To Spend a Stone with Six Birds: Currency-Constraint Duality and Shadow Prices Across Closure Layers

Across disciplines, stable descriptive layers rely on small sets of variables that budget feasibility, accumulate along protocols, or appear as exchange rates between incompatible objectives. We call such variables currencies. This paper gives a layer-relative definition of currency within the Six Birds closure calculus and proposes a currency-constraint principle: lower-layer currencies become higher-layer constraints because packaged higher-level objects can persist only under bounded lower-layer spending, while higher-layer currencies emerge as the shadow prices of those budgets.

In finite Markov laboratories this principle becomes measurable. Path-reversal asymmetry and cycle affinities furnish lower-layer audits; constrained maximum-entropy closure yields dual prices; and across a resolution ladder we observe five robust signatures: honest coarse-grained directionality, monotone price emergence with a slack regime, growth of currency dimension with resolution, failure of proxy currencies under prediction and dual stability, and improved packaging coherence under stronger budget enforcement.

A Lean formalization proves deterministic-pushforward monotonicity for a finite KL form, providing a formal anchor for the audit side of the story. The result is a practical and conceptual account of why temperatures, prices, regularizers, and attention weights recur across sciences as the same mathematical kind of object.