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 \ (xₙ^ (r) = n + n² + 10ʳ \), \ (n Z^+ \), \ (r = 0, 1, 2, 3, \), defined as the positive roots of \ (x² - 2nx - 10ʳ = 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.