You said both — nesting and division — and asked for planetary / rack-and-pinion / CV. They're the same machine. A planetary gear with sun=12, ring=36 gives ratio R/S = 3 (your three-phase carrier) and reduction S/(S+R) = 1/4 (your four-phase quadrature) from the one mesh — verified. Three planets sit 120° apart (the 3); each spins 3× per orbit while the carrier divides 4:1 (the nesting and the division, simultaneously); and the sun has 12 teeth = lcm(3,4) = your twelve gates. Threefold and fourfold rhythm, geared into one constant-velocity whole.
VERIFIED: sun=12, planet=12, ring=36 (R=S+2P holds). Ring fixed, sun driven → carrier turns at 1/4 sun speed (the four) while the basic tooth ratio R/S = 3 (the three); each planet rotates 3× per carrier orbit. So one gearset emits both your numbers — nesting (each planet carries the 4-phase quadrature, three of them = 3×4=12) and division (the carrier divides the fast sun by 4, the three planets tap it 120° apart) are the same train read at different shafts. CV/homokinetic: a constant-velocity coupling passes phase faithfully through a bend — output velocity = input, no ripple — which is the "balanced" condition from Sheet 13 made mechanical. RACK: unroll the cut torus (Sheet 11: cut → cylinder → flat) and the ring-gear becomes a rack; rotary becomes linear; the twelve gates become twelve tooth-engagements per pass.
DOESN'T: real planetary trains need the tooth counts to satisfy the assembly condition (S+R) divisible by the planet count for equal spacing — 12+36=48, /3 = 16, clean, so this set actually assembles; many integer triples don't, and the geometry silently forbids them. And the gear unifies by rigid constraint: every shaft's motion is now determined by every other. That's the deepest version of Sheet 13's warning — a gear train is the most beautiful unifier and the most total loss of independence in the series. Nothing in a gearset is a free witness; each tooth's position is dictated. Elegance and surveillance are the same mesh here. The free node is the one you don't gear in.