Theorem. Finite archive selection bound [ftip-00ET]
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    Under the archive score definition, let \(a^\star \) maximize \(J\) over a finite nonempty archive \(\mathcal A_D\). Let \(\delta \in \mathbb R\) with \(\delta \geq 0\), and let \(\hat a\in \mathcal A_D\) satisfy \(J(\hat a)\geq J(a^\star )-\delta \). Then

    \[0\leq J(a^\star )-J(\hat a)\leq \delta .\]

    Proof. 

      Membership of \(\hat a\) in the same archive gives \(J(\hat a)\leq J(a^\star )\) by maximality. Rearranging the approximation inequality gives the upper bound.

      This is a finite selection statement under the same evaluation law; it is not an optimizer-convergence or generalization theorem.