What does this equation mean?

Every mechanism described above shares a single underlying structure, and naming it precisely is more useful than treating “safety” and “usefulness” as opposed virtues balanced by instinct. A classifier, or a trained refusal policy, reduces to a decision rule: given some score s(x) computed from a request — and, for output classifiers, a response — block if s(x) exceeds a threshold $\tau$ . Two error rates follow directly from that single rule: $\text{FPR}(\tau) = P(\text{block} \mid \text{benign}), \qquad \text{FNR}(\tau) = P(\text{allow} \mid \text{harmful})$. Over-refusal is the false positive rate: benign requests wrongly blocked. Under-refusal is the false negative rate: harmful requests wrongly allowed through. Because both are read off the same score against the same threshold, they move in opposite directions together — raising $\tau$ to let more borderline traffic through necessarily lets in more of the genuinely harmful traffic sitting just past that border, and lowering it to catch more harmful traffic necessarily catches more benign traffic that merely resembles it. This is a simplification of what Anthropic’s own cascade architecture actually does — a two-stage system with a cheap screening threshold and a separate, more careful second threshold is not fully captured by one scalar $\tau$ — but the simplification exposes the real constraint underneath the engineering: nothing described in this article moves one error rate without moving the other, and any published number for one without the other is only half a measurement.