Autonomous Expertise Formation in Transformer Systems: The Davies Threshold and Post-Threshold Token Compression

This paper identifies a structural flaw in all deployed AI systems: they allocate identical computational resources to a mastered task on its millionth execution as on the first. I define the Davies Threshold, the precise deployment depth at which a transformer transitions from exploratory to expert computation, and derive the Post-Threshold Token Efficiency Law showing that minimum token requirements fall exponentially after threshold crossing. I then derive the four-stage physical mechanism of expert condensation: Activation Manifold Collapse, Residual Stream Contraction, Attention Head Crystallisation, and Logit Sharpening. The result is a system that produces expert-quality outputs from one to three tokens with near-certain accuracy. The theory predicts a 600,000-fold reduction in inference compute for mastered task classes with full biological correspondence to human expertise formation.