The golden vein: Generating sets K are complete additive bases of order 2. On the representation of natural numbers as sums of pairs of 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 Covering Theorem that K4 (and indeed every K2𝑚) is a complete additive basis of order 2. The author proves that for every integer 𝑚 ≥ 2 the set K2𝑚 = { 𝑘 ∈ N | at least one of 2𝑚𝑘+1, 2𝑚𝑘-1 is prime } is a complete additive basis of order 2. That is, there exists a constant 𝑁0(𝑚) such that every integer 𝑁 > 𝑁0 (𝑚) can be written as 𝑁 = 𝑘1 + 𝑘2 with 𝑘1, 𝑘2 ∈ K2𝑚. For 𝑚 = 2 this proves the binary Goldbach conjecture. The proof proceeds by contradiction: assuming a smallest non-representable 𝑁 > 𝑁0(𝑚), we construct a recursive process that forces a bijection between K2𝑚 and the set of idle numbers G up to a small central set. This bijection implies that |G(𝑁) | - |K2𝑚(𝑁) | is zero or bounded by a constant (1 or 2). On the other hand, Dirichlet’s theorem on primes in arithmetic progressions gives the asymptotic |K2𝑚(𝑁) | ∼ 4𝑁/log 𝑁, and hence |G(𝑁) | ∼ 𝑁, so the difference grows without bound – a contradiction. Small values can be verified directly. As a by-product we uncover a golden vein of additive bases: the same proof works for the whole family K2𝑚 (𝑚 ≥ 2), showing that zero asymptotic density does not prevent a set from being a complete basis of order 2. All these sets K2𝑚 have zero asymptotic density, yet each of them is a complete basis – a counterintuitive phenomenon we call the golden vein of additive bases.