Structured Residual Minorants and Finite Certificates for Complete-Sieve Additive Survivor Ladders

This paper develops a finite certificate framework for complete-sieve survivor nonemptiness in additive prime-sum ladders. The framework separates an exact small-prime layer from a residual layer and constructs pointwise minorants for complete-sieve survivor indicators using residual obstruction depth. Positive total mass of such a minorant over the complete-sieve interior gives a finite certificate of nonemptiness, and therefore a prime representation in the corresponding ladder rung. The additive ladder includes the unary base rung, which recovers complete-sieve primality, the binary rung, which corresponds to the Goldbach two-cloud survivor problem, and higher additive prime-sum rungs. The paper proves the general split residual minorant theorem, formulates Bonferroni and linear-programming certificate constructions, and gives a parametric template-exhaustion reduction. The computational section records finite certificate data from Minorant Lab, including Bonferroni certificates and a factored degree-three LP polynomial certificate for the binary Goldbach complete-sieve setting. These computations are included as finite validation and do not constitute a proof of Goldbach. The final part isolates the Universal Certificate Positivity Problem as the next frontier: proving suitable residual minorant positivity uniformly, or in averaged form, would imply corresponding complete-sieve survivor nonemptiness results for the classes of systems covered. This paper does not prove Goldbach, the additive Goldbach ladder, Hardy--Littlewood, Bateman--Horn, or prime-tuple conjectures.