Deca Metallic Ratios

This paper introduces and develops, in complete generality, the Deca-Metallic Ratios δₙ = n + √ (n² + 10), n ∈ ℤ⁺, defined as the positive roots of x² − 2nx − 10 = 0. By anchoring the algebraic norm of the family at 10 — the base of the decimal numeral system universally used in human arithmetic — the Deca-Metallic Ratios exhibit a self-similarity that is not merely algebraic but arithmetically visible in ordinary decimal notation. The central discovery of this theory is the decimal-shift phenomenon: for every n ≥ 1 and every k ≥ 0, the fractional part of δₙ satisfies δₙ = 10/δₙ, and more generally δₙ^ (2k+1) = 10^ (2k+1) /δₙ^ (2k+1) for every odd power, while for even powers the fractional parts satisfy the complementary law δₙ^ (2k) + 10^ (2k) /δₙ^ (2k) = 1. These two laws — preservation under odd powers and complementation under even powers — are shown to arise from a single algebraic mechanism governed by the sign of the conjugate root, and together they constitute the defining arithmetic signature of the family. The paper further establishes that every Deca-Metallic Ratio arises canonically from a primitive Pythagorean triple (a, b, c) via the parameter n = 2a/ (c − b), admits the trigonometric representation n = 2cot (θ/2), and satisfies the Crown Identity δₙ = 2a/ (c − b) + √ (8c/ (c − b) + 6), which stands in exact structural parallel to the classical metallic-means identity μₙ = a/ (c − b) + √ (2c/ (c − b) ). A complete classification of admissible triples into three arithmetic families (Pythagoras, Plato, Socrates) is given via the Euclid parametrization, together with a bijective inverse construction. The special case n = 5 yields the Decimal Golden Ratio δ₅ = 5 + √35, distinguished as the unique member for which the floor equals the decimal base, ⌊δ₅⌋ = 10, giving the exact decimal analogue 1/δ₅ = δ₅/10 of the classical identity 1/φ = φ − 1. Continued-fraction expansions, a matrix generator, the associated Deca-Fibonacci and Deca-Lucas integer sequences, a Cassini-type identity, and an exact transcendental representation δₙ = √10 exp (arcsinh (n/√10) ) embedding the family as integer lattice points on a hyperbolic manifold are all developed. Together these results forge a unified algebraic-geometric-arithmetic-transcendental framework that substantially extends the classical theory of metallic means and reveals a deep and previously unsuspected harmony between right-triangle geometry, the decimal numeral system, and hyperbolic space.