Nothing Binds a Twin

We read the integers prime to 6 as the positions of a world of clocks. Each prime p ≥ 5 is a clock that marks the two centres k with 6k ≡ ±1 (mod p), and a centre that no clock marks is a full zero; below the reach of the clocks a full zero is exactly a twin prime pair 6k ± 1. We prove