Omni Metallic Ratios : Chetansing Rajput

We introduce the Omni-Metallic Family ℧m,r(n,N), the unique positive real root of the equation x^m - 2n xᵐ⁻¹ - N^r = 0. This four-parameter family unifies and vastly generalizes all previously studied metallic means. For the quadratic case m=2, we establish the Integer Value Theorem, which yields a complete prime characterization, uniform representation counts, and a natural lattice structure in the (n,N)-plane. The theory rests on the fundamental Mirror Identity, which implies the Shift Theorem, Cassini-type identities, nested continued fractions, and deep connections to primitive Pythagorean triples via the Crown Identity. We further develop the analytic theory, deriving a closed-form Omni-Metallic zeta function, a new Euler constant, and novel representations for π² and e. Additional results include sharp Pisot characterizations (for m=2,3), high-order Diophantine approximations, and rich complex dynamics on the associated rational maps. This framework reveals a previously hidden unity between algebraic numbers, prime distribution, analytic number theory, and dynamical systems, opening new avenues in multiple branches of mathematics.