Sequential E-Value Merging Under Bounded Changes in Input Validity

Several sources may produce sequential evidence even though only an unknown source is conditionally valid at each time. We study this problem when the valid source can change predictably at most a prescribed number of times. Other sources are unrestricted, and dependence is arbitrary. For predictable affine betting factors shared across candidate source paths, we prove that the minimum accumulated wealth over all the allowed paths is an e-process. If every betting fraction is strictly positive, each fixed-time rule is admissible among all the finite-valued nonnegative Borel rules under pointwise domination. The proof constructs, through every prescribed data history, a finitely supported null law that attains expectation one; no continuity assumption is imposed, and histories containing zeros are included. A dynamic program evaluates these envelopes using a state consisting of the current source and the number of changes. Mixtures of betting strategies require a different analysis. We give a finite-state pruning procedure that preserves optional-stopping validity and can strictly improve the unpruned mixture envelope. Exact counterexamples show that ignoring the change budget can destroy validity and the safe pruning procedure need not yield an admissible rule. A final-step refinement compares feasible continuations across different current sources. An alternating-source example gives an exact pointwise sharp measure of the evidence that is lost when the allowed number of changes increases.