The Universal Metallic Family: Quadratic Irrationals n + √(n²+N), Base-N Shift Property, and Primitive Pythagorean Triples

This paper introduces the Universal Metallic Family, an infinite two-parameter family of quadratic irrationals α(n,N) = n + √(n²+N), defined as the positive roots of x²−2nx−N=0. The family simultaneously unifies and classifies the classical Metallic Means, the Deca-Metallic Ratios, the Decimal-Metallic Ratios, and every primitive Pythagorean triple as natural special cases governed by the single parameter N. Ten results are proved uniformly for every positive integer N, including the Base-N Shift Theorem, the Universal Crown Identity, the Universal Cassini Identity, a transcendental hyperbolic representation, and a complete Parity Classification Theorem (necessary and sufficient). The family admits a definitive two-class partition: Class A (perfect square N) and Class B (non-square N), with the Decimal-Metallic family identified as the canonical base-10 subfamily.