A Conditional Pathway to the Strong Goldbach Conjecture via Inherited Primes, Nodes, and Short Windows

We introduce the Inherited Prime Conjecture (CH3), a structural condition that, if true,implies the Strong Goldbach Conjecture (SGC). For n ≥ 14, the ternary part of CH3 isa direct consequence of the Weak Goldbach Conjecture (Helfgott), thus the substantivecontent lies in the existence of an inherited prime p such that n − p is also prime (thebinary component). By integrating a deterministic framework based on nodes 2p (doublesof primes), a baseline bound, and short windows centered at n/2, we formulate ananalytical scheme and an explicit conditional theorem. This theorem reduces the proofof the SGC to a computational threshold N0 under a Short Window Hypothesis (SWH) andthe standard Hardy-Littlewood heuristics for the binary case. Additionally, we introducethe Double Equilibrium as a substructure that, when present, directly implies CH3 andwhose absence is explained by specific arithmetic families, all within the framework of anexplicit conditional proof under SWH.