Close the Bessel-basis denominator gap for Jacobi phase constants

Close the gap between the lower and upper bounds for the 2-adic valuation of the Bessel-basis denominator of the Jacobi phase constant \(\kappa_r\), namely establish whether the valuation can be determined within the interval \(2r+\nu_2((r-1)!)\le \nu_2(\operatorname{den}_\mu\kappa_r)\le 3r-1+\nu_2((r-1)!)\).

Background

The paper studies the denominators of the additive phase constants $\kappa_r$ arising in the asymptotic phase expansion of the zeros of Jacobi polynomials. It establishes a coarse 2-adic upper bound and, assuming the verified top-layer identity, a lower bound obtained from an extremal coefficient in the Bessel basis.

These two bounds differ by exactly $r-1$: the lower bound is $2r+\nu_2((r-1)!)$, while the upper bound is $3r-1+\nu_2((r-1)!)$. The authors explicitly leave unresolved the problem of determining the exact Bessel-basis denominator valuation or otherwise eliminating this gap. The separate sharp law in the physical $(A,B)$-basis is reported as proved in the companion paper, but the Bessel-basis gap remains listed as open in the paper's status summary.