The Omni-Metallic Framework: Arithmetic, Dynamics, and Analytic Theory

We introduce the Omni-Metallic Family ℧m,r(n,N), defined as the unique positive real root of x^m - 2n xᵐ⁻¹ - N^r = 0, where m ≥ 2 and n,N,r ∈ ℕ. This four-parameter family contains previously studied metallic families as special cases. For the quadratic case m=2, we prove the Integer Value Theorem: ℧2,1(n,N) = k iff N = k(k-2n), yielding a uniform representation count r(k) = floor((k-1)/2), a complete prime characterization, and a universal square-root generator. The theory rests upon the Mirror Identity: ℧m,r(n,N) - N^r/℧m,r(n,N)ᵐ⁻¹ = 2n, which implies the Shift Theorem, the Cassini Identity, and the nested continued fraction expansion. For m=2,3, we characterize when ℧m,r(n,N) is a Pisot number: N^r ≤ 2n. The associated zeta function satisfies ζ_𝒪(s) = 1/2 ζ(s-1) - 1/2(1+2⁻ˢ)ζ(s), yielding the Omni-Metallic Euler constant γ_𝒪 = γ/2 - 5π^2/48, and new representations for π^2 and e. The Bridge Identity connects the analytic and algebraic theories. Additional results include the Crown Identity connecting the family to primitive Pythagorean triples, a family of Diophantine approximations with error O(q_k⁻⁽ᵐ⁺¹⁾), and connections to units and class numbers in real quadratic fields. Several open problems, including the sharp Pisot boundary for m ≥ 4, are stated.