Papers I and II moved a current you supplied. Now the toroid turns to face time and the world: it holds energy across time in its field (the dual of memory), and it lets the world's current write straight into its flux (the dual of sensing). And doing so reveals the half of the dual law the series quietly held still.
The same confession David made, in the mirror. Every instrument so far ran on v = L·di/dt — true only when the inductance L is constant. The honest, complete voltage is the rate of change of flux linkage, Φ = L·I, and by the product rule that's two terms:
v = L·di/dt + I·dL/dt
The first term is the whole story when the core sits still and only the current moves — Papers I and II. The second term is what happens when the inductance changes: move the core, change its reluctance, let an outside current thread the ring. That's how a toroid becomes a current sensor, a clamp meter, a magnetometer — the world doesn't change the current you drive, it changes the coupling, and dL/dt (or a neighbour's di/dt) carries the message in. The exact mirror of the cap's V·dC/dt.
The dual of memory. The cap's gift in Paper III was holding a voltage; the toroid holds a current — its field is inertia, and it will fight to keep the current flowing even after you stop driving it. Charge the choke, then open the source: the field collapses to push the current onward (this is the flyback, the boost converter, every switching supply). But the ideal coil was a polite fiction — real windings have resistance, and the stored energy bleeds away as heat: I = I₀·e^(−t·R/L). The dual of DRAM's leak — a held current that forgets, unless something keeps re-driving it.
The ideal coil would hold its current forever. The real coil forgets as heat — so a magnetic memory, like the electric one, is not storage but a refusal to stop re-driving.
I = I₀·e^(−tR/L) (the L/R time constant), slowed enormously to watch. Real chokes also lose energy to core hysteresis and eddy currents, and saturate at high current (L falls, the field can hold no more). A superconducting coil is the limit where R→0 and the current really does persist — which is exactly where Paper IV goes.
Now the second term, made physical — the mirror of the condenser mic. There, the world moved a plate and changed C. Here, an outside wire carries a current you don't control, threads it through the ring, and its changing field writes a flux in the core — inducing a voltage in the winding with no electrical connection to the measured wire at all. That's a current transformer (the clamp meter on an electrician's belt) and a Rogowski coil: the toroid is an ear for current. The world's di/dt in the wire becomes your signal in the coil.
You never drove the coil. The world's current threaded the ring and the toroid turned it into a voltage. That is the second term of the dual law — and it is how a passive ring becomes a sense for current.
the world's changing current writes the flux, and that induces v — is exact Faraday; the burden and calibration are the engineering not drawn.
Across three papers the same passive ring spoke in three directions — the exact mirror of the capacitor — and it never stopped being a change-resister. It just kept finding new kinds of change to fight or to feel.
v = L·di/dt. the seed.I·dL/dt term.The full dual law, v = L·di/dt + I·dL/dt, was always the whole toroid. The first term moves the currents we make; the second lets the world speak in flux. A toroid was never a lump of wound wire — it was an instrument for resisting change and for feeling it, wherever it came from: the wire, the gap, the field, or a stranger's current threading the ring.
If the series goes on as the cap's did: the coldest loop — a superconducting ring where R→0 and the current persists forever, paired with a junction to make a flux qubit, the dual of the transmon. And then the two halves, married.