UF0: τ(0)⊕0=τ(1)⊕1 A One-Line Axiomatization of the Birth and Dissolution of Mathematics

We present UF0(Unified Foundation 0), a minimal formal system in which the entire edifice of natural-numberarithmetic is generated and dissolved by a single equation. • 0 is the pre-numerical, non-dual point.• τ is a unary “tear” operator enacting the primal split. • ⊕ is a non-repeating union.The sole axiomτ (0) ⊕ 0 = τ (1) ⊕ 1simultaneously encodes1. the birth of mathematics (0 → 1), and 2. the fundamental duality (1 → 2),while collapsing both into identity when 0 = 1 = 2. No external arithmetic, set theory, or category theory is presupposed. UF0 is therefore the first fully formal, one-sentence foundation that captures both the emergence and the illusoriness of number.