Toward a Goldbach Proof: An Explicit Conditional Reduction Through U2* Signed Contamination and Bilateral Prime Synchronization

Source abstract

The binary Goldbach conjecture asserts that every even integer E>2 is a sum of two primes. This paper does not claim an unconditional proof. It establishes an explicit reduction in which the proved part consists of: (i) exact destructive-congruence and collision identities; (ii) the separation of repeated-factor concentration from large-factor perfect dispersion; (iii) an exact Chen-Liouville prime/semiprime identity on Chen support; (iv) the unconditional Goldbach Space criterion; (v) the U2 quotient and destructor-renewal principles; (vi) the U2* identity D_-(E)=R_P(E)+R_{P3}(E)+R_{P5}(E)+... and the exact implication D_-(E)>U_odd>=3(E) => R_P(E)>0; and (vii) the bilateral symmetric sieve geometry, for which an odd prime r not dividing E removes exactly two deviation classes and contributes the local factor 1-2/r. The reduction leaves one explicit analytic lock. For E=2m and a nonnegative Rabbit/U2* weight W_E, one must obtain a local signed anti-saturation estimate for T_E(W)=sum_d W_E(d) lambda(m-d)lambda(m+d), with a quantitative margin strong enough to dominate the U2* contamination from complements with odd Omega at least 3. Mangerel's unconditional global anti-saturation theorem for the reflected Liouville convolution does not by itself imply this weighted polylogarithmic local estimate. Consequently the proof of Goldbach is complete within the framework conditional on this R3-local anti-saturation estimate together with the stated U2* contamination bound; those uniform estimates are not proved here. The contribution is therefore an explicit conditional reduction of binary Goldbach to a sharply formulated local signed inequality, together with exact structural lemmas identifying the only remaining analytic obstruction.