The golden vein: Generating sets K are complete additive bases of order 2. On the representation of natural numbers as sums of pairs of K-indices of primes

In the previous paper (DOI 10.5281/zenodo.19356728) the author proved the Covering Lemma ”On the Covering of the Set of Natural Numbers N by the Generating Set K” for a special configuration of pairs (𝑘, 𝑁-𝑘), namely the case where every number in the upper interval [𝑁/2, 𝑁] (the “right” parts of the pairs) lies outside K4, in other words the case where the interval [1, 𝑁 − 1] contains a lacuna (gap, a block of consecutive idle numbers) of length 𝑁/2. That proof is rigorous, but covers only that specific configuration. The present paper completes the Covering Lemma by providing a proof that works for all possible configurations of pairs (𝑘, 𝑁-𝑘). Together, the two works form a complete proof of the…