Hidden-State Privacy Has an Empty Middle

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 Σ^\star_{diag}(K) = (2K/d),diag(1/F_{ii}) is the unique minimax-optimal diagonal mechanism at first-order KL budget 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 100% top-1 under the adaptive Mahalanobis attacker, and a full-trajectory sequence inverter recovers 94% of clean GPT-2 prefixes but 0% under Σ_{diag}. A split-memory transformer trained from scratch reaches G_{Mah} \in [20, 33] at 90M and maintains a 6--24\times advantage over same-budget GPT baselines from 30M to 1B at a fixed-token language-modeling loss penalty; pretrained models top out at 9.3. These results reframe hidden-state release from mechanism-design within the Gaussian class to architecture or release co-design.