  We derive simple closed-form estimates for the test risk and other
generalization metrics of kernel ridge regression (KRR). Relative to prior
work, our derivations are greatly simplified and our final expressions are more
readily interpreted. These improvements are enabled by our identification of a
sharp conservation law which limits the ability of KRR to learn any orthonormal
basis of functions. Test risk and other objects of interest are expressed
transparently in terms of our conserved quantity evaluated in the kernel
eigenbasis. We use our improved framework to: i) provide a theoretical
explanation for the "deep bootstrap" of Nakkiran et al (2020), ii) generalize a
previous result regarding the hardness of the classic parity problem, iii)
fashion a theoretical tool for the study of adversarial robustness, and iv)
draw a tight analogy between KRR and a well-studied system in statistical
physics.
