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We prove a complementary Fisher-ball lower bound: every full-rank Gaussian release at $O(1)$ Fisher utility admits a direction whose Mahalanobis signal grows linearly in hidden width, ruling out uniform Gaussian safety in the class and matching the empirical empty middle. The diagonal inverse-Fisher release $\\Sigma^\\star_{\\mathrm{diag}}(\\mathcal{K}) = (2\\mathcal{K}/d)\\,\\mathrm{diag}(1/F_{ii})$ is the unique minimax-optimal diagonal mechanism at first-order KL budget $\\mathcal{K}$ and the only release with worst-attacker top-1 $\\le 0.001$ at every point of a 32 model-layer grid, but it sits on a privacy/utility edge rather than filling the middle. A generalized-eigen mechanism reaching $13\\times$ Pareto reduction under Euclidean retrieval collapses to $","title":"Hidden-State Privacy Has an Empty Middle","url":"https://arxiv.org/abs/2605.24042","vendor":"arxiv_cs_ai"},"summary":"arXiv:2605.24042v1 Announce Type: cross \nAbstract: Of $1{,}536$ Gaussian release covariances we tested for single-layer hidden-state privacy, zero achieve both moderate utility and moderate privacy against an adaptive retrieval attacker. We prove a complementary Fisher-ball lower bound: every full-rank Gaussian release at $O(1)$ Fisher utility admits a direction whose Mahalanobis signal grows linearly in hidden width, ruling out uniform Gaussian safety in the class and matching the empirical empty middle. The diagonal inverse-Fisher release $\\Sigma^\\star_{\\mathrm{diag}}(\\mathcal{K}) = (2\\mathcal{K}/d)\\,\\mathrm{diag}(1/F_{ii})$ is the unique minimax-optimal diagonal mechanism at first-order KL budget $\\mathcal{K}$ and the only release with worst-attacker top-1 $\\le 0.001$ at every point of a 32 model-layer grid, but it sits on a privacy/utility edge rather than filling the middle. 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