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

This paper introduces the Decimal Metallic Ratios, an infinite hierarchy of quadratic irrationals x_n^(r) = n + √(n² + 10^r), defined as the positive roots of x² − 2nx − 10^r = 0 for n ≥ 1 and r ≥ 0. The hierarchy unifies two previously distinct families: the classical Metallic Means and the Deca-Metallic Ratios. The case r = 0 is proved to coincide exactly with the even-indexed classical Metallic Means, while r = 1 recovers the Deca-Metallic Ratios. Higher values of r generate infinitely many new families governed by powers of ten. A complete theory is developed, including reciprocal identities, odd-power decimal-shift phenomena, generalized continued fractions, matrix generators, associated Fibonacci- and Lucas-type sequences, Cassini identities, transcendental hyperbolic representations, and canonical constructions from primitive Pythagorean triples through a generalized Crown Identity. The results reveal a unified algebraic, arithmetic, geometric, and hyperbolic framework connecting Metallic Means, Deca-Metallic Ratios, and primitive Pythagorean triples within a single infinite decimal hierarchy.