One Bit to Certify the Whole Update: A Closed-Form PAC-Bayes Certificate for a Single Post-Training Round

Post-training is rarely a single event: a deployed model is adapted again whenever the task in front of it moves, so what repeats is a round — a parameter state, a rule that fits an update to it from a little labelled data, and a new state to deploy. This paper asks what one round can be certified for. TaskLine-PB splits the round's labelled budget in two. The first half learns the whole update exactly as ordinary parameter-efficient adaptation would — nothing compressed, projected or discarded. The second is allowed one decision: the odds with which a stochastic predictor draws the state the round produced rather than the one it began from, and that predictor's class-balanced risk carries the guarantee. Relative entropy survives the map from the mixing coordinate to the parameter vector, so the complexity charged is that of the deployed state, not of a proxy; the best certificate over every posterior on the resulting line has a closed form, which at the two named ends reduces to a formula in two error counts; and the round is charged one bit, whatever the dimension of the update, with nothing left to estimate: no Monte-Carlo term, no quadrature, no free parameter. That bit buys a two-sided guarantee: the certificate exceeds neither single-branch certificate — never updating, updating fully — by more than that branch's relative entropy against the prior, and on the first this holds however badly the round's fitting data is corrupted. It is a theorem about the guarantee that can be stated, not a claim that the updated model is never the worse one. Sixteen labelled examples per class put every certificate on eight tasks of a frozen vision-language model below random guessing, at a cost of 0.0066 on average and 0.0144 at worst against a certificate of the fitted prompt alone; and under corruption of the fitting split the certificate spans 0.064 over a CIFAR-10 ladder on which the fitted-endpoint one spans 0.743. Nothing relates one round to the next: what is certified is a round, against the distribution its calibration split was drawn from.