The Mathematics of Market Impact and Order Execution: Impact Kernels, Feasible Policies, and Execution Certificates

An execution policy must complete a parent mandate while choosing quantities, order instructions and destinations under uncertain liquidity. This monograph connects that operational problem to impact models and conditional bounds on execution cost. A controlled, tagged order-book model preserves the parent ledger, live-order reservations and the trader's information. Feasible signature policies admit exact quadratic reductions in an exogenous setting; partially observed endogenous books require separate planning and observation-reduction certificates. The structural analysis covers impact kernels, manipulation, nonlinear size laws, cross-impact, information value and venue allocation. The control analysis separates physical feasibility, approximation error, model error and numerical optimization error. A baseline-relative extension answers a further deployment question: how much of a proposed schedule change can be accepted without increasing cost under any law in a declared uncertainty family? For diagonal impact uncertainty and ellipsoidal linear charges, the worst-case comparison along a feasible schedule segment is an explicit quadratic. Its minimizer gives both an acceptance fraction and a guaranteed saving. A capacity condition preserves completion at each update. A joint fill–completion calculation shows how dependence alone can reverse a routing decision while preserving both marginal laws. A discrete implementation layer adds reservation-preserving order updates, robust completion masks and exact integer allocation. Finite-sample row coverage and a policy-transfer bound separate calibration uncertainty from the loss of the emitted implementation. Reproducible synthetic experiments, including exact rational cases, test the identities and certificates. The results concern supplied models and explicit uncertainty bounds. They do not establish market calibration, counterfactual identification or live-market outperformance.