Structural Failure Mode Analysis of the Binary Goldbach Conjecture

This paper analyzes the Binary Goldbach Conjecture (bGC) through a deterministic structural lens, employing a Failure Mode Analysis (FMA) framework to map prime and composite inventories onto the Left-Right Partition Table (LRPT). The FMA framework identifies the specific structural conditions—categorized into three distinct Tiers—that render the existence of a counterexample (a “Failure State”) structurally inadmissible under standard density constraints. We establish structural identities governing the conservation of partition elements, demonstrating that the count of Prime-Prime (iP P/i) pairs functions as a necessary deterministic residual. The analysis identifies tiered inadmissible failure states where, in each Tier, the exhaustion of composite inventories mathematically forces prime-prime partitions into existence to preserve information conservation. Numerical analysis for N up to 10sup6/sup shows how the tiered FMA framework quantifies the structural mechanisms through which the bGC remains valid mathematically. Furthermore, as explained in Appendix I, by leveraging the midpoint symmetry of Goldbach primes, the FMA approach yields a ”Mirror Search” mechanism for distal primes that demonstrates superior discovery efficiency compared to sequential scanning methods guided by the Prime Number Theorem. The analysis also reveals, as detailed in Appendix II, that the failure state (iP P/i (iN/i) = 0) implies a deterministic dependency between partition components, allowing the primality characteristic function on the interval 3, 2N − 3 to be determined by testing π(N) fewer odd integers.