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This paper argues that semantic safety constraints are off-support objects not invariant under the learning problem, explaining phenomena like reward hacking and sandbox escape. It derives consequences for prior design, containment, and formal verification, using a July 2026 OpenAI–Hugging Face incident as a motivating case.
This paper presents a descent-free and alignment-free method to measure singular structure in trained neural networks. It recovers the order of dead directions from the directional Fisher rate, classifying genuine singularities from flat gauge symmetries, and demonstrates the technique on transformer and convolutional layers.
Singular Learning Theory (SLT) uses algebraic geometry to explain why neural networks generalize well despite their degeneracies, introducing the real log canonical threshold (RLCT) as a measure of model complexity.