M05 The RTL Calculus: Tower Algebra, Growth Speedometers, and the Taxonomy of Hyperoperation Speeds

This monograph introduces the RTL calculus, a hierarchy of growth measures for quantifying the speed of recursive depth in hyperoperations and related iterative systems. Generalising the classical Tower Level (TTL) used for tetration, the RTL calculus defines a family of measures RTLm that track how many times the (m−1)‑th operation is nested within a recursive formula. This separates the wrapper rank that governs nesting from the internal growth rank describing how the free variable appears inside the formula. A precise notion of degeneration is introduced, together with the associated active measure, allowing operations to be classified by the highest level at which unbounded growth occurs. Within this framework, right‑caterpillar hyperoperations exhibit immediate activation, while left‑caterpillar operations display delayed activation through a cascading sequence of internal growth regimes, explaining the observed four‑rank delay. The monograph develops the RTL lattice, assigning each operation coordinates(m,r) that provide a uniform taxonomy for classical hyperoperations, their left‑ and right‑caterpillar variants, and new operations generated by explicit formulas at fixed wrapper rank. The RTL calculus functions as a growth “speedometer” for recursive systems, offering a systematic language for comparing hyperoperations by their depth dynamics rather than symbolic rank alone. The purpose of this deposit is to document the framework and conceptual priority of the RTL calculus within the hyperoperation theory series.