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

What do the Golden Ratio, the Silver Ratio, the Bronze Ratio, primitive Pythagorean triples, and the decimal number system have in common? This paper shows they are all faces of a single, infinite two-parameter family of numbers: α(n,N) = n + √(n² + N). By letting N range freely over the positive integers — not just powers of ten — this family unifies and classifies the classical Metallic Means of de Spinadel, the author's previously-introduced Deca-Metallic and Decimal-Metallic Ratios, and every primitive Pythagorean triple, all within one coherent algebraic, geometric, and hyperbolic framework. Two central theorems anchor the paper. The Base-N Shift Theorem shows that when N is a power of a base b, the fractional digits of α(n,N) reappear after an exact digit shift in that base — decimal shift for base 10, binary for base 2, and so on for any base. The Universal Crown Identity derives every member of the family directly from a primitive Pythagorean triple via a single geometric formula. A complete Classification Theorem partitions all positive integers N into two classes — perfect squares and non-squares — governed by a newly-proved Parity Classification Theorem describing exactly which Metallic Means appear, and how, at each layer of any base-b hierarchy. Along the way, the paper uncovers striking patterns: the Golden Ratio reappears scaled by 10, 100, 1000, 10000, ... at every even power of ten, and at N=100 alone the entire Metallic Means family — Golden, Silver, Bronze, and beyond — emerges scaled uniformly by ten. The family also generates generalized Fibonacci and Lucas sequences satisfying Binet formulas and a universal Cassini identity, connecting this work to the broader theory of integer sequences.