The volatility input σ_imp that, plugged into the Black-Scholes formula, returns the observed market price: C^BS(S, K, T, r, σ_imp) = C^mkt. Solved numerically because BSM has no closed-form inverse in σ (Newton-Raphson on Vega, or Jäckel's 'Let's Be Rational' algorithm for tail-precision). The IV surface across (K, T) is the market's nonparametric encoding of jump risk, stochastic vol, and skew — the workhorse interpolation/extrapolation object on every options desk.