= Finite Yang-Mills action forbids nontrivial translation symmetry
A smooth <gauge field> of finite positive Euclidean <Yang-Mills action> cannot be invariant under a nonzero translation, even up to <gauge transformation>. The positive gauge-invariant action density would be periodic along that translation. A ball on which the density is bounded below by a positive constant has infinitely many disjoint translated copies, making the action infinite. Continuous translation invariance is a special case. Consequently a nonflat translation-invariant field satisfying the <Anti-self-dual Yang-Mills equations> on $\mathbb R^3\times\mathbb R$ is not a finite-action <Yang-Mills instanton>. Flat connections are the zero-action exception. Translations still act on instanton moduli, sending a localized instanton to a distinct centered configuration.
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