A Structural Resolution of the Strong Goldbach Conjecture via Indexed Modular Diagonals

We present a structural approach toward a resolution of the strong Goldbach conjecture based on a canonically anchored modular framework. The analysis begins with an exhaustive classification of all additive contributions involving the primes $2$ and $3$, after which the problem is reduced to a finite indexed diagonal that parametrizes all remaining candidates. Divisibility by primes $p>3$ translates into rigid congruential obstructions acting on the index axis. These obstructions are canonically determined, admit no freedom of choice, and assemble exactly via the Chinese Remainder Theorem. Using archimedean non–coverage principles for anchored indexed systems (including SALEN), we show that such obstructions cannot completely cover the indexed diagonal for arbitrarily large even integers. As a consequence, at least one admissible index always survives, yielding a pair of genuine primes whose sum equals the given even integer. The argument is finite, deterministic, and purely structural: no analytic estimates, probabilistic heuristics, or density arguments are employed.