AbstractWe investigate a structural reparameterization of the prime sequence via normalized forward gaps, assigning to each odd prime half the distance to its successor. Algebraic identities induced by this labeling are examined together with statistical properties of the resulting integer sequence for primes up to 10⁹. The induced representation yields affine geometric families under a difference-of-squares transformation and exhibits pronounced oscillatory structure, slow growth of distinct values consistent with Cramér’s heuristic, and relatively low empirical Shannon entropy. Observed gap frequencies are compared with predictions of the Hardy–Littlewood prime pair conjecture and display close numerical agreement within the tested computational range. These results provide a descriptive framework for the forward-gap sequence.
Jian Jun Hu (Wed,) studied this question.