Chapter 10: The Ultrametric and First Disagreement Depth

the relevant chapter defined cylinders as clopen sets determined by ABCD prefix constraints and proved that they form a basis for the canonical topology on τ³. This chapter metrizes that combinatorial structure. We define a distance function on τ³ by measuring the first disagreement depth between CRT reductions, then show that this distance satisfies the strong (ultrametric) triangle inequality. The resulting ultrametric space is canonical: no choices are involved—it is forced by the primorial tower and the CRT structure inherited from Book I. The main payoff: cylinders are precisely ultrametric balls, and the space is complete.