A Proof of the Collatz Conjecture via the Collatz Convergence Axiom

This paper presents a proof of the Collatz conjecture. The proof rests on three structural elements: the +1 in the odd rule prohibits all loops other than {4, 2, 1}; the contraction ratio 3/4 < 1 defines a scale-invariant descent vector; and the Collatz Convergence Axiom — that every sequence σᵏ(N) converges to 1 — is proposed as a foundational premise independent of ZFC, in the tradition of the Axiom of Choice. Proof by contradiction: non-convergence violates the axiom. Formalized in Lean 4 with CI passing.