This paper develops local certificates for population-risk increments around a current model. For a local candidate set \(\mathcal D\), the certificate is a two-sided confidence band for \(P({\ell_{θ+v}-\ell_θ})\) over \(v\in\mathcal D\). As an application, the upper endpoint of this band yields a risk-controlled update rule: an update is accepted only when its certified upper endpoint is nonpositive; otherwise the current model is retained.

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