PIHR-GOLDBACH XXXVI (Gold-36) Centered Prime-Modulus Sparsity, Exhaustive HB6 Packet Closure, and the Final Candidate Proof Chain for Binary Goldbach

Source abstract

We develop a stand-alone continuation of the PIHR-Goldbach programme beyond Gold-35. The purpose is to place the strongest presently available proof chain for the binary Goldbach conjecture into a single theorem-level architecture and to distinguish rigorously between exact identities, proved finite combinatorial reductions, published analytic inputs, and the remaining certification boundary. This is the first stage of the PIHR-Goldbach programme in which no uncovered exponent-polytope cell and no unidentified analytic mechanism remain in the proposed asymptotic chain. Nevertheless, because the binary Goldbach conjecture is a major open problem, the distinction between a complete proof architecture and an independently certified proof is essential. The present manuscript therefore states the final exact certification theorem explicitly. An unconditional Q.E.D. is justified only after the original long-divisor operator, every packet normalization, every published-theorem hypothesis, and the complete reassembly identity have been checked in the exact weighted form stated below. The subsequent all-even conclusion additionally requires an effective analytic threshold or finite computational overlap.