is hyperfinite (see [@MR1900547 Theorem 1.16]), and one may therefore represent $ \cber[\mathcal{C}] $ as an increasing union of finite equivalence relations: $ \cber[\mathcal{C}] = \bigcup_{n} \fber[m] $.

We are going to construct a spiral of cross sections $ (C_{n}, h_{n}) $, $ n \in \mathbb{N} $, $ \mathcal{C}_{0} = \mathcal{C} $, such that $ h_{n} : \mathcal{C}_{n} \to B\bigl(2^{-n-1}\bigr) $. Let $ \fber[m][n] $ denote the equivalence relation $ \fber[m] $ transferred onto $ \mathcal{C}_{n} $ via $ \phi_{0,n} $: $$x \fber*[m][n] y \iff \phi^{-1}_{0,n}(x) \fber*[m] \phi^{-1}_{0,n}(y).$$ The spiral will satisfy the following two conditions:

1.  \[item:hn-constant-on-Cn-classes\] $ h_{n} $ is constant on $ \fber[n-1][n] $-equivalence classes: $$x \fber*[n-1][n] y \implies h_{n}(x) = h_{n}(y).$$

2.  \[item:Fn-class-on-rational-grid\] Every $ \fber[n-1][n] $ class in $ \mathcal{C}_{n} $ is “on a rational grid:” $$x \fber*[n-1][n] y \implies \rho(x,y) \in \mathbb{Q}^{d}.$$

To this end pick a Borel linear ordering $ \prec $ on $ X $. For the base of construction we set $ \mathcal{C}_{0} = \mathcal{C} $; let $ s_{0} : \mathcal{C}_{0} \to \mathcal{C}_{0} $ be the Borel selector that picks the $ \prec $-minimal element within $ \fber[0][0] $-classes, and define $ h_{0} : \mathcal{C}_{0} \to B(1/2) $ to be $$h_{0}(x) 