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). We establish structural identities governing the conservation of partition elements, demonstrating that the count of Prime-Prime (PP) 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 106 validates these findings, showing that the boundary of failure admissibility, parameterized by the ratio \( \hat{\lambda}_L(N) \), converges toward a global structural ceiling. Furthermore, 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 that the failure state (PP(N)=0) precipitates an information-theoretic paradox: it implies that the global prime counting function π(2N) can be fully reconstructed from the local modular geometry of a subset of composites, violating the established algorithmic irreducibility of the prime sequence.