My test is if I can feel a bump on the cut area after the tire has been inflated to full pressure.
If there's no bump, I'll stick a piece of duct tape on the inside like 10 wheels said.
If there's a bump, however small, like dsbrantjr, I assume a cord (or more) has been cut so I replace the tire while it's convenient rather than wait for it to fail on the road.
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My greatest fear is all of my kids standing around my coffin and talking about "how sensible" dad was.