Centered Prime Segments, Modular Covering, and the Remaining Goldbach Proof Barrier

This expanded review audits the centered prime-segment approach to binary Goldbach, modular destructive residues, pairwise gcd constraints, complete and cube-root sieves, Chinese remainder constructions, circle-method correlations, and the Chen–Liouville parity selector. Every elementary proposition is accompanied by a proof or an exact derivation; three long appendices provide symbol-only derivations. A CRT construction proves arbitrarily deep fixed peripheral deserts and a logarithmic lower-bound obstruction along an unbounded sequence in the extended-value least-summand convention. None of these statements resolves the universal non-coverage lemma L. We separate established literature, exact equivalences, computational observations, conditional implications, and open estimates, and give explicit success criteria for future attempts to overcome the parity barrier.