Prime Polarity: Primes as Finite Witnesses of Infinity
Every prime carries a canonical bipolar polarization via γ/η dominance, rooting all local-global gluing in τ.
Overview
The Prime Polarity Theorem (I.T05, Hinge Theorem 2) assigns every prime p in the τ-index set a canonical bipolar polarization: each prime is either γ-dominant (EM sector) or η-dominant (strong sector). This polarization is provable in ZFC and is the root structure from which holomorphic behaviour, the lemniscate boundary, the split-complex algebra, and ultimately the Central Theorem are derived. Primes act as finite witnesses to the infinite boundary structure of τ³.
Detail
The τ-index set τ-Idx consists of primes organised by their ρ-orbit positions. Each prime p ∈ τ-Idx falls into one of two canonical polarity classes determined by the γ/η dominance of its orbit: γ-dominant primes index even-orbit positions (EM sector contributions) and η-dominant primes index odd-orbit positions (strong sector contributions). The Prime Polarity Theorem I.T05 proves that this polarization is unique (no prime is neutral) and canonical (no choice is involved — orbit parity is fixed by ρ). This bipolar structure at the prime level mirrors the two-lobe structure of the lemniscate boundary L = S¹ ∨ S¹, which is the central object of Book II. All holomorphic functions on τ³ (Book II) decompose into γ-even and η-odd parts by the prime polarity assignment. Prime Polarity is thus the number-theoretic foundation of all holomorphic structure in the series.
Result Statement
Every prime p ∈ τ-Idx carries a canonical bipolar polarization via γ/η dominance (I.T05, Hinge Theorem 2). This is the root of all local-global gluing in τ and provable in ZFC.