Chapter 27: The Prime Polarity Theorem

This chapter proves the Prime Polarity Theorem, the second hinge theorem of the Panta Rhei series. Every prime in ℙ_τ carries a canonical polarity: it is either B-dominant (exponent-primary, γ-channel) or C-dominant (tetration-primary, η-channel). Both polarity classes are infinite. The proof proceeds by analyzing the tower-atom divisibility structure and exploiting the growth-rate separation between exponentiation and tetration. The bipolar partition of the primes is the arithmetic origin of the two lobes of the algebraic lemniscate 𝕃 (whose geometric form S¹ ∨ S¹ is earned in Book II).