| _version_ | 1866901512344567808 |
|---|---|
| author | Tsiokos, Ioannis |
| author_facet | Tsiokos, Ioannis |
| contents | <p>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 <em>currencies</em>. This paper gives a layer-relative definition of currency within the Six Birds closure calculus and proposes a <em>currency-constraint principle</em>: 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.</p> <p>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.</p> <p>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.</p> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_18926771 |
| institution | Zenodo |
| language | eng |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | To Spend a Stone with Six Birds: Currency-Constraint Duality and Shadow Prices Across Closure Layers Tsiokos, Ioannis <p>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 <em>currencies</em>. This paper gives a layer-relative definition of currency within the Six Birds closure calculus and proposes a <em>currency-constraint principle</em>: 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.</p> <p>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.</p> <p>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.</p> |
| title | To Spend a Stone with Six Birds: Currency-Constraint Duality and Shadow Prices Across Closure Layers |
| url | https://doi.org/10.5281/zenodo.18926771 |