= Solution
\b[<Spacetime> geometry gives an acceptable structural explanation of <length contraction>, provided the relation between material <rods>, measurement and the <metric tensor> is included.] It is not a force that squeezes a rod, and the mere existence of a quadratic form cannot guarantee how arbitrary matter behaves. The explanatory issue is whether a common geometrical constraint is sufficient for the phenomenon at issue, or whether a <constructive theory> of the rod is also required.
For an equilibrium <rod> with <proper length> $L_0$, use its <rest frame> $S'$, where its endpoint histories are $x'_A=0$ and $x'_B=L_0$. In a laboratory frame use
$$
x'=\gamma(x-vt),\qquad \gamma=(1-v^2/c^2)^{-1/2}.
$$
A laboratory measurement compares endpoints at the same laboratory <time>. Subtract the two endpoint equations at $\Delta t=0$ to obtain
$$
\boxed{L=\Delta x=L_0/\gamma.}
$$
The two frames measure different pairs of <spacetime events>. There is no contradiction in each assigning the other's moving <rods> smaller <lengths>; their <simultaneity> relations differ. This is neither an optical distortion nor an invariant compression diagnosed by an attached strain gauge. A photograph also involves unequal emission <times> and propagation of <light>, so it is not this simultaneous endpoint measurement.
The geometrical explanation has genuine unifying content. Once all suitable equilibrium <rods> measure the spatial distances induced by the same <Minkowski metric>, the result is independent of chemical composition. It relates <length contraction> to <time dilation> and <relativity of simultaneity>, rather than postulating separate mechanisms for every device. A structural explanation need not be a temporal causal story: the result compares descriptions of an existing <world tube>, not successive states of a rod being accelerated.
The main objection asks what warrants the italicized measurement premise. Matter on a mathematical <Minkowski spacetime> could obey equations involving an additional preferred <vector field>, as in <3>. Such matter need not form devices with the usual transformation behaviour. <Harvey R. Brown> and <Oliver Pooley> argue that a constructive understanding must attend to the <Lorentz-invariant dynamics> of matter.<Minkowski space-time: a glorious non-entity by Harvey R. Brown and Oliver Pooley>[§§3–5, critique of assigning constructive explanatory work to bare geometry]{ref} This objection has force against treating a bare <metric tensor> as an independently acting agent. It does not invalidate the endpoint derivation once the physical metric interpretation is established.
Acceleration introduces a different explanatory task. A rod pushed at one end can undergo stresses, heating and oscillations; its final <proper length> can change. A dynamical calculation is needed to determine whether it settles into an equivalent equilibrium state. The comparison $L=L_0/\gamma$ assumes a specified equilibrium <proper length>, and does not assert that every acceleration history preserves it. Geometry constrains the measurement of that state; material <dynamics> explain its formation and stability. Those explanatory roles can coexist without making either empty.
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