The Omni-Metallic Family: Integer Representations, Analytic Framework, and Complex Dynamics

We introduce the Omni-Metallic Family α_{m,r}(n, N), defined as the unique positive real root of x^m − 2n x^{m−1} − N^r = 0 (m ≥ 2, n, N, r ∈ ℕ). This four-parameter family provides a unified framework that contains all previously studied metallic means and their generalizations as exact special cases. The central result is the Integer Value Theorem: α_{2,1}(n, N) = k ⇐⇒ N = k(k − 2n), which yields a uniform representation count ⌊(k − 1)/2⌋, a complete prime characterization, and a universal square-root generator. The entire theory rests upon the Mirror Identity α_{m,r}(n, N) − N^r / α_{m,r}(n, N)^{m−1} = 2n. This single relation generates a wealth of algebraic identities, including reciprocal, scaling, Pell-type, and Cassini analogues. We develop an associated analytic framework featuring the Omni-Metallic zeta function ζ_O(s) = ½ ζ(s−1) − ½(1 + 2^{-s}) ζ(s), an Euler constant γ_O = γ/2 − 5π²/48, and new series representations for π² and e derived from the representation function r(k) = ⌊(k − 1)/2⌋. Additional structural results include a sharp Pisot boundary for cubic members, nested continued fractions with exponentiation, and a dynamical dichotomy linking algebraic Pisot numbers to ordered dynamics and non-Pisot cases to chaotic Julia sets. A complete unification theorem establishes equivalence between five seemingly independent conditions: the Pisot condition, continued fraction convergence, connectedness of the Julia set, negative Lyapunov exponent, and attractivity of the fixed point. The Omni-Metallic Family reveals deep connections between metallic numbers, Diophantine approximation, complex dynamics, and ergodic theory. Keywords Omni-Metallic Family; metallic means; primitive Pythagorean triples; integer representations; prime characterization; Pell equations; Lucas sequences; zeta functions; prime counting; Pisot numbers; nested continued fractions; Julia sets; complex dynamics; Toeplitz matrices