The Möbius Origin of φ

From the simplest Möbius transformation to the full cyclotomic dynamics of J = 1 + ζ₅². We derive the golden ratio φ = (1 + √5)/2 as the unique stable fixed point of the simplestnontrivial Möbius transformation f(x) = 1 + 1/x, the composition of inversion and translationby 1. Its matrix representation is the Fibonacci matrix F = ⎡1 1⎤ ⎣1 0⎦ with entries in {0, 1}. We then show that F is the real shadow of the 4×4 multiplicationmatrix M_J ∈ SL(4,ℤ) associated to the cyclotomic unit J = 1 + ζ₅² in the fifth cyclotomicfield ℚ(ζ₅), established in [1]. The matrix M_J has entries in {−1, 0, 1} (Binary Theorem)and eigenvalue moduli {φ, φ, φ⁻¹, φ⁻¹}, producing the stable/unstable splitting ℝ⁴ = E⁺ ⊕ E⁻with Lyapunov exponents ±ln φ. Adding the Thue–Morse binary sequence as an aperiodicdriver completes the kernel: three integer engines (Fibonacci for geometry, M_J for structure,Thue–Morse for dynamics), all implementable using only integer additions and sign flips(matrix–vector products over ℤ with coefficients in {−1, 0, 1} require no integer multiplication),together generate all cyclotomic, icosahedral, and spectral data of the TWIST‑J framework.The icosahedral invariants (V, E, F) = (12, 30, 20) and group orders |A₅| = 60, |2I| = 120arise from Dedekind zeta values of the same field [2]. No free parameter is introduced at any step. This paper is not a proposal of new dynamics, not a phenomenological model, and not a numerical simulation. It is a structural derivation: the minimal algebraic chain from which TWIST-J dynamics necessarily follow. Every step is a theorem. No step introduces a free parameter. The physical interpretation is secondary and explicitly labeled throughout. MSC 2020: 11R18 (Cyclotomic extensions) 37D20 (Uniformly hyperbolic systems) 51M10 (Hyperbolic and elliptic geometries)