5 · Tally a stretch four ways

Eight pin slots on a stretch of the strip. Click a slot to cycle it: clean on the line, doubled, or a mirror pair.

Try this. Make a few pins doubled and a few into mirror pairs, and watch which counters move apart. The theorems bound three of the counters against the first.

Doubled pins and mirror pairs are synthetic: every zero anyone has computed is clean and on the line. The counters here move exactly the way the paper's four counting functions do.