The minimal-walk functions are coherent and finite-to-one

Fix $n<\omega$ and set $D=\{\alpha <\beta :e_{\beta }(\alpha )\le n\text{ or }{e}_{\beta }(\alpha )\mathrm{\ne }{e}_{{\beta }^{\mathrm{\prime }}}(\alpha )\}$. We prove that $D$ has no limit point at or below $\beta$.

Let $0<\delta \le \beta$ be a limit ordinal. The two traces $\mathrm{Tr}(\delta ,\beta )$ and $\mathrm{Tr}(\delta ,{\beta }^{\mathrm{\prime }})$ are finite. Local finiteness makes each $C_{\zeta }\cap \delta$ finite for a trace node $\zeta >\delta$. Choose ${\delta }_0<\delta$ above every member of all these intersections, and let $N$ be the maximum of $n$ and their finitely many cardinalities. Cofinality of $C_{\delta }$ permits enlarging ${\delta }_0$ so that $\mathrm{\mid }{C}_{\delta }\cap \alpha \mathrm{\mid }>N$ whenever ${\delta }_0<\alpha <\delta$.

For ${\delta }_0<\alpha <\delta$, no trace node above $\delta$ has a $C$-point in $[\alpha ,\delta )$. Hence the walks toward $\alpha$ first follow the walks toward $\delta$ and then the walk from $\delta$ to $\alpha$; this is the same splice calculation as [F2]. Moreover, every local weight on either upper segment is its stabilized value $\mathrm{\mid }{C}_{\zeta }\cap \delta \mathrm{\mid }\le N$, whereas the lower segment contains the weight $\mathrm{\mid }{C}_{\delta }\cap \alpha \mathrm{\mid }>N$.

Taking the maxima in [F1] therefore gives $e_{\beta }(\alpha )=e_{\delta }(\alpha )=e_{{\beta }^{\mathrm{\prime }}}(\alpha )>n$ for every ${\delta }_0<\alpha <\delta$. Such $\alpha$ is not in $D$, so $\delta$ is not a limit point of $D$. Zero and successor ordinals are not limit points from below, so $D$ has no limit point at or below $\beta$.

If $D$ were infinite, its well-order would recursively give a strictly increasing $\omega$-sequence from $D$. Its supremum is a nonzero limit ordinal $\delta \le \beta$ and every final segment below $\delta$ meets $D$, contradicting step 3.1. Hence $D$ is finite.

The set $\{\alpha <\beta :e_{\beta }(\alpha )=n\}$ lies in $D$, so every fiber of $e_{\beta }$ is finite. Taking, for example, $n=0$, the disagreement set between $e_{\beta }$ and $e_{{\beta }^{\mathrm{\prime }}}\upharpoonright \beta$ also lies in $D$ and is finite.

Step 5.1 proves finite-to-one behavior and coherence simultaneously. It also covers $\beta =0$, where the domain and disagreement set are empty, and $\beta ={\beta }^{\mathrm{\prime }}$, where disagreement is empty.