Paper I gave the one behaviour: a toroid resists change in current, ignores steady flow. Now watch that single trick do three jobs — pass the DC, block the wiggle, choose which changes pass, and send a signal across a gap with no connection at all — each one the magnetic mirror of a capacitor job.
The dual of the coupling cap. Put a toroid in series with a signal and it does the mirror of what the cap did: it passes the part that isn't moving and fights the part that is. Steady current (DC, the supply rail) sails through — to DC the coil is wire. Fast wiggles (ripple, noise, RF) hit its rising impedance and are choked off. That's a choke: the part in every power supply that lets the clean DC through and strands the noise. Pair it with a resistor and the dividing line becomes tunable — the LR filter, cutoff fc = R/(2πL).
The cap stripped the bias and kept the message. The toroid keeps the bias and strips the wiggle — the same corner, walked the other way. One is the other's mirror.
Here is the dual of Maxwell's gap. The cap's signature was displacement current: a changing electric field carrying a signal across empty space, no charge crossing. The toroid's signature is its exact mirror — Faraday's law: a changing magnetic field carrying a signal across a gap, no current crossing. Wind a second coil on the same ring. A changing current in the first makes a changing flux in the core; that changing flux induces a voltage in the second — v₂ = M·di₁/dt — and the two windings are not electrically connected at all. The message leaps the gap on the field, and only the change makes it across.
No current crosses between the coils. The changing field carries the message — Faraday, the exact mirror of Maxwell's displacement current. The wire between was never needed; the change was.
v₂ = M·di₁/dt and the turns-ratio law v₂/v₁ = N₂/N₁ are exact; the front-end and loading are the engineering not drawn.
Four instruments, one equation. The core held a state; the inductor proved voltage follows the slope; the choke passed the steady and stripped the wiggle; the LR filter chose which changes; and the transformer carried a message across a gap with no connection. Every one was v = L·di/dt pointed at a different job — the magnetic mirror of David's communications kit.
Next: the cap turned to face time and the world (memory, sensing); the toroid does the dual — energy held against the leak, and the world's current written straight into the flux. Then the coldest loop, and the marriage.