Coherence Types
A Refinement Calculus where Convention Coherence is the Declarations Forcing Every Reconciliation
The coherence fee is the obstruction to global type coherence: a well-typed composition with fee 0 reaches no convention-stuck state (Coherence Soundness), and disclosure-normal-form elaboration inserts exactly fee = dim H¹ coercions
A well-typed λ_∇ term (consistent, fee 0) that reaches a convention-stuck reconciliation state — refutes Coherence Soundness
Abstract
The Hoare rule for sequential composition is sound only when the seam predicate means the same on both sides; open agent and tool compositions break that, because the conventions components attach to shared quantities are transported across interfaces and need not agree where a feedback loop closes. Since coherence cannot be inferred bottom-up (a companion proves it SQ- and LPN-hard), it must be carried top-down as a typing invariant. This paper builds λ_∇, a refinement calculus where types carry convention bundles, a coercion is an admissible disclosure, and global type coherence is sheaf gluing whose obstruction is the coherence fee = dim H¹. It proves Coherence Soundness (progress and preservation), realizes the receipt as a graded modality whose grade is the repair cost, and derives the minimum coercion set as a disclosure normal form of size exactly the fee.