The Harmonic Prime Clock: Eliminating Cyclic Redundancy from Wheel Sieves via Unified Indexing

A new approach to prime sieving is presented by shifting the computational domain from valuespace to position space for candidates of the form 6k ± 1. By defining a bijection f: N → {6k ± 1},it is shown that composites in this sequence can be crossed out using a deterministic “tape” ofalternating steps. Closed-form formulas for these steps are given in terms of the anchor (the indexof the prime), and it is proved that for any prime p, excluding its multiples is governed by exactlytwo alternating increments, step_a and step_b, whose sum is exactly 2p.