Earliest arrivals and coordinate-varying examples

On a finite time-expanded DAG, earliest arrivals are obtained by the recurrence

including all waits and availability restrictions. Topological induction proves the recurrence exactly. The value is $+\infty$ when no path exists.

Two closed-form comparisons can be made without changing the prescribed local Minkowski geometry. In a one-dimensional regular rod coordinate $x$, the time reparametrization $\tau =a\log (1+u)$, $a>0$, gives

Therefore

This is time-varying, not spatially varying.

For a genuinely spatially varying coordinate speed, set $x=a\log (1+q)$, $q>-1$. Then

Both formulas follow by integrating the rod-distance bound; monotone null motion attains equality. They are coordinate descriptions of the installed regular geometry, not derived gravitational dynamics.