= Solution
To prove the <vanishing component theorem for a Lorentz four-vector>, first suppose $Y^0=0$ in every <inertial frame>. An infinitesimal boost along spatial direction $i$ gives $Y'^0=Y^0-\beta Y^i+O(\beta^2)$. It too vanishes for every $\beta$, so each $Y^i=0$. If instead a specified spatial component vanishes in every frame, spatial rotations first imply that all three spatial components vanish in a chosen frame, and a boost in the specified direction then forces $Y^0=0$. Therefore
$$
\boxed{\text{one specified component zero in every Lorentz frame}\ \Longrightarrow\ Y^\mu=0.}
$$
A physical consequence is that <conservation of energy> in every <inertial frame> implies full <four-momentum conservation>: apply the result to the difference between outgoing and incoming total <four-momenta>.
The statement fails for general <tensors>. A nonzero antisymmetric rank-two <tensor> has $F^{00}=0$ in every frame, because antisymmetry is preserved. For an off-diagonal component, the nonzero <Minkowski metric> is another counterexample: $\eta^{01}=0$ in all orthonormal <Lorentz frames>, while $\eta$ is certainly not zero. The transformation law can preserve structural zero components without forcing the entire <tensor> to vanish.
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