Algebraic Method for Goldbach's Conjecture: Application of the Cyclic Coset Structure (LCC)

This study explores the potential relationship between the Algebraic Structureof Cyclic Cosets (LCC) in cyclic rings and Goldbach’s Conjecture. Wecomputationally verified that the LCC structure in ℤ63 acts as a perfect collector,encompassing the entire set of prime numbers ℙ < 64. Based on this inclusion,and by leveraging the LCC structure to effectively bound Jacobi Sums bygrouping Galois conjugate characters (Theorem 4.1), we prove that Goldbach’sConjecture is true for all even numbers 𝑁 > 𝑁0. Quantitative calculation shows that 𝑁0 is reduced to a threshold of N0 ≈ 10^15, which falls within thecomputationally verified range, comple