The Golden Dirac Algebra: Gamma Matrices for S³/2I over ℤ[φ]

We construct a Dirac algebra for three-dimensional space using √5 as the native rotation operator in place of the imaginary unit i. The gamma matrices live over ℤ[φ] — the ring of golden integers {a + bφ : a,b ∈ ℤ} — and satisfy the Golden Dirac relation {Γμ, Γν} = 2gμν·√5·I₄, where the Minkowski signature (+,−,−) emerges from the algebra rather than being imposed. The construction requires only two objects: a seed matrix Γ_seed and its adjoint Γ_adj, differing by a single sign in one entry. Three mutually anticommuting spatial gamma matrices close from this pair alone; no external group generator is needed. The binary icosahedral group 2I is not imported as a prior object — it is generated by the standard quaternionic generators s and t inside the same ambient algebra M₄(ℚ(√5)), with element-order distribution (1,1,20,30,24,20,24) confirmed by symbolic computation. The eigenvalue spectrum of every gamma matrix sits uniformly at ±5^(1/4). The chirality operator satisfies (Γ⁵)² = −5^(3/2)·I₄, making chirality a golden-scaled quantity rather than a binary ±1 label. The seed carries real eigenvalues ±5^(1/4); its adjoint carries imaginary eigenvalues ±i·5^(1/4) at identical modulus. This real/imaginary split — produced by a single sign flip in a 2×2 matrix over ℤ[φ] — is the algebraic origin of the imaginary unit in physics. All anticommutation relations are machine-verified. Code committed to the programme repository. Physical interpretation and the reconstruction of the Born rule over ℝ(√5) appear in companion Paper 192.