Always-valid p-value

An always-valid p-value is a p-value defined for every moment at once, so you can watch it fall as data arrives and act the instant it crosses your line — no penalty for looking. A classical p-value comes with fine print: it is only honest if you decided the sample size in advance and looked exactly once. Peek early, peek often, or stop when it dips below 0.05, and the number quietly lies. An always-valid p-value removes the fine print.

For monitoring, this is what lets a threshold actually mean what it says. Set the alarm at, say, 0.01 and the guarantee holds however many times you refresh the page, because the validity was built to survive continuous checking.

Mechanically, an always-valid p-value is just the reciprocal of a running e-process, capped at one. That is why the same object powers both an alarm (the e-value) and a familiar p-value — two views of the same anytime-valid evidence.