Chapter 36: The Constructive Reals: Ordered Field and Completeness

the relevant chapter defined ℝ_τ as the Cauchy completion of ℚ_τ with explicit modulus and sketched its field properties. This chapter carries out the formal development: ordered field axioms (Proposition [prop:ordered-field-axioms]), the Archimedean property distinguishing ℝ_τ from ℤ_τ , Cauchy completeness via diagonal extraction, and the canonical metric making ℝ_τ a complete metric space.