Chapter 17: Tower Atoms and the Greedy Peel

the relevant chapter gave us primes and the Fundamental Theorem of Arithmetic on τ-Idx. The prime factorization of n is a flat product of prime powers p^e. But the arithmetic of Part III includes not only multiplication and exponentiation, but also tetration . A tower atom is a prime raised to a power that is itself an iterated exponential — a tetration tower. This chapter defines tower atoms, explains why the nesting X = ((A ↑↑ C)^{B}) · D is the only binding compatible with the diagonal discipline, and introduces the greedy peel-off algorithm that extracts these atoms from any object.