Closed-form inversion of the local-volatility function from observed call prices: σ_loc^2(K, T) = dfracpartial C/partial T + r K,partial C/partial Ktfrac12 K^2,partial^2 C/partial K^2. Equivalently restated in implied-vol space (Gatheral). The 1-factor Markovian completion of an arbitrage-free option surface, exact by construction but a poor forward-vol predictor — local-vol future skew flattens, opposite to empirical behaviour.