Power Tower Generator (PTG): A Representational and Computational Framework for Finite Tetration

This work presents the Power Tower Generator (PTG), a structured notational framework for representing finite tetration over the natural numbers. While existing tetration notations such as Goodstein’s notation, Knuth’s up-arrow notation, and Conway’s chained arrow notation are mathematically equivalent, they often emphasize expressive generality over practical readability and computational alignment. PTG explicitly separates the base and height of a power tower, improving structural clarity without redefining tetration. Complementing the notation, this study introduces PTG Calc, a computational tool based on arbitrary-precision arithmetic that evaluates finite tetration expressions within feasible resource limits on standard computing hardware. The scope of this work is restricted to finite tetration over the natural numbers, and no claims are made regarding analytic extensions or algorithmic optimality. This manuscript documents work originally completed in 2024 and is made publicly available in its revised form.