Closing the gap: maximal prime gaps are Θ(𝑙𝑛𝑙𝑛(𝑝)∙ln(𝑝)) via additive basis properties of K-indices of primes

It has been proved by the author earlier, that the generating set of K-indices of primes K = {𝑘 ∈ N | 4𝑘 − 1 ∈ P or 4𝑘 + 1 ∈ P} is a complete additive basis of order 2. From this property, we derive an upper bound for the maximal gap between consecutive primes: 𝑔p(𝑝𝑛) = 𝑝𝑛 - 𝑝𝑛-1 = 𝑂(4lnln𝑝𝑛∙ln𝑝𝑛). In particular, every sufficiently large gap satisfies 𝑝𝑛 - 𝑝𝑛-1 ≤ 4𝐶(lnln𝑝𝑛∙ln𝑝𝑛) with an absolute constant 𝐶 > 0. Idea of the proof. Assuming, for contradiction, that a gap 𝑘𝑖 - 𝑘𝑖-1 in K exceeds some function 𝑓(𝑘) that must grow faster than the gaps themselves, we choose 𝑓(𝑘) = 𝐶 lnln𝑘𝑖∙ln𝑘𝑖 and consider the number 𝑁 = 2𝑘𝑖 + 1. Using the fact that K is an additive basis of order 2, 𝑁 must be representable as 𝑎 + 𝑏 with 𝑎, 𝑏 ∈ K. A case-by-case analysis of the possible positions of 𝑎 and 𝑏 relative to 𝑘𝑖-1 and 𝑘𝑖 shows that such a representation is impossible. Hence, all gaps in K are 𝑂(lnln𝑘𝑖∙ln𝑘𝑖), and via the form 𝑝 = 4𝑘 ± 1 we obtain the bound for prime gaps 𝑔p(𝑝𝑛) = 𝑂(4lnln𝑝𝑛∙ln𝑝𝑛). The proof is elementary; it uses no analytic number theory; it relies only on the additive basis property of K and a combinatorial analysis of gaps. Consequences. Since the bound 𝑝𝑛 - 𝑝𝑛-1 = 𝑂(4lnln𝑝𝑛∙ln𝑝𝑛) grows slower than the lengths of the intervals in Legendre’s, Brocard’s and Oppermann’s conjectures, these conjectures follow for all sufficiently large indices. In comparison with known results, our estimate is asymptotically stronger than the unconditional bound 𝑂(𝑝0.525) (Baker–Harman–Pintz) and also stronger than the conjectural Cramer bound 𝑂(ln2𝑝). Numerical verification. Numerical verification using known record prime gaps (including the 2022 record gap of 7’186’572) shows that the renormalized quantity 𝐶(𝑝) = 𝑔(𝑝)/(lnln𝑝∙ln𝑝) attains a maximum of about 10 and then slowly decreases, approaching a value near 1.1467 - 0.871 for the largest available unconfirmed prime gaps. This behavior is consistent with the theoretical bound and suggests that the true asymptotic value is 2𝑒−𝛾 ≈ 1.1229.