Two-Sided Constraints on a Valued Lattice

Abstract

A declaration on a finite bounded lattice is a pair of subsets: descriptions a state must reach, and descriptions it must not exceed. When both cannot hold, no state satisfies it, and a state is charged its shortfall against the demand and its overshoot beyond the limit. That charge is not chosen: two axioms fix it up to two weights, and requiring that neither response to a conflict — retreat, or overshoot — be vacuous fixes the weights up to scale. One identity then carries the theory. For every valuation obeying a data-processing inequality, and only for those, the advantage of overshooting over retreating is exactly the residual information of demand and limit — their mutual information beyond their common part. Across all monotone valuations that advantage vanishes identically exactly when the valuation is supermodular; among the data-processing ones, exactly when it is modular. On the partition lattice under Shannon entropy its maximum is log2(n) − 2 + 2/n, attained at a path; measured against the planned cost rather than in bits it grows instead as n/2. On a product of two subsystems, the declaration demanding one and forbidding the other has advantage exactly I(X;Y) — mutual information as the value of a single declaration, evaluated once. We also characterise when auditing a declaration one pair of descriptions at a time is sound for every monotone valuation.