9 · Expose the doubles with energy

Ten pin locations; the slider doubles some of them. The energy is measured, the count is measured, and the trick deduces a floor on the clean pins.

Try this. Double more pins and watch the energy climb: a double pays four for its two of count. The deduced floor, twice the count minus the energy, tracks the true number of clean pins exactly.

In the paper the count and the energy come from the primes, and the scalar trick becomes the rank-trace inequality so it survives saddles and finite microphones. The arithmetic shown here is the trick's soul: a double pays four for two.