The Universal Metallic Family: Quadratic Irrationals \( n + √n^2 + N \), Base-N Shift Property and Primitive Pythagorean Triples

This paper introduces the two-parameter family of quadratic irrationals α(n, N) = n + √(n² + N), n, N ∈ ℤ⁺, defined as the positive roots of x² − 2nx − N = 0. The family contains the classical Metallic Means (at N = 4, up to scaling), the even-indexed Metallic Means (at N = 1), the Deca-Metallic Ratios (at N = 10), and the Decimal-Metallic hierarchy (at N = 10ʳ) as special cases. A connection to primitive Pythagorean triples is established via the Crown Identity. The family satisfies the characteristic relation, the reciprocal identity, and the integer power property. The Base-N Shift Theorem is proved: for every n, N ∈ ℤ⁺ and k ≥ 0, {α(n, N)²ᵏ⁺¹} = {N²ᵏ⁺¹ / α(n, N)²ᵏ⁺¹}, where {·} denotes the fractional part. For N = bʳ, this gives an exact r(2k+1)-digit shift in base b for odd powers. For every primitive Pythagorean triple (a, b, c) with governing parameter n = 2a/(c−b), the Crown Identity gives α(n, N) = 2a/(c−b) + √[8c/(c−b) + (N−4)]. It is shown that for any real T > 0, with continued fraction convergents pₖ/qₖ, there exist integers N₁, N₂ satisfying N₁/N₂ ≈ T² such that α(pₖ, N₁)/α(qₖ, N₂) → T as k → ∞, with error O(qₖ⁻²). For T = √D with D ∈ ℤ⁺, a symmetric averaging sum gives O(qₖ⁻⁴) convergence. The set A = {α(p, N₁)/α(q, N₂) : p, q, N₁, N₂ ∈ ℤ⁺} is dense in ℝ⁺. Additional topics include Signature Dynamics, fundamental unit theory via the Trace-Power Theorem, connections to the modular surface, and a Parity Classification Theorem.