Localized Partial-Gauss Transfer, Semiprime Kloosterman Coverage, and the Remaining Structured-Sieve Ledger in the Binary Goldbach Problem (PIHR-Goldach-XX) Progress Beyond PIHR–GOLDBACH XIX: From Endpoint Coefficient-Preserving Transfer to a Localized Partial-Gauss Closure Candidate

We report the developments obtained after PIHR–GOLDBACH XIX (Gold-19) in the continuing attempt to isolate a rigorous analytic route toward the binary Goldbach conjecture.

Gold-19 returned from the endpoint Möbius-variance formulation of Gold-18 to the signed determinant geometry of Gold-17. Its central candidate theorem, Endpoint Coefficient-Preserving Kloosterman Transfer (ECPKT), asked that the parity-sensitive four-weight determinant block be converted, without destructive absolute-value majorization, into bilinear Kloosterman forms with two -controlled coefficient vectors. The post-Gold-19 analysis substantially sharpens that proposal.

First, a false shortcut has been eliminated. The centered endpoint large-sieve/Ramaré block does not itself contain the reciprocal geometry required by the strongest Kloosterman estimates. Opening the square produces Ramanujan sums rather than Kloosterman sums, so the modular inverse must already come from the underlying determinant or congruence geometry.

Second, the algebraic core of ECPKT can be carried out explicitly. If the character diagonalization of Gold-17 is stopped one Gauss transformation earlier, a two-weight arithmetic block admits an exact Partial-Gauss Kloosterman Transfer