Welcome to EnigMatematica! In this deep-dive episode, our hosts Andrea (the mathematician) and Alex (the philosopher) explore the staggering world of Tetration—the fourth hyper-operation.

How can the "tail" of a number larger than the observable universe be perfectly predictable? Through an exclusive collaboration with researcher Marco Ripà, we analyze his latest papers and preprints to reveal the Congruence Speed V(a). We break down how the trailing digits of power towers "freeze" into stability using p-adic valuations of trivial base manipulations (such as a−1, a+1, or a

2

+1).

In this episode, you will learn:

• Why Base 3 tetration is the perfect model for understanding stability (V(3)=1).

• The exact solution to the stable digits of Graham’s Number: slog

3

(G)−1.

• The Square-free Radix mystery: Why stability is a general property of non-trivial bases (those not divisible by the radical of the radix, rad(r)).

• How perfect powers act as "order amplifiers" in the chaos of infinity.

A special thanks to Marco Ripà for his groundbreaking research and for joining forces with EnigMatematica to bring these concepts to light.

Keywords: Tetration, Marco Ripà, Graham's Number, Congruence Speed, p-adic Valuation, Theory of Numbers, Pure Mathematics, EnigMatematica, Stable Digits, Infinite Sequences.