Language generation in the limit is a theoretical framework for studying how a generator can learn to produce new valid strings from a stream of positive examples. In this model, an adversary chooses an unknown language from a countable family and enumerates its elements in an arbitrary order, while the generator must eventually output only elements of the language that have not yet appeared in the enumeration. Reliable generation is thus formalized through two eventual guarantees: validity and novelty relative to the observed data. To further quantify the breadth of the generator's outputs, Kleinberg and Wei (FOCS 2025, STOC 2026) introduced lower density as a measure of output coverage. Given an order representing the importance or relevance of possible outputs, lower density is the asymptotic lower bound, as $n$ grows, on the fraction of the first $n$ elements of the target language that the generator outputs before they appear in the data. Kleinberg and Wei showed that $1/2$ is the optimal lower-density guarantee for deterministic algorithms. We develop a simple and unified framework for obtaining optimal lower-density guarantees. We first give a deterministic algorithm that recovers the optimal guarantee of $1/2$ with a significantly simpler analysis than prior work. We then demonstrate the flexibility of our framework through two extensions. First, against an oblivious adversary, randomization raises the optimal guarantee to $1-1/e$. Second, for any finite collection of orders, the optimal deterministic and randomized guarantees can be achieved simultaneously with respect to every order, so accommodating multiple notions of importance or relevance entails no loss in the optimal guarantee.
At a fixed privacy budget, curvature is exactly the object the budget makes unavailable. A recent answer estimates it from synthetic probes, at neither privacy cost nor distribution shift, and validates a probe by the eigenvalue decay it reproduces. That test reads the spectrum, not the basis, and what a probe reconstructs is unknown. This paper gives a random-matrix account of data-free preconditioning, organized by one dichotomy. Of the two Kronecker factors, the error-signal factor $ G $ and the architecture-determined bulk of the activation factor $ A $ are reconstructible without data up to an explicit error-energy bound; only the data-induced spikes of $ A $ are not. We verify that bound with zero violations along a full training trajectory, though it stays one to nearly three orders of magnitude above the gap it bounds. We locate the failure boundary at a Baik-Ben Arous-Péché threshold: a bulk-only probe suffers an exact damage $s^2/(1-s^2)$ from a spike of strength $s^2$, invisible once its amplitude exceeds the scaled-bulk threshold $( \sqrt {c}/(1+ \sqrt {c}))^{1/2}$, $c = d/n$. The test in use is basis-blind: rotating only the probe eigenbasis at a near-exact matched spectrum inflates the residual by $2.3 \times $ to $89.8 \times $, and a closed-form volume law fixes the within-spectrum ordering of the critical volume while a two-oracle construction proves that no functional of the probe spectrum fixes its magnitude. Over forty seeds on three modalities, per-layer tuning does not transfer to accuracy, no data-free arm beats the flat probe, public-data curvature does on both modalities tested, and the ordering inverts on PathMNIST under real DP noise. The account yields a probe-design protocol and falsifiable predictions for the regimes where data-free curvature can be trusted.