Golden Ratio and Metallic Means from Pythagorean Triples

**Version 1 (29 April 2026)** This paper establishes a canonical, complete, and bijective correspondence between the family of primitive Pythagorean triples and the classical metallic means (including the golden ratio φ = δ₁). **Main results**:- **Canonical Bijection Theorem**: For every integer n ≥ 5 there exists a unique primitive Pythagorean triple (a, b, c) with c − b ∈ {1, 2, 8} such that the n-th metallic mean δₙ = cot(θ/4), where θ is the smaller acute angle of the triple.- All admissible n are explicitly constructed and classified into the Pythagoras, Plato, and Socrates families via the Euclid parametrization and the 1-2-8 rule.- Unified trigonometric representation valid across all three families: δₙ = cot(arctan(4n/(n² − 4))/4).- Two independent constructions showing that every Fermat-family triple (b − a = 1) generates metallic means both from its integer sides and from the difference of its acute angles.- Hierarchical structure: when primitive triples are composed via the Brahmagupta–Fibonacci identity, configurations of two, three, and four triples generate the golden ratio φ, its cube φ³ = δ₄, and higher metallic means through cotangent normalisation of aggregated acute angles. The algebraic mechanism is interpreted via Möbius transformations on cotangent values and via Gaussian-integer multiplication in ℤ[i], confirming that the emergence of quadratic irrationals from rational triangle data is structurally inevitable. This standalone preprint complements the author’s unified Deca-Metallic Ratios manuscript (which incorporates these geometric results into the base-10 Deca-Metallic framework). **Keywords**: Golden ratio, metallic means, primitive Pythagorean triples, cotangent normalisation, Brahmagupta–Fibonacci identity, Euclid parametrization, angle aggregation, Möbius transformations, Gaussian integers, quadratic irrationals.