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

An execution policy chooses quantities, instructions and destinations under uncertain fills and a completion mandate. This monograph develops conditional certificates that compare a feasible implemented policy with the optimum in a stated information class, with all errors expressed in execution cost. The main construction connects a controlled, tagged order-flow and clearing model to a conserved parent ledger, observable feedback policies and bounds on the effects of primitive model error. For an exogenous signal augmented by exponential-kernel coordinates, a completion-constrained signature rate policy has an exact quadratic objective and an explicit kernel, moment and optimization-error certificate. For endogenous partially observed books, a bounded-loss transcript argument gives an existential policy-class approximation result under finite actions and exact feasibility masks. A separate binary hidden-liquidity construction makes the global planning gap computable by a lower relaxation and physical hidden-state evaluation of a finite controller. Conditional action-law envelopes extend that certificate to policies using richer reports than the controller retains. Their backward recursions bound observation-reduction loss without assuming a Markov summary; binary posterior intervals reduce each robust step to two endpoint evaluations. Singleton envelopes establish exact value sufficiency; action comparisons can certify optimal decisions even when the conditional value is not determined by the summary. Structural foundations include analytic cost expansions, quartic corrections, size-law crossovers and the relation between permanent cross-impact and inventory-loop area. Information bounds, marginal-cost routing, quantity reservations and distinct NYSE and Nasdaq auction representations connect these results to execution decisions. Synthetic experiments check the certificates, including exact rational cases with a strictly positive cost of discarded reports. The claims concern supplied models and explicit conditional bounds. They do not establish market calibration, compact signature representations, or live-market outperformance.