= Solution
\b[The ideal <slow clock transport synchronization> procedure reproduces <Einstein synchronisation> in an <inertial frame>, as a limiting result about <proper time>.] It does not determine distant simultaneity without assumptions about <clocks> and their <dynamics>. Begin with identical ideal <clocks> together at $x=0$, with matching rates and readings. Leave one there and transport the other to a laboratory position $L$ at small constant <speed> $u$, neglecting <gravity>. Use Einstein-synchronised laboratory coordinates to calculate the result, rather than assuming that the transported <clock> already displays coordinate <time>.
The journey takes laboratory <time> $T=L/u$. The transported <clock>'s <proper time> is $\tau=T\sqrt{1-u^2/c^2}$, giving the lag
$$
\delta=T-\tau=\frac{L}{u}\left(1-\sqrt{1-u^2/c^2}\right)
=\frac{Lu}{c^2\left(1+\sqrt{1-u^2/c^2}\right)}.
$$
Consequently
$$
\boxed{\delta\sim\frac{Lu}{2c^2}\longrightarrow0\quad(u\longrightarrow0).}
$$
The small <time dilation> accumulates over a long journey, but its quadratic dependence on $u$ dominates the journey's inverse dependence on $u$. Saying that the journey is arbitrarily long is therefore not an objection to this limit. For a nonuniform journey with $|v|\leq u$, $\delta=\int(1-\sqrt{1-v^2/c^2})dt$; a simple monotonic path of fixed <length> has the same vanishing bound. Acceleration stages can be arranged with vanishing displacement and <proper time> deficit, assuming the <clock hypothesis>.<On the Electrodynamics of Moving Bodies by Albert Einstein>[§4, transported clocks and the leading quadratic retardation]{ref}
This procedure establishes synchronization operationally for stationary laboratory <clocks>, once ideal-clock behaviour has empirical support. It does not show that every actual device is insensitive to transport: thermal shifts, <acceleration>-dependent mechanisms or differing <gravitational potentials> can spoil the idealization. Nor does the same construction define global synchronization for rotating laboratories, where path dependence is significant.
The conventionality question is separate. Resynchronize the same laboratory with $t_\kappa=t+\kappa x/c$, where $|\kappa|<1$. At the arrival event the transported <clock> approaches the reading $t$, whereas a stationary <clock> displaying $t_\kappa$ reads $t+\kappa L/c$. Slow transport has not eliminated that difference. The transformed <metric tensor> has an off-diagonal term, and the <proper time> calculation still gives the same physical <clock> reading: changing labels cannot change its ticks. This is the <invariance of reunion ages under clock resynchronization>, applied to transport comparisons.
<Slow clock transport synchronization> is thus a useful physical realization of a preferred conventional coordinate choice given <Lorentz-invariant dynamics> and ideal <clocks>. It does not independently prove that the one-way <speed of light> must be isotropic in every admissible synchronization. It supplies a reproducible convention with a strong physical rationale, rather than a convention-free observation of an absolute present.
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