The semi-local framework conjecture for the Weil inequality

Let $q$ be the parameter appearing in the support interval, and let

$$S:=\{\infty\}\cup\{p\mid p<q\}.$$

The preceding discussion identifies Weil positivity with the operator-theoretic formulation for test functions satisfying the two vanishing conditions

$$\int_0^\infty h_1(x)x^{\pm 1/2}d^*x=0.$$

Semi-local framework conjecture. The semi-local operator theoretic framework with $S:=\{\infty\}\cup \{p\mid p<q\}$ suffices to prove the Weil inequality for all test functions with support in the interval $(q^{-1/2},q^{1/2})$. This would provide an operator-theoretic route to Weil's criterion for the Riemann hypothesis using only the indicated finite set of places together with the archimedean place; the source does not establish whether the proposed framework is sufficient.