Each comb is one layer: a row of teeth with fixed strengths. As it turns, it rests on the sheet below, sums tooth strength times groove depth, and cuts a groove on the next sheet with that pressure. If the pressure comes out negative, the stylus lifts and nothing is cut. That lift is ReLU. Stack the combs and you have a convolutional network, cut ring by ring.
| Sheet | Crushed to 0 | Matches a full pass |
|---|
Blank the paper and watch the grooves settle outward. Sheet 2 is only right once sheet 1 is, so depth costs revolutions.
Nothing is kept but the input record. Click any groove on the disc to ask for it; each comb regrows the grooves it needs from the record, one tooth at a time, then forgets them.
The trick inside Williams' proof, run exactly. Three registers are already full of someone else's grooves. We need P(v) for a polynomial P, and we have nowhere clean to write v. So we write v on top of the junk, rotated through m = 8 phases, evaluate P each time and add the results. Every power of the junk picks up a phase that sums to zero around the wheel. Only P(v) survives, and each register is handed back exactly as it was. Arithmetic is in the finite field of 257 elements, so nothing is rounded.
| phase i | ωⁱ | P sees | P(…) | running sum |
|---|
The same groove, three schedules. Numbers follow the lathe settings above, for one groove on the outermost sheet.
| Schedule | Paper held at once | Presses for one groove | What it costs you |
|---|---|---|---|
| Keep every sheet | – | – | Paper grows with depth. Every intermediate groove sits on its own sheet. |
| Regrow, depth first | – | – | Presses grow as teeth to the power of depth. This comb's rule is a running sum, so each level holds one running number. |
| Cook–Mertz, general rule | – | more still | For a rule that needs all its inputs at once, each level would hold d values of b bits. Cook–Mertz folds those levels into one shared register set, plus a small bookmark per level. Each node's rule must first be rewritten as a polynomial, and for a b-bit rule that costs about 2ᵇ. |
The combs never change. Williams changes the schedule: which grooves stay on paper and which are cut again when needed. Every saving in paper is paid for in turns of the lathe.