Dual Decomposition in ℤ[φ]

This is an expository note. It assembles named results from three separate communities — algebraic number theory, the theory of numeration systems, and the arithmetic of Mahler measure — and places them relative to the 2I programme. It claims no new theorem. The observation it organises is that the positive integers ℤ⁺ admit two unique decompositions, and one real-quadratic ring supplies the data for both — but in two different senses, which we are careful to keep apart. Multiplicatively, the ring factorises: the Fundamental Theorem of Arithmetic (FTA) is recovered from the prime ideals of ℤ[φ], φ = (1+√5)/2. Additively, the fundamental unit numerates: Zeckendorf's theorem is the digit system on the powers of φ (Binet). The common object is ℤ⁺, decomposed two ways; the ring supplies one decomposition, the unit the other. Both legs are named in the literature — the additive one end-to-end, the multiplicative one trivially — and the golden field's distinguished role is also named, and more sharply than this note's own phrasing: by Schinzel's theorem (1973), the golden ratio realises the minimal Mahler measure above 1 among totally real algebraic integers. The single thing we found unstated across the accessible literature is the assembly — the two decompositions set side by side as one ring's two faces — and that, framed honestly as synthesis, is this note's only contribution. We assemble; we do not prove.