Omni-Metallic Family

We introduce the Omni-Metallic Family ℧m,r(n, N), defined as the unique positive real root of xm −2n xm−1 −N^r = 0, m ≥2, 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 ⇐⇒ N = k(k −2n), yielding a uniform representation count r(k) = ⌊(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 −5π²/48, and new representations for π² and e. The Bridge Identity connects the analytic and algebraic theories. Additional results include the Crown Identity connecting the family to primitive Pythagorean triples, Diophantine approximations, optimality results, connections to units and class numbers, and the Omni-Prime counting function π_O(x) ∼ x²/(4 log x) with error controlled by the Riemann zeta function.