When Thoughts Become Memory: Unifying In-Context Learning and Fine-Tuning

We are the first to give a certified rule for whether unseen data belong in a pretrained model's context or its weights, built on an exact law for their information value. That value, the code length saved on held-out queries, equals in expectation the bits stored about the query, plus a prior repair, minus a decoding loss, for any model, architecture, dataset and task. The law's ceilings certify the choice from one measurement, and an anytime-valid test makes it from held-out queries, with error controlled whenever it stops. Every certificate the ceilings issue on a trained transformer, TabPFN, an occupancy network and a stereo network is correct. The test picks TabPFN over XGBoost and a trained surrogate over an in-context learner on airfoil pressure, calls the better of GPT-2's cache and fine-tune in 61,948 of 64,000 query streams over 32 comparisons, and errs in at most 3.3% of null streams. The law also diagnoses when learning in context helps and when it does not. Deleting part of a context is held-out validation of that part in any model, and adding parts one at a time traces the learning curve in context: one patch that sees a shape feature cuts an occupancy network's code length there a hundredfold, while eight that miss it nearly double it, and one camera that sees the surface beats four in a stereo network. Ranked by this label-free valuation, a kriging learner's three best support cars beat four drawn at random, and on a yes-or-no claim the valuation recovers a published evidence adjudicator, with AUROC 0.91.