Adjacent-Shift Proper Position for Jensen Polynomials

This paper studies adjacent shifts of Jensen polynomials associated with positive sequences and their relation to real-rootedness, interlacing, and proper position. An exact transfer principle is established between hyperbolicity of a Jensen polynomial of degree d+1d+1 and proper position of the corresponding adjacent degree-dd Jensen polynomials. The resulting framework is expressed through Wronskian identities and finite positive-semidefinite Gram certificates. In low degrees, these certificates recover Turán-type inequalities and provide explicit algebraic criteria for adjacent-shift interlacing. The general results are applied to arithmetic sequences, with particular emphasis on the Taylor coefficients of the Riemann ξ\\xi-function. Published effective hyperbolicity results for ξ\\xi-Jensen polynomials are transferred to adjacent-shift proper-position statements, including an unconditional range obtained by combining known Jensen-polynomial results with rigorous verification of the Riemann hypothesis through height 3×10123\\times10^{12}. A recent stronger asymptotic hyperbolicity wedge is also discussed separately. Multi-shift consequences give interlacing staircases and uniform bounds for the variation of zero-counting functions across consecutive Jensen shifts. A further application to the partition function illustrates the connection with higher-order Turán inequalities. The paper does not claim a proof of the Riemann hypothesis.