Quoting Under a Misspecified Fill Law: Order-Book Queues, Adverse Selection, and Inventory-Control Certificates

A market maker chooses prices and messages using an uncertain law of fills, queue movements and post-fill price changes. This paper connects errors in that law to the value of a deployed quoting policy. A marked-wealth identity separates spread capture, carried-inventory drift and simultaneous fill–price jumps. A finite event model retains tagged queue priority, partial fills, inventory reservations and orders that remain executable while cancellation is pending. A finite-stage latency approximation has an explicit leading value error and remainder. The control certificates account for primitive model errors, continuation values and numerical residuals. Smooth quote adjustments admit quadratic bounds; discrete messages require action gaps and state-dependent occupation bounds. A baseline-relative extension turns an enclosure of the current policy's value into a selective improvement rule. It changes a message only when its advantage is certified nonnegative throughout the uncertainty set, and supplies a computable lower bound on the resulting discounted gain. A marginal-coupling calculation gives a sharp participation interval when fill and price-jump dependence is unknown. A finite-message implementation layer encloses rounding and score errors, retains cancellation tails and states a joint-event calibration contract. The monograph's finite-sample interface transfers supported calibration uncertainty and compiled-policy error to the same objective. A continuing 74-state queue-and-inventory experiment verifies both loss and improvement certificates with rational outward enclosures and independent policy evaluation. Additional lower bounds show the cost of omitting tagged rank and the difficulty of calibration near a replacement boundary. These are conditional mathematical guarantees, supported by synthetic calculations; they do not establish empirical fill calibration or trading profitability.