The Universal Metallic Family n + √(n² + N): Base-N Shift Laws, Decimal-Metallic Layers, and Primitive Pythagorean Triples

The Universal Metallic Family α(n,N) = n + √(n² + N) unifies metallic means, Deca-metallic ratios and all decimal layers under a single two-sided shift law and a universal Crown identity realised by the same Pythagorean triples in every layer. We study the two-parameter family of quadratic irrationals α(n,N) = n + √(n² + N) (n,N ∈ ℤ⁺), the positive roots of x² − 2nx − N = 0. It contains the even-indexed metallic means at N = 1, every metallic mean (doubled) at N = 4, the Deca-metallic ratios at N = 10, and the Decimal-Metallic layers N = 10ʳ. We prove a two-sided shift law valid for every N: the fractional part of αᵐ equals that of Nᵐ/αᵐ for odd m, and the two fractional parts are complementary for even m. For N = bʳ this is an exact rm-digit shift in base b; in base ten, even powers produce the nines’ complement. The members with N ≤ 2n are exactly the Pisot numbers of the family. Through the parameter n = 2a/(c − ℓ) of a primitive Pythagorean triple we obtain a universal Crown identity α(n,N) = 2a/(c − ℓ) + √(8c/(c − ℓ) + N − 4), and we show that the integer values of 2a/(c − ℓ) are exactly the integers n ≥ 5, each realised by a unique triple, the primitive reduction of (4n, n² − 4, n² + 4), so the geometric realisation is the same in every layer. We determine exactly when α(n,N) is a rational multiple of a classical metallic mean (if and only if N = s² and s | 2n), derive a parity law for the layers of any non-square base, and prove that α(n,N) is an integer if and only if N = g(g − 2n) with g > 2n. In the decimal layers we obtain the decimal grid 10ᵏ Mₚ of all metallic means, the golden row 10φ, 100φ, 1000φ, …, scaled Deca-metallic ratios in the odd layers, and a unique decimal golden member in each layer. The family is the quadratic case m = 2, r = 1 of the Omni-Metallic framework.