Part VI: The Prime Polarity Theorem

Part Overview

The Hyperfactorization Theorem (Part V) established that every object of Category τ has a unique ABCD address. The four coordinates are independent, complete, and faithful. But the coordinate system carries more structure than mere addressing.

This Part proves the Prime Polarity Theorem, the second hinge theorem of the Panta Rhei series. The theorem discovers that the internal primes ℙ_τ carry a canonical bipolar structure: every prime is either B-dominant (its tower atoms primarily grow through the exponent, the γ-channel) or C-dominant (its tower atoms primarily grow through tetration height, the η-channel). Both polarity classes are infinite.

The Prime Polarity Theorem is a purely finite-regime result: it concerns individual primes and their decidable, computable polarity. The geometric consequence of this bipolar partition — the algebraic lemniscate 𝕃 — will be earned in Part VII, where omega-germs and spectral splitting transform the arithmetic polarity data into an emergent topological boundary.

The chapter sequence: the relevant chapter poses the spectral question — what global structure do the primes reveal? the relevant chapter states and proves the Prime Polarity Theorem, including the stabilization mechanism that determines each prime’s polarity.

Chapters

• Chapter 26: The Spectral Question: What Do Primes See?
• Chapter 27: The Prime Polarity Theorem