Concatenation and limit control for minimal-walk traces

If $\alpha =\beta$ or $\beta =\gamma$, one upper and lower trace is empty by [F1], so both concatenation identities reduce to equality with the other trace. Hence suppose $\alpha <\beta <\gamma$ and $L(\beta ,\gamma )<L(\alpha ,\beta )$.

Let $\delta$ be a nonzero limit and $0<\xi <\delta$. The first lower-trace value for the walk from $\delta$ to $\xi$ is $\max (C_{\delta }\cap \xi )$, and all later values are running maxima containing that first intersection. Hence $\min L(\xi ,\delta )=\max (C_{\delta }\cap \xi )$. The same equality is harmless at $\xi =0$ under the explicit zero convention, although limits only concern a final tail.

Every member of $L(\alpha ,\beta )$ is below $\alpha$, so the separation hypothesis puts every member of $L(\beta ,\gamma )$ below $\alpha$. If $\zeta$ is a node of $\mathrm{Tr}(\beta ,\gamma )$, the running maximum that records $C_{\zeta }\cap \beta$ occurs in $L(\beta ,\gamma )$ or is bounded by a later recorded maximum. Thus $C_{\zeta }\cap \beta \subseteq \alpha$, and therefore $C_{\zeta }\cap \alpha =C_{\zeta }\cap \beta$. In particular $C_{\zeta }$ has no point in $[\alpha ,\beta )$, so $\min (C_{\zeta }\setminus \alpha )=\min (C_{\zeta }\setminus \beta )$.

Given $\eta <\delta$, cofinality of $C_{\delta }$ gives $c\in C_{\delta }$ with $\eta <c<\delta$. Since $\delta$ is a limit, ${\xi }_0=c+1<\delta$. Whenever ${\xi }_0<\xi <\delta$, one has $c\in C_{\delta }\cap \xi$, and step 1.2 gives $\eta <c\le \min L(\xi ,\delta )<\delta$. This is exactly the ordinal-limit assertion in clause 2.

Step 2.1 says that the walk aimed at $\alpha$ makes exactly the same choices as the walk aimed at $\beta$ until it reaches $\beta$. From that node onward its recursion is the walk from $\beta$ to $\alpha$. This proves the asserted upper-trace concatenation, with no repeated $\beta$ because the first trace excludes its terminal point and the second includes its starting point.

Along the first segment, step 2.1 identifies every initial intersection and hence every running maximum with the corresponding value in $L(\beta ,\gamma )$. All those values are below every value of $L(\alpha ,\beta )$. Consequently, after the walk reaches $\beta$, taking running maxima for the continued walk produces exactly the values of $L(\alpha ,\beta )$; no earlier value suppresses or changes one. The lower trace is therefore the ordered union $L(\beta ,\gamma )\cup L(\alpha ,\beta )$.

Steps 3.1 and 4.1 prove the two concatenation identities, including the endpoint reductions in step 1.1, and step 2.2 proves the exact limit-target control from step 1.2.