The Immense Meets the Infinite: A Cardinal Classification of Generative Regimes in RAYOBETH Theory

Generative systems produce items that can be combined, so what they could produce outruns what they do produce. This article makes that statement exact for stocks governed by the core law of RAYOBETH Theory. At every finite stage the space of combinations is finite but exponentially larger than anything realizable; over a system’s whole history at most countably many items can be realized; and whenever that set is infinite, its combinations have the cardinality of the continuum. A classification theorem shows that extinction is the only regime that keeps the world of possibilities finite, and a cardinal duality theorem shows that a stock’s world of possibilities is finite exactly when the stock can be eliminated. The article defines the possibility gap, a measure in bits of the unexplored share of possibilities, states which of its predictions can be tested, and gives the name RAYOBETH a precise mathematical meaning.