Square-Root Barriers in Prime-Log First-Return Problems: Moment, Product, Lattice, and Padé Obstructions

We study a class of prime-log first-return problems arising from weighted sums over logarithms of prime numbers. The objective is not to claim a proof of the Riemann Hypothesis, but to identify precisely which structural mechanisms succeed and which barriers prevent currently available arguments from reaching the critical square-root scale. We develop and compare moment, product, lattice, and Padé-based approaches and isolate the quantitative obstruction responsible for the remaining gap. The results establish several unconditional structural statements for the arithmetic model and reduce the unresolved part to a specific signed-Laplace lower-bound problem. Particular attention is given to separating proved statements, asymptotic arguments, standard inputs, rejected routes, and genuinely open steps. The accompanying supplementary material provides detailed derivations, proof audits, intermediate calculations, and the precise formulation of the remaining open problem. This work should therefore be viewed as a rigorous barrier analysis and reduction framework rather than as a proof of the Riemann Hypothesis.