Strong Golbach's conjecture proof using a structural approach to numbers

This paper provides an elementary proof of Strong Goldbach’s conjecture stating that every even natural number greater than two can be written as the sum of two primes. The conjecture addresses one of the oldest and most fundamental problems in number theory, which is understanding the distribution and properties of prime numbers.The proof employs a structural approach to numbers. On one hand, it builds upon the understanding of the structure of primes, specifically focusing on their 6k±1 patterns (primes≥5) as discussed in the opening section of the document, and on the other hand, it explores the underlying structure of even numbers, classifying them into three distinct categories based on their relationship to multiples of 3. Additionally, the proof employs a sieve logic encapsulated within an ad-hoc notation introduced in this paper (ad-hoc sieve notation), to provide the description for prime numbers into which an even number, according to its structure, can be decomposed. Finally, the proof covers formally the even numbers greater or equal to 10. The even numbers 4,6 and 8 are cases accepted as outliers in this approach since they involve the particular primes 2 and 3 that are not fitting any of 6k±1 prime patterns (as: 4=2+2,6=3+3 and 8=3+5). At the conclusion of this paper, several practical and direct implications of this approach are discussed. These implications involve the identification of twin primes, new primality testing, identifying sequences of primes, and the articulation with the Goldbach’s Comet.