Exact certification of a paired Jacobsthal bound and the dimension-2 sieve barrier: a reproducible computational note on a tightened Goldbach statement

Working note (v0.2). No proof of Goldbach's conjecture is claimed, and novelty of the individual items is not established. The note re-derives an elementary reduction of a tightened Goldbach statement to a bound on a paired Jacobsthal function (Ziller and Morack, arXiv:1706.03668), certifies the published values h2(k) for k=3..11 by an exhaustive solver-free search (k=11 from one implementation only), records a check of the tightened statement for even N up to 2e7, and explains why a standard dimension-2 sieve cannot reach the required range (using source-reported constants, not re-derived). Rows k>=12 of the source table are transcribed, not verified. All data, figures and code are reproducible from the repository. AI assistance: code, drafting and checks were produced with Claude (Anthropic); the author directed the project. See the paper for limitations.