When Verification Design Factorises: Reads, Routing, and the Joint Second Best

Abstract

A designer of verification chooses two things: what a verifier reads from disclosed evidence, and how far the information reaching the verifier can be held apart from the information reaching the party whose conduct verification is meant to discipline. This paper asks when these two margins can be designed separately. In a Bayesian persuasion model with a meanreading deterrence audience and a verifier who applies a coherent risk measure, the sender's value is a contest between two envelopes-a concave envelope serving deterrence and a convex envelope serving liability-whose gap carries all interaction between the margins and equals the sender's willingness to pay for audience separation. Directional factorisation is exact: which reads are gaming-proof is decided by the read's belief-curvature alone, independent of routing. Calibration and value factorisation fail generically, but the failure is confined to two explicit terms-a product-structure term, in which the read's responsiveness and the seal enter only through their product, and a band term activated by disclosure mandates-each of which vanishes to first order, at a saturated deterrence margin, or under level-insensitive reads. Finally, the routing margin's own invariance is a curvature pairing, not a consequence of coherence: equilibrium deterrence is unmoved by the seal, for every prior and every stake, if and only if a belief-convex read is paired with a concave compliance response; off the pairing, the seal moves deterrence through sheltering when the read is gameable and through retreat below an explicit saturation threshold even when the read is coherent.