The Decimal Hierarchy: An Infinite Family of Quadratic Irrationals Unifying Metallic Means, Deca-Metallic Ratios, and Primitive Pythagorean Triples

This paper introduces and develops the Decimal Hierarchy, an infinite parametric family of quadratic irrationals \( x_n⁽ʳ⁾ = n + √n^2 + 10^r \), \( n ∈ Z^+ \), \( r = 0, 1, 2, 3, \), defined as the positive roots of \( x^2 - 2nx - 10^r = 0 \). The parameter \( r \) governs a precise decimal shift phenomenon. Every member arises canonically from a primitive Pythagorean triple via a generalised Crown Identity and admits an exact transcendental representation via inverse hyperbolic sine, embedding the hierarchy as integer lattice points on a one-parameter family of hyperbolic manifolds. Special cases include: \( r=0 \) recovers even-indexed classical Metallic Means, and \( r=1 \) recovers the Deca-Metallic Ratios. The framework unifies algebraic, geometric, arithmetic, and hyperbolic properties across all powers of ten.