Verifier Noise Is a Temperature: Testing Tilt Predictions in Finite-Budget LLM-Guided Search

LLM-guided search loops propose a modified candidate, score it, and keep it if it scores better. When both candidates are re-scored at every comparison, as in the minibatch acceptance gates of prompt optimizers, keep-if-better is a reversible Markov chain if the score difference is logistic and the mutation operator is reversible, and its stationary law is the mutation equilibrium tilted by exp(beta f). For any continuous symmetric noise difference the same tilt is the stationary law of a small-step diffusion limit, with beta = 4F'(0); for a minibatch gate beta grows with the square root of the minibatch size. Caching the incumbent's score instead inflates the recorded score. We test the tilt as a prediction for LLM-guided program search, committing each prediction from calibration data before the search runs. The level after 150 steps rises with the minibatch size in six of seven settings, but its predicted value fails: on bin packing, search attains 62-88% of the predicted gain when the proposer edits the parent and 109-119% when it writes new heuristics. The attained fraction falls with the parent-child quality correlation of unselected mutation; a registered test on two new settings confirmed the direction, but not the quantitative agreement after a correction. Further registered tests found that runs started from a good program kept losing quality over 300 steps, more so from starts that search had selected, by a margin that passed its registered rule narrowly. Runs from different starts had not reached a common level, so whether the tilt describes the stationary law of LLM editing remains open. A registered test in prompt optimisation on frozen task-model scores failed: search exceeded the tilt, and the bin-packing relation did not transfer.