Infinite-reservoir stem forcing ($\mathbb{P}=\{(n,s,A):s\in {\omega }^n,A\in [\omega {]}^{\omega }\}$)

A condition consists of a finite sequence and an infinite reservoir of allowed future values. A stronger condition extends the sequence, shrinks the reservoir, and takes all newly appended values from the old reservoir. Stem values may repeat. The union of the generic stems is an unbounded real over a model: for every ground-model function $g$ and every threshold $K$, the conditions whose stem already has some $k\ge K$ with $g(k)<s(k)$ form a dense subset of a forcing order. An infinite subset of $\omega$ always supplies a sufficiently large value.