Temporal Out-of-Sample Predictability of Primes Through Convergences in the Prime-Gap Sequence

This work investigates a deterministic structure derived from the sequence of prime gaps. Consecutive-prime gaps are accumulated to construct a natural representation in which distinct rows may converge to the same value V. These convergences generate advance announcements associated with a subsequent prime candidate. The study develops a computational mechanism for generating such announcements, applying a causal discard procedure based only on prime information available at the time of the announcement, and subsequently evaluating primality without using the future primality label during prediction. A computational experiment involving the first 100,000,000 primes produced 5,121,444 announcements. A predefined chronological out-of-sample experiment used 1,536,434 future announcements. Their observed primality rate was 11.165855481%, compared with 9.918030973% for the matched control, corresponding to a difference of 1.247824508 percentage points and a lift of 1.125813734. A one-sided paired permutation test gave p = 0.00009999, while the paired bootstrap 95% confidence interval for the difference was [1.179289185, 1.315970943] percentage points. The results provide computational evidence of chronological out-of-sample predictive enrichment under the stated protocol. The work does not claim a universal theorem or a complete characterization of the prime numbers. The deposited materials include the manuscript, computational code, experimental results, and reproducibility protocol.