= Infinite-reservoir stem forcing
{title2=$\mathbb P=\{(n,s,A):s\in\omega^n,\ A\in[\omega]^\omega\}$}
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 $k\geq K$ with $g(k)<s(k)$ form a <dense subset of a forcing order>. An infinite subset of $\omega$ always supplies a sufficiently large value.
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