Invariance of reunion ages under clock resynchronization ($s=t+\kappa x/c$)

Consider a change of distant-time convention $s=t+\kappa x/c$, where $\kappa$ is constant and $|\kappa |<1$. This changes time labels and simultaneity slices while leaving the world lines and the metric tensor as geometric objects unchanged. In one spatial dimension,Substitute $dt=ds-\kappa dx/c$ into the Minkowski interval to obtain the formula. For a timelike curve, the proper time calculated from this transformed metric tensor equals the old value. At shared departure and reunion events with $x=0$, the labels also obey $s=t$. Thus a distant synchronization convention cannot alter the twin paradox's reunited clock readings. It would be an error to change time labels while retaining the old diagonal metric components.