On the Infinitude of Twin Primes via the Additive Basis Property of Generating set K (k-indices in 4k ± 1 of primes)

We prove that the set of twin primes is infinite. The proof is based on the previously established fact that the set K = {k ∈ N | 4k − 1 ∈ P or 4k + 1 ∈ P} is a complete additive basis of order 2. The proof relies on modular analysis and construction of a sequence of numbers that cannot be represented without indices of twin primes. Assuming finiteness of twin primes, we construct a modified set K′ by “removing” all indices that participate in pure twin primes or steptwin primes configurations. Using the Chinese Remainder Theorem, we construct a blocking sequence - an arithmetic progression 𝑁𝑖 = M∙i + A such that for every residue class modulo M, at least one of the associated expressions 4k1 + 1 or 4 k2 − 1 is divisible by a small prime. Consequently, no sufficiently large 𝑁𝑖 can be expressed as a sum of two elements of K′ = K − ”All Twin Primes Indices”. Since K is an additive basis of order 2, the remaining finitely many twin indices cannot cover all 𝑁𝑖 - a contradiction. i.e. this blocking sequence contains numbers that are not representable as sums of two elements from K, leading to a contradiction if twin primes were finite. Hence, there are infinitely many twin primes (p, p + 2). The argument is elementary, using only additive basis property of Generating set K and the Chinese Remainder Theorem, without any unproven conjectures.