That's what I guessed, but you said "So we abandoned this idea," implying that the rest of the article was completely different, while the rest of the article was about still proof lengths, just smoothing thresholds into weights, so I became skeptical of the guess. I don't have any suggestions for how to talk about false starts and how they relate, but I think it might be useful for insight into my confusion. (Actually, in this particular case, I do have a suggestion, which is to use the term "proof length" much earlier.)