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To Spend a Stone with Six Birds: Currency-Constraint Duality and Shadow Prices Across Closure Layers
Why temperatures, prices and attention weights are the same kind of thing.
In plain words
Every stable layer has a few variables that budget what it can do. Call them currencies.
The principle: a lower layer's currency becomes a higher layer's constraint, because higher objects only persist under bounded lower spending. The higher layer's currency is the shadow price of that constraint.
In finite Markov labs this is measurable. Five signatures appear across a resolution ladder, including monotone price emergence and failure of proxy currencies. A Lean anchor proves the audit side.
What it shows
- A layer relative definition of currency.
- The currency constraint principle with measurable signatures.
- A formal anchor for pushforward monotonicity.
What it does not claim
No claim about any specific economy or physical system.
Cite
Tsiokos, I. (2026). To Spend a Stone with Six Birds: Currency-Constraint Duality and Shadow Prices Across Closure Layers. Zenodo. https://doi.org/10.5281/zenodo.18926771