Why can saying no be cheap when saying yes cannot?

There is a familiar idea in cryptography: some things are hard to make and easy to check. A signature takes real effort to forge and almost none to verify. People assume that shape is always available.

The shape depends on a check that does not redo the work. A signature has that check. A leg, the long run of work inside the object, does not. There is no short summary of a leg that lets you inspect the result and pronounce it genuine without paying. Confirming that an object came from a particular sequence of work means doing enough of that sequence again. That replay is the only way to say yes.

Saying no can stop sooner. A wrong length, a malformed header, a digest that does not match the bytes it claims, or a commitment that does not sit where it says in public time: those inspections fail before the replay starts. The public order those commitments sit in is the subject of How do you prove when the work happened? The inspections are allowed to say no. They are never allowed to say yes. A genuine object would have passed them on the way in, so failing one is enough to refuse. Passing all of them still leaves the work unconfirmed.

So the shape of the cost is not maker against checker. It is a cheap refuse against a full replay.

The cheap refuse says almost nothing. It does not mean the object is fake. It means this one did not survive an inspection a genuine object would have passed trivially. The long look is the full replay: the legs the check asks for, taken again, to establish the leg results the object claims.

The long look only ever runs on something that already looks real. Everything that costs almost nothing has had its chance to refuse first. By the time a verifier commits to taking those legs again, the full replay is the remaining check, not a toll on arrival. Yes is rare, because most arrivals never earn the long look.

What the replay establishes is a measurement: this object is what that sequence of work produces. Verify measures the object. Admit is the operator's policy about whether that fact is enough. Neither substitutes for the other. Where the evidence for that fact sits is Can you check a stranger without asking anyone?