Magnetise iron, then switch the field off — and it stays magnetised. That leftover is remanence: the iron REMEMBERS the field it felt. Drive it back and forth and the state traces a loop, not a line: for the same field there are two answers, depending on where you came from. The loop encloses an area, and that area is energy lost to the past. This is the eponym — the shape of every gap that won’t close. Slide the drive and watch the state lag.
Hysteresis: the state B lags the drive H, so B(H) is TWO-VALUED — an up-sweep branch and a down-sweep branch. At H=0 the two branches sit apart: that gap is the REMANENCE (the field that remains when the cause is gone). The enclosed loop area equals the energy dissipated per cycle, ∮H dB > 0. There is no single fixed point: the present state depends on the whole history of the drive. Model: B₁₀₀=tanh((H−Hᶜ)/a) rising, tanh((H+Hᶜ)/a) falling. A fail-loud self-check throws unless the branches differ at H=0 and the loop area is positive. ◆ real physics, node-verified.
The tanh two-branch model is the idealisation (real ferromagnets have Barkhausen jumps and minor loops); remanence, coercivity, and loss-as-area are exact features of it. REMANENT is named for the first of these.