From Immensity to the Infinite: Deriving the RAYOBETH Equation That Connects the Immense Finite to the Beth Hierarchy

RAYOBETH Theory takes its name from two mathematical objects: Rayo’s number, a finite number too large to be listed or computed, and the Beth hierarchy of infinite cardinals. The name promises a passage from the immense finite to the infinite, yet no equation connecting the two has existed in the theory or anywhere else. This article supplies it. For a generative system whose active repertoire obeys the RAYOBETH core law, it proves four results. A dynamic identity shows that the logarithm of the possibility space equals the active repertoire plus attrition accumulated over time. A discontinuity theorem shows that the limit of the finite possibility spaces is the countable infinity ℶ₀, while the possibility space of the limit is the uncountable ℶ₁, so accumulation reaches the first infinity and only completion reaches the second. A closed form, the RAYOBETH equation, gives the cardinality of the possibility space for every parameter value: it equals 2 raised to N₀ times the series of R₀ to the power k divided by 1 + k(β − 1) when β ≥ 1 and the initial generative ratio R₀ is below one, and ℶ₁ otherwise. The finite value diverges at the boundary R₀ = 1 as a power law when β = 1 and logarithmically when β > 1, so every finite benchmark, Rayo’s number included, is exceeded arbitrarily close to the boundary and the infinite begins on it. A Rayo clock theorem dates the crossing of any finite benchmark in each regime, and a stochastic version shows that the possibility space is ℶ₁ with probability 1 − 𝑅−𝑁₀. Every closed form is verified numerically. Applied to 152,635 commits in three code and formal mathematics repositories and to seven series from the RAYOBETH Atlas, the evidence places all ten systems in the BETH region. Elasticities of generation are below one in all three repositories (CPython 0.10, Lean 3 mathlib 0.64, Lean 4 Mathlib −0.60), the gap between the possible and the realised widens one for one with novelty (slopes of 1.000 in all three repositories, every 95% interval within 0.997 to 1.003), and the realised share of possibilities is below 10−1237 in every repository and below 10−48,335 in United States patents. Lean 3 mathlib, closed in 2023, is a measured finite world whose possibility space has exactly 24123 elements, a number of 1,242 digits.