The Additive Basis Property of 𝕂-Indices of Primes in a Combinatorial Approach to Goldbach's Conjecture

We propose using 𝕂-indices of primes as the foundation for an elementary approach to solving Goldbach’s binary conjecture. The approach is based on a combinatorial covering argument and does not rely on analytic methods or heavy machinery. We introduce the set of 𝑘-indices of primes 𝕂 = {𝑘 ∈ ℕ : 4𝑘 ±1 is prime} and formulate the 𝕂-Basis conjecture: 𝕂 + 𝕂 ⊇ ℕ\{1}, that is, 𝕂 is a complete additive basis of order 2 for natural numbers greater than 1. We prove the Equivalence Theorem: Goldbach’s binary conjecture holds if and only if 𝕂 is such a basis. This equivalence reduces the classical problem to a purely combinatorial covering question about the set 𝕂. We prove a Covering Lemma in a non-trivial combinatorial setting – for a special “lacunary” configuration of the set 𝕂. The proof is elementary and proceeds by contradiction, using Bertrand’s postulate (the Bertrand - Chebyshev theorem) to rule out the existence of a counterexample. From this lemma and the Equivalence Theorem, Goldbach’s binary conjecture follows directly in the restricted case. The discovery of a sparse complete additive basis arising from the 𝑘-indices of primes and the proposal of the 𝕂-Basis conjecture, together with its partial proof in this setting, reveal a new structural feature of the primes and provide a fresh combinatorial framework for approaching classical additive problems.