Pricing the Risk of Runtime Compression: Anytime-Valid Admission and a Served-Output Law for Compressed Serving State

Abstract

Runtime compression of serving state trades quality for capacity with no priced guarantee: systems adapt precision on load signals with no soundness statement, and certified approaches budget request-level risk by a union bound over a pre-declared event count. We show the union budget exhausts on every long request in a production serving stack (100% of requests), and replace it with an anytime-valid, physically accounted ledger whose bound holds at every one of 352,333 admission calls on live traffic and which, in a pre-registered held-out confirmatory round, halves the exact-fallback rate at matched risk (0.30 -> 0.14) -- coverage is bought at a price the account states. We then price the remaining distance from the certified witness to what a user experiences: a machine-checked design law (TV <= tanh(a_q w_thr)) turns the served-TV target into a threshold knob, and a three-layer audit of its instantiation -- an operator-norm query envelope measured 1.5x from tight, a measured-ellipsoid replacement for the Cauchy-Schwarz ball that buys nothing (0.89x, held-out sound), and the gate&#39;s operating point (~700x) -- localizes the entire 1064x gap to the operating point, a price the law now states rather than an unknown. A priced bound is worth nothing on a request one has not seen, so the third link is the quantifier: exchangeable extrapolation across 80 serving histories replaces binary conformal prediction&#39;s vacuous certificates with order-statistic bounds that discriminate (0.41 against 0.51 calibration risk). All probabilistic kernels are Lean 4-checked (228 exported theorems, no sorry); which object deserves this machinery at all is settled empirically in a companion paper that adjudicates -- and rejects -- the natural alternative of certifying routing. What ships is an account: risk you can spend, a gap you can read off a law, and a bound that survives the request you have not seen.