A Candidate Proof Architecture for Binary Goldbach: Quartic Sieve Reassembly, the Liouville Parity Obstruction, Quantitative Dual Reserve, and the Remaining Signed Prime-Correlation Theorem (PIHR-Goldbach-XLI)

Source abstract

We present a self-contained account of the PIHR-Goldbach research programme through Gold-41. The objective is to prove the binary Goldbach conjecture through a weighted prime-pair inequality derived from a quartic divisor construction. The framework employs exact Möbiusweighted divisor identities, character weights modulo eight, short arithmetic-progression estimates, cutoff transport, and a decomposition into rough semiprime, three-prime, four-prime, five-prime, and mixed-factor classes. The Gold-40 analysis reports a quartic lower balance in which the remaining obstruction is a single signed correlation involving the actual shifted-prime sequence. A subsequent five-prime estimate reduces the unresolved operator to the semiprime, three-prime, and mixed-factor classes. We examine the Liouville parity comparison, which demonstrates that the previously available short distribution estimates do not, by themselves, imply the required uniform correlation lower bound. We then formulate the Quantitative Dual Reserve (QDR) method and establish its exact finite-dimensional primal-dual structure. The strongest candidate completion is a signed parity-separation theorem for the residual arithmetic operator after subtraction of controlled short-row contributions. We formulate its precise logical role, discuss possible bilinear and character-theoretic approaches, and identify the remaining requirements for an effective Goldbach proof. The manuscript distinguishes established classical theorems, algebraic identities verified here, results reported in the preceding Gold-40 manuscripts, conditional implications, numerical experiments, and unproved hypotheses.