (generating functions, summatory functions, Dirichlet series, Omni-Metallic zeta function), algebraic structure, geometric properties, dynamical aspects, and further connections across number theory, algebra, and dynamics. Comments and feedback are welcome.