The fourth fundamental circuit element — and its unmistakable fingerprint, the pinched hysteresis loop that always crosses through the origin.
Chua 1971 (postulated) · Strukov / Snider / Stewart / Williams, HP Labs, Nature 2008 (realized as a TiO₂ thin film)
A thin TiO₂ film has a doped (oxygen-vacancy) region of width w and an undoped region, total thickness D. Normalising the state to x = w/D ∈ [0,1], the device is a series resistor whose two limits are Ron (fully doped) and Roff (fully undoped):
Drive it with v(t) = A·sin(2πft) and integrate x each timestep (this page uses RK4). The current lags the voltage through the moving state — but because i = v/M and M is always finite and positive, i = 0 exactly whenever v = 0. That is the pinch at the origin: the signature no linear R, L, or C can imitate, and no combination of them can fake either.
Why it collapses at high f. The state excursion over one drive period scales roughly like Δx ∝ 1/f (a fixed charge budget spread over a shorter time moves the boundary less). As f → ∞, x barely moves, M is effectively constant, and the loop degenerates to the single straight line i = v/M — an ordinary linear resistor. Slide the frequency up and watch the lobes shrink toward zero.
REAL PHYSICS — Leon Chua's 1971 postulate of the memristor as the fourth fundamental two-terminal element (the missing φ–q relation, alongside R, L, C) is established circuit theory. Its defining pinched-hysteresis signature is a real, published criterion. The HP linear-drift model (Strukov et al., Nature 453, 2008) is a real physical model of the realized TiO₂ device, and it is exactly what this page simulates: M(x), i = v/M, and dx/dt = μvRoni/D² are implemented as written, integrated numerically (RK4), with the state clamped to [0,1]. The origin-pinch and the frequency→line collapse you see are genuine consequences of those equations, not painted-on decoration.
DESIGN / DISCLOSURE — A “ROOT0 memristor” is a disclosure that stands on Chua and HP. Nothing here is a fabricated part, a device I have built, or a granted patent. The simulation uses illustrative normalised units (Ron = 100 Ω, a scaled dopant mobility, D = 10 nm) chosen so the loop is legible — not a datasheet for a real component.
TOY FLAG — The pure linear-drift model has a known weakness: hard clamping at the boundaries. Real devices are better modelled with a window function (Joglekar–Wolf, Biolek) that softens the drift near x = 0 and x = 1. This page uses the plain clamp for clarity, so treat the near-rail behaviour as a teaching cartoon, not a boundary-accurate model.